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Add McL full verification
Co-authored-by: Codex <noreply@openai.com>
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‎docs/.nav.yml‎

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- "U6(2):2": examples/atlas/leech-lattice/co2/2300.md
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- "McLaughlin group McL":
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- examples/atlas/leech-lattice/mcl/index.md
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- "Trivial": examples/atlas/leech-lattice/mcl/898128000.md
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- "M22": examples/atlas/leech-lattice/mcl/2025.md
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- "M21 = L3(4)": examples/atlas/leech-lattice/mcl/44550.md
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- "S6": examples/atlas/leech-lattice/mcl/1247400.md
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- "2^4:S4": examples/atlas/leech-lattice/mcl/2338875.md
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- "3^2:D8": examples/atlas/leech-lattice/mcl/12474000.md
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- "(D8 * D8):2": examples/atlas/leech-lattice/mcl/14033250.md
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- "2 x S4": examples/atlas/leech-lattice/mcl/18711000.md
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- "S4": examples/atlas/leech-lattice/mcl/37422000.md
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- "Monster sections":
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- "Held group He":
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- examples/atlas/monster-sections/he/index.md

‎docs/examples/atlas/index.md‎

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<a href="leech-lattice/co2">Co<sub>2</sub></a>
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</td>
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<td class="no-presentation-in-atlas">Co<sub>3</sub></td>
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<td class="work-in-progress"><a href="leech-lattice/mcl">McL</a></td>
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<td class="correct-size-alt"><a href="leech-lattice/mcl">McL</a></td>
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<td class="no-presentation-in-atlas">Suz</td>
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</tr>
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</table>
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The presentation defines a group of the correct size.
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</td>
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</tr>
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<tr>
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<td class="correct-size-alt">
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The source of the presentation is not the ATLAS and the
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presentation defines a group of the correct size.
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</td>
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</tr>
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<tr>
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<td class="correct-index">
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The presentation has at least 1 subgroup of the correct index.

‎docs/examples/atlas/leech-lattice/mcl/1247400.md‎

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# $S_6$ as a subgroup of the McLaughlin group McL
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The presentation and subgroup generators on this page come from presentation
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(11.1) in [Đoković's paper](https://doi.org/10.1017/S1446788700030068).
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## Claim
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If
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$$
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H=\langle a,b,c,e,f\rangle \cong S_6,
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$$
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then
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$$
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[\mathrm{McL}:H]=1,247,400.
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$$
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We verify the claimed index, but not the claimed isomorphism type. The shared
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presentation adds ten relations listed as consequences in the source. We do not
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verify that these relations are redundant here.
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## The code
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The presentation helper is shown on the [entire enumeration page](898128000.md).
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The recorded run took about 8.5 seconds.
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```python
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from McL_dokovic_i import mcl_dokovic_i
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from libsemigroups_pybind11 import ToddCoxeter, congruence_kind
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p = mcl_dokovic_i()
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tc = ToddCoxeter(congruence_kind.onesided, p)
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tc.strategy(ToddCoxeter.options.strategy.felsch).use_relations_in_extra(True)
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tc.add_generating_pair("a", "")
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tc.add_generating_pair("b", "")
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tc.add_generating_pair("c", "")
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tc.add_generating_pair("e", "")
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tc.add_generating_pair("f", "")
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tc.run()
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print(f"The index of the subgroup is {tc.number_of_classes()}")
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```
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## The output
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??? info "Truncated output from the Python script"
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```
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++++++++++++++++++++++++++++++++
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#0: ToddCoxeter: RUN 0 START (strategy() = felsch)
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#0: ToddCoxeter: |A| = 6, |R| = 28, |u| + |v| ∈ [2, 21], ∑(|u| + |v|) = 191
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[... lines omitted ...]
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#0: ToddCoxeter: RUN 0 STOP (finished)
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#0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch
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#0: ToddCoxeter: num. phases | 0 | 0 | 0 | 1
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#0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | - (0%) | 8.467s (100%)
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#0: ToddCoxeter: phase 0.1 = 8.472s | run 0 = 8.472s | all runs = 8.472s | elapsed = 8.472s
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The index of the subgroup is 1247400
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```
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:material-checkbox-marked-circle-outline: The computed index is the claimed
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index: $1,247,400$.

‎docs/examples/atlas/leech-lattice/mcl/12474000.md‎

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# $3^2:D_8$ as a subgroup of the McLaughlin group McL
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The presentation and subgroup generators on this page come from presentation
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(11.1) in [Đoković's paper](https://doi.org/10.1017/S1446788700030068).
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## Claim
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If
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$$
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H=\langle b,d,f\rangle \cong 3^2:D_8,
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$$
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then
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$$
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[\mathrm{McL}:H]=12,474,000.
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$$
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We verify the claimed index, but not the claimed isomorphism type. The shared
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presentation adds ten relations listed as consequences in the source. We do not
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verify that these relations are redundant here.
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## The code
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The presentation helper is shown on the [entire enumeration page](898128000.md).
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The recorded run took about 2 minutes 13 seconds.
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```python
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from McL_dokovic_i import mcl_dokovic_i
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from libsemigroups_pybind11 import ToddCoxeter, congruence_kind
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p = mcl_dokovic_i()
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tc = ToddCoxeter(congruence_kind.onesided, p)
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tc.strategy(ToddCoxeter.options.strategy.felsch).use_relations_in_extra(True)
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tc.add_generating_pair("b", "")
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tc.add_generating_pair("d", "")
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tc.add_generating_pair("f", "")
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tc.run()
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print(f"The index of the subgroup is {tc.number_of_classes()}")
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```
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## The output
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??? info "Truncated output from the Python script"
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```
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++++++++++++++++++++++++++++++++
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#0: ToddCoxeter: RUN 0 START (strategy() = felsch)
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#0: ToddCoxeter: |A| = 6, |R| = 28, |u| + |v| ∈ [2, 21], ∑(|u| + |v|) = 191
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[... lines omitted ...]
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#0: ToddCoxeter: RUN 0 STOP (finished)
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#0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch
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#0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1
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#0: ToddCoxeter: time spent in phases | 30.994s (23%) | - (0%) | - (0%) | 1min42s (77%)
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#0: ToddCoxeter: phase 0.2 = 31.064s | run 0 = 2min13s | all runs = 2min13s | elapsed = 2min13s
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The index of the subgroup is 12474000
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```
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:material-checkbox-marked-circle-outline: The computed index is the claimed
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index: $12,474,000$.

‎docs/examples/atlas/leech-lattice/mcl/14033250.md‎

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# $(D_8 * D_8):2$ as a subgroup of the McLaughlin group McL
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The presentation and subgroup generators on this page come from presentation
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(11.1) in [Đoković's paper](https://doi.org/10.1017/S1446788700030068).
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## Claim
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If
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$$
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H=\langle c,d,f\rangle \cong (D_8 * D_8):2,
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$$
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then
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$$
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[\mathrm{McL}:H]=14,033,250.
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$$
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We verify the claimed index, but not the claimed isomorphism type. The shared
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presentation adds ten relations listed as consequences in the source. We do not
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verify that these relations are redundant here.
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## The code
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The presentation helper is shown on the [entire enumeration page](898128000.md).
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The recorded run took about 2 minutes 27 seconds.
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```python
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from McL_dokovic_i import mcl_dokovic_i
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from libsemigroups_pybind11 import ToddCoxeter, congruence_kind
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p = mcl_dokovic_i()
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tc = ToddCoxeter(congruence_kind.onesided, p)
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tc.strategy(ToddCoxeter.options.strategy.felsch).use_relations_in_extra(True)
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tc.add_generating_pair("c", "")
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tc.add_generating_pair("d", "")
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tc.add_generating_pair("f", "")
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tc.run()
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print(f"The index of the subgroup is {tc.number_of_classes()}")
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```
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## The output
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??? info "Truncated output from the Python script"
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```
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++++++++++++++++++++++++++++++++
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#0: ToddCoxeter: RUN 0 START (strategy() = felsch)
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#0: ToddCoxeter: |A| = 6, |R| = 28, |u| + |v| ∈ [2, 21], ∑(|u| + |v|) = 191
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[... lines omitted ...]
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#0: ToddCoxeter: RUN 0 STOP (finished)
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#0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch
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#0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1
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#0: ToddCoxeter: time spent in phases | 35.108s (24%) | - (0%) | - (0%) | 1min52s (76%)
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#0: ToddCoxeter: phase 0.2 = 35.199s | run 0 = 2min27s | all runs = 2min27s | elapsed = 2min27s
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The index of the subgroup is 14033250
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```
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:material-checkbox-marked-circle-outline: The computed index is the claimed
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index: $14,033,250$.

‎docs/examples/atlas/leech-lattice/mcl/18711000.md‎

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# $2\times S_4$ as a subgroup of the McLaughlin group McL
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The presentation and subgroup generators on this page come from presentation
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(11.1) in [Đoković's paper](https://doi.org/10.1017/S1446788700030068).
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## Claim
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If
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$$
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H=\langle a,b,d,e\rangle \cong 2\times S_4,
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$$
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then
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$$
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[\mathrm{McL}:H]=18,711,000.
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$$
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We verify the claimed index, but not the claimed isomorphism type. The shared
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presentation adds ten relations listed as consequences in the source. We do not
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verify that these relations are redundant here.
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## The code
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The presentation helper is shown on the [entire enumeration page](898128000.md).
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The recorded run took about 2 minutes 34 seconds.
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```python
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from McL_dokovic_i import mcl_dokovic_i
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from libsemigroups_pybind11 import ToddCoxeter, congruence_kind
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p = mcl_dokovic_i()
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tc = ToddCoxeter(congruence_kind.onesided, p)
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tc.strategy(ToddCoxeter.options.strategy.felsch).use_relations_in_extra(True)
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tc.add_generating_pair("a", "")
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tc.add_generating_pair("b", "")
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tc.add_generating_pair("d", "")
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tc.add_generating_pair("e", "")
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tc.run()
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print(f"The index of the subgroup is {tc.number_of_classes()}")
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```
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## The output
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??? info "Truncated output from the Python script"
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```
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++++++++++++++++++++++++++++++++
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#0: ToddCoxeter: RUN 0 START (strategy() = felsch)
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#0: ToddCoxeter: |A| = 6, |R| = 28, |u| + |v| ∈ [2, 21], ∑(|u| + |v|) = 191
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[... lines omitted ...]
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#0: ToddCoxeter: RUN 0 STOP (finished)
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#0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch
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#0: ToddCoxeter: num. phases | 0 | 0 | 0 | 1
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#0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | - (0%) | 2min34s (100%)
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#0: ToddCoxeter: phase 0.1 = 2min34s | run 0 = 2min34s | all runs = 2min34s | elapsed = 2min34s
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The index of the subgroup is 18711000
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```
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:material-checkbox-marked-circle-outline: The computed index is the claimed
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index: $18,711,000$.

‎docs/examples/atlas/leech-lattice/mcl/2025.md‎

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# $\mathrm{M}_{22}$ as a subgroup of the McLaughlin group McL
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The presentation and subgroup generators on this page come from presentation
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(11.1) in [Đoković's paper](https://doi.org/10.1017/S1446788700030068).
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## Claim
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If
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$$
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H=\langle a,b,c,d,e\rangle \cong \mathrm{M}_{22},
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$$
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then
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$$
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[\mathrm{McL}:H]=2,025.
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$$
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We verify the claimed index, but not the claimed isomorphism type. The shared
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presentation adds ten relations listed as consequences in the source. We do not
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verify that these relations are redundant here.
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## The code
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The presentation helper is shown on the [entire enumeration page](898128000.md).
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The recorded run took about 7 milliseconds.
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```python
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from McL_dokovic_i import mcl_dokovic_i
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from libsemigroups_pybind11 import ToddCoxeter, congruence_kind
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p = mcl_dokovic_i()
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tc = ToddCoxeter(congruence_kind.onesided, p)
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tc.strategy(ToddCoxeter.options.strategy.felsch).use_relations_in_extra(True)
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tc.add_generating_pair("a", "")
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tc.add_generating_pair("b", "")
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tc.add_generating_pair("c", "")
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tc.add_generating_pair("d", "")
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tc.add_generating_pair("e", "")
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tc.run()
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print(f"The index of the subgroup is {tc.number_of_classes()}")
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```
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## The output
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??? info "Truncated output from the Python script"
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```
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++++++++++++++++++++++++++++++++
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#0: ToddCoxeter: RUN 0 START (strategy() = felsch)
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#0: ToddCoxeter: |A| = 6, |R| = 28, |u| + |v| ∈ [2, 21], ∑(|u| + |v|) = 191
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[... lines omitted ...]
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#0: ToddCoxeter: RUN 0 STOP (finished)
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#0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch
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#0: ToddCoxeter: num. phases | 0 | 0 | 0 | 1
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#0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | - (0%) | 7ms (99%)
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#0: ToddCoxeter: phase 0.1 = 7ms | run 0 = 7ms | all runs = 7ms | elapsed = 7ms
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The index of the subgroup is 2025
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```
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:material-checkbox-marked-circle-outline: The computed index is the claimed
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index: $2,025$.

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