C₃²×C₃⋊(C₃⋊D₃) #92413

order	486
isoclinism	isoclinic class of C₃⋊(C₃⋊D₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃⁴

Aliases

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

Fitting, socle, Sylow	C₃⁵ ⋊ C₂
derived	C₃³ ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₃²	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃³	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃⁵	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₃²

1584 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 258);