order	54
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂×AGL(2,𝔽₃) (order 864) · Inn(G) = C₃⋊D₃ · Out(G) = C₂×S₄
holomorph	G ⋊ Aut(G) = D₃×#72210382
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃

Aliases

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

Splittings

Fitting, socle, Sylow	C₃³ ⋊ C₂
derived	C₃² ⋊ C₂×C₃

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

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 3 generators on 9 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(54, 13);