C₂²×#12833 #72322717

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

Aliases

direct product

C₂²×#12833

Subgroups

	subgroup	G/H
center, socle	C₂³	#1807
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

6637 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)(5,6,10,13,9,14,24,28,11,15,25,29,20,7,16,18)(8,12,23,27,21,19,33,34,22,26,35,36,31,17,30,32);(5,7,9,15,11,14,20,6)(8,19,21,12,22,17,31,26)(10,13,24,18,25,29,16,28)(23,36,33,34,35,27,30,32);(1,2);(1,2)(3,4)(5,8,20,31,11,22,9,21)(6,12,7,17,15,26,14,19)(10,23,16,30,25,35,24,33)(13,27,18,32,29,36,28,34);(3,4)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#495899, #72322802, #72322883
maximal normal subgroup of	C₂²×#495899, #72322802, #72322883

GAP

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

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

C22×#12833 #72322717