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ΓL(1,𝔽8)×#97 #10620535

order672
automorphism?
homologyAbelianization (H1) = C3×C42 · Schur multiplier (H2) = C2

Aliases

direct productΓL(1,𝔽8)×#97
unambiguous semidirect product(C7×#97)⋊C3

Splittings

Fitting, HallC7×#97 ⋊ C3
SylowC7 ⋊ C3×#97

Subgroups

subgroupG/H
center, FrattiniC2×C4C22×ΓL(1,𝔽8)
commutatorC2×C7C3×C42
FittingC7×#97C3
socleC22×C7C2×C3×C4
prime radicalC22×ΓL(1,𝔽8)C2×C4
cosocleC2×C4×C7C22×C3
hypercenter#97ΓL(1,𝔽8)
hypocenterC7C3×#97
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #97

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89261⋊C3, #89259⋊C3, #89260⋊C3, …
maximal normal subgroup of#89261⋊C3, #89259⋊C3, #89260⋊C3, …
maximal quotient ofΓL(1,𝔽8)×#322, ΓL(1,𝔽8)×#336, ΓL(1,𝔽8)×#347, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 117);