order	486
isoclinism	isoclinic class of C₃⋊(C₃⋊D₉)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃³

Aliases

direct product	C₃×C₃⋊(C₃⋊D₉)
unambiguous semidirect product	C₃⋊(C₃×C₃⋊D₉)

Splittings

Fitting, Sylow	C₃³×C₉ ⋊ C₂
derived	C₃²×C₉ ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₃	C₃⋊(C₃⋊D₉)
commutator, cosocle, hypocenter	C₃²×C₉	C₂×C₃
Frattini	C₃	C₃×C₃⋊(C₃⋊D₃)
Fitting	C₃³×C₉	C₂
socle	C₃⁴	D₃
LCS	G ▷ C₃²×C₉
UCS	C₁ ◁ C₃

348 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 4 generators on 18 points:

(1,2,3)(5,6)(8,9)(11,13)(12,16)(14,15)(17,18);(7,8,9);(4,5,6);(10,18,13,16,15,14,12,11,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 240);