C₂×C₃×D₃ #160

order	36
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂²×D₃ (order 24) · Inn(G) = D₃ · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = D₃×#3811
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

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

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

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 8 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36, 12);