C₂×C₃²×S₄ #89113

order	432
isoclinism	isoclinic class of S₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product	C₂×C₃²×S₄
unambiguous semidirect product	A₄⋊(C₂²×C₃²)
unambiguous semidirect product	(C₂×A₄)⋊(C₂×C₃²)

Fitting, socle	C₂³×C₃² ⋊ D₃
derived	A₄ ⋊ C₂²×C₃²

	subgroup	G/H
center, hypercenter	C₂×C₃²	S₄
commutator, cosocle, hypocenter	A₄	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂³×C₃²	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂×C₃²

230 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 760);