C₃²×C₃⋊D₃ #4216

order	162
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = GL(2,𝔽₃)×AGL(2,𝔽₃) (order 20736) · Inn(G) = C₃⋊D₃ · Out(G) = S₄×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃²

Aliases

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

Fitting, socle, Sylow	C₃⁴ ⋊ C₂
derived	C₃² ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₃²	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃⁴	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₃²

160 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 3 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 52);