C₃²×S₄ #6326

order	216
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = S₄×GL(2,𝔽₃) (order 1152) · Inn(G) = S₄ · Out(G) = GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

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

Fitting, socle	C₂²×C₃² ⋊ D₃
derived	A₄ ⋊ C₂×C₃²

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

82 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 10 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 163);