C₂×C₃²×D₃ #1115

order	108
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×D₃×GL(2,𝔽₃) (order 576) · Inn(G) = D₃ · Out(G) = C₂×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₂×C₃²×D₃
unambiguous semidirect product	C₃⋊(C₂²×C₃²)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₃²)

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

76 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 42);