order	216
isoclinism	isoclinic class of A₄×C₃⋊D₃
automorphism	Aut(G) = S₄×AGL(2,𝔽₃) (order 10368) · Inn(G) = G · Out(G) = C₂×S₄
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	A₄×C₃⋊D₃
unambiguous semidirect product	C₃⋊(D₃×A₄)

Splittings

C₂²×C₃² ⋊ C₂×C₃

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

88 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 10 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 167);