diff --git a/docs/.nav.yml b/docs/.nav.yml
index 22914c7..971c112 100644
--- a/docs/.nav.yml
+++ b/docs/.nav.yml
@@ -25,12 +25,13 @@ nav:
               - "Mathieu group M22": examples/atlas/mathieu/m22.md
               - "Mathieu group M23":
                   - examples/atlas/mathieu/m23/index.md
+                  - "Presentation 1": examples/atlas/mathieu/m23/p1.md
+                  - "Presentation 2": examples/atlas/mathieu/m23/p2.md
                   - "M11, P1": examples/atlas/mathieu/m23/p1_m11.md
                   - "M22, P1": examples/atlas/mathieu/m23/p1_m22.md
                   - "M11, P2": examples/atlas/mathieu/m23/p2_m11.md
                   - "M22, P2": examples/atlas/mathieu/m23/p2_m22.md
-              - "Mathieu group M24":
-                  - examples/atlas/mathieu/m24/index.md
+              - "Mathieu group M24": examples/atlas/mathieu/m24.md
           - "Leech lattice groups":
               - examples/atlas/leech-lattice/hs.md
               - "Janko group J2, Hall-Janko group HJ":
@@ -39,11 +40,23 @@ nav:
                   - "P2": examples/atlas/leech-lattice/j2/j2-p2.md
               - "Conway group Co2":
                   - examples/atlas/leech-lattice/co2/index.md
+                  - "U6(2):2": examples/atlas/leech-lattice/co2/2300.md
               - "McLaughlin group McL":
                   - examples/atlas/leech-lattice/mcl/index.md
           - "Monster sections":
               - "Held group He":
                   - examples/atlas/monster-sections/he/index.md
+                  - "S4(4):2": examples/atlas/monster-sections/he/2058.md
+                  - "2^2.L3(4).S3": examples/atlas/monster-sections/he/8330.md
+                  - "2^6:3.S6 (I)": examples/atlas/monster-sections/he/29155-i.md
+                  - "2^6:3.S6 (II)": examples/atlas/monster-sections/he/29155-ii.md
+                  - "2^(1+6).L3(2)": examples/atlas/monster-sections/he/187425.md
+                  - "7^2:2.L2(7)": examples/atlas/monster-sections/he/244800.md
+                  - "3.S7": examples/atlas/monster-sections/he/266560.md
+                  - "7^(1+2):(3 x S3)": examples/atlas/monster-sections/he/652800.md
+                  - "S4 x L3(2)": examples/atlas/monster-sections/he/999600.md
+                  - "7:3 x L3(2)": examples/atlas/monster-sections/he/1142400.md
+                  - "5^2:4A4": examples/atlas/monster-sections/he/3358656.md
               - "Fischer group Fi22":
                   - examples/atlas/monster-sections/fi22/index.md
           - "Pariahs":
diff --git a/docs/examples/atlas/index.md b/docs/examples/atlas/index.md
index 5150c6e..b6df7ee 100644
--- a/docs/examples/atlas/index.md
+++ b/docs/examples/atlas/index.md
@@ -78,7 +78,7 @@ see more information.
           <td class="correct-size">
             <a href="mathieu/m22">M<sub>22</sub></a>
           </td>
-          <td class="some-correct-index">
+          <td class="correct-size">
             <a href="mathieu/m23">M<sub>23</sub></a>
           </td>
           <td class="correct-size">
@@ -97,13 +97,13 @@ see more information.
           <td class="correct-size">
             <a href="leech-lattice/j2">J<sub>2</sub></a>
           </td>
-          <td class="no-presentation">Co<sub>1</sub></td>
-          <td class="work-in-progress">
+          <td class="no-presentation-known">Co<sub>1</sub></td>
+          <td class="correct-index">
             <a href="leech-lattice/co2">Co<sub>2</sub></a>
           </td>
-          <td class="no-presentation">Co<sub>3</sub></td>
+          <td class="no-presentation-in-atlas">Co<sub>3</sub></td>
           <td class="work-in-progress"><a href="leech-lattice/mcl">McL</a></td>
-          <td class="no-presentation">Suz</td>
+          <td class="no-presentation-in-atlas">Suz</td>
         </tr>
       </table>
     </div>
@@ -113,16 +113,16 @@ see more information.
           <th colspan="8">Monster sections</th>
         </tr>
         <tr>
-          <td class="work-in-progress"><a href="monster-sections/he">He</a></td>
-          <td class="no-presentation">HN</td>
-          <td class="no-presentation">Th</td>
+          <td class="correct-index"><a href="monster-sections/he">He</a></td>
+          <td class="no-presentation-in-atlas">HN</td>
+          <td class="no-presentation-in-atlas">Th</td>
           <td class="work-in-progress">
             <a href="monster-sections/fi22">Fi<sub>22</sub></a>
           </td>
-          <td class="no-presentation">Fi<sub>23</sub></td>
-          <td class="no-presentation">Fi<sub>24</sub>'</td>
-          <td class="no-presentation">B</td>
-          <td class="no-presentation">M</td>
+          <td class="no-presentation-in-atlas">Fi<sub>23</sub></td>
+          <td class="no-presentation-in-atlas">Fi<sub>24</sub>'</td>
+          <td class="no-presentation-in-atlas">B</td>
+          <td class="no-presentation-in-atlas">M</td>
         </tr>
       </table>
     </div>
@@ -135,7 +135,7 @@ see more information.
           <td class="correct-size">
             <a href="pariahs/j1">J<sub>1</sub></a>
           </td>
-          <td class="no-presentation">O'N</td>
+          <td class="no-presentation-in-atlas">O'N</td>
           <td class="correct-size">
             <a href="pariahs/j3">J<sub>3</sub></a>
           </td>
@@ -143,7 +143,7 @@ see more information.
           <td class="work-in-progress">
             <a href="pariahs/j4">J<sub>4</sub></a>
           </td>
-          <td class="no-presentation">Ly</td>
+          <td class="no-presentation-in-atlas">Ly</td>
         </tr>
       </table>
     </div>
@@ -181,13 +181,14 @@ see more information.
             presentation has at least 1 subgroup of the correct index.
           </td>
         </tr>
-        </tr>
+        <!--
         <tr>
           <td class="some-correct-index">
             The presentation has at least 1 subgroup of the correct index, and
             at least 1 subgroup whose index could not be verified.
           </td>
         </tr>
+        -->
         <!-- <tr>
           <td class="incorrect-size">
             The presentation defines a group of the incorrect size
@@ -197,7 +198,10 @@ see more information.
           <td class="work-in-progress">Work in progress.</td>
         </tr>
         <tr>
-          <td class="no-presentation">No presentation is provided.</td>
+          <td class="no-presentation-in-atlas">No presentation in ATLAS.</td>
+        </tr>
+        <tr>
+          <td class="no-presentation-known">No presentation known.</td>
         </tr>
       </table>
     </div>
diff --git a/docs/examples/atlas/leech-lattice/co2/2300.md b/docs/examples/atlas/leech-lattice/co2/2300.md
new file mode 100644
index 0000000..da3b0ec
--- /dev/null
+++ b/docs/examples/atlas/leech-lattice/co2/2300.md
@@ -0,0 +1,183 @@
+# $U_6(2):2$ as a subgroup of Conway group Co~2~
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co2/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co2/).
+
+## Claim
+
+$$
+    \mathrm{Co}_2 = \langle a, b \mid a^2 = b^5 = (ab^2)^9
+    = [a,b]^4 = [a,b^2]^4 = [a,bab]^3 = [a,bab^2ab]^2
+    = [a,bab^{-2}]^3 = [a,b^{-2}abab^{-2}]^2
+    = (abab^2ab^{-1}ab^{-2})^7 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$U_6(2):2$. More precisely, if
+
+$$
+    H = \langle a,\ (ab)^2 \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{Co}_2 : H] = 2,300.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for Co~2~, adds the generating pairs that define the maximal subgroup, and runs
+the Todd-Coxeter algorithm. It uses the additional relation
+$[a,b^2ab^2]^3 = 1$, which the source code marks as redundant but useful. We do
+not verify that this relation is redundant here.
+
+The code took about 1 minute 8 seconds to run on a 2024 MacBook Pro M4 with 48GB
+of RAM:
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^5"), "")
+presentation.add_rule(p, parse("(ab^2)^9"), "")
+presentation.add_rule(p, parse("(a,b)^4"), "")
+presentation.add_rule(p, parse("(a,b^2)^4"), "")
+presentation.add_rule(p, parse("(a,bab)^3"), "")
+presentation.add_rule(p, parse("(a,bab^2ab)^2"), "")
+presentation.add_rule(p, parse("(a,baB^2)^3"), "")
+presentation.add_rule(p, parse("(a,B^2abaB^2)^2"), "")
+presentation.add_rule(p, parse("(a,b^2ab^2)^3"), "")  # Redundant, but useful.
+presentation.add_rule(p, parse("(abab^2aBaB^2)^7"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("Bab")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# Add generating pairs for the maximal subgroup
+tc.add_generating_pair("a", "")
+tc.add_generating_pair("abab", "")
+
+print(f"The index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The truncated output of the enumeration is below:
+
+??? info "Truncated output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 15, |u| + |v| ∈ [2, 70], ∑(|u| + |v|) = 302
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |         defined
+    #0: ToddCoxeter: nodes            |            3 |               0 |               3
+    #0: ToddCoxeter:                  |       active |         missing |      % complete
+    #0: ToddCoxeter: edges            |            5 |               4 |           55.6%
+    #0: ToddCoxeter: time             | run 0 = 82µs | all runs = 82µs | elapsed = 132µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      1,836,440 |            37,239 |        1,873,692
+    #1: ToddCoxeter: diff 0.0.0         |     +1,836,437 |           +37,239 |       +1,873,689
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      4,163,654 |         1,345,666 |            75.6%
+    #1: ToddCoxeter: diff 0.0.0         |     +4,163,649 |        +1,345,662 |           +20.0%
+    #1: ToddCoxeter: phase 0.0 = 1.000s | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.000s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      3,697,682 |            73,383 |        3,771,084
+    #1: ToddCoxeter: diff 0.0.1         |     +1,861,242 |           +36,144 |       +1,897,392
+    #1: ToddCoxeter: diff 0.0.0         |     +3,697,679 |           +73,383 |       +3,771,081
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      8,383,347 |         2,709,699 |            75.6%
+    #1: ToddCoxeter: diff 0.0.1         |     +4,219,693 |        +1,364,033 |            -0.0%
+    #1: ToddCoxeter: diff 0.0.0         |     +8,383,342 |        +2,709,695 |           +20.0%
+    #1: ToddCoxeter: phase 0.0 = 2.005s | run 0 = 2.005s | all runs = 2.005s | elapsed = 2.005s
+    [... lines omitted ...]
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.68      |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      5,372,361 |       106,952,651 |      112,324,634
+    #1: ToddCoxeter: diff 0.0.67        |    -31,972,961 |       +31,973,060 |               +0
+    #1: ToddCoxeter: diff 0.0.0         |     +5,372,358 |      +106,952,651 |     +112,324,631
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |              0 |        16,117,083 |             0.0%
+    #1: ToddCoxeter: diff 0.0.67        |             +0 |       -95,918,883 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |             -5 |       +16,117,079 |           -55.6%
+    #1: ToddCoxeter: phase 0.0 = 1min8s | run 0 = 1min8s | all runs = 1min8s | elapsed = 1min8s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 STOP
+    #0: ToddCoxeter: FELSCH 0.0.69      |         active |            killed |          defined
+    #0: ToddCoxeter: nodes              |          4,600 |       112,320,034 |      112,324,634
+    #0: ToddCoxeter: diff 0.0.68        |     -5,367,761 |        +5,367,383 |               +0
+    #0: ToddCoxeter: diff 0.0.0         |         +4,597 |      +112,320,034 |     +112,324,631
+    #0: ToddCoxeter:                    |         active |           missing |       % complete
+    #0: ToddCoxeter: edges              |         13,800 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.0.68        |        +13,800 |       -16,117,083 |          +100.0%
+    #0: ToddCoxeter: diff 0.0.0         |        +13,795 |                -4 |           +44.4%
+    #0: ToddCoxeter: phase 0.0 = 1min8s | run 0 = 1min8s | all runs = 1min8s | elapsed = 1min8s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
+    #0: ToddCoxeter: LOOKAHEAD 0.1.0  |         active |            killed |          defined
+    #0: ToddCoxeter: nodes            |          4,600 |       112,320,034 |      112,324,634
+    #0: ToddCoxeter:                  |         active |           missing |       % complete
+    #0: ToddCoxeter: edges            |         13,800 |                 0 |           100.0%
+    #0: ToddCoxeter: time             | run 0 = 1min8s | all runs = 1min8s | elapsed = 1min8s
+    #0: ToddCoxeter: triggered because there are skipped definitions (4,600 active nodes)!
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 STOP
+    #0: ToddCoxeter: LOOKAHEAD 0.1.1   |         active |            killed |          defined
+    #0: ToddCoxeter: nodes             |          2,300 |       112,322,334 |      112,324,634
+    #0: ToddCoxeter: diff 0.1.0        |         -2,300 |            +2,300 |               +0
+    #0: ToddCoxeter:                   |         active |           missing |       % complete
+    #0: ToddCoxeter: edges             |          6,900 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.1.0        |         -6,900 |                +0 |            +0.0%
+    #0: ToddCoxeter: phase 0.1 = 705µs | run 0 = 1min8s | all runs = 1min8s | elapsed = 1min8s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 4,600, m = 10,000) (-4,990,000)
+    #0: ToddCoxeter: because f x a < n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 2,300
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.2      |         active |            killed |          defined
+    #0: ToddCoxeter: nodes             |          2,300 |       112,322,334 |      112,324,634
+    #0: ToddCoxeter: diff 0.2.1        |             +0 |                +0 |               +0
+    #0: ToddCoxeter: diff 0.2.0        |         -2,300 |            +2,300 |               +0
+    #0: ToddCoxeter:                   |         active |           missing |       % complete
+    #0: ToddCoxeter: edges             |          6,900 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.2.1        |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: diff 0.2.0        |         -6,900 |                +0 |            +0.0%
+    #0: ToddCoxeter: phase 0.2 = 730µs | run 0 = 1min8s | all runs = 1min8s | elapsed = 1min8s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |      lookahead |        lookbehind |              hlt |        felsch
+    #0: ToddCoxeter: num. phases          |              1 |                 0 |                0 |             1
+    #0: ToddCoxeter: time spent in phases |     727µs (0%) |            - (0%) |           - (0%) | 1min8s (100%)
+    #0: ToddCoxeter: phase 0.2 = 752µs    | run 0 = 1min8s | all runs = 1min8s | elapsed = 1min8s
+    The index of the subgroup is 2300
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $2,300$.
diff --git a/docs/examples/atlas/leech-lattice/co2/index.md b/docs/examples/atlas/leech-lattice/co2/index.md
index 8f49820..0c59de2 100644
--- a/docs/examples/atlas/leech-lattice/co2/index.md
+++ b/docs/examples/atlas/leech-lattice/co2/index.md
@@ -1,10 +1,31 @@
 # Conway group Co~2~
 
-> Order: $42,305,421,312,000$
+On this page, we provide links to verifications that the generators of maximal
+subgroups of the Conway group Co~2~ from the ATLAS generate subgroups of the
+correct index.
 
-> Presentation:
-> $\langle a, b \mid a^2 = b^5 = (abb)^9 = [a, b]^4 = [a, b^2]^4 = [a, bab]^3 = [a, bab^2ab]^2 = [a, bab^{−2}]^3 = [a, b^{−2}abab^{−2}]^2 = (abab^2ab^{−1}ab^{−2})^7 = 1 \rangle$
+<div class="center-table" markdown>
 
-!!! info "Work in Progress"
+| Subgroup | Subgroup order | Index in Co~2~ | Verification |
+| :------- | -------------: | -------------: | :----------: |
+| [$U_6(2):2$](2300.md) | $18,393,661,440$ | $2,300$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
+| $2^{10}:\mathrm{M}_{22}:2$ | $908,328,960$ | $46,575$ | :material-close-circle-outline:{ style="color: #f44336" } |
+| $\mathrm{McL}$ | $898,128,000$ | $47,104$ | :material-close-circle-outline:{ style="color: #f44336" } |
+| $2^{1+8}.S_6(2)$ | $743,178,240$ | $56,925$ | :material-help-circle-outline: |
+| $\mathrm{HS}:2$ | $88,704,000$ | $476,928$ | :material-help-circle-outline: |
+| $(2^4 \times 2^{1+6}).A_8$ | $41,287,680$ | $1,024,650$ | :material-help-circle-outline: |
+| $U_4(3):D_8$ | $26,127,360$ | $1,619,200$ | :material-help-circle-outline: |
+| $2^{4+10}.(S_5 \times S_3)$ | $11,796,480$ | $3,586,275$ | :material-help-circle-outline: |
+| $\mathrm{M}_{23}$ | $10,200,960$ | $4,147,200$ | :material-help-circle-outline: |
+| $3^{1+4}.2^{1+4}.S_5$ | $933,120$ | $45,337,600$ | :material-help-circle-outline: |
+| $5^{1+2}:4S_4$ | $12,000$ | $3,525,451,776$ | :material-help-circle-outline: |
 
-    This page is a work in progress.
+**Key**
+
+| Symbol | Meaning |
+| :----: | :------ |
+| :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | We could verify the index. |
+| :material-close-circle-outline:{ style="color: #f44336" } | The computation did not terminate, so we were unable to verify the index. |
+| :material-help-circle-outline: | We have not yet attempted the computation. |
+
+</div>
diff --git a/docs/examples/atlas/mathieu/m23/index.md b/docs/examples/atlas/mathieu/m23/index.md
index 270bc16..ba46a36 100644
--- a/docs/examples/atlas/mathieu/m23/index.md
+++ b/docs/examples/atlas/mathieu/m23/index.md
@@ -1,8 +1,15 @@
 # Mathieu group M~23~
 
-On this page, we provide links to verifications that the maximal subgroups of
-the Mathieu group M~23~ define groups of the correct index. Two presentations
-are given for M~23~, and two maximal subgroups have generating pairs specified.
+On this page, we provide links to verifications of two presentations of the
+Mathieu group M~23~, and to attempts to verify the indices of two of its maximal
+subgroups.
+
+## Presentations
+
+- [Presentation 1](p1.md)
+  :material-checkbox-marked-circle-outline:{ style="color: #00c853" }
+- [Presentation 2](p2.md)
+  :material-checkbox-marked-circle-outline:{ style="color: #00c853" }
 
 ## Maximal subgroups
 
diff --git a/docs/examples/atlas/mathieu/m23/p1.md b/docs/examples/atlas/mathieu/m23/p1.md
new file mode 100644
index 0000000..2963a77
--- /dev/null
+++ b/docs/examples/atlas/mathieu/m23/p1.md
@@ -0,0 +1,167 @@
+# Mathieu group M~23~ (Presentation 1)
+
+The claims on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/v3/pres/M23G1-P1](https://brauer.maths.qmul.ac.uk/Atlas/v3/pres/M23G1-P1).
+
+The Python script includes the additional relation
+$(abab^2ab^2)^6 = 1$, which its source marks as redundant and useful. We do
+not verify that this relation is redundant here.
+
+## Claim
+
+$$
+    \mathrm{M}_{23} = \langle a, b \mid a^2 = b^4 = (ab)^{23}
+    = (ab^2)^6 = [a, b]^6 = (abab^{-1}ab^2)^4
+    = (ab)^3ab^{-1}ab^2(abab^{-1})^2(ab)^3(ab^{-1})^3
+    = (abab^2)^3(ab^2ab^{-1})^2abab^2abab^{-1}ab^2 = 1 \rangle
+$$
+
+with
+
+$$
+    |\mathrm{M}_{23}| = 10,200,960.
+$$
+
+On this page, we verify that the presentation used by the script defines a
+group of order $10,200,960$.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for M~23~ and runs the Todd-Coxeter algorithm.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse_relations("a^2"), "")
+presentation.add_rule(p, parse_relations("b^4"), "")
+presentation.add_rule(p, parse_relations("(ab)^23"), "")
+presentation.add_rule(p, parse_relations("(ab^2)^6"), "")
+presentation.add_rule(p, parse_relations("(a,b)^6"), "")
+presentation.add_rule(p, parse_relations("(abaBab^2)^4"), "")
+presentation.add_rule(p, parse_relations("(ab)^3aBab^2(abaB)^2(ab)^3(aB)^3"), "")
+presentation.add_rule(
+    p, parse_relations("(abab^2ab^2)^6"), ""
+)  # Is redundant, but very useful.
+presentation.add_rule(p, parse_relations("(abab^2)^3(ab^2aB)^2abab^2abaBab^2"), "")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("Bab")
+
+# Run the Todd-Coxeter algorithm
+tc = ToddCoxeter(congruence_kind.twosided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# Takes approx. 2.5 minutes
+
+print(f"The size of the group defined by the presentation is {tc.number_of_classes()}")
+```
+
+## The output
+
+The truncated output of the enumeration is below:
+
+??? info "Truncated output from the Python script"
+
+    ```
+    Bab
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 13, |u| + |v| ∈ [2, 48], ∑(|u| + |v|) = 246
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |        defined
+    #0: ToddCoxeter: nodes            |            1 |               0 |              1
+    #0: ToddCoxeter:                  |       active |         missing |     % complete
+    #0: ToddCoxeter: edges            |            0 |               3 |           0.0%
+    #0: ToddCoxeter: time             | run 0 = 22µs | all runs = 22µs | elapsed = 66µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      1,822,250 |            36,043 |        1,858,318
+    #1: ToddCoxeter: diff 0.0.0         |     +1,822,249 |           +36,043 |       +1,858,317
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      4,253,812 |         1,212,938 |            77.8%
+    #1: ToddCoxeter: diff 0.0.0         |     +4,253,812 |        +1,212,935 |           +77.8%
+    #1: ToddCoxeter: phase 0.0 = 1.000s | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.000s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      3,655,607 |            83,517 |        3,739,139
+    #1: ToddCoxeter: diff 0.0.1         |     +1,833,357 |           +47,474 |       +1,880,821
+    #1: ToddCoxeter: diff 0.0.0         |     +3,655,606 |           +83,517 |       +3,739,138
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      8,538,514 |         2,428,307 |            77.9%
+    #1: ToddCoxeter: diff 0.0.1         |     +4,284,702 |        +1,215,369 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |     +8,538,514 |        +2,428,304 |           +77.9%
+    #1: ToddCoxeter: phase 0.0 = 2.001s | run 0 = 2.001s | all runs = 2.001s | elapsed = 2.001s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.3       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      5,466,932 |           137,547 |        5,604,495
+    #1: ToddCoxeter: diff 0.0.2         |     +1,811,325 |           +54,030 |       +1,865,356
+    #1: ToddCoxeter: diff 0.0.0         |     +5,466,931 |          +137,547 |       +5,604,494
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     12,774,618 |         3,626,178 |            77.9%
+    #1: ToddCoxeter: diff 0.0.2         |     +4,236,104 |        +1,197,871 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +12,774,618 |        +3,626,175 |           +77.9%
+    #1: ToddCoxeter: phase 0.0 = 3.006s | run 0 = 3.006s | all runs = 3.006s | elapsed = 3.006s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.4       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      7,294,223 |           190,832 |        7,485,071
+    #1: ToddCoxeter: diff 0.0.3         |     +1,827,291 |           +53,285 |       +1,880,576
+    #1: ToddCoxeter: diff 0.0.0         |     +7,294,222 |          +190,832 |       +7,485,070
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     17,048,446 |         4,834,223 |            77.9%
+    #1: ToddCoxeter: diff 0.0.3         |     +4,273,828 |        +1,208,045 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +17,048,446 |        +4,834,220 |           +77.9%
+    #1: ToddCoxeter: phase 0.0 = 4.011s | run 0 = 4.011s | all runs = 4.011s | elapsed = 4.011s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.5       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      9,065,055 |           235,680 |        9,300,749
+    #1: ToddCoxeter: diff 0.0.4         |     +1,770,832 |           +44,848 |       +1,815,678
+    #1: ToddCoxeter: diff 0.0.0         |     +9,065,054 |          +235,680 |       +9,300,748
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    [... lines omitted ...]
+    #0: ToddCoxeter: diff 0.1.51         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.1.0          |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                     |          active |            missing |        % complete
+    #0: ToddCoxeter: edges               |      30,602,880 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.1.51         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.1.0          |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.1 = 51.591s | run 0 = 2min29s | all runs = 2min29s | elapsed = 2min29s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 20,401,920, m = 10,000) (+15,401,920)
+    #0: ToddCoxeter: because a > n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 10,200,960
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.53       |          active |             killed |           defined
+    #0: ToddCoxeter: nodes               |      10,200,960 |         33,207,284 |        43,408,244
+    #0: ToddCoxeter: diff 0.2.52         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.2.0          |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                     |          active |            missing |        % complete
+    #0: ToddCoxeter: edges               |      30,602,880 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.2.52         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.2.0          |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.2 = 51.591s | run 0 = 2min29s | all runs = 2min29s | elapsed = 2min29s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |       lookahead |         lookbehind |               hlt |        felsch
+    #0: ToddCoxeter: num. phases          |               1 |                  0 |                 0 |             1
+    #0: ToddCoxeter: time spent in phases |   51.591s (35%) |             - (0%) |            - (0%) | 1min37s (65%)
+    #0: ToddCoxeter: phase 0.2 = 51.708s  | run 0 = 2min29s | all runs = 2min29s | elapsed = 2min29s
+    The size of the group defined by the presentation is 10200960
+    ```
+
+:material-checkbox-marked-circle-outline: The computed size of the group
+matches the claimed size: $10,200,960$.
diff --git a/docs/examples/atlas/mathieu/m23/p2.md b/docs/examples/atlas/mathieu/m23/p2.md
new file mode 100644
index 0000000..9b35075
--- /dev/null
+++ b/docs/examples/atlas/mathieu/m23/p2.md
@@ -0,0 +1,172 @@
+# Mathieu group M~23~ (Presentation 2)
+
+The claims on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/v3/pres/M23G1-P2](https://brauer.maths.qmul.ac.uk/Atlas/v3/pres/M23G1-P2).
+
+The Python script includes the additional relation
+$[a,b]^6 = 1$, which does not appear in the presentation displayed on the
+ATLAS page. We do not verify that this relation is redundant here.
+
+## Claim
+
+$$
+    \mathrm{M}_{23} = \langle a, b \mid a^2 = b^4 = (ab^2)^6
+    = (abab^{-1}ab^2)^4
+    = abababab^{-1}ab^2abab^{-1}abab^{-1}abababab^{-1}ab^{-1}ab^{-1}
+    = abab^2abab^2abab^2ab^2ab^{-1}ab^2ab^{-1}abab^2abab^{-1}ab^2
+    = abab^2ab^2abab^2ab^2abab^2ab^2abab^2ab^2ab^{-1}ab^2ab^2ab^{-1}ab^2ab^2
+    = ababababab^2abab^{-1}abab^2ababab^{-1}ab^2abab^2ab^2abab^{-1}ab^{-1}abab^2
+    = 1 \rangle
+$$
+
+with
+
+$$
+    |\mathrm{M}_{23}| = 10,200,960.
+$$
+
+On this page, we verify that the presentation used by the script defines a
+group of order $10,200,960$.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for M~23~ and runs the Todd-Coxeter algorithm.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse_relations("a^2"), "")
+presentation.add_rule(p, parse_relations("b^4"), "")
+presentation.add_rule(p, parse_relations("(ab^2)^6"), "")
+presentation.add_rule(p, parse_relations("(a,b)^6"), "")
+presentation.add_rule(p, parse_relations("(abaBab^2)^4"), "")
+presentation.add_rule(p, parse_relations("abababaBab^2abaBabaBabababaBaBaB"), "")
+presentation.add_rule(
+    p, parse_relations("abab^2abab^2abab^2ab^2aBab^2aBabab^2abaBab^2"), ""
+)
+presentation.add_rule(
+    p,
+    parse_relations("abab^2ab^2abab^2ab^2abab^2ab^2abab^2ab^2aBab^2ab^2aBab^2ab^2"),
+    "",
+)
+presentation.add_rule(
+    p, parse_relations("ababababab^2abaBabab^2ababaBab^2abab^2ab^2abaBaBabab^2"), ""
+)
+
+# Run the Todd-Coxeter algorithm
+tc = ToddCoxeter(congruence_kind.twosided, p)
+tc.strategy(tc.options.strategy.felsch)
+
+print(f"The size of the group defined by the presentation is {tc.number_of_classes()}")
+```
+
+## The output
+
+The truncated output of the enumeration is below:
+
+??? info "Truncated output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 4, |R| = 13, |u| + |v| ∈ [2, 48], ∑(|u| + |v|) = 248
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |        defined
+    #0: ToddCoxeter: nodes            |            1 |               0 |              1
+    #0: ToddCoxeter:                  |       active |         missing |     % complete
+    #0: ToddCoxeter: edges            |            0 |               4 |           0.0%
+    #0: ToddCoxeter: time             | run 0 = 26µs | all runs = 26µs | elapsed = 72µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      1,275,486 |            25,934 |        1,301,428
+    #1: ToddCoxeter: diff 0.0.0         |     +1,275,485 |           +25,934 |       +1,301,427
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      3,967,993 |         1,133,951 |            77.8%
+    #1: ToddCoxeter: diff 0.0.0         |     +3,967,993 |        +1,133,947 |           +77.8%
+    #1: ToddCoxeter: phase 0.0 = 1.005s | run 0 = 1.005s | all runs = 1.005s | elapsed = 1.005s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      2,506,185 |            60,138 |        2,566,339
+    #1: ToddCoxeter: diff 0.0.1         |     +1,230,699 |           +34,204 |       +1,264,911
+    #1: ToddCoxeter: diff 0.0.0         |     +2,506,184 |           +60,138 |       +2,566,338
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      7,801,046 |         2,223,694 |            77.8%
+    #1: ToddCoxeter: diff 0.0.1         |     +3,833,053 |        +1,089,743 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |     +7,801,046 |        +2,223,690 |           +77.8%
+    #1: ToddCoxeter: phase 0.0 = 2.010s | run 0 = 2.010s | all runs = 2.010s | elapsed = 2.010s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.3       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      3,727,047 |            96,273 |        3,823,324
+    #1: ToddCoxeter: diff 0.0.2         |     +1,220,862 |           +36,135 |       +1,256,985
+    #1: ToddCoxeter: diff 0.0.0         |     +3,727,046 |           +96,273 |       +3,823,323
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     11,606,914 |         3,301,274 |            77.9%
+    #1: ToddCoxeter: diff 0.0.2         |     +3,805,868 |        +1,077,580 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +11,606,914 |        +3,301,270 |           +77.9%
+    #1: ToddCoxeter: phase 0.0 = 3.011s | run 0 = 3.011s | all runs = 3.011s | elapsed = 3.011s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.4       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      4,908,390 |           131,357 |        5,039,763
+    #1: ToddCoxeter: diff 0.0.3         |     +1,181,343 |           +35,084 |       +1,216,439
+    #1: ToddCoxeter: diff 0.0.0         |     +4,908,389 |          +131,357 |       +5,039,762
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     15,288,906 |         4,344,654 |            77.9%
+    #1: ToddCoxeter: diff 0.0.3         |     +3,681,992 |        +1,043,380 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +15,288,906 |        +4,344,650 |           +77.9%
+    #1: ToddCoxeter: phase 0.0 = 4.015s | run 0 = 4.015s | all runs = 4.015s | elapsed = 4.015s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.5       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      6,098,758 |           169,799 |        6,268,568
+    #1: ToddCoxeter: diff 0.0.4         |     +1,190,368 |           +38,442 |       +1,228,805
+    #1: ToddCoxeter: diff 0.0.0         |     +6,098,757 |          +169,799 |       +6,268,567
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     18,999,097 |         5,395,935 |            77.9%
+    [... lines omitted ...]
+    #0: ToddCoxeter: diff 0.1.53         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.1.0          |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                     |          active |            missing |        % complete
+    #0: ToddCoxeter: edges               |      40,803,840 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.1.53         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.1.0          |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.1 = 53.174s | run 0 = 4min38s | all runs = 4min38s | elapsed = 4min38s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 20,401,920, m = 10,000) (+15,401,920)
+    #0: ToddCoxeter: because a > n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 10,200,960
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.55       |          active |             killed |           defined
+    #0: ToddCoxeter: nodes               |      10,200,960 |         15,538,858 |        25,739,818
+    #0: ToddCoxeter: diff 0.2.54         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.2.0          |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                     |          active |            missing |        % complete
+    #0: ToddCoxeter: edges               |      40,803,840 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.2.54         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.2.0          |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.2 = 53.174s | run 0 = 4min38s | all runs = 4min38s | elapsed = 4min38s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |       lookahead |         lookbehind |               hlt |        felsch
+    #0: ToddCoxeter: num. phases          |               1 |                  0 |                 0 |             1
+    #0: ToddCoxeter: time spent in phases |   53.174s (19%) |             - (0%) |            - (0%) | 3min45s (81%)
+    #0: ToddCoxeter: phase 0.2 = 53.410s  | run 0 = 4min38s | all runs = 4min38s | elapsed = 4min38s
+    The size of the group defined by the presentation is 10200960
+    ```
+
+:material-checkbox-marked-circle-outline: The computed size of the group
+matches the claimed size: $10,200,960$.
diff --git a/docs/examples/atlas/mathieu/m24/index.md b/docs/examples/atlas/mathieu/m24.md
similarity index 99%
rename from docs/examples/atlas/mathieu/m24/index.md
rename to docs/examples/atlas/mathieu/m24.md
index 1157692..081cb6f 100644
--- a/docs/examples/atlas/mathieu/m24/index.md
+++ b/docs/examples/atlas/mathieu/m24.md
@@ -5,8 +5,8 @@ The claims on this page come from
 This is a link to a Magma file that contains a presentation (amongst other
 things). This presentation differs from the presentation provided in plain-text
 on the ATLAS webpage due to the inclusion of an additional relation. The
-comments in the Magma file claim that this additional relation is redundant, and
-we do not verify this claim.
+comments in the Magma file claim that this additional relation is redundant. We
+do not verify that this relation is redundant here.
 
 ## Claim
 
diff --git a/docs/examples/atlas/monster-sections/he/1142400.md b/docs/examples/atlas/monster-sections/he/1142400.md
new file mode 100644
index 0000000..eda06b7
--- /dev/null
+++ b/docs/examples/atlas/monster-sections/he/1142400.md
@@ -0,0 +1,183 @@
+# $7:3 \times L_3(2)$ as a subgroup of Held group He
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/).
+
+## Claim
+
+$$
+    \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17}
+    = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab]
+    = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$7:3 \times L_3(2)$. More precisely, if
+
+$$
+    H = \langle (q^{-1})^{11}(ab^2)^4q^{11},\ b^{-1}a(ab^2)^4ab,\ \text{ where }q=ababb \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{He} : H] = 1,142,400.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for He, adds the generating pairs that define the maximal subgroup, and runs the
+Todd-Coxeter algorithm. The source code marks three of the presentation
+relations as redundant. We do not verify that these relations are redundant
+here.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^7"), "")
+presentation.add_rule(p, parse("(ab)^17"), "")
+presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
+presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
+presentation.add_rule(p, parse("(a,babaBabab)"), "")
+presentation.add_rule(p, parse("(a,b^3)^5"), "")
+presentation.add_rule(p, parse("(a,b)^6"), "")
+presentation.add_rule(
+    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
+)  # redundant, maybe useful.
+presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("abB")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# The subgroup generators here are taken from one of the straight line
+# programmes from:
+# https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/
+
+# takes approx 7.5 minutes
+q = "ababb"
+Q = ("".join(reversed(q))).replace("b", "B")
+tc.add_generating_pair(parse(f"({Q})^11(ab^2)^4({q})^11"), "")
+tc.add_generating_pair(parse("Ba(ab^2)^4ab"), "")
+
+print(f"The index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The truncated output of the enumeration is below:
+
+??? info "Truncated output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |         defined
+    #0: ToddCoxeter: nodes            |           75 |              61 |             136
+    #0: ToddCoxeter:                  |       active |         missing |      % complete
+    #0: ToddCoxeter: edges            |          136 |              89 |           60.4%
+    #0: ToddCoxeter: time             | run 0 = 73µs | all runs = 73µs | elapsed = 295µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      2,480,272 |            49,631 |        2,529,943
+    #1: ToddCoxeter: diff 0.0.0         |     +2,480,197 |           +49,570 |       +2,529,807
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      5,256,084 |         2,184,732 |            70.6%
+    #1: ToddCoxeter: diff 0.0.0         |     +5,255,948 |        +2,184,643 |           +10.2%
+    #1: ToddCoxeter: phase 0.0 = 1.005s | run 0 = 1.005s | all runs = 1.005s | elapsed = 1.005s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      4,956,987 |           103,095 |        5,060,108
+    #1: ToddCoxeter: diff 0.0.1         |     +2,476,715 |           +53,464 |       +2,530,165
+    #1: ToddCoxeter: diff 0.0.0         |     +4,956,912 |          +103,034 |       +5,059,972
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     10,505,388 |         4,365,573 |            70.6%
+    #1: ToddCoxeter: diff 0.0.1         |     +5,249,304 |        +2,180,841 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +10,505,252 |        +4,365,484 |           +10.2%
+    #1: ToddCoxeter: phase 0.0 = 2.006s | run 0 = 2.006s | all runs = 2.006s | elapsed = 2.006s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.3       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      7,477,894 |           158,644 |        7,636,556
+    #1: ToddCoxeter: diff 0.0.2         |     +2,520,907 |           +55,549 |       +2,576,448
+    #1: ToddCoxeter: diff 0.0.0         |     +7,477,819 |          +158,583 |       +7,636,420
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     15,849,138 |         6,584,544 |            70.6%
+    #1: ToddCoxeter: diff 0.0.2         |     +5,343,750 |        +2,218,971 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +15,849,002 |        +6,584,455 |           +10.2%
+    #1: ToddCoxeter: phase 0.0 = 3.011s | run 0 = 3.011s | all runs = 3.011s | elapsed = 3.011s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.4       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      9,900,800 |           211,433 |       10,112,265
+    #1: ToddCoxeter: diff 0.0.3         |     +2,422,906 |           +52,789 |       +2,475,709
+    #1: ToddCoxeter: diff 0.0.0         |     +9,900,725 |          +211,372 |      +10,112,129
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     20,985,770 |         8,716,630 |            70.7%
+    #1: ToddCoxeter: diff 0.0.3         |     +5,136,632 |        +2,132,086 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +20,985,634 |        +8,716,541 |           +10.2%
+    #1: ToddCoxeter: phase 0.0 = 4.016s | run 0 = 4.016s | all runs = 4.016s | elapsed = 4.016s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.5       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |     12,440,550 |           269,907 |       12,710,479
+    #1: ToddCoxeter: diff 0.0.4         |     +2,539,750 |           +58,474 |       +2,598,214
+    #1: ToddCoxeter: diff 0.0.0         |    +12,440,475 |          +269,846 |      +12,710,343
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     26,370,604 |        10,951,046 |            70.7%
+    [... lines omitted ...]
+    #0: ToddCoxeter: diff 0.1.2         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.1.0         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                    |          active |            missing |        % complete
+    #0: ToddCoxeter: edges              |       3,427,200 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.1.2         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.1.0         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.1 = 2.053s | run 0 = 7min30s | all runs = 7min30s | elapsed = 7min30s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 2,284,800, m = 10,000) (-2,715,200)
+    #0: ToddCoxeter: because f x a < n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 1,142,400
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.4       |          active |             killed |           defined
+    #0: ToddCoxeter: nodes              |       1,142,400 |        780,127,944 |       781,270,344
+    #0: ToddCoxeter: diff 0.2.3         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.2.0         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                    |          active |            missing |        % complete
+    #0: ToddCoxeter: edges              |       3,427,200 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.2.3         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.2.0         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.2 = 2.053s | run 0 = 7min30s | all runs = 7min30s | elapsed = 7min30s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |       lookahead |         lookbehind |               hlt |         felsch
+    #0: ToddCoxeter: num. phases          |               1 |                  0 |                 0 |              1
+    #0: ToddCoxeter: time spent in phases |     2.053s (0%) |             - (0%) |            - (0%) | 7min28s (100%)
+    #0: ToddCoxeter: phase 0.2 = 2.061s   | run 0 = 7min30s | all runs = 7min30s | elapsed = 7min30s
+    The index of the subgroup is 1142400
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $1,142,400$.
diff --git a/docs/examples/atlas/monster-sections/he/187425.md b/docs/examples/atlas/monster-sections/he/187425.md
new file mode 100644
index 0000000..6ca21f8
--- /dev/null
+++ b/docs/examples/atlas/monster-sections/he/187425.md
@@ -0,0 +1,184 @@
+# $2^{1+6}.L_3(2)$ as a subgroup of Held group He
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/).
+
+## Claim
+
+$$
+    \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17}
+    = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab]
+    = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$2^{1+6}.L_3(2)$. More precisely, if
+
+$$
+    H = \langle ab^3,\ (pb^{-1}pb)^2(pbpb^{-1})^2,\ \text{ where }p=(ab^3)^4 \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{He} : H] = 187,425.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for He, adds the generating pairs that define the maximal subgroup, and runs the
+Todd-Coxeter algorithm. The source code marks three of the presentation
+relations as redundant. We do not verify that these relations are redundant
+here.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^7"), "")
+presentation.add_rule(p, parse("(ab)^17"), "")
+presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
+presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
+presentation.add_rule(p, parse("(a,babaBabab)"), "")
+presentation.add_rule(p, parse("(a,b^3)^5"), "")
+presentation.add_rule(p, parse("(a,b)^6"), "")
+presentation.add_rule(
+    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
+)  # redundant, maybe useful.
+presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("abB")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# The subgroup generators here are taken from one of the straight line
+# programmes from:
+# https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/
+
+# takes approx 4.5 minutes
+
+p = parse("(ab^3)^4")
+tc.add_generating_pair(parse("ab^3"), "")
+tc.add_generating_pair(parse(f"({p}B{p}b)^2({p}b{p}B)^2"), "")
+tc.use_relations_in_extra(True)
+
+print(f"The index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The truncated output of the enumeration is below:
+
+??? info "Truncated output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |        active |           killed |         defined
+    #0: ToddCoxeter: nodes            |           261 |               37 |             298
+    #0: ToddCoxeter:                  |        active |          missing |      % complete
+    #0: ToddCoxeter: edges            |           304 |              479 |           38.8%
+    #0: ToddCoxeter: time             | run 0 = 104µs | all runs = 104µs | elapsed = 342µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      2,426,227 |            52,073 |        2,478,337
+    #1: ToddCoxeter: diff 0.0.0         |     +2,425,966 |           +52,036 |       +2,478,039
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      5,141,618 |         2,137,063 |            70.6%
+    #1: ToddCoxeter: diff 0.0.0         |     +5,141,314 |        +2,136,584 |           +31.8%
+    #1: ToddCoxeter: phase 0.0 = 1.003s | run 0 = 1.003s | all runs = 1.003s | elapsed = 1.003s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      4,813,554 |           108,463 |        4,922,040
+    #1: ToddCoxeter: diff 0.0.1         |     +2,387,327 |           +56,390 |       +2,443,703
+    #1: ToddCoxeter: diff 0.0.0         |     +4,813,293 |          +108,426 |       +4,921,742
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     10,203,878 |         4,236,784 |            70.7%
+    #1: ToddCoxeter: diff 0.0.1         |     +5,062,260 |        +2,099,721 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +10,203,574 |        +4,236,305 |           +31.8%
+    #1: ToddCoxeter: phase 0.0 = 2.004s | run 0 = 2.004s | all runs = 2.004s | elapsed = 2.004s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.3       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      7,279,489 |           169,041 |        7,448,552
+    #1: ToddCoxeter: diff 0.0.2         |     +2,465,935 |           +60,578 |       +2,526,512
+    #1: ToddCoxeter: diff 0.0.0         |     +7,279,228 |          +169,004 |       +7,448,254
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     15,432,998 |         6,405,469 |            70.7%
+    #1: ToddCoxeter: diff 0.0.2         |     +5,229,120 |        +2,168,685 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +15,432,694 |        +6,404,990 |           +31.8%
+    #1: ToddCoxeter: phase 0.0 = 3.009s | run 0 = 3.009s | all runs = 3.009s | elapsed = 3.009s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.4       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      9,597,950 |           232,139 |        9,830,108
+    #1: ToddCoxeter: diff 0.0.3         |     +2,318,461 |           +63,098 |       +2,381,556
+    #1: ToddCoxeter: diff 0.0.0         |     +9,597,689 |          +232,102 |       +9,829,810
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     20,349,896 |         8,443,954 |            70.7%
+    #1: ToddCoxeter: diff 0.0.3         |     +4,916,898 |        +2,038,485 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +20,349,592 |        +8,443,475 |           +31.8%
+    #1: ToddCoxeter: phase 0.0 = 4.014s | run 0 = 4.014s | all runs = 4.014s | elapsed = 4.014s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.5       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |     12,053,029 |           297,869 |       12,350,930
+    #1: ToddCoxeter: diff 0.0.4         |     +2,455,079 |           +65,730 |       +2,520,822
+    #1: ToddCoxeter: diff 0.0.0         |    +12,052,768 |          +297,832 |      +12,350,632
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     25,556,108 |        10,602,979 |            70.7%
+    [... lines omitted ...]
+    #0: ToddCoxeter: LOOKAHEAD 0.1.1  |          active |             killed |           defined
+    #0: ToddCoxeter: nodes            |         187,425 |        513,212,960 |       513,400,385
+    #0: ToddCoxeter: diff 0.1.0       |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                  |          active |            missing |        % complete
+    #0: ToddCoxeter: edges            |         562,275 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.1.0       |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.1 = 74ms | run 0 = 4min37s | all runs = 4min37s | elapsed = 4min37s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 374,850, m = 10,000) (-4,625,150)
+    #0: ToddCoxeter: because f x a < n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 187,425
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.2     |          active |             killed |           defined
+    #0: ToddCoxeter: nodes            |         187,425 |        513,212,960 |       513,400,385
+    #0: ToddCoxeter: diff 0.2.1       |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.2.0       |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                  |          active |            missing |        % complete
+    #0: ToddCoxeter: edges            |         562,275 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.2.1       |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.2.0       |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.2 = 74ms | run 0 = 4min37s | all runs = 4min37s | elapsed = 4min37s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |       lookahead |         lookbehind |               hlt |         felsch
+    #0: ToddCoxeter: num. phases          |               1 |                  0 |                 0 |              1
+    #0: ToddCoxeter: time spent in phases |       74ms (0%) |             - (0%) |            - (0%) | 4min37s (100%)
+    #0: ToddCoxeter: phase 0.2 = 75ms     | run 0 = 4min37s | all runs = 4min37s | elapsed = 4min37s
+    The index of the subgroup is 187425
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $187,425$.
diff --git a/docs/examples/atlas/monster-sections/he/2058.md b/docs/examples/atlas/monster-sections/he/2058.md
new file mode 100644
index 0000000..bd11ae5
--- /dev/null
+++ b/docs/examples/atlas/monster-sections/he/2058.md
@@ -0,0 +1,177 @@
+# $S_4(4):2$ as a subgroup of Held group He
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/spor/He/mag/HeG1-P1.M](https://brauer.maths.qmul.ac.uk/Atlas/spor/He/mag/HeG1-P1.M).
+
+## Claim
+
+$$
+    \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17}
+    = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab]
+    = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$S_4(4):2$. More precisely, if
+
+$$
+    H = \langle a,\ b^{-2}ab^2,\ b^2ab^{-2} \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{He} : H] = 2,058.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for He, adds the generating pairs that define the maximal subgroup, and runs the
+Todd-Coxeter algorithm. The source code marks three of the presentation
+relations as redundant. We do not verify that these relations are redundant
+here.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^7"), "")
+presentation.add_rule(p, parse("(ab)^17"), "")
+presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
+presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
+presentation.add_rule(p, parse("(a,babaBabab)"), "")
+presentation.add_rule(p, parse("(a,b^3)^5"), "")
+presentation.add_rule(p, parse("(a,b)^6"), "")
+presentation.add_rule(
+    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
+)  # redundant, maybe useful.
+presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("abB")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# takes approx 1.6s
+
+# The generators for this subgroup are taken from:
+# https://brauer.maths.qmul.ac.uk/Atlas/spor/He/mag/HeG1-P1.M
+
+# The generators on:
+# https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/HeG1-max1W1
+# are:
+# a, (abb)-7(bababababbababb)7(abb)7
+# tc.add_generating_pair(parse("(BBa)^7(bababababbababb)^7(abb)^7"), "")
+# and in the straightline programme at:
+# https://brauer.maths.qmul.ac.uk/Atlas/spor/He/words/HeG1-max1W1
+# are:
+# a, and (abababbab)^10
+# which is altogether rather confusing...
+
+tc.add_generating_pair("a", "")
+tc.add_generating_pair("BBabb", "")
+tc.add_generating_pair("bbaBB", "")
+
+print(f"The index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The output of the enumeration is below:
+
+??? info "Output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |         defined
+    #0: ToddCoxeter: nodes            |            5 |               4 |               9
+    #0: ToddCoxeter:                  |       active |         missing |      % complete
+    #0: ToddCoxeter: edges            |           11 |               4 |           73.3%
+    #0: ToddCoxeter: time             | run 0 = 63µs | all runs = 63µs | elapsed = 111µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      2,389,517 |            86,354 |        2,475,902
+    #1: ToddCoxeter: diff 0.0.0         |     +2,389,512 |           +86,350 |       +2,475,893
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      5,071,468 |         2,097,083 |            70.7%
+    #1: ToddCoxeter: diff 0.0.0         |     +5,071,457 |        +2,097,079 |            -2.6%
+    #1: ToddCoxeter: phase 0.0 = 1.002s | run 0 = 1.002s | all runs = 1.002s | elapsed = 1.002s
+    #0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = large_collapse()!
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 STOP
+    #0: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #0: ToddCoxeter: nodes              |          2,058 |         3,621,394 |        3,623,452
+    #0: ToddCoxeter: diff 0.0.1         |     -2,387,459 |        +3,535,040 |       +1,147,550
+    #0: ToddCoxeter: diff 0.0.0         |         +2,053 |        +3,621,390 |       +3,623,443
+    #0: ToddCoxeter:                    |         active |           missing |       % complete
+    #0: ToddCoxeter: edges              |          6,174 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.0.1         |     -5,065,294 |        -2,097,083 |           +29.3%
+    #0: ToddCoxeter: diff 0.0.0         |         +6,163 |                -4 |           +26.7%
+    #0: ToddCoxeter: phase 0.0 = 1.564s | run 0 = 1.564s | all runs = 1.564s | elapsed = 1.564s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
+    #0: ToddCoxeter: LOOKAHEAD 0.1.0  |         active |            killed |          defined
+    #0: ToddCoxeter: nodes            |          2,058 |         3,621,394 |        3,623,452
+    #0: ToddCoxeter:                  |         active |           missing |       % complete
+    #0: ToddCoxeter: edges            |          6,174 |                 0 |           100.0%
+    #0: ToddCoxeter: time             | run 0 = 1.564s | all runs = 1.564s | elapsed = 1.564s
+    #0: ToddCoxeter: triggered because there are skipped definitions (2,058 active nodes)!
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 STOP
+    #0: ToddCoxeter: LOOKAHEAD 0.1.1   |         active |            killed |          defined
+    #0: ToddCoxeter: nodes             |          2,058 |         3,621,394 |        3,623,452
+    #0: ToddCoxeter: diff 0.1.0        |             +0 |                +0 |               +0
+    #0: ToddCoxeter:                   |         active |           missing |       % complete
+    #0: ToddCoxeter: edges             |          6,174 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.1.0        |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: phase 0.1 = 412µs | run 0 = 1.565s | all runs = 1.565s | elapsed = 1.565s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 4,116, m = 10,000) (-4,990,000)
+    #0: ToddCoxeter: because f x a < n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 2,058
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.2      |         active |            killed |          defined
+    #0: ToddCoxeter: nodes             |          2,058 |         3,621,394 |        3,623,452
+    #0: ToddCoxeter: diff 0.2.1        |             +0 |                +0 |               +0
+    #0: ToddCoxeter: diff 0.2.0        |             +0 |                +0 |               +0
+    #0: ToddCoxeter:                   |         active |           missing |       % complete
+    #0: ToddCoxeter: edges             |          6,174 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.2.1        |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: diff 0.2.0        |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: phase 0.2 = 435µs | run 0 = 1.565s | all runs = 1.565s | elapsed = 1.565s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |      lookahead |        lookbehind |              hlt |        felsch
+    #0: ToddCoxeter: num. phases          |              1 |                 0 |                0 |             1
+    #0: ToddCoxeter: time spent in phases |     433µs (0%) |            - (0%) |           - (0%) | 1.564s (100%)
+    #0: ToddCoxeter: phase 0.2 = 456µs    | run 0 = 1.565s | all runs = 1.565s | elapsed = 1.565s
+    The index of the subgroup is 2058
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $2,058$.
diff --git a/docs/examples/atlas/monster-sections/he/244800.md b/docs/examples/atlas/monster-sections/he/244800.md
new file mode 100644
index 0000000..c9e32dc
--- /dev/null
+++ b/docs/examples/atlas/monster-sections/he/244800.md
@@ -0,0 +1,163 @@
+# $7^2:2.L_2(7)$ as a subgroup of Held group He
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/spor/He/words/HeG1-max6W1](https://brauer.maths.qmul.ac.uk/Atlas/spor/He/words/HeG1-max6W1).
+
+## Claim
+
+$$
+    \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17}
+    = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab]
+    = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$7^2:2.L_2(7)$. More precisely, if
+
+$$
+    H = \langle (ab)^5b^2a,\ b \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{He} : H] = 244,800.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for He, adds the generating pairs that define the maximal subgroup, and runs the
+Todd-Coxeter algorithm. The source code marks three of the presentation
+relations as redundant. We do not verify that these relations are redundant
+here.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^7"), "")
+presentation.add_rule(p, parse("(ab)^17"), "")
+presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
+presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
+presentation.add_rule(p, parse("(a,babaBabab)"), "")
+presentation.add_rule(p, parse("(a,b^3)^5"), "")
+presentation.add_rule(p, parse("(a,b)^6"), "")
+presentation.add_rule(
+    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
+)  # redundant, maybe useful.
+presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("abB")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# takes approx 1.6s
+
+# The words are from:
+# https://brauer.maths.qmul.ac.uk/Atlas/spor/He/words/HeG1-max6W1
+tc.add_generating_pair(parse("(ab)^5b^2a"), "")
+tc.add_generating_pair("b", "")
+
+print(f"The by index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The output of the enumeration is below:
+
+??? info "Output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |        active |           killed |         defined
+    #0: ToddCoxeter: nodes            |            12 |                1 |              13
+    #0: ToddCoxeter:                  |        active |          missing |      % complete
+    #0: ToddCoxeter: edges            |            16 |               20 |           44.4%
+    #0: ToddCoxeter: time             | run 0 = 104µs | all runs = 104µs | elapsed = 314µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |        999,860 |         1,081,844 |        2,081,604
+    #1: ToddCoxeter: diff 0.0.0         |       +999,848 |        +1,081,843 |       +2,081,591
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      2,169,279 |           830,301 |            72.3%
+    #1: ToddCoxeter: diff 0.0.0         |     +2,169,263 |          +830,281 |           +27.9%
+    #1: ToddCoxeter: phase 0.0 = 1.005s | run 0 = 1.005s | all runs = 1.005s | elapsed = 1.005s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 STOP
+    #0: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #0: ToddCoxeter: nodes              |        244,800 |         1,839,761 |        2,084,561
+    #0: ToddCoxeter: diff 0.0.1         |       -755,060 |          +757,917 |           +2,957
+    #0: ToddCoxeter: diff 0.0.0         |       +244,788 |        +1,839,760 |       +2,084,548
+    #0: ToddCoxeter:                    |         active |           missing |       % complete
+    #0: ToddCoxeter: edges              |        734,400 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.0.1         |     -1,434,879 |          -830,301 |           +27.7%
+    #0: ToddCoxeter: diff 0.0.0         |       +734,384 |               -20 |           +55.6%
+    #0: ToddCoxeter: phase 0.0 = 1.515s | run 0 = 1.516s | all runs = 1.516s | elapsed = 1.516s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
+    #0: ToddCoxeter: LOOKAHEAD 0.1.0  |         active |            killed |          defined
+    #0: ToddCoxeter: nodes            |        244,800 |         1,839,761 |        2,084,561
+    #0: ToddCoxeter:                  |         active |           missing |       % complete
+    #0: ToddCoxeter: edges            |        734,400 |                 0 |           100.0%
+    #0: ToddCoxeter: time             | run 0 = 1.516s | all runs = 1.516s | elapsed = 1.516s
+    #0: ToddCoxeter: triggered because there are skipped definitions (244,800 active nodes)!
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 STOP
+    #0: ToddCoxeter: LOOKAHEAD 0.1.1   |         active |            killed |          defined
+    #0: ToddCoxeter: nodes             |        244,800 |         1,839,761 |        2,084,561
+    #0: ToddCoxeter: diff 0.1.0        |             +0 |                +0 |               +0
+    #0: ToddCoxeter:                   |         active |           missing |       % complete
+    #0: ToddCoxeter: edges             |        734,400 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.1.0        |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: phase 0.1 = 100ms | run 0 = 1.616s | all runs = 1.616s | elapsed = 1.616s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 489,600, m = 10,000) (-4,510,400)
+    #0: ToddCoxeter: because f x a < n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 244,800
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.2      |         active |            killed |          defined
+    #0: ToddCoxeter: nodes             |        244,800 |         1,839,761 |        2,084,561
+    #0: ToddCoxeter: diff 0.2.1        |             +0 |                +0 |               +0
+    #0: ToddCoxeter: diff 0.2.0        |             +0 |                +0 |               +0
+    #0: ToddCoxeter:                   |         active |           missing |       % complete
+    #0: ToddCoxeter: edges             |        734,400 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.2.1        |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: diff 0.2.0        |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: phase 0.2 = 100ms | run 0 = 1.616s | all runs = 1.616s | elapsed = 1.616s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |      lookahead |        lookbehind |              hlt |       felsch
+    #0: ToddCoxeter: num. phases          |              1 |                 0 |                0 |            1
+    #0: ToddCoxeter: time spent in phases |     100ms (6%) |            - (0%) |           - (0%) | 1.515s (94%)
+    #0: ToddCoxeter: phase 0.2 = 101ms    | run 0 = 1.617s | all runs = 1.617s | elapsed = 1.617s
+    The by index of the subgroup is 244800
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $244,800$.
diff --git a/docs/examples/atlas/monster-sections/he/266560.md b/docs/examples/atlas/monster-sections/he/266560.md
new file mode 100644
index 0000000..f13663b
--- /dev/null
+++ b/docs/examples/atlas/monster-sections/he/266560.md
@@ -0,0 +1,193 @@
+# $3.S_7$ as a subgroup of Held group He
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/).
+
+## Claim
+
+$$
+    \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17}
+    = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab]
+    = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$3.S_7$. More precisely, if
+
+$$
+    H = \langle a,\ b^3ab^2ab^{-1}ab^3 \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{He} : H] = 266,560.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for He, adds the generating pairs that define the maximal subgroup, and runs the
+Todd-Coxeter algorithm. The source code marks three of the presentation
+relations as redundant. We do not verify that these relations are redundant
+here.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^7"), "")
+presentation.add_rule(p, parse("(ab)^17"), "")
+presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
+presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
+presentation.add_rule(p, parse("(a,babaBabab)"), "")
+presentation.add_rule(p, parse("(a,b^3)^5"), "")
+presentation.add_rule(p, parse("(a,b)^6"), "")
+presentation.add_rule(
+    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
+)  # redundant, maybe useful.
+presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("abB")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# takes approx 5s
+
+# These generators are from the webpages of the ATLAS
+tc.add_generating_pair("a", "")
+tc.add_generating_pair(parse("b^3ab^2aBab^3"), "")
+
+print(f"The index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The output of the enumeration is below:
+
+??? info "Output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |         defined
+    #0: ToddCoxeter: nodes            |           12 |               0 |              12
+    #0: ToddCoxeter:                  |       active |         missing |      % complete
+    #0: ToddCoxeter: edges            |           14 |              22 |           38.9%
+    #0: ToddCoxeter: time             | run 0 = 52µs | all runs = 52µs | elapsed = 194µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      2,441,115 |            72,800 |        2,513,948
+    #1: ToddCoxeter: diff 0.0.0         |     +2,441,103 |           +72,800 |       +2,513,936
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      5,177,771 |         2,145,574 |            70.7%
+    #1: ToddCoxeter: diff 0.0.0         |     +5,177,757 |        +2,145,552 |           +31.8%
+    #1: ToddCoxeter: phase 0.0 = 1.005s | run 0 = 1.005s | all runs = 1.005s | elapsed = 1.005s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      4,760,613 |           184,070 |        4,944,702
+    #1: ToddCoxeter: diff 0.0.1         |     +2,319,498 |          +111,270 |       +2,430,754
+    #1: ToddCoxeter: diff 0.0.0         |     +4,760,601 |          +184,070 |       +4,944,690
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     10,102,170 |         4,179,669 |            70.7%
+    #1: ToddCoxeter: diff 0.0.1         |     +4,924,399 |        +2,034,095 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +10,102,156 |        +4,179,647 |           +31.8%
+    #1: ToddCoxeter: phase 0.0 = 2.009s | run 0 = 2.009s | all runs = 2.009s | elapsed = 2.009s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.3       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      7,003,728 |           378,125 |        7,381,870
+    #1: ToddCoxeter: diff 0.0.2         |     +2,243,115 |          +194,055 |       +2,437,168
+    #1: ToddCoxeter: diff 0.0.0         |     +7,003,716 |          +378,125 |       +7,381,858
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     14,866,661 |         6,144,523 |            70.8%
+    #1: ToddCoxeter: diff 0.0.2         |     +4,764,491 |        +1,964,854 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +14,866,647 |        +6,144,501 |           +31.9%
+    #1: ToddCoxeter: phase 0.0 = 3.014s | run 0 = 3.014s | all runs = 3.014s | elapsed = 3.014s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.4       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      8,202,622 |         1,337,951 |        9,540,372
+    #1: ToddCoxeter: diff 0.0.3         |     +1,198,894 |          +959,826 |       +2,158,502
+    #1: ToddCoxeter: diff 0.0.0         |     +8,202,610 |        +1,337,951 |       +9,540,360
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     17,363,740 |         7,244,126 |            70.6%
+    #1: ToddCoxeter: diff 0.0.3         |     +2,497,079 |        +1,099,603 |            -0.2%
+    #1: ToddCoxeter: diff 0.0.0         |    +17,363,726 |        +7,244,104 |           +31.7%
+    #1: ToddCoxeter: phase 0.0 = 4.019s | run 0 = 4.019s | all runs = 4.019s | elapsed = 4.020s
+    #0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = large_collapse()!
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 STOP
+    #0: ToddCoxeter: FELSCH 0.0.5       |         active |            killed |          defined
+    #0: ToddCoxeter: nodes              |        266,560 |         9,273,871 |        9,540,431
+    #0: ToddCoxeter: diff 0.0.4         |     -7,936,062 |        +7,935,920 |              +59
+    #0: ToddCoxeter: diff 0.0.0         |       +266,548 |        +9,273,871 |       +9,540,419
+    #0: ToddCoxeter:                    |         active |           missing |       % complete
+    #0: ToddCoxeter: edges              |        799,680 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.0.4         |    -16,564,060 |        -7,244,126 |           +29.4%
+    #0: ToddCoxeter: diff 0.0.0         |       +799,666 |               -22 |           +61.1%
+    #0: ToddCoxeter: phase 0.0 = 4.668s | run 0 = 4.668s | all runs = 4.668s | elapsed = 4.668s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
+    #0: ToddCoxeter: LOOKAHEAD 0.1.0  |         active |            killed |          defined
+    #0: ToddCoxeter: nodes            |        266,560 |         9,273,871 |        9,540,431
+    #0: ToddCoxeter:                  |         active |           missing |       % complete
+    #0: ToddCoxeter: edges            |        799,680 |                 0 |           100.0%
+    #0: ToddCoxeter: time             | run 0 = 4.668s | all runs = 4.668s | elapsed = 4.668s
+    #0: ToddCoxeter: triggered because there are skipped definitions (266,560 active nodes)!
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 STOP
+    #0: ToddCoxeter: LOOKAHEAD 0.1.1   |         active |            killed |          defined
+    #0: ToddCoxeter: nodes             |        266,560 |         9,273,871 |        9,540,431
+    #0: ToddCoxeter: diff 0.1.0        |             +0 |                +0 |               +0
+    #0: ToddCoxeter:                   |         active |           missing |       % complete
+    #0: ToddCoxeter: edges             |        799,680 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.1.0        |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: phase 0.1 = 113ms | run 0 = 4.781s | all runs = 4.781s | elapsed = 4.781s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 533,120, m = 10,000) (-4,466,880)
+    #0: ToddCoxeter: because f x a < n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 266,560
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.2      |         active |            killed |          defined
+    #0: ToddCoxeter: nodes             |        266,560 |         9,273,871 |        9,540,431
+    #0: ToddCoxeter: diff 0.2.1        |             +0 |                +0 |               +0
+    #0: ToddCoxeter: diff 0.2.0        |             +0 |                +0 |               +0
+    #0: ToddCoxeter:                   |         active |           missing |       % complete
+    #0: ToddCoxeter: edges             |        799,680 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.2.1        |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: diff 0.2.0        |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: phase 0.2 = 113ms | run 0 = 4.781s | all runs = 4.781s | elapsed = 4.781s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |      lookahead |        lookbehind |              hlt |       felsch
+    #0: ToddCoxeter: num. phases          |              1 |                 0 |                0 |            1
+    #0: ToddCoxeter: time spent in phases |     113ms (2%) |            - (0%) |           - (0%) | 4.668s (98%)
+    #0: ToddCoxeter: phase 0.2 = 114ms    | run 0 = 4.783s | all runs = 4.783s | elapsed = 4.783s
+    The index of the subgroup is 266560
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $266,560$.
diff --git a/docs/examples/atlas/monster-sections/he/29155-i.md b/docs/examples/atlas/monster-sections/he/29155-i.md
new file mode 100644
index 0000000..8870ecd
--- /dev/null
+++ b/docs/examples/atlas/monster-sections/he/29155-i.md
@@ -0,0 +1,182 @@
+# $2^6:3.S_6$ (I) as a subgroup of Held group He
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/).
+
+## Claim
+
+$$
+    \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17}
+    = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab]
+    = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$2^6:3.S_6$. More precisely, if
+
+$$
+    H = \langle a,\ ab^{-2}ab^3ab^{-3}ab^{-1} \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{He} : H] = 29,155.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for He, adds the generating pairs that define the maximal subgroup, and runs the
+Todd-Coxeter algorithm. The source code marks three of the presentation
+relations as redundant. We do not verify that these relations are redundant
+here.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^7"), "")
+presentation.add_rule(p, parse("(ab)^17"), "")
+presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
+presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
+presentation.add_rule(p, parse("(a,babaBabab)"), "")
+presentation.add_rule(p, parse("(a,b^3)^5"), "")
+presentation.add_rule(p, parse("(a,b)^6"), "")
+presentation.add_rule(
+    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
+)  # redundant, maybe useful.
+presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("abB")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# The subgroup generators here are taken from one of the straight line
+# programmes from:
+# https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/
+
+# takes approx 20s
+
+tc.add_generating_pair("a", "")
+tc.add_generating_pair("aBBabbbaBBBaB", "")
+
+print(f"The index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The truncated output of the enumeration is below:
+
+??? info "Truncated output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |         defined
+    #0: ToddCoxeter: nodes            |           12 |               0 |              12
+    #0: ToddCoxeter:                  |       active |         missing |      % complete
+    #0: ToddCoxeter: edges            |           14 |              22 |           38.9%
+    #0: ToddCoxeter: time             | run 0 = 59µs | all runs = 59µs | elapsed = 110µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      2,473,292 |            58,553 |        2,531,861
+    #1: ToddCoxeter: diff 0.0.0         |     +2,473,280 |           +58,553 |       +2,531,849
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      5,243,526 |         2,176,350 |            70.7%
+    #1: ToddCoxeter: diff 0.0.0         |     +5,243,512 |        +2,176,328 |           +31.8%
+    #1: ToddCoxeter: phase 0.0 = 1.005s | run 0 = 1.005s | all runs = 1.005s | elapsed = 1.005s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      4,907,283 |           123,428 |        5,030,726
+    #1: ToddCoxeter: diff 0.0.1         |     +2,433,991 |           +64,875 |       +2,498,865
+    #1: ToddCoxeter: diff 0.0.0         |     +4,907,271 |          +123,428 |       +5,030,714
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     10,405,718 |         4,316,131 |            70.7%
+    #1: ToddCoxeter: diff 0.0.1         |     +5,162,192 |        +2,139,781 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +10,405,704 |        +4,316,109 |           +31.8%
+    #1: ToddCoxeter: phase 0.0 = 2.010s | run 0 = 2.010s | all runs = 2.010s | elapsed = 2.010s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.3       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      7,356,730 |           202,010 |        7,558,764
+    #1: ToddCoxeter: diff 0.0.2         |     +2,449,447 |           +78,582 |       +2,528,038
+    #1: ToddCoxeter: diff 0.0.0         |     +7,356,718 |          +202,010 |       +7,558,752
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     15,602,237 |         6,467,953 |            70.7%
+    #1: ToddCoxeter: diff 0.0.2         |     +5,196,519 |        +2,151,822 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +15,602,223 |        +6,467,931 |           +31.8%
+    #1: ToddCoxeter: phase 0.0 = 3.015s | run 0 = 3.015s | all runs = 3.015s | elapsed = 3.015s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.4       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      9,645,930 |           298,259 |        9,944,212
+    #1: ToddCoxeter: diff 0.0.3         |     +2,289,200 |           +96,249 |       +2,385,448
+    #1: ToddCoxeter: diff 0.0.0         |     +9,645,918 |          +298,259 |       +9,944,200
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     20,461,214 |         8,476,576 |            70.7%
+    #1: ToddCoxeter: diff 0.0.3         |     +4,858,977 |        +2,008,623 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +20,461,200 |        +8,476,554 |           +31.8%
+    #1: ToddCoxeter: phase 0.0 = 4.020s | run 0 = 4.020s | all runs = 4.020s | elapsed = 4.020s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.5       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |     12,015,424 |           393,537 |       12,408,973
+    #1: ToddCoxeter: diff 0.0.4         |     +2,369,494 |           +95,278 |       +2,464,761
+    #1: ToddCoxeter: diff 0.0.0         |    +12,015,412 |          +393,537 |      +12,408,961
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     25,491,801 |        10,554,471 |            70.7%
+    [... lines omitted ...]
+    #0: ToddCoxeter: LOOKAHEAD 0.1.1  |          active |             killed |           defined
+    #0: ToddCoxeter: nodes            |          29,155 |         44,559,134 |        44,588,289
+    #0: ToddCoxeter: diff 0.1.0       |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                  |          active |            missing |        % complete
+    #0: ToddCoxeter: edges            |          87,465 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.1.0       |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.1 = 8ms  | run 0 = 20.191s | all runs = 20.191s | elapsed = 20.192s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 58,310, m = 10,000) (-4,941,690)
+    #0: ToddCoxeter: because f x a < n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 29,155
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.2     |          active |             killed |           defined
+    #0: ToddCoxeter: nodes            |          29,155 |         44,559,134 |        44,588,289
+    #0: ToddCoxeter: diff 0.2.1       |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.2.0       |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                  |          active |            missing |        % complete
+    #0: ToddCoxeter: edges            |          87,465 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.2.1       |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.2.0       |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.2 = 8ms  | run 0 = 20.191s | all runs = 20.191s | elapsed = 20.192s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |       lookahead |         lookbehind |               hlt |         felsch
+    #0: ToddCoxeter: num. phases          |               1 |                  0 |                 0 |              1
+    #0: ToddCoxeter: time spent in phases |        8ms (0%) |             - (0%) |            - (0%) | 20.183s (100%)
+    #0: ToddCoxeter: phase 0.2 = 8ms      | run 0 = 20.192s | all runs = 20.192s | elapsed = 20.192s
+    The index of the subgroup is 29155
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $29,155$.
diff --git a/docs/examples/atlas/monster-sections/he/29155-ii.md b/docs/examples/atlas/monster-sections/he/29155-ii.md
new file mode 100644
index 0000000..3c317e5
--- /dev/null
+++ b/docs/examples/atlas/monster-sections/he/29155-ii.md
@@ -0,0 +1,206 @@
+# $2^6:3.S_6$ (II) as a subgroup of Held group He
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/).
+
+## Claim
+
+$$
+    \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17}
+    = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab]
+    = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$2^6:3.S_6$. More precisely, if
+
+$$
+    H = \langle a,\ ab^{-2}ab^2ab^{-3}ab^{-1} \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{He} : H] = 29,155.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for He, adds the generating pairs that define the maximal subgroup, and runs the
+Todd-Coxeter algorithm. The source code marks three of the presentation
+relations as redundant. We do not verify that these relations are redundant
+here.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^7"), "")
+presentation.add_rule(p, parse("(ab)^17"), "")
+presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
+presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
+presentation.add_rule(p, parse("(a,babaBabab)"), "")
+presentation.add_rule(p, parse("(a,b^3)^5"), "")
+presentation.add_rule(p, parse("(a,b)^6"), "")
+presentation.add_rule(
+    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
+)  # redundant, maybe useful.
+presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("abB")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# The subgroup generators here are taken from one of the straight line
+# programmes from:
+# https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/
+
+# takes approx 5s
+
+tc.add_generating_pair("a", "")
+tc.add_generating_pair("aBBabbaBBBaB", "")
+
+print(f"The index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The output of the enumeration is below:
+
+??? info "Output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |         defined
+    #0: ToddCoxeter: nodes            |           11 |               0 |              11
+    #0: ToddCoxeter:                  |       active |         missing |      % complete
+    #0: ToddCoxeter: edges            |           13 |              20 |           39.4%
+    #0: ToddCoxeter: time             | run 0 = 55µs | all runs = 55µs | elapsed = 103µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      2,450,610 |            56,933 |        2,507,577
+    #1: ToddCoxeter: diff 0.0.0         |     +2,450,599 |           +56,933 |       +2,507,566
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      5,195,501 |         2,156,329 |            70.7%
+    #1: ToddCoxeter: diff 0.0.0         |     +5,195,488 |        +2,156,309 |           +31.3%
+    #1: ToddCoxeter: phase 0.0 = 1.005s | run 0 = 1.005s | all runs = 1.005s | elapsed = 1.005s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      4,857,043 |           122,296 |        4,979,354
+    #1: ToddCoxeter: diff 0.0.1         |     +2,406,433 |           +65,363 |       +2,471,777
+    #1: ToddCoxeter: diff 0.0.0         |     +4,857,032 |          +122,296 |       +4,979,343
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     10,299,694 |         4,271,435 |            70.7%
+    #1: ToddCoxeter: diff 0.0.1         |     +5,104,193 |        +2,115,106 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +10,299,681 |        +4,271,415 |           +31.3%
+    #1: ToddCoxeter: phase 0.0 = 2.010s | run 0 = 2.010s | all runs = 2.010s | elapsed = 2.010s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.3       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      7,241,525 |           226,832 |        7,468,390
+    #1: ToddCoxeter: diff 0.0.2         |     +2,384,482 |          +104,536 |       +2,489,036
+    #1: ToddCoxeter: diff 0.0.0         |     +7,241,514 |          +226,832 |       +7,468,379
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     15,360,734 |         6,363,841 |            70.7%
+    #1: ToddCoxeter: diff 0.0.2         |     +5,061,040 |        +2,092,406 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +15,360,721 |        +6,363,821 |           +31.3%
+    #1: ToddCoxeter: phase 0.0 = 3.015s | run 0 = 3.015s | all runs = 3.015s | elapsed = 3.015s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.4       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      9,527,319 |           309,104 |        9,836,440
+    #1: ToddCoxeter: diff 0.0.3         |     +2,285,794 |           +82,272 |       +2,368,050
+    #1: ToddCoxeter: diff 0.0.0         |     +9,527,308 |          +309,104 |       +9,836,429
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     20,212,108 |         8,369,849 |            70.7%
+    #1: ToddCoxeter: diff 0.0.3         |     +4,851,374 |        +2,006,008 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +20,212,095 |        +8,369,829 |           +31.3%
+    #1: ToddCoxeter: phase 0.0 = 4.020s | run 0 = 4.020s | all runs = 4.020s | elapsed = 4.020s
+    #0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = large_collapse()!
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.5       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |         45,650 |        11,465,859 |       11,511,509
+    #1: ToddCoxeter: diff 0.0.4         |     -9,481,669 |       +11,156,755 |       +1,675,069
+    #1: ToddCoxeter: diff 0.0.0         |        +45,639 |       +11,465,859 |      +11,511,498
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |         70,390 |            66,560 |            51.4%
+    #1: ToddCoxeter: diff 0.0.4         |    -20,141,718 |        -8,303,289 |           -19.3%
+    #1: ToddCoxeter: diff 0.0.0         |        +70,377 |           +66,540 |           +12.0%
+    #1: ToddCoxeter: phase 0.0 = 5.025s | run 0 = 5.025s | all runs = 5.025s | elapsed = 5.025s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 STOP
+    #0: ToddCoxeter: FELSCH 0.0.6       |         active |            killed |          defined
+    #0: ToddCoxeter: nodes              |         29,155 |        11,482,367 |       11,511,522
+    #0: ToddCoxeter: diff 0.0.5         |        -16,495 |           +16,508 |              +13
+    #0: ToddCoxeter: diff 0.0.0         |        +29,144 |       +11,482,367 |      +11,511,511
+    #0: ToddCoxeter:                    |         active |           missing |       % complete
+    #0: ToddCoxeter: edges              |         87,465 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.0.5         |        +17,075 |           -66,560 |           +48.6%
+    #0: ToddCoxeter: diff 0.0.0         |        +87,452 |               -20 |           +60.6%
+    #0: ToddCoxeter: phase 0.0 = 5.029s | run 0 = 5.029s | all runs = 5.029s | elapsed = 5.029s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
+    #0: ToddCoxeter: LOOKAHEAD 0.1.0  |         active |            killed |          defined
+    #0: ToddCoxeter: nodes            |         29,155 |        11,482,367 |       11,511,522
+    #0: ToddCoxeter:                  |         active |           missing |       % complete
+    #0: ToddCoxeter: edges            |         87,465 |                 0 |           100.0%
+    #0: ToddCoxeter: time             | run 0 = 5.029s | all runs = 5.029s | elapsed = 5.029s
+    #0: ToddCoxeter: triggered because there are skipped definitions (29,155 active nodes)!
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 STOP
+    #0: ToddCoxeter: LOOKAHEAD 0.1.1  |         active |            killed |          defined
+    #0: ToddCoxeter: nodes            |         29,155 |        11,482,367 |       11,511,522
+    #0: ToddCoxeter: diff 0.1.0       |             +0 |                +0 |               +0
+    #0: ToddCoxeter:                  |         active |           missing |       % complete
+    #0: ToddCoxeter: edges            |         87,465 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.1.0       |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: phase 0.1 = 9ms  | run 0 = 5.038s | all runs = 5.038s | elapsed = 5.039s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 58,310, m = 10,000) (-4,941,690)
+    #0: ToddCoxeter: because f x a < n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 29,155
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.2     |         active |            killed |          defined
+    #0: ToddCoxeter: nodes            |         29,155 |        11,482,367 |       11,511,522
+    #0: ToddCoxeter: diff 0.2.1       |             +0 |                +0 |               +0
+    #0: ToddCoxeter: diff 0.2.0       |             +0 |                +0 |               +0
+    #0: ToddCoxeter:                  |         active |           missing |       % complete
+    #0: ToddCoxeter: edges            |         87,465 |                 0 |           100.0%
+    #0: ToddCoxeter: diff 0.2.1       |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: diff 0.2.0       |             +0 |                +0 |            +0.0%
+    #0: ToddCoxeter: phase 0.2 = 9ms  | run 0 = 5.039s | all runs = 5.039s | elapsed = 5.039s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |      lookahead |        lookbehind |              hlt |        felsch
+    #0: ToddCoxeter: num. phases          |              1 |                 0 |                0 |             1
+    #0: ToddCoxeter: time spent in phases |       9ms (0%) |            - (0%) |           - (0%) | 5.029s (100%)
+    #0: ToddCoxeter: phase 0.2 = 9ms      | run 0 = 5.039s | all runs = 5.039s | elapsed = 5.039s
+    The index of the subgroup is 29155
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $29,155$.
diff --git a/docs/examples/atlas/monster-sections/he/3358656.md b/docs/examples/atlas/monster-sections/he/3358656.md
new file mode 100644
index 0000000..47ba429
--- /dev/null
+++ b/docs/examples/atlas/monster-sections/he/3358656.md
@@ -0,0 +1,179 @@
+# $5^2:4A_4$ as a subgroup of Held group He
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/).
+
+## Claim
+
+$$
+    \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17}
+    = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab]
+    = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$5^2:4A_4$. More precisely, if
+
+$$
+    H = \langle a,\ abab^2ab^{-3}ab \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{He} : H] = 3,358,656.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for He, adds the generating pairs that define the maximal subgroup, and runs the
+Todd-Coxeter algorithm. The source code marks three of the presentation
+relations as redundant. We do not verify that these relations are redundant
+here.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^7"), "")
+presentation.add_rule(p, parse("(ab)^17"), "")
+presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
+presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
+presentation.add_rule(p, parse("(a,babaBabab)"), "")
+presentation.add_rule(p, parse("(a,b^3)^5"), "")
+presentation.add_rule(p, parse("(a,b)^6"), "")
+presentation.add_rule(
+    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
+)  # redundant, maybe useful.
+presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("abB")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# takes approx 54s
+
+# These generators are from the webpages of the ATLAS
+tc.add_generating_pair("a", "")
+tc.add_generating_pair(parse("abab^2aB^3ab"), "")
+
+print(f"The index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The truncated output of the enumeration is below:
+
+??? info "Truncated output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |         defined
+    #0: ToddCoxeter: nodes            |           10 |               0 |              10
+    #0: ToddCoxeter:                  |       active |         missing |      % complete
+    #0: ToddCoxeter: edges            |           12 |              18 |           40.0%
+    #0: ToddCoxeter: time             | run 0 = 54µs | all runs = 54µs | elapsed = 231µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      2,432,372 |            62,383 |        2,494,757
+    #1: ToddCoxeter: diff 0.0.0         |     +2,432,362 |           +62,383 |       +2,494,747
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      5,157,707 |         2,139,409 |            70.7%
+    #1: ToddCoxeter: diff 0.0.0         |     +5,157,695 |        +2,139,391 |           +30.7%
+    #1: ToddCoxeter: phase 0.0 = 1.004s | run 0 = 1.004s | all runs = 1.004s | elapsed = 1.004s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      4,842,108 |           140,945 |        4,983,074
+    #1: ToddCoxeter: diff 0.0.1         |     +2,409,736 |           +78,562 |       +2,488,317
+    #1: ToddCoxeter: diff 0.0.0         |     +4,842,098 |          +140,945 |       +4,983,064
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     10,271,116 |         4,255,208 |            70.7%
+    #1: ToddCoxeter: diff 0.0.1         |     +5,113,409 |        +2,115,799 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +10,271,104 |        +4,255,190 |           +30.7%
+    #1: ToddCoxeter: phase 0.0 = 2.009s | run 0 = 2.009s | all runs = 2.009s | elapsed = 2.009s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.3       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      7,158,599 |           221,358 |        7,379,979
+    #1: ToddCoxeter: diff 0.0.2         |     +2,316,491 |           +80,413 |       +2,396,905
+    #1: ToddCoxeter: diff 0.0.0         |     +7,158,589 |          +221,358 |       +7,379,969
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     15,187,405 |         6,288,392 |            70.7%
+    #1: ToddCoxeter: diff 0.0.2         |     +4,916,289 |        +2,033,184 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +15,187,393 |        +6,288,374 |           +30.7%
+    #1: ToddCoxeter: phase 0.0 = 3.011s | run 0 = 3.011s | all runs = 3.011s | elapsed = 3.011s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.4       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      9,255,773 |           422,754 |        9,678,545
+    #1: ToddCoxeter: diff 0.0.3         |     +2,097,174 |          +201,396 |       +2,298,566
+    #1: ToddCoxeter: diff 0.0.0         |     +9,255,763 |          +422,754 |       +9,678,535
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     19,641,556 |         8,125,763 |            70.7%
+    #1: ToddCoxeter: diff 0.0.3         |     +4,454,151 |        +1,837,371 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +19,641,544 |        +8,125,745 |           +30.7%
+    #1: ToddCoxeter: phase 0.0 = 4.016s | run 0 = 4.016s | all runs = 4.016s | elapsed = 4.016s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.5       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      6,914,229 |         3,237,712 |       10,151,941
+    #1: ToddCoxeter: diff 0.0.4         |     -2,341,544 |        +2,814,958 |         +473,396
+    #1: ToddCoxeter: diff 0.0.0         |     +6,914,219 |        +3,237,712 |      +10,151,931
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     14,845,037 |         5,897,650 |            71.6%
+    [... lines omitted ...]
+    #0: ToddCoxeter: diff 0.1.0          |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                     |          active |            missing |        % complete
+    #0: ToddCoxeter: edges               |      10,075,968 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.1.10         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.1.0          |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.1 = 10.090s | run 0 = 54.796s | all runs = 54.796s | elapsed = 54.796s
+    #0: ToddCoxeter: lookahead_next() is now n x f = 10,000,000 (+5,000,000)
+    #0: ToddCoxeter: because: l < (l + a) / t = 839,664
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 3,358,656
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         l = nodes killed in lookahead    = 0
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    #0: ToddCoxeter:         t = lookahead_growth_threshold() = 4
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.12       |          active |             killed |           defined
+    #0: ToddCoxeter: nodes               |       3,358,656 |          7,414,887 |        10,773,543
+    #0: ToddCoxeter: diff 0.2.11         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.2.0          |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                     |          active |            missing |        % complete
+    #0: ToddCoxeter: edges               |      10,075,968 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.2.11         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.2.0          |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.2 = 10.090s | run 0 = 54.796s | all runs = 54.796s | elapsed = 54.796s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |       lookahead |         lookbehind |               hlt |        felsch
+    #0: ToddCoxeter: num. phases          |               1 |                  0 |                 0 |             1
+    #0: ToddCoxeter: time spent in phases |   10.090s (18%) |             - (0%) |            - (0%) | 44.705s (82%)
+    #0: ToddCoxeter: phase 0.2 = 10.147s  | run 0 = 54.852s | all runs = 54.852s | elapsed = 54.852s
+    The index of the subgroup is 3358656
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $3,358,656$.
diff --git a/docs/examples/atlas/monster-sections/he/652800.md b/docs/examples/atlas/monster-sections/he/652800.md
new file mode 100644
index 0000000..8acb322
--- /dev/null
+++ b/docs/examples/atlas/monster-sections/he/652800.md
@@ -0,0 +1,183 @@
+# $7^{1+2}:(3 \times S_3)$ as a subgroup of Held group He
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/).
+
+## Claim
+
+$$
+    \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17}
+    = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab]
+    = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$7^{1+2}:(3 \times S_3)$. More precisely, if
+
+$$
+    H = \langle (b^{-1}a)^7(ab^2)^6(ab)^7,\ (q^{-1})^5(ab^2)^2q^5,\ \text{ where }q=ababb \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{He} : H] = 652,800.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for He, adds the generating pairs that define the maximal subgroup, and runs the
+Todd-Coxeter algorithm. The source code marks three of the presentation
+relations as redundant. We do not verify that these relations are redundant
+here.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^7"), "")
+presentation.add_rule(p, parse("(ab)^17"), "")
+presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
+presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
+presentation.add_rule(p, parse("(a,babaBabab)"), "")
+presentation.add_rule(p, parse("(a,b^3)^5"), "")
+presentation.add_rule(p, parse("(a,b)^6"), "")
+presentation.add_rule(
+    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
+)  # redundant, maybe useful.
+presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("abB")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# The subgroup generators here are taken from one of the straight line
+# programmes from:
+# https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/
+
+# takes approx 2.5 minutes
+q = "ababb"
+Q = ("".join(reversed(q))).replace("b", "B")
+tc.add_generating_pair(parse("(Ba)^7(ab^2)^6(ab)^7"), "")
+tc.add_generating_pair(parse(f"({Q})^5(ab^2)^2({q})^5"), "")
+
+print(f"The index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The truncated output of the enumeration is below:
+
+??? info "Truncated output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |         defined
+    #0: ToddCoxeter: nodes            |           57 |              43 |             100
+    #0: ToddCoxeter:                  |       active |         missing |      % complete
+    #0: ToddCoxeter: edges            |          100 |              71 |           58.5%
+    #0: ToddCoxeter: time             | run 0 = 81µs | all runs = 81µs | elapsed = 302µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      2,498,917 |            52,325 |        2,551,269
+    #1: ToddCoxeter: diff 0.0.0         |     +2,498,860 |           +52,282 |       +2,551,169
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      5,296,218 |         2,200,533 |            70.6%
+    #1: ToddCoxeter: diff 0.0.0         |     +5,296,118 |        +2,200,462 |           +12.2%
+    #1: ToddCoxeter: phase 0.0 = 1.005s | run 0 = 1.005s | all runs = 1.005s | elapsed = 1.005s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      4,989,394 |           110,077 |        5,099,485
+    #1: ToddCoxeter: diff 0.0.1         |     +2,490,477 |           +57,752 |       +2,548,216
+    #1: ToddCoxeter: diff 0.0.0         |     +4,989,337 |          +110,034 |       +5,099,385
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     10,576,096 |         4,392,086 |            70.7%
+    #1: ToddCoxeter: diff 0.0.1         |     +5,279,878 |        +2,191,553 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +10,575,996 |        +4,392,015 |           +12.2%
+    #1: ToddCoxeter: phase 0.0 = 2.010s | run 0 = 2.010s | all runs = 2.010s | elapsed = 2.010s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.3       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      7,541,248 |           170,690 |        7,711,948
+    #1: ToddCoxeter: diff 0.0.2         |     +2,551,854 |           +60,613 |       +2,612,463
+    #1: ToddCoxeter: diff 0.0.0         |     +7,541,191 |          +170,647 |       +7,711,848
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     15,986,986 |         6,636,758 |            70.7%
+    #1: ToddCoxeter: diff 0.0.2         |     +5,410,890 |        +2,244,672 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +15,986,886 |        +6,636,687 |           +12.2%
+    #1: ToddCoxeter: phase 0.0 = 3.015s | run 0 = 3.015s | all runs = 3.015s | elapsed = 3.015s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.4       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      9,990,510 |           226,874 |       10,217,405
+    #1: ToddCoxeter: diff 0.0.3         |     +2,449,262 |           +56,184 |       +2,505,457
+    #1: ToddCoxeter: diff 0.0.0         |     +9,990,453 |          +226,831 |      +10,217,305
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     21,180,244 |         8,791,286 |            70.7%
+    #1: ToddCoxeter: diff 0.0.3         |     +5,193,258 |        +2,154,528 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +21,180,144 |        +8,791,215 |           +12.2%
+    #1: ToddCoxeter: phase 0.0 = 4.020s | run 0 = 4.020s | all runs = 4.020s | elapsed = 4.020s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.5       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |     12,531,004 |           287,403 |       12,818,441
+    #1: ToddCoxeter: diff 0.0.4         |     +2,540,494 |           +60,529 |       +2,601,036
+    #1: ToddCoxeter: diff 0.0.0         |    +12,530,947 |          +287,360 |      +12,818,341
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     26,567,136 |        11,025,876 |            70.7%
+    [... lines omitted ...]
+    #0: ToddCoxeter: LOOKAHEAD 0.1.1   |          active |             killed |           defined
+    #0: ToddCoxeter: nodes             |         652,800 |        251,782,228 |       252,435,028
+    #0: ToddCoxeter: diff 0.1.0        |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                   |          active |            missing |        % complete
+    #0: ToddCoxeter: edges             |       1,958,400 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.1.0        |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.1 = 572ms | run 0 = 2min24s | all runs = 2min24s | elapsed = 2min24s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 1,305,600, m = 10,000) (-3,694,400)
+    #0: ToddCoxeter: because f x a < n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 652,800
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.2      |          active |             killed |           defined
+    #0: ToddCoxeter: nodes             |         652,800 |        251,782,228 |       252,435,028
+    #0: ToddCoxeter: diff 0.2.1        |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.2.0        |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                   |          active |            missing |        % complete
+    #0: ToddCoxeter: edges             |       1,958,400 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.2.1        |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.2.0        |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.2 = 572ms | run 0 = 2min24s | all runs = 2min24s | elapsed = 2min24s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |       lookahead |         lookbehind |               hlt |         felsch
+    #0: ToddCoxeter: num. phases          |               1 |                  0 |                 0 |              1
+    #0: ToddCoxeter: time spent in phases |      572ms (0%) |             - (0%) |            - (0%) | 2min23s (100%)
+    #0: ToddCoxeter: phase 0.2 = 576ms    | run 0 = 2min24s | all runs = 2min24s | elapsed = 2min24s
+    The index of the subgroup is 652800
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $652,800$.
diff --git a/docs/examples/atlas/monster-sections/he/8330.md b/docs/examples/atlas/monster-sections/he/8330.md
new file mode 100644
index 0000000..4842f8b
--- /dev/null
+++ b/docs/examples/atlas/monster-sections/he/8330.md
@@ -0,0 +1,154 @@
+# $2^2.L_3(4).S_3$ as a subgroup of Held group He
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/spor/He/words/HeG1-max2W1](https://brauer.maths.qmul.ac.uk/Atlas/spor/He/words/HeG1-max2W1).
+
+## Claim
+
+$$
+    \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17}
+    = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab]
+    = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$2^2.L_3(4).S_3$. More precisely, if
+
+$$
+    H = \langle a,\ bab^2 \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{He} : H] = 8,330.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for He, adds the generating pairs that define the maximal subgroup, and runs the
+Todd-Coxeter algorithm. The source code marks three of the presentation
+relations as redundant. We do not verify that these relations are redundant
+here.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^7"), "")
+presentation.add_rule(p, parse("(ab)^17"), "")
+presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
+presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
+presentation.add_rule(p, parse("(a,babaBabab)"), "")
+presentation.add_rule(p, parse("(a,b^3)^5"), "")
+presentation.add_rule(p, parse("(a,b)^6"), "")
+presentation.add_rule(
+    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
+)  # redundant, maybe useful.
+presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("abB")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# The subgroup generators here are taken from:
+# https://brauer.maths.qmul.ac.uk/Atlas/spor/He/words/HeG1-max2W1
+
+# takes approx 231ms
+
+tc.add_generating_pair("a", "")
+tc.add_generating_pair("babb", "")
+
+print(f"The index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The output of the enumeration is below:
+
+??? info "Output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |         defined
+    #0: ToddCoxeter: nodes            |            4 |               0 |               4
+    #0: ToddCoxeter:                  |       active |         missing |      % complete
+    #0: ToddCoxeter: edges            |            6 |               6 |           50.0%
+    #0: ToddCoxeter: time             | run 0 = 60µs | all runs = 60µs | elapsed = 106µs
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 STOP
+    #0: ToddCoxeter: FELSCH 0.0.1      |        active |           killed |         defined
+    #0: ToddCoxeter: nodes             |         8,330 |          506,595 |         514,925
+    #0: ToddCoxeter: diff 0.0.0        |        +8,326 |         +506,595 |        +514,921
+    #0: ToddCoxeter:                   |        active |          missing |      % complete
+    #0: ToddCoxeter: edges             |        24,990 |                0 |          100.0%
+    #0: ToddCoxeter: diff 0.0.0        |       +24,984 |               -6 |          +50.0%
+    #0: ToddCoxeter: phase 0.0 = 222ms | run 0 = 222ms | all runs = 222ms | elapsed = 222ms
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
+    #0: ToddCoxeter: LOOKAHEAD 0.1.0  |        active |           killed |         defined
+    #0: ToddCoxeter: nodes            |         8,330 |          506,595 |         514,925
+    #0: ToddCoxeter:                  |        active |          missing |      % complete
+    #0: ToddCoxeter: edges            |        24,990 |                0 |          100.0%
+    #0: ToddCoxeter: time             | run 0 = 222ms | all runs = 222ms | elapsed = 222ms
+    #0: ToddCoxeter: triggered because there are skipped definitions (8,330 active nodes)!
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: LOOKAHEAD 0.1 STOP
+    #0: ToddCoxeter: LOOKAHEAD 0.1.1  |        active |           killed |         defined
+    #0: ToddCoxeter: nodes            |         8,330 |          506,595 |         514,925
+    #0: ToddCoxeter: diff 0.1.0       |            +0 |               +0 |              +0
+    #0: ToddCoxeter:                  |        active |          missing |      % complete
+    #0: ToddCoxeter: edges            |        24,990 |                0 |          100.0%
+    #0: ToddCoxeter: diff 0.1.0       |            +0 |               +0 |           +0.0%
+    #0: ToddCoxeter: phase 0.1 = 1ms  | run 0 = 224ms | all runs = 224ms | elapsed = 224ms
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 16,660, m = 10,000) (-4,983,340)
+    #0: ToddCoxeter: because f x a < n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 8,330
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.2     |        active |           killed |         defined
+    #0: ToddCoxeter: nodes            |         8,330 |          506,595 |         514,925
+    #0: ToddCoxeter: diff 0.2.1       |            +0 |               +0 |              +0
+    #0: ToddCoxeter: diff 0.2.0       |            +0 |               +0 |              +0
+    #0: ToddCoxeter:                  |        active |          missing |      % complete
+    #0: ToddCoxeter: edges            |        24,990 |                0 |          100.0%
+    #0: ToddCoxeter: diff 0.2.1       |            +0 |               +0 |           +0.0%
+    #0: ToddCoxeter: diff 0.2.0       |            +0 |               +0 |           +0.0%
+    #0: ToddCoxeter: phase 0.2 = 2ms  | run 0 = 224ms | all runs = 224ms | elapsed = 224ms
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |     lookahead |       lookbehind |             hlt |       felsch
+    #0: ToddCoxeter: num. phases          |             1 |                0 |               0 |            1
+    #0: ToddCoxeter: time spent in phases |      2ms (1%) |           - (0%) |          - (0%) |  222ms (99%)
+    #0: ToddCoxeter: phase 0.2 = 2ms      | run 0 = 224ms | all runs = 224ms | elapsed = 224ms
+    The index of the subgroup is 8330
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $8,330$.
diff --git a/docs/examples/atlas/monster-sections/he/999600.md b/docs/examples/atlas/monster-sections/he/999600.md
new file mode 100644
index 0000000..82880f1
--- /dev/null
+++ b/docs/examples/atlas/monster-sections/he/999600.md
@@ -0,0 +1,183 @@
+# $S_4 \times L_3(2)$ as a subgroup of Held group He
+
+The subgroup generators used on this page come from
+[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/).
+
+## Claim
+
+$$
+    \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17}
+    = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab]
+    = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle.
+$$
+
+The generators are claimed to generate a maximal subgroup isomorphic to
+$S_4 \times L_3(2)$. More precisely, if
+
+$$
+    H = \langle (q^{-1})^2(ab^2)^6q^2,\ (ab)^6(ab^2)^4(b^{-1}a)^6,\ \text{ where }q=ababb \rangle,
+$$
+
+then
+
+$$
+    [\mathrm{He} : H] = 999,600.
+$$
+
+On this page, we verify that the subgroup generated by these words has the
+claimed index.
+
+## The code
+
+In [libsemigroups_pybind11][], the following script constructs the presentation
+for He, adds the generating pairs that define the maximal subgroup, and runs the
+Todd-Coxeter algorithm. The source code marks three of the presentation
+relations as redundant. We do not verify that these relations are redundant
+here.
+
+```python
+from libsemigroups_pybind11 import (
+    Presentation,
+    ToddCoxeter,
+    congruence_kind,
+    presentation,
+)
+from libsemigroups_pybind11.words import parse_relations as parse
+
+
+# Setup the presentation object with the empty and inverses, so it can represent a group
+p = Presentation("abAB")
+p.contains_empty_word(True)
+presentation.add_inverse_rules(p, "ABab")
+
+# Add the defining relations
+presentation.add_rule(p, parse("a^2"), "")
+presentation.add_rule(p, parse("b^7"), "")
+presentation.add_rule(p, parse("(ab)^17"), "")
+presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
+presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
+presentation.add_rule(p, parse("(a,babaBabab)"), "")
+presentation.add_rule(p, parse("(a,b^3)^5"), "")
+presentation.add_rule(p, parse("(a,b)^6"), "")
+presentation.add_rule(
+    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
+)  # redundant, maybe useful.
+presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")
+
+presentation.balance(p, "abAB", "ABab")
+presentation.replace_subword(p, "A", "a")
+p.alphabet("abB")
+
+tc = ToddCoxeter(congruence_kind.onesided, p)
+tc.strategy(ToddCoxeter.options.strategy.felsch)
+
+# The subgroup generators here are taken from one of the straight line
+# programmes from:
+# https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/
+
+# takes approx 2.5 minutes
+q = "ababb"
+Q = ("".join(reversed(q))).replace("b", "B")
+tc.add_generating_pair(parse(f"({Q})^2(ab^2)^6({q})^2"), "")
+tc.add_generating_pair(parse("(ab)^6(ab^2)^4(Ba)^6"), "")
+
+print(f"The index of the subgroup is {tc.number_of_classes()}")
+```
+
+## The output
+
+The truncated output of the enumeration is below:
+
+??? info "Truncated output from the Python script"
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = felsch)
+    #0: ToddCoxeter: |A| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.0 START
+    #0: ToddCoxeter: FELSCH 0.0.0     |       active |          killed |         defined
+    #0: ToddCoxeter: nodes            |           48 |              25 |              73
+    #0: ToddCoxeter:                  |       active |         missing |      % complete
+    #0: ToddCoxeter: edges            |           74 |              70 |           51.4%
+    #0: ToddCoxeter: time             | run 0 = 70µs | all runs = 70µs | elapsed = 264µs
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      2,473,159 |            53,957 |        2,527,133
+    #1: ToddCoxeter: diff 0.0.0         |     +2,473,111 |           +53,932 |       +2,527,060
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |      5,242,117 |         2,177,360 |            70.7%
+    #1: ToddCoxeter: diff 0.0.0         |     +5,242,043 |        +2,177,290 |           +19.3%
+    #1: ToddCoxeter: phase 0.0 = 1.005s | run 0 = 1.005s | all runs = 1.005s | elapsed = 1.005s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      4,992,438 |           112,926 |        5,105,379
+    #1: ToddCoxeter: diff 0.0.1         |     +2,519,279 |           +58,969 |       +2,578,246
+    #1: ToddCoxeter: diff 0.0.0         |     +4,992,390 |          +112,901 |       +5,105,306
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     10,583,240 |         4,394,074 |            70.7%
+    #1: ToddCoxeter: diff 0.0.1         |     +5,341,123 |        +2,216,714 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +10,583,166 |        +4,394,004 |           +19.3%
+    #1: ToddCoxeter: phase 0.0 = 2.010s | run 0 = 2.010s | all runs = 2.010s | elapsed = 2.010s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.3       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |      7,553,726 |           174,059 |        7,727,802
+    #1: ToddCoxeter: diff 0.0.2         |     +2,561,288 |           +61,133 |       +2,622,423
+    #1: ToddCoxeter: diff 0.0.0         |     +7,553,678 |          +174,034 |       +7,727,729
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     16,014,216 |         6,646,962 |            70.7%
+    #1: ToddCoxeter: diff 0.0.2         |     +5,430,976 |        +2,252,888 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +16,014,142 |        +6,646,892 |           +19.3%
+    #1: ToddCoxeter: phase 0.0 = 3.015s | run 0 = 3.015s | all runs = 3.015s | elapsed = 3.015s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.4       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |     10,022,736 |           236,866 |       10,259,624
+    #1: ToddCoxeter: diff 0.0.3         |     +2,469,010 |           +62,807 |       +2,531,822
+    #1: ToddCoxeter: diff 0.0.0         |    +10,022,688 |          +236,841 |      +10,259,551
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     21,250,224 |         8,817,984 |            70.7%
+    #1: ToddCoxeter: diff 0.0.3         |     +5,236,008 |        +2,171,022 |            +0.0%
+    #1: ToddCoxeter: diff 0.0.0         |    +21,250,150 |        +8,817,914 |           +19.3%
+    #1: ToddCoxeter: phase 0.0 = 4.020s | run 0 = 4.020s | all runs = 4.020s | elapsed = 4.020s
+    ++++++++++++++++++++++++++++++++
+    #1: ToddCoxeter: FELSCH 0.0.5       |         active |            killed |          defined
+    #1: ToddCoxeter: nodes              |     12,591,904 |           299,790 |       12,891,716
+    #1: ToddCoxeter: diff 0.0.4         |     +2,569,168 |           +62,924 |       +2,632,092
+    #1: ToddCoxeter: diff 0.0.0         |    +12,591,856 |          +299,765 |      +12,891,643
+    #1: ToddCoxeter:                    |         active |           missing |       % complete
+    #1: ToddCoxeter: edges              |     26,698,162 |        11,077,550 |            70.7%
+    [... lines omitted ...]
+    #0: ToddCoxeter: diff 0.1.1         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.1.0         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                    |          active |            missing |        % complete
+    #0: ToddCoxeter: edges              |       2,998,800 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.1.1         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.1.0         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.1 = 1.339s | run 0 = 2min31s | all runs = 2min31s | elapsed = 2min31s
+    #0: ToddCoxeter: lookahead_next() is now max(f x a = 1,999,200, m = 10,000) (-3,000,800)
+    #0: ToddCoxeter: because f x a < n
+    #0: ToddCoxeter: where:  a = number_of_nodes_active()     = 999,600
+    #0: ToddCoxeter:         f = lookahead_growth_factor()    = 2
+    #0: ToddCoxeter:         m = lookahead_min()              = 10,000
+    #0: ToddCoxeter:         n = lookahead_next()             = 5,000,000
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: FELSCH 0.2.3       |          active |             killed |           defined
+    #0: ToddCoxeter: nodes              |         999,600 |        287,397,473 |       288,397,073
+    #0: ToddCoxeter: diff 0.2.2         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter: diff 0.2.0         |              +0 |                 +0 |                +0
+    #0: ToddCoxeter:                    |          active |            missing |        % complete
+    #0: ToddCoxeter: edges              |       2,998,800 |                  0 |            100.0%
+    #0: ToddCoxeter: diff 0.2.2         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: diff 0.2.0         |              +0 |                 +0 |             +0.0%
+    #0: ToddCoxeter: phase 0.2 = 1.339s | run 0 = 2min31s | all runs = 2min31s | elapsed = 2min31s
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |       lookahead |         lookbehind |               hlt |        felsch
+    #0: ToddCoxeter: num. phases          |               1 |                  0 |                 0 |             1
+    #0: ToddCoxeter: time spent in phases |     1.339s (1%) |             - (0%) |            - (0%) | 2min29s (99%)
+    #0: ToddCoxeter: phase 0.2 = 1.346s   | run 0 = 2min31s | all runs = 2min31s | elapsed = 2min31s
+    The index of the subgroup is 999600
+    ```
+
+:material-checkbox-marked-circle-outline: The computed index is the same as
+the claimed index: $999,600$.
diff --git a/docs/examples/atlas/monster-sections/he/index.md b/docs/examples/atlas/monster-sections/he/index.md
index 75b5801..506d69f 100644
--- a/docs/examples/atlas/monster-sections/he/index.md
+++ b/docs/examples/atlas/monster-sections/he/index.md
@@ -1,10 +1,23 @@
 # Held group He
 
-> Order: $4,030,387,200$
+On this page, we provide links to verifications that the generators of maximal
+subgroups of the Held group He from the ATLAS generate subgroups of the correct
+index.
 
-> Presentation:
-> $\langle a, b \mid a^2 = b^7 = (ab)^{17} = [a, b]^6 = [a, b^3]^5 = [a, babab^{−1}abab] = (ab)^4ab^2ab^{−3}ababab^{−1}ab^3ab^{−2}ab^2 = 1 \rangle$
+<div class="center-table" markdown>
 
-!!! info "Work in Progress"
+| Subgroup | Subgroup order | Index in He | Verification |
+| :------- | -------------: | ----------: | :----------: |
+| [$S_4(4):2$](2058.md) | $1,958,400$ | $2,058$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
+| [$2^2.L_3(4).S_3$](8330.md) | $483,840$ | $8,330$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
+| [$2^6:3.S_6$](29155-i.md) | $138,240$ | $29,155$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
+| [$2^6:3.S_6$](29155-ii.md) | $138,240$ | $29,155$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
+| [$2^{1+6}.L_3(2)$](187425.md) | $21,504$ | $187,425$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
+| [$7^2:2.L_2(7)$](244800.md) | $16,464$ | $244,800$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
+| [$3.S_7$](266560.md) | $15,120$ | $266,560$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
+| [$7^{1+2}:(3 \times S_3)$](652800.md) | $6,174$ | $652,800$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
+| [$S_4 \times L_3(2)$](999600.md) | $4,032$ | $999,600$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
+| [$7:3 \times L_3(2)$](1142400.md) | $3,528$ | $1,142,400$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
+| [$5^2:4A_4$](3358656.md) | $1,200$ | $3,358,656$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } |
 
-    This page is a work in progress.
+</div>
diff --git a/docs/examples/cong-latt.md b/docs/examples/cong-latt.md
deleted file mode 100644
index e69de29..0000000
diff --git a/docs/examples/gis.md b/docs/examples/gis.md
deleted file mode 100644
index e69de29..0000000
diff --git a/docs/examples/stephen.md b/docs/examples/stephen.md
deleted file mode 100644
index e69de29..0000000
diff --git a/docs/examples/trans-rep.md b/docs/examples/trans-rep.md
deleted file mode 100644
index e69de29..0000000
diff --git a/docs/stylesheets/extra.css b/docs/stylesheets/extra.css
index dea2783..889644e 100644
--- a/docs/stylesheets/extra.css
+++ b/docs/stylesheets/extra.css
@@ -26,6 +26,15 @@
   text-decoration: none;
 }
 
+.center-table {
+  text-align: center;
+}
+
+.center-table th:not(:last-child),
+.center-table td:not(:last-child) {
+  border-right: 0.05rem solid var(--md-typeset-table-color);
+}
+
 .correct-size {
   background-color: #4dac26;
 }
@@ -50,6 +59,10 @@
   background-color: #dddddd;
 }
 
-.no-presentation {
+.no-presentation-in-atlas {
   background-color: grey;
 }
+
+.no-presentation-known {
+  background-color: #F28C28;
+}