C₂²×#13007 #72320190

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

Aliases

direct product

C₂²×#13007

Subgroups

	subgroup	G/H
center, socle	C₂³	#1874
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#472	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

10737 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)(6,12)(7,18)(8,20)(9,24)(10,25)(14,26)(15,17)(16,28)(19,31)(21,29)(22,27)(23,36)(30,32)(34,35);(1,2);(3,4);(5,6,8,14,11,17,20,16)(7,19,21,13,23,31,22,33)(9,26,24,12,10,28,25,15)(18,35,27,34,36,32,29,30);(1,2)(3,4)(5,7,20,22,11,23,8,21)(6,13,16,19,17,33,14,31)(9,27,25,18,10,29,24,36)(12,30,26,32,15,34,28,35)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#495737
maximal normal subgroup of	C₂²×#495737
maximal quotient of	C₂×#72319991

GAP

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

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

C22×#13007 #72320190