C₃×C₃⋊#272 #92398

order	486
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃²

Aliases

direct product	C₃×C₃⋊#272
unambiguous semidirect product	C₃⋊(C₃×#272)

Fitting, Sylow	C₃²×H(3,𝔽₃) ⋊ C₂
derived	C₃×H(3,𝔽₃) ⋊ C₂×C₃

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

417 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 243);