C₂²×#11003 #72280496

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

Aliases

direct product

C₂²×#11003

Subgroups

	subgroup	G/H
center	C₂²×C₄	#392
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);(6,11)(7,16)(8,19)(12,25)(13,18)(14,23)(15,24)(20,34)(22,27)(28,36)(29,30)(31,35);(6,12)(8,19)(9,21)(11,22)(13,18)(14,27)(15,31)(17,32)(23,25)(24,36)(28,30)(29,35);(5,6,13,29)(7,11,9,15)(8,12,26,36)(10,14,18,24)(16,23,21,30)(17,22,20,28)(19,27,33,35)(25,34,31,32);(3,4)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72239645, C₂×#72239625, C₂×#72239632, …
maximal normal subgroup of	C₂×#72239645, C₂×#72239625, C₂×#72239632, …

GAP

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

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

C22×#11003 #72280496