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ΓL(1,𝔽8)×#101 #10620539

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

Aliases

direct productΓL(1,𝔽8)×#101
unambiguous semidirect product(C7×#101)⋊C3

Splittings

Fitting, HallC7×#101 ⋊ C3
SylowC7 ⋊ C3×#101

Subgroups

subgroupG/H
centerC2C22⋊C4×ΓL(1,𝔽8)
commutatorC22×C7C2×C3×C4
FrattiniC2×C4C22×ΓL(1,𝔽8)
FittingC7×#101C3
socleC2×C7C3×C22⋊C4
prime radicalC22×ΓL(1,𝔽8)C2×C4
cosocleC2×C4×C7C22×C3
hypercenter#101ΓL(1,𝔽8)
hypocenterC7C3×#101
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

76 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)(8,9,11,15,12,14,20,22)(10,16,17,23,18,13,19,21);(1,2,3,5,4,6,7)(8,10,12,18)(9,13,14,16)(11,19,20,17)(15,23,22,21)

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89272⋊C3, #89273⋊C3, ΓL(1,𝔽8)×#355, …
maximal normal subgroup of#89272⋊C3, #89273⋊C3, ΓL(1,𝔽8)×#355, …
maximal quotient ofΓL(1,𝔽8)×#324, ΓL(1,𝔽8)×#330, ΓL(1,𝔽8)×#332, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 121);