order	162
isoclinism	isoclinic class of (C₃⋊D₃)⋊C₃
automorphism	Aut(G) = D₃×((C₃⋊D₃)⋊C₃)⋊C₂ (order 648) · Inn(G) = (C₃⋊D₃)⋊C₃ · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃²

Aliases

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

Splittings

Fitting, Sylow	C₃×H(3,𝔽₃) ⋊ C₂
derived	C₃² ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₃	(C₃⋊D₃)⋊C₃
commutator, cosocle, hypocenter	C₃²	C₂×C₃²
Frattini	C₃	C₃²×D₃
Fitting	C₃×H(3,𝔽₃)	C₂
socle	C₃²	C₃×D₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₃

58 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 34);