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6466eb2
Add Held group
codex Aug 18, 2026
8c0c019
Make M23 pages consistent with Held
james-d-mitchell Aug 18, 2026
764a5f7
Revert "Make M23 pages consistent with Held"
james-d-mitchell Aug 18, 2026
915c0e6
M24 move file for consistency
james-d-mitchell Aug 18, 2026
7072a46
M23 complete enum added
james-d-mitchell Aug 18, 2026
8fd1e4e
Rm empty files
james-d-mitchell Aug 18, 2026
dc4492a
Tweak the key
james-d-mitchell Aug 18, 2026
f597bab
Add Co2 group
james-d-mitchell Aug 19, 2026
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Tweaks from JDE
james-d-mitchell Aug 19, 2026
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17 changes: 15 additions & 2 deletions docs/.nav.yml
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Original file line number Diff line number Diff line change
Expand Up @@ -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":
Expand All @@ -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":
Expand Down
36 changes: 20 additions & 16 deletions docs/examples/atlas/index.md
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Expand Up @@ -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">
Expand All @@ -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>
Expand All @@ -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>
Expand All @@ -135,15 +135,15 @@ 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>
<td class="correct-index-alt"><a href="pariahs/ru">Ru</a></td>
<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>
Expand Down Expand Up @@ -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
Expand All @@ -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>
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183 changes: 183 additions & 0 deletions docs/examples/atlas/leech-lattice/co2/2300.md
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# $U_6(2):2$ 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: $2,300$.
31 changes: 26 additions & 5 deletions docs/examples/atlas/leech-lattice/co2/index.md
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# 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>
13 changes: 10 additions & 3 deletions docs/examples/atlas/mathieu/m23/index.md
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# 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

Expand Down
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