C₂²×#13079 #72310842

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#13079

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

10197 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)(10,24)(12,26)(14,20)(16,27)(17,33)(18,22)(19,21)(25,36)(28,35)(29,31)(30,32);(1,2);(3,4);(1,2)(3,4)(5,6,8,11)(7,16,18,12)(9,21,19,13)(10,14,20,24)(15,27,22,26)(17,30,25,28)(23,31,34,29)(32,36,35,33);(5,7,14,30,11,16,10,25,8,18,24,28,6,12,20,17)(9,22,29,36,21,27,23,32,19,15,31,33,13,26,34,35)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500001, C₂²×#495573, C₂²×#500164
maximal normal subgroup of	C₂²×#500001, C₂²×#495573, C₂²×#500164

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

C22×#13079 #72310842