C₃²×(C₃⋊D₃)⋊C₃ #92377

order	486
isoclinism	isoclinic class of (C₃⋊D₃)⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product	C₃²×(C₃⋊D₃)⋊C₃
unambiguous semidirect product	(C₃⋊D₃)⋊C₃³
unambiguous semidirect product	(C₃×C₃⋊D₃)⋊C₃²

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

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

360 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 222);