C₂×C₃×S₄ #3845

order	144
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₂²×S₄ (order 96) · Inn(G) = S₄ · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = D₃×#72215293
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×C₃×S₄
unambiguous semidirect product	A₄⋊(C₂²×C₃)
unambiguous semidirect product	(C₂×A₄)⋊(C₂×C₃)

Fitting, socle	C₂³×C₃ ⋊ D₃
derived	A₄ ⋊ C₂²×C₃

	subgroup	G/H
center, hypercenter	C₂×C₃	S₄
commutator, cosocle, hypocenter	A₄	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₃	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂×C₃

70 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 9 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 188);