C₂²×C₃×S₄ #64385

order	288
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₂×S₄² (order 1152) · Inn(G) = S₄ · Out(G) = C₂×S₄
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product	C₂²×C₃×S₄
unambiguous semidirect product	A₄⋊(C₂³×C₃)
unambiguous semidirect product	(C₂×A₄)⋊(C₂²×C₃)

Fitting, socle	C₂⁴×C₃ ⋊ D₃
derived	A₄ ⋊ C₂³×C₃

	subgroup	G/H
center, hypercenter	C₂²×C₃	S₄
commutator, cosocle, hypocenter	A₄	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂²×C₃

272 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1033);