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## G = D63order 126 = 2·32·7

### Dihedral group

Aliases: D63, C9⋊D7, C7⋊D9, C631C2, C3.D21, C21.1S3, sometimes denoted D126 or Dih63 or Dih126, SmallGroup(126,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C63 — D63
 Chief series C1 — C3 — C21 — C63 — D63
 Lower central C63 — D63
 Upper central C1

Generators and relations for D63
G = < a,b | a63=b2=1, bab=a-1 >

63C2
21S3
9D7
7D9
3D21

Smallest permutation representation of D63
On 63 points
Generators in S63
```(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)
(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)```

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

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

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

D63 is a maximal subgroup of   D7×D9  D189  C9⋊F7  C92F7  C95F7  D63⋊C3  C3⋊D63
D63 is a maximal quotient of   Dic63  D189  C3⋊D63

33 conjugacy classes

 class 1 2 3 7A 7B 7C 9A 9B 9C 21A ··· 21F 63A ··· 63R order 1 2 3 7 7 7 9 9 9 21 ··· 21 63 ··· 63 size 1 63 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

33 irreducible representations

 dim 1 1 2 2 2 2 2 type + + + + + + + image C1 C2 S3 D7 D9 D21 D63 kernel D63 C63 C21 C9 C7 C3 C1 # reps 1 1 1 3 3 6 18

Matrix representation of D63 in GL2(𝔽127) generated by

 66 27 100 39
,
 63 23 87 64
`G:=sub<GL(2,GF(127))| [66,100,27,39],[63,87,23,64] >;`

D63 in GAP, Magma, Sage, TeX

`D_{63}`
`% in TeX`

`G:=Group("D63");`
`// GroupNames label`

`G:=SmallGroup(126,5);`
`// by ID`

`G=gap.SmallGroup(126,5);`
`# by ID`

`G:=PCGroup([4,-2,-3,-7,-3,369,341,434,1347]);`
`// Polycyclic`

`G:=Group<a,b|a^63=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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