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Q₈×S₄ #5918

order	192
automorphism	Aut(G) = C₂×S₄² (order 1152) · Inn(G) = C₂²×S₄ · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Q₈×S₄

Splittings

C₂²×Q₈ ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×S₄
commutator, cosocle	C₂×A₄	C₂³
Fitting	C₂²×Q₈	D₃
socle	C₂³	C₂²×D₃
prime radical	C₂×S₄	C₂²
hypercenter	Q₈	S₄
hypocenter	A₄	C₂×Q₈
LCS	G ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂ ◁ Q₈

156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 3 generators on 12 points:

(1,2,4,3);(2,3,4)(5,7,8,12)(6,9,10,11);(5,6,8,10)(7,11,12,9)

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×S₄, #85461, Q₁₆×S₄, …
maximal normal subgroup of	SD₁₆×S₄, #85461, Q₁₆×S₄, …
maximal quotient of	#85302, #85303, #85304, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1477);