C₂²×C₃×D₃ #651

order	72
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×D₃×S₄ (order 288) · Inn(G) = D₃ · Out(G) = C₂×S₄
holomorph	G ⋊ Aut(G) = D₃×#72210357
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂²×C₃×D₃
unambiguous semidirect product	C₃⋊(C₂³×C₃)
unambiguous semidirect product	(C₂×C₃)⋊(C₂²×C₃)

	subgroup	G/H
center, hypercenter	C₂²×C₃	D₃
commutator, cosocle, hypocenter	C₃	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃²	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃

69 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 24

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 48);