C₂²×#13374 #72311379

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

Aliases

direct product

C₂²×#13374

Subgroups

	subgroup	G/H
center, socle	C₂³	#2023
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

6609 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#499269, C₂²×#500940, C₂²×#500003
maximal normal subgroup of	C₂²×#499269, C₂²×#500940, C₂²×#500003

GAP

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

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

C22×#13374 #72311379