C₂²×#12873 #72310826

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#12873

Subgroups

	subgroup	G/H
center, socle	C₂³	#1823
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

7361 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500349, C₂²×#494058, C₂²×#500003
maximal normal subgroup of	C₂²×#500349, C₂²×#494058, C₂²×#500003

GAP

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

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

C22×#12873 #72310826