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ΓL(1,𝔽8)×#108 #10620546

order672
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×#108
unambiguous semidirect product(C7×#108)⋊C3

Splittings

Fitting, HallC7×#108 ⋊ C3
SylowC7 ⋊ C3×#108

Subgroups

subgroupG/H
centerC4D4×ΓL(1,𝔽8)
commutatorC4×C7C2×C3×C4
FrattiniC2×C4C22×ΓL(1,𝔽8)
FittingC7×#108C3
socleC2×C7C3×C4⋊C4
prime radicalC22×ΓL(1,𝔽8)Q8
cosocleC2×C4×C7C22×C3
hypercenter#108ΓL(1,𝔽8)
hypocenterC7C3×#108
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ #108

60 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)(8,9,12,15,13,16,11,21)(10,17,19,22,20,23,18,14);(1,2,3,5,4,6,7)(8,10,11,18,13,20,12,19)(9,14,15,17,16,22,21,23)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89288⋊C3, #89289⋊C3, #89290⋊C3, …
maximal normal subgroup of#89288⋊C3, #89289⋊C3, #89290⋊C3, …
maximal quotient ofΓL(1,𝔽8)×#334, ΓL(1,𝔽8)×#335, ΓL(1,𝔽8)×#341, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 128);