\(S_4\) as a subgroup of the McLaughlin group McL

The presentation and subgroup generators on this page come from presentation (11.1) in Đoković's paper.

Claim

If

\[ H=\langle c,d,e\rangle \cong S_4, \]

then

\[ [\mathrm{McL}:H]=37,422,000. \]

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

The code

The presentation helper is shown on the entire enumeration page. The recorded run took about 6 minutes 47 seconds.

Code

from McL_dokovic_i import mcl_dokovic_i
from libsemigroups_pybind11 import ToddCoxeter, congruence_kind

p = mcl_dokovic_i()

tc = ToddCoxeter(congruence_kind.onesided, p)
tc.strategy(ToddCoxeter.options.strategy.felsch).use_relations_in_extra(True)

tc.add_generating_pair("c", "")
tc.add_generating_pair("d", "")
tc.add_generating_pair("e", "")

tc.run()

print(f"The index of the subgroup is {tc.number_of_classes()}")

The output

Truncated output from the Python script

++++++++++++++++++++++++++++++++
#0: ToddCoxeter: RUN 0 START (strategy() = felsch)
#0: ToddCoxeter: |A| = 6, |R| = 28, |u| + |v| ∈ [2, 21], ∑(|u| + |v|) = 191
[... 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 |   1min39s (24%) |             - (0%) |            - (0%) | 5min8s (76%)
#0: ToddCoxeter: phase 0.2 = 1min39s  | run 0 = 6min47s | all runs = 6min47s | elapsed = 6min47s
The index of the subgroup is 37422000

The computed index is the claimed index: \(37,422,000\).