order	486
isoclinism	isoclinic class of C₃⋊((C₃⋊D₃)⋊C₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃³

Aliases

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

Splittings

Fitting, Sylow	C₃²×H(3,𝔽₃) ⋊ C₂
derived	C₃³ ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₃	C₃⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle, hypocenter	C₃³	C₂×C₃²
Frattini	C₃	C₃²×C₃⋊D₃
Fitting	C₃²×H(3,𝔽₃)	C₂
socle	C₃³	C₃×D₃
LCS	G ▷ C₃³
UCS	C₁ ◁ C₃

333 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 229);