C₂²×#11000 #72280715

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#11000

Subgroups

	subgroup	G/H
center	C₂²×C₄	#390
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂³	#1438
prime radical	C₂²×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

41849 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);(5,6,9,16,10,15,22,31)(7,12,18,27,19,13,28,29)(8,14,21,30,20,11,23,25)(17,24,33,35,32,26,34,36);(6,11)(8,20)(9,22)(12,26)(13,24)(14,15)(16,30)(17,32)(18,28)(25,31)(27,35)(29,36);(6,12,15,13)(8,20)(11,24,14,26)(16,27,31,29)(17,32)(21,23)(25,35,30,36)(33,34);(3,4)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72239645, C₂×#72237174, C₂×#72239632, …
maximal normal subgroup of	C₂×#72239645, C₂×#72237174, C₂×#72239632, …

GAP

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

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

C22×#11000 #72280715