C₂²×C₃²×D₃ #6337

order	216
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = D₃×S₄×GL(2,𝔽₃) (order 6912) · Inn(G) = D₃ · Out(G) = S₄×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product	C₂²×C₃²×D₃
unambiguous semidirect product	C₃⋊(C₂³×C₃²)
unambiguous semidirect product	(C₂×C₃)⋊(C₂²×C₃²)

	subgroup	G/H
center, hypercenter	C₂²×C₃²	D₃
commutator, cosocle, hypocenter	C₃	C₂³×C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃³	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃²

232 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 13 points:

(6,7)(8,9,10);(3,4)(5,6,7)(11,12,13);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 174);