C₂²×#13400 #72311389

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

Aliases

direct product

C₂²×#13400

Subgroups

	subgroup	G/H
center, socle	C₂³	#2034
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#3060	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

7773 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#499269, C₂²×#500168, C₂²×#500473
maximal normal subgroup of	C₂²×#499269, C₂²×#500168, C₂²×#500473

GAP

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

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

C22×#13400 #72311389