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## G = D65order 130 = 2·5·13

### Dihedral group

Aliases: D65, C13⋊D5, C5⋊D13, C651C2, sometimes denoted D130 or Dih65 or Dih130, SmallGroup(130,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C65 — D65
 Chief series C1 — C13 — C65 — D65
 Lower central C65 — D65
 Upper central C1

Generators and relations for D65
G = < a,b | a65=b2=1, bab=a-1 >

65C2
13D5
5D13

Smallest permutation representation of D65
On 65 points
Generators in S65
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65)
(1 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)```

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

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

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

D65 is a maximal subgroup of   C13⋊F5  C652C4  D5×D13  D65⋊C3  D195
D65 is a maximal quotient of   Dic65  D195

34 conjugacy classes

 class 1 2 5A 5B 13A ··· 13F 65A ··· 65X order 1 2 5 5 13 ··· 13 65 ··· 65 size 1 65 2 2 2 ··· 2 2 ··· 2

34 irreducible representations

 dim 1 1 2 2 2 type + + + + + image C1 C2 D5 D13 D65 kernel D65 C65 C13 C5 C1 # reps 1 1 2 6 24

Matrix representation of D65 in GL2(𝔽131) generated by

 36 102 54 113
,
 32 7 41 99
`G:=sub<GL(2,GF(131))| [36,54,102,113],[32,41,7,99] >;`

D65 in GAP, Magma, Sage, TeX

`D_{65}`
`% in TeX`

`G:=Group("D65");`
`// GroupNames label`

`G:=SmallGroup(130,3);`
`// by ID`

`G=gap.SmallGroup(130,3);`
`# by ID`

`G:=PCGroup([3,-2,-5,-13,49,1082]);`
`// Polycyclic`

`G:=Group<a,b|a^65=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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