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Q₈×C₇⋊C₈ #89795

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

Q₈×C₇⋊C₈

Splittings

C₇ ⋊ C₈×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator	C₂×C₇	C₂²×C₈
Fitting	C₄×C₇×Q₈	C₂
socle, prime radical	C₂²×C₇	C₂²×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₄×Q₈	D₇
hypocenter	C₇	C₈×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11725966, #11725967, #11725968, …
maximal normal subgroup of	#11725966, #11725967, #11725968, …
maximal quotient of	C₄⋊C₄×C₇⋊C₈, #11727236, #11727238, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 557);