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Q₈×Dic₁₁ #66998

order	352
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Q₈×Dic₁₁

Splittings

C₁₁ ⋊ C₄×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₁
commutator	C₂×C₁₁	C₂²×C₄
Fitting	C₂×Q₈×C₁₁	C₂
socle, prime radical, cosocle	C₂²×C₁₁	C₂³
hypercenter	C₂×Q₈	D₁₁
hypocenter	C₁₁	C₄×Q₈
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 352

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

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10622401, #10622402, #10622403, …
maximal normal subgroup of	#10622401, #10622402, #10622403, …
maximal quotient of	C₄⋊C₄×Dic₁₁, #10622594, #10622598, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 140);