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ΓL(1,𝔽8)×M32 #10620548

order672
automorphism?
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×M32
unambiguous semidirect product(C7×M32)⋊C3

Splittings

Fitting, HallC7×M32 ⋊ C3
SylowC7 ⋊ C3×M32

Subgroups

subgroupG/H
center, FrattiniC8C22×ΓL(1,𝔽8)
commutator, socleC2×C7C2×C3×C8
FittingC7×M32C3
prime radicalC22×ΓL(1,𝔽8)C8
cosocleC7×C8C22×C3
hypercenterM32ΓL(1,𝔽8)
hypocenterC7C3×M32
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C8 ◁ M32

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 2 generators on 23 points:

(2,3,4)(5,7,6)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23);(1,2,3,5,4,6,7)(9,17)(11,19)(13,21)(15,23)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89306⋊C3, #89307⋊C3, #89308⋊C3, …
maximal normal subgroup of#89306⋊C3, #89307⋊C3, #89308⋊C3, …
maximal quotient ofΓL(1,𝔽8)×#346, ΓL(1,𝔽8)×C22⋊C16, ΓL(1,𝔽8)×C4⋊C16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 130);