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## G = D61order 122 = 2·61

### Dihedral group

Aliases: D61, C61⋊C2, sometimes denoted D122 or Dih61 or Dih122, SmallGroup(122,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C61 — D61
 Chief series C1 — C61 — D61
 Lower central C61 — D61
 Upper central C1

Generators and relations for D61
G = < a,b | a61=b2=1, bab=a-1 >

Smallest permutation representation of D61
On 61 points: primitive
Generators in S61
```(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)
(1 61)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)```

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

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

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

D61 is a maximal subgroup of   C61⋊C4  C61⋊C6  D183
D61 is a maximal quotient of   Dic61  D183

32 conjugacy classes

 class 1 2 61A ··· 61AD order 1 2 61 ··· 61 size 1 61 2 ··· 2

32 irreducible representations

 dim 1 1 2 type + + + image C1 C2 D61 kernel D61 C61 C1 # reps 1 1 30

Matrix representation of D61 in GL2(𝔽367) generated by

 62 366 1 0
,
 62 366 173 305
`G:=sub<GL(2,GF(367))| [62,1,366,0],[62,173,366,305] >;`

D61 in GAP, Magma, Sage, TeX

`D_{61}`
`% in TeX`

`G:=Group("D61");`
`// GroupNames label`

`G:=SmallGroup(122,1);`
`// by ID`

`G=gap.SmallGroup(122,1);`
`# by ID`

`G:=PCGroup([2,-2,-61,481]);`
`// Polycyclic`

`G:=Group<a,b|a^61=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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