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Q₈×#168 #65312

order	320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

C₅ ⋊ C₈×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator	C₂×C₅	C₂²×C₈
Fitting	C₄×C₅×Q₈	C₂
socle, prime radical	C₂²×C₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	C₈×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Q₈⋊#674, #10599005, #10599006, …
maximal normal subgroup of	Q₈⋊#674, #10599005, #10599006, …
maximal quotient of	C₄⋊C₄×#168, #10600275, #10600277, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 650);