C₉⋊#272 #92337

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

Aliases

	subgroup	G/H
center, hypercenter	C₃	C₃⋊(C₃⋊D₉)
commutator, Fitting, cosocle, hypocenter	C₉×H(3,𝔽₃)	C₂
Frattini, socle	C₃²	C₃⋊(C₃⋊D₃)
LCS	G ▷ C₉×H(3,𝔽₃)
UCS	C₁ ◁ C₃

172 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 54

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

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

Polycyclic presentation:

2,3^5|b^3=e^2,b^a=b^2e,c^a=c^2,d^a=d^2,e^a=e^2,d^c=df

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 182);