C₂²×#13396 #72310848

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

Aliases

direct product

C₂²×#13396

Subgroups

	subgroup	G/H
center, socle	C₂³	#2034
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#3058	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

8309 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)(6,11)(7,15)(8,21)(10,14)(12,25)(13,22)(16,30)(17,24)(18,34)(19,27)(20,33)(26,35)(28,32)(29,36);(1,2);(1,2)(3,4)(5,6,9,11)(7,16,20,32)(8,13,22,21)(10,24,17,14)(12,26,19,34)(15,30,33,28)(18,25,35,27)(23,36,31,29);(3,4);(1,2)(3,4)(5,7,17,35,11,19,14,28,9,20,10,18,6,12,24,30)(8,15,31,16,21,25,29,34,22,33,23,32,13,27,36,26)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500349, C₂²×#499269, C₂²×#500743
maximal normal subgroup of	C₂²×#500349, C₂²×#499269, C₂²×#500743

GAP

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

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

C22×#13396 #72310848