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## G = D59order 118 = 2·59

### Dihedral group

Aliases: D59, C59⋊C2, sometimes denoted D118 or Dih59 or Dih118, SmallGroup(118,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C59 — D59
 Chief series C1 — C59 — D59
 Lower central C59 — D59
 Upper central C1

Generators and relations for D59
G = < a,b | a59=b2=1, bab=a-1 >

Smallest permutation representation of D59
On 59 points: primitive
Generators in S59
```(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)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)```

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

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

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

D59 is a maximal subgroup of   D177
D59 is a maximal quotient of   Dic59  D177

31 conjugacy classes

 class 1 2 59A ··· 59AC order 1 2 59 ··· 59 size 1 59 2 ··· 2

31 irreducible representations

 dim 1 1 2 type + + + image C1 C2 D59 kernel D59 C59 C1 # reps 1 1 29

Matrix representation of D59 in GL2(𝔽709) generated by

 82 708 1 0
,
 82 708 342 627
`G:=sub<GL(2,GF(709))| [82,1,708,0],[82,342,708,627] >;`

D59 in GAP, Magma, Sage, TeX

`D_{59}`
`% in TeX`

`G:=Group("D59");`
`// GroupNames label`

`G:=SmallGroup(118,1);`
`// by ID`

`G=gap.SmallGroup(118,1);`
`# by ID`

`G:=PCGroup([2,-2,-59,465]);`
`// Polycyclic`

`G:=Group<a,b|a^59=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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