C₂²×#13077 #72311388

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

Aliases

direct product

C₂²×#13077

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

9701 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500168, C₂²×#495573, C₂²×#500003
maximal normal subgroup of	C₂²×#500168, C₂²×#495573, C₂²×#500003

GAP

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

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

C22×#13077 #72311388