C₂×D₃×C₃⋊D₃ #6334

order	216
isoclinism	isoclinic class of D₃×C₃⋊D₃
automorphism	Aut(G) = C₂²×D₃×AGL(2,𝔽₃) (order 10368) · Inn(G) = D₃×C₃⋊D₃ · Out(G) = C₂²×S₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃

Aliases

C₂×D₃×C₃⋊D₃

Fitting, socle	C₂×C₃³ ⋊ C₂²
derived, Sylow	C₃³ ⋊ C₂³

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

232 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);(3,4,5)(10,11);(1,2)(6,7,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 171);