C₂×C₃²×A₄ #6336

order	216
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = S₄×AGL(2,𝔽₃) (order 10368) · Inn(G) = A₄ · Out(G) = C₂×AGL(2,𝔽₃)
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product	C₂×C₃²×A₄
unambiguous semidirect product	C₂³⋊C₃³
unambiguous semidirect product	(C₂³×C₃)⋊C₃²

	subgroup	G/H
center, hypercenter	C₂×C₃²	A₄
commutator, cosocle, hypocenter	C₂²	C₂×C₃³
Frattini	C₁	G
Fitting, socle	C₂³×C₃²	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂×C₃²

136 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 173);