A₄×S₄ #64376

order	288
isoclinism	isoclinic class of A₄×S₄
automorphism	Aut(G) = S₄² (order 576) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	A₄×S₄
unambiguous semidirect product	A₄⋊(C₂×A₄)
unambiguous semidirect product	(C₂²×S₄)⋊C₃

Fitting, socle	C₂⁴ ⋊ C₃×D₃
derived	C₂²×A₄ ⋊ C₂×C₃

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²×A₄	C₂×C₃
Fitting, socle	C₂⁴	C₃×D₃
LCS	G ▷ C₂²×A₄
UCS	C₁

86 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 2 generators on 8 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1024);