C₂³×C₃×3₋¹⁺² #10620120

order	648
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃³ · Schur multiplier (H₂) = C₂³×C₃²

Aliases

C₂³×C₃×3₋¹⁺²

	subgroup	G/H
center, socle	C₂³×C₃²	C₃²
commutator, Frattini, cosocle	C₃	C₂³×C₃³
prime radical	C₂³×C₃³	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

608 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 18 points:

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

264 conjugacy classes. The character table is too large to display.

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 639);