C₂²×#13076 #72311277

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

Aliases

direct product

C₂²×#13076

Subgroups

	subgroup	G/H
center, socle	C₂³	#1893
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#3060	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

9661 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#499752, C₂²×#495573, C₂²×#500351
maximal normal subgroup of	C₂²×#499752, C₂²×#495573, C₂²×#500351

GAP

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

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

C22×#13076 #72311277