C₂×C₃×C₃⋊D₃ #1116

order	108
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂²×AGL(2,𝔽₃) (order 1728) · Inn(G) = C₃⋊D₃ · Out(G) = C₂²×S₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

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

	subgroup	G/H
center, hypercenter	C₂×C₃	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃³	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₃

76 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 43);