C₂²×S₄ #1019

order	96
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = S₄² (order 576) · Inn(G) = S₄ · Out(G) = S₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product	C₂²×S₄
unambiguous semidirect product	A₄⋊C₂³
unambiguous semidirect product	(C₂×A₄)⋊C₂²

Fitting, socle	C₂⁴ ⋊ D₃
derived	A₄ ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂²	S₄
commutator, cosocle, hypocenter	A₄	C₂³
Frattini	C₁	G
Fitting, socle	C₂⁴	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂²

131 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 3 generators on 8 points:

(5,6,8,7);(3,4)(6,7,8);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 226);