C₂²×#13032 #72320213

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

Aliases

direct product

C₂²×#13032

Subgroups

	subgroup	G/H
center, socle	C₂³	#1879
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

10049 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)(8,23)(9,24)(10,25)(13,19)(14,26)(15,17)(16,28)(20,27)(21,29)(22,35)(30,32)(33,36);(1,2);(3,4);(5,6,8,14,11,17,23,16)(7,19,20,34,22,13,21,31)(9,26,24,12,10,28,25,15)(18,33,29,36,35,30,27,32);(1,2)(3,4)(5,7)(6,13)(8,20)(9,27)(10,29)(11,22)(12,30)(14,31)(15,33)(16,34)(17,19)(18,24)(21,23)(25,35)(26,36)(28,32)

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×#72319998

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

C22×#13032 #72320213