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\(3^3:(2 \times S_4)\) as a subgroup of Conway group Co3

The presentation and subgroup generators on this page come from presentation (13.1) in Đoković's paper.

Claim

If

\[ H=\langle c,d,e,g\rangle \cong 3^3:(2 \times S_4), \]

then

\[ [\mathrm{Co}_3:H]=382,536,000. \]

We verify the claimed index, but not the claimed isomorphism type. The shared presentation adds twelve relations listed as consequences in the source. We do not verify that these relations are redundant here.

The code

The recorded run took about 1 day, 10 hours, and 23 minutes.

Code 
from libsemigroups_pybind11 import (
    Presentation,
    ToddCoxeter,
    congruence_kind,
    presentation,
)

p = Presentation("abcdefg")
p.contains_empty_word(True)

for letter in "abcdefg":
    presentation.add_rule(p, letter * 2, "")
for word in ("ac", "ad", "ae", "bd", "ce", "dg", "eg"):
    presentation.add_rule(p, word * 2, "")
for word, exponent in (
    ("ab", 3),
    ("cd", 3),
    ("de", 3),
    ("cfg", 3),
    ("bcfcf", 3),
):
    presentation.add_rule(p, word * exponent, "")
presentation.add_rule(p, "f" + "ag" * 3, "")
presentation.add_rule(p, "g" + "bf" * 3, "")
presentation.add_rule(p, "e" + "abc" * 3, "")
presentation.add_rule(p, "e" + "acg" * 4, "")
presentation.add_rule(p, "e" + "abcf" * 7, "")

# Consequences listed in (13.1), included as useful redundant relators.
for word, exponent in (
    ("af", 2),
    ("be", 2),
    ("bg", 2),
    ("df", 2),
    ("ef", 2),
    ("fg", 2),
    ("bc", 4),
    ("cf", 4),
    ("ag", 6),
    ("eg", 6),
    ("bcde", 4),
):
    presentation.add_rule(p, word * exponent, "")
presentation.add_rule(p, "a" + "cdef" * 4, "")

tc = ToddCoxeter(congruence_kind.onesided, p)
tc.strategy(ToddCoxeter.options.strategy.felsch).use_relations_in_extra(True)
for word in ("c", "d", "e", "g"):
    tc.add_generating_pair(word, "")

tc.run()
actual_index = tc.number_of_classes()
print(f"The index of the subgroup is {actual_index}")
if actual_index != 382_536_000:
    raise RuntimeError(f"expected index 382536000, got {actual_index}")

The output

The output file is about 120 MB, so a truncated version is shown below.

Truncated output from the Python script 
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: RUN 0 START (strategy() = felsch)
#0: ToddCoxeter: |A| = 7, |R| = 36, |u| + |v| ∈ [2, 29], ∑(|u| + |v|) = 247
[... lines omitted ...]
#1: ToddCoxeter: nodes                |           382,536,000 |              796,892,522 |           1,179,428,522
#1: ToddCoxeter: edges                |         2,677,752,000 |                        0 |                  100.0%
#1: ToddCoxeter: phase 0.2 = 36min40s | run 0 = 1d10h23min25s | all runs = 1d10h23min25s | elapsed = 1d10h23min25s
++++++++++++++++++++++++++++++++
#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 |         36min11s (2%) |                   - (0%) |                  - (0%) | 1d9h46min44s (98%)
#0: ToddCoxeter: phase 0.2 = 36min41s | run 0 = 1d10h23min26s | all runs = 1d10h23min26s | elapsed = 1d10h23min26s
The index of the subgroup is 382536000

The computed index is the claimed index: \(382,536,000\).