C₂²×#13074 #72310837

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

Aliases

direct product

C₂²×#13074

Subgroups

	subgroup	G/H
center, socle	C₂³	#1893
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

10237 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)(8,19)(10,14)(12,24)(13,20)(16,30)(17,25)(18,29)(22,23)(26,28)(27,33)(31,36)(32,35);(1,2);(3,4);(1,2)(3,4)(5,6,9,11)(7,12,18,17)(8,19,20,13)(10,14,22,23)(15,24,29,25)(16,26,32,31)(21,33,34,27)(28,35,36,30);(5,7,10,16,11,17,23,31,9,18,22,32,6,12,14,26)(8,15,21,28,19,24,33,35,20,29,34,36,13,25,27,30)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500349, C₂²×#495573, C₂²×#499742
maximal normal subgroup of	C₂²×#500349, C₂²×#495573, C₂²×#499742

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

C22×#13074 #72310837