C₂×C₃×A₄ #650

order	72
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = D₃×S₄ (order 144) · Inn(G) = A₄ · Out(G) = C₂×D₃
holomorph	G ⋊ Aut(G) = C₂×#72213328
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

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

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

32 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 47);