C₃×3₋¹⁺²×AGL(1,ℤ/8) #72254691

order	2592
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃³ · Schur multiplier (H₂) = C₂²×C₃²

Aliases

	subgroup	G/H
center, socle	C₂×C₃²	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₃×C₄	C₂³×C₃³
prime radical	C₃³×AGL(1,ℤ/8)	C₃
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃×C₄×3₋¹⁺² ◁ G

1292 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 20 points:

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

363 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((2,5)(3,7)(6,8)(13,15,18)(16,20,19), (2,6)(5,8)(12,13,16,14,15,19,17,18,20), (1,2,3,5,4,6,7,8)(9,10,11));