C₂²×#13417 #72311003

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#13417

Subgroups

	subgroup	G/H
center, socle	C₂³	#2041
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#3155	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

9253 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)(7,15,19,29)(9,21,16,32)(10,24,20,14)(12,22,25,17)(13,28,26,23)(18,35,33,31)(27,34,30,36);(1,2);(6,11)(7,12,19,25)(13,26)(14,24)(15,22,29,17)(18,27,33,30)(21,32)(31,36,35,34);(3,4);(5,7,16,17,20,33,23,36,8,19,9,22,10,18,28,34)(6,12,26,15,24,30,21,35,11,25,13,29,14,27,32,31)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500650, C₂²×#499269, C₂²×#500164
maximal normal subgroup of	C₂²×#500650, C₂²×#499269, C₂²×#500164

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

C22×#13417 #72311003