C₂²×#13334 #72311000

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#13334

Subgroups

	subgroup	G/H
center, socle	C₂³	#2011
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

12385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6,8,11)(7,15,19,32)(9,22,20,28)(10,24,21,14)(12,27,25,17)(13,16,26,23)(18,36,34,31)(29,35,30,33);(1,2);(6,11)(7,12,19,25)(13,26)(14,24)(15,27,32,17)(18,29,34,30)(22,28)(31,33,36,35);(3,4);(5,7,16,33,10,18,9,17,8,19,23,35,21,34,20,27)(6,12,28,36,14,29,13,15,11,25,22,31,24,30,26,32)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500650, C₂²×#499269, C₂²×#499742
maximal normal subgroup of	C₂²×#500650, C₂²×#499269, C₂²×#499742

GAP

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

gap> G := Group((1,2)(3,4)(5,6,8,11)(7,15,19,32)(9,22,20,28)(10,24,21,14)(12,27,25,17)(13,16,26,23)(18,36,34,31)(29,35,30,33), (1,2), (6,11)(7,12,19,25)(13,26)(14,24)(15,27,32,17)(18,29,34,30)(22,28)(31,33,36,35), (3,4), (5,7,16,33,10,18,9,17,8,19,23,35,21,34,20,27)(6,12,28,36,14,29,13,15,11,25,22,31,24,30,26,32));

C22×#13334 #72311000