C₂²×#12743 #72311304

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#12743

Subgroups

	subgroup	G/H
center, socle	C₂³	#1786
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#576	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

32189 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#499752, C₂²×#494058, C₂²×#500164, …
maximal normal subgroup of	C₂²×#499752, C₂²×#494058, C₂²×#500164, …
maximal quotient of	C₂×#72280443

GAP

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

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

C22×#12743 #72311304