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C2×#496723 #72321393

order1024
nilpotency class3
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#496723

Subgroups

subgroupG/H
centerC2×C8#535
commutatorC22×C4C24×C4
Frattini, cosocleC22×C8C25
socleC22#21973
prime radicalC22×#41C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

2523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 66
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 66 points:

(3,4,8,15,10,16,19,14,11,17,33,49,9,34,7,31)(5,18,21,51,22,12,36,39,24,40,56,48,23,41,32,38)(6,25,28,58,29,46,52,57,30,13,42,45,35,47,27,44)(20,50,53,65,55,37,61,64,26,59,60,66,54,43,62,63);(1,2);(3,5,19,36,9,23,8,21,11,24,7,32,10,22,33,56)(4,12,14,48,34,18,15,39,17,41,31,51,16,40,49,38)(6,26,52,62,35,55,28,60,30,20,27,61,29,54,42,53)(13,43,44,65,46,59,45,63,25,37,57,66,47,50,58,64);(3,6,11,30)(4,13,17,25)(5,20,24,26)(7,27,19,52)(8,28,33,42)(9,35,10,29)(12,37,41,43)(14,44,31,57)(15,45,49,58)(16,47,34,46)(18,50,40,59)(21,53,56,60)(22,55,23,54)(32,62,36,61)(38,63,39,64)(48,66,51,65);(3,7,9,33,11,19,10,8)(4,14,34,15,17,31,16,49)(5,21,22,36,24,56,23,32)(6,27,35,42,30,52,29,28)(12,38,40,51,41,39,18,48)(13,44,46,45,25,57,47,58)(20,53,55,61,26,60,54,62)(37,63,59,65,43,64,50,66)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72224951, C2×#72224958, C2×#72224961
maximal normal subgroup ofC2×#72224951, C2×#72224958, C2×#72224961

GAP

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

gap> G := Group((3,4,8,15,10,16,19,14,11,17,33,49,9,34,7,31)(5,18,21,51,22,12,36,39,24,40,56,48,23,41,32,38)(6,25,28,58,29,46,52,57,30,13,42,45,35,47,27,44)(20,50,53,65,55,37,61,64,26,59,60,66,54,43,62,63), (1,2), (3,5,19,36,9,23,8,21,11,24,7,32,10,22,33,56)(4,12,14,48,34,18,15,39,17,41,31,51,16,40,49,38)(6,26,52,62,35,55,28,60,30,20,27,61,29,54,42,53)(13,43,44,65,46,59,45,63,25,37,57,66,47,50,58,64), (3,6,11,30)(4,13,17,25)(5,20,24,26)(7,27,19,52)(8,28,33,42)(9,35,10,29)(12,37,41,43)(14,44,31,57)(15,45,49,58)(16,47,34,46)(18,50,40,59)(21,53,56,60)(22,55,23,54)(32,62,36,61)(38,63,39,64)(48,66,51,65), (3,7,9,33,11,19,10,8)(4,14,34,15,17,31,16,49)(5,21,22,36,24,56,23,32)(6,27,35,42,30,52,29,28)(12,38,40,51,41,39,18,48)(13,44,46,45,25,57,47,58)(20,53,55,61,26,60,54,62)(37,63,59,65,43,64,50,66));