C₂²×#13310 #72310861

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

Aliases

direct product

C₂²×#13310

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

12129 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)(6,11)(7,15)(9,13)(10,17)(12,27)(14,21)(16,32)(18,35)(19,22)(20,26)(24,28)(25,29)(30,33)(31,34);(1,2);(1,2)(3,4)(5,6,8,11)(7,16,19,33)(9,13,20,26)(10,17,21,14)(12,28,25,35)(15,32,22,30)(18,27,24,29)(23,34,36,31);(3,4);(1,2)(3,4)(5,7,17,30,11,25,21,18,8,19,14,32,6,12,10,24)(9,22,34,35,26,29,36,33,20,15,31,28,13,27,23,16)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500001, C₂²×#499269, C₂²×#500743
maximal normal subgroup of	C₂²×#500001, C₂²×#499269, C₂²×#500743

GAP

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

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

C22×#13310 #72310861