C₃²×#12062 #72269709

order	2304
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₁₆ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#12062

Subgroups

	subgroup	G/H
center	C₂×C₃²×C₈	C₂×D₄
commutator	C₄	C₂²×C₃²×C₁₆
Frattini, cosocle	C₄×C₈	C₂³×C₃²
socle	C₂²×C₃²	C₈×D₄
prime radical	C₂×C₃²×D₈	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₈ ◁ C₃²×C₄×C₁₆ ◁ G

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 1152

A minimal faithful permutation representation — 3 generators on 30 points:

(2,5)(3,7)(6,8)(9,10,12,16,13,17,19,23,14,18,22,11,15,20,24,21)(25,26,27);(2,6)(5,8)(9,11,19,10,15,23,12,20,14,16,24,18,13,21,22,17)(28,29,30);(1,2,3,5,4,6,7,8)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃²×#122963
maximal normal subgroup of	C₃²×#122963
maximal quotient of	C₃²×#122963

GAP

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)(9,10,12,16,13,17,19,23,14,18,22,11,15,20,24,21)(25,26,27), (2,6)(5,8)(9,11,19,10,15,23,12,20,14,16,24,18,13,21,22,17)(28,29,30), (1,2,3,5,4,6,7,8));

C32×#12062 #72269709