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\(2.S_6(2)\) 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

\[ H=\langle a,b,c,e,f,g\rangle \cong 2.S_6(2), \]

then

\[ [\mathrm{Co}_3:H]=170,775. \]

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 1.5 seconds.

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 ("a", "b", "c", "e", "f", "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 != 170_775:
    raise RuntimeError(f"expected index 170775, got {actual_index}")

The output

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 ...]
#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 |      79ms (5%) |            - (0%) |           - (0%) | 1.454s (95%)
#0: ToddCoxeter: phase 0.2 = 80ms     | run 0 = 1.536s | all runs = 1.536s | elapsed = 1.536s
The index of the subgroup is 170775

The computed index is the claimed index: \(170,775\).