Finite Group Databasebeta

← Prev (#72322072) Next (#72322074) →

C2×#506440 #72322073

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#506440

Subgroups

subgroupG/H
centerC2×C4#3058
commutator, Frattini, cosocleC22×C8C25
socleC22#28267
prime radicalC22×#508C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#2960 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

2587 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:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72217287, C2×#72224999, C2×#72225007
maximal normal subgroup ofC2×#72217287, C2×#72224999, C2×#72225007

GAP

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

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