C₂²×#13318 #72311284

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#13318

Subgroups

	subgroup	G/H
center, socle	C₂³	#1999
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

11593 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#499752, C₂²×#499269, C₂²×#500473
maximal normal subgroup of	C₂²×#499752, C₂²×#499269, C₂²×#500473

GAP

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

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

C22×#13318 #72311284