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## G = D50order 100 = 22·52

### Dihedral group

Aliases: D50, C2×D25, C50⋊C2, C25⋊C22, C5.D10, C10.2D5, sometimes denoted D100 or Dih50 or Dih100, SmallGroup(100,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — D50
 Chief series C1 — C5 — C25 — D25 — D50
 Lower central C25 — D50
 Upper central C1 — C2

Generators and relations for D50
G = < a,b | a50=b2=1, bab=a-1 >

Character table of D50

 class 1 2A 2B 2C 5A 5B 10A 10B 25A 25B 25C 25D 25E 25F 25G 25H 25I 25J 50A 50B 50C 50D 50E 50F 50G 50H 50I 50J size 1 1 25 25 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 -1 1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 2 2 0 0 2 2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ6 2 -2 0 0 2 2 -2 -2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ7 2 -2 0 0 2 2 -2 -2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ8 2 2 0 0 2 2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ9 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 ζ2516+ζ259 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2514+ζ2511 ζ2519+ζ256 ζ2524+ζ25 ζ2517+ζ258 ζ2521+ζ254 -ζ2514-ζ2511 -ζ2519-ζ256 -ζ2524-ζ25 -ζ2517-ζ258 -ζ2513-ζ2512 -ζ2518-ζ257 -ζ2523-ζ252 -ζ2522-ζ253 -ζ2521-ζ254 -ζ2516-ζ259 orthogonal faithful ρ10 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 ζ2517+ζ258 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2518+ζ257 ζ2522+ζ253 ζ2513+ζ2512 ζ2521+ζ254 ζ2523+ζ252 -ζ2518-ζ257 -ζ2522-ζ253 -ζ2513-ζ2512 -ζ2521-ζ254 -ζ2519-ζ256 -ζ2516-ζ259 -ζ2524-ζ25 -ζ2514-ζ2511 -ζ2523-ζ252 -ζ2517-ζ258 orthogonal faithful ρ11 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ2516+ζ259 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2514+ζ2511 ζ2519+ζ256 ζ2524+ζ25 ζ2517+ζ258 ζ2521+ζ254 ζ2514+ζ2511 ζ2519+ζ256 ζ2524+ζ25 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2521+ζ254 ζ2516+ζ259 orthogonal lifted from D25 ρ12 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 ζ2523+ζ252 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2517+ζ258 ζ2518+ζ257 ζ2522+ζ253 ζ2524+ζ25 ζ2513+ζ2512 -ζ2517-ζ258 -ζ2518-ζ257 -ζ2522-ζ253 -ζ2524-ζ25 -ζ2514-ζ2511 -ζ2521-ζ254 -ζ2519-ζ256 -ζ2516-ζ259 -ζ2513-ζ2512 -ζ2523-ζ252 orthogonal faithful ρ13 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 ζ2521+ζ254 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2516+ζ259 ζ2514+ζ2511 ζ2519+ζ256 ζ2523+ζ252 ζ2524+ζ25 -ζ2516-ζ259 -ζ2514-ζ2511 -ζ2519-ζ256 -ζ2523-ζ252 -ζ2522-ζ253 -ζ2517-ζ258 -ζ2513-ζ2512 -ζ2518-ζ257 -ζ2524-ζ25 -ζ2521-ζ254 orthogonal faithful ρ14 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 ζ2522+ζ253 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2513+ζ2512 ζ2523+ζ252 ζ2517+ζ258 ζ2514+ζ2511 ζ2518+ζ257 ζ2513+ζ2512 ζ2523+ζ252 ζ2517+ζ258 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2518+ζ257 ζ2522+ζ253 orthogonal lifted from D25 ρ15 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ2514+ζ2511 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2519+ζ256 ζ2524+ζ25 ζ2521+ζ254 ζ2518+ζ257 ζ2516+ζ259 ζ2519+ζ256 ζ2524+ζ25 ζ2521+ζ254 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2516+ζ259 ζ2514+ζ2511 orthogonal lifted from D25 ρ16 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ2521+ζ254 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2516+ζ259 ζ2514+ζ2511 ζ2519+ζ256 ζ2523+ζ252 ζ2524+ζ25 ζ2516+ζ259 ζ2514+ζ2511 ζ2519+ζ256 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2524+ζ25 ζ2521+ζ254 orthogonal lifted from D25 ρ17 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 ζ2524+ζ25 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2521+ζ254 ζ2516+ζ259 ζ2514+ζ2511 ζ2513+ζ2512 ζ2519+ζ256 -ζ2521-ζ254 -ζ2516-ζ259 -ζ2514-ζ2511 -ζ2513-ζ2512 -ζ2518-ζ257 -ζ2523-ζ252 -ζ2522-ζ253 -ζ2517-ζ258 -ζ2519-ζ256 -ζ2524-ζ25 orthogonal faithful ρ18 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 ζ2518+ζ257 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2522+ζ253 ζ2513+ζ2512 ζ2523+ζ252 ζ2516+ζ259 ζ2517+ζ258 ζ2522+ζ253 ζ2513+ζ2512 ζ2523+ζ252 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2517+ζ258 ζ2518+ζ257 orthogonal lifted from D25 ρ19 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 ζ2514+ζ2511 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2519+ζ256 ζ2524+ζ25 ζ2521+ζ254 ζ2518+ζ257 ζ2516+ζ259 -ζ2519-ζ256 -ζ2524-ζ25 -ζ2521-ζ254 -ζ2518-ζ257 -ζ2523-ζ252 -ζ2522-ζ253 -ζ2517-ζ258 -ζ2513-ζ2512 -ζ2516-ζ259 -ζ2514-ζ2511 orthogonal faithful ρ20 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 ζ2518+ζ257 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2522+ζ253 ζ2513+ζ2512 ζ2523+ζ252 ζ2516+ζ259 ζ2517+ζ258 -ζ2522-ζ253 -ζ2513-ζ2512 -ζ2523-ζ252 -ζ2516-ζ259 -ζ2524-ζ25 -ζ2514-ζ2511 -ζ2521-ζ254 -ζ2519-ζ256 -ζ2517-ζ258 -ζ2518-ζ257 orthogonal faithful ρ21 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 ζ2517+ζ258 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2518+ζ257 ζ2522+ζ253 ζ2513+ζ2512 ζ2521+ζ254 ζ2523+ζ252 ζ2518+ζ257 ζ2522+ζ253 ζ2513+ζ2512 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2523+ζ252 ζ2517+ζ258 orthogonal lifted from D25 ρ22 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 ζ2522+ζ253 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2513+ζ2512 ζ2523+ζ252 ζ2517+ζ258 ζ2514+ζ2511 ζ2518+ζ257 -ζ2513-ζ2512 -ζ2523-ζ252 -ζ2517-ζ258 -ζ2514-ζ2511 -ζ2521-ζ254 -ζ2519-ζ256 -ζ2516-ζ259 -ζ2524-ζ25 -ζ2518-ζ257 -ζ2522-ζ253 orthogonal faithful ρ23 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ2519+ζ256 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2524+ζ25 ζ2521+ζ254 ζ2516+ζ259 ζ2522+ζ253 ζ2514+ζ2511 ζ2524+ζ25 ζ2521+ζ254 ζ2516+ζ259 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2514+ζ2511 ζ2519+ζ256 orthogonal lifted from D25 ρ24 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 ζ2523+ζ252 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2517+ζ258 ζ2518+ζ257 ζ2522+ζ253 ζ2524+ζ25 ζ2513+ζ2512 ζ2517+ζ258 ζ2518+ζ257 ζ2522+ζ253 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2513+ζ2512 ζ2523+ζ252 orthogonal lifted from D25 ρ25 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 ζ2513+ζ2512 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2523+ζ252 ζ2517+ζ258 ζ2518+ζ257 ζ2519+ζ256 ζ2522+ζ253 -ζ2523-ζ252 -ζ2517-ζ258 -ζ2518-ζ257 -ζ2519-ζ256 -ζ2516-ζ259 -ζ2524-ζ25 -ζ2514-ζ2511 -ζ2521-ζ254 -ζ2522-ζ253 -ζ2513-ζ2512 orthogonal faithful ρ26 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ2524+ζ25 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2521+ζ254 ζ2516+ζ259 ζ2514+ζ2511 ζ2513+ζ2512 ζ2519+ζ256 ζ2521+ζ254 ζ2516+ζ259 ζ2514+ζ2511 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2519+ζ256 ζ2524+ζ25 orthogonal lifted from D25 ρ27 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 ζ2513+ζ2512 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2523+ζ252 ζ2517+ζ258 ζ2518+ζ257 ζ2519+ζ256 ζ2522+ζ253 ζ2523+ζ252 ζ2517+ζ258 ζ2518+ζ257 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2522+ζ253 ζ2513+ζ2512 orthogonal lifted from D25 ρ28 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 ζ2519+ζ256 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2524+ζ25 ζ2521+ζ254 ζ2516+ζ259 ζ2522+ζ253 ζ2514+ζ2511 -ζ2524-ζ25 -ζ2521-ζ254 -ζ2516-ζ259 -ζ2522-ζ253 -ζ2517-ζ258 -ζ2513-ζ2512 -ζ2518-ζ257 -ζ2523-ζ252 -ζ2514-ζ2511 -ζ2519-ζ256 orthogonal faithful

Smallest permutation representation of D50
On 50 points
Generators in S50
```(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)
(1 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)```

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

`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), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26) );`

`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)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26)]])`

D50 is a maximal subgroup of   D100  C25⋊D4
D50 is a maximal quotient of   Dic50  D100  C25⋊D4

Matrix representation of D50 in GL2(𝔽101) generated by

 64 51 89 52
,
 4 6 48 97
`G:=sub<GL(2,GF(101))| [64,89,51,52],[4,48,6,97] >;`

D50 in GAP, Magma, Sage, TeX

`D_{50}`
`% in TeX`

`G:=Group("D50");`
`// GroupNames label`

`G:=SmallGroup(100,4);`
`// by ID`

`G=gap.SmallGroup(100,4);`
`# by ID`

`G:=PCGroup([4,-2,-2,-5,-5,434,250,1283]);`
`// Polycyclic`

`G:=Group<a,b|a^50=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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