C₂²×C₅²×3₋¹⁺² #72247337

order	2700
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₅² · Schur multiplier (H₂) = C₂×C₅

Aliases

C₂²×C₅²×3₋¹⁺²

	subgroup	G/H
center, socle	C₂²×C₃×C₅²	C₃²
commutator, Frattini, cosocle	C₃	C₂²×C₃²×C₅²
prime radical	C₂²×C₃²×C₅²	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅² ◁ G

320 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 900

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2)(6,8,11)(9,13,12)(14,15,16,17,18);(3,4)(5,6,9,7,8,12,10,11,13)(19,20,21,22,23)

1100 conjugacy classes. The character table is too large to display.

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2)(6,8,11)(9,13,12)(14,15,16,17,18), (3,4)(5,6,9,7,8,12,10,11,13)(19,20,21,22,23));