order	96
isoclinism	isoclinic class of C₄²⋊C₃
automorphism	Aut(G) = #4626⋊C₂ (order 384) · Inn(G) = C₄²⋊C₃ · Out(G) = D₄
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₄

Aliases

direct product	C₂×C₄²⋊C₃
unambiguous semidirect product	(C₂×C₄²)⋊C₃

Splittings

Fitting, Sylow	C₂×C₄² ⋊ C₃
derived	C₄² ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₂	C₄²⋊C₃
commutator, cosocle, hypocenter	C₄²	C₂×C₃
Frattini	C₂²	C₂×A₄
Fitting	C₂×C₄²	C₃
socle	C₂³	A₄
LCS	G ▷ C₄²
UCS	C₁ ◁ C₂

28 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 68);