C₂²×#13460 #72311371

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#13460

Subgroups

	subgroup	G/H
center, socle	C₂³	#2058
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂⁵	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5365 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#499269, C₂²×#500940, C₂²×#500351
maximal normal subgroup of	C₂²×#499269, C₂²×#500940, C₂²×#500351

GAP

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

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

C22×#13460 #72311371