C₂²×#13215 #72311009

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

direct product

C₂²×#13215

Subgroups

	subgroup	G/H
center, socle	C₂³	#1961
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#3155	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

9469 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500650, C₂²×#495573, C₂²×#500743
maximal normal subgroup of	C₂²×#500650, C₂²×#495573, C₂²×#500743

GAP

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

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

C22×#13215 #72311009