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## G = D62order 124 = 22·31

### Dihedral group

Aliases: D62, C2×D31, C62⋊C2, C31⋊C22, sometimes denoted D124 or Dih62 or Dih124, SmallGroup(124,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C31 — D62
 Chief series C1 — C31 — D31 — D62
 Lower central C31 — D62
 Upper central C1 — C2

Generators and relations for D62
G = < a,b | a62=b2=1, bab=a-1 >

Smallest permutation representation of D62
On 62 points
Generators in S62
```(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)
(1 62)(2 61)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)```

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

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

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

D62 is a maximal subgroup of   D124  C31⋊D4
D62 is a maximal quotient of   Dic62  D124  C31⋊D4

34 conjugacy classes

 class 1 2A 2B 2C 31A ··· 31O 62A ··· 62O order 1 2 2 2 31 ··· 31 62 ··· 62 size 1 1 31 31 2 ··· 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D31 D62 kernel D62 D31 C62 C2 C1 # reps 1 2 1 15 15

Matrix representation of D62 in GL3(𝔽311) generated by

 310 0 0 0 239 211 0 100 100
,
 1 0 0 0 239 211 0 198 72
`G:=sub<GL(3,GF(311))| [310,0,0,0,239,100,0,211,100],[1,0,0,0,239,198,0,211,72] >;`

D62 in GAP, Magma, Sage, TeX

`D_{62}`
`% in TeX`

`G:=Group("D62");`
`// GroupNames label`

`G:=SmallGroup(124,3);`
`// by ID`

`G=gap.SmallGroup(124,3);`
`# by ID`

`G:=PCGroup([3,-2,-2,-31,1082]);`
`// Polycyclic`

`G:=Group<a,b|a^62=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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