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## G = D25order 50 = 2·52

### Dihedral group

Aliases: D25, C25⋊C2, C5.D5, sometimes denoted D50 or Dih25 or Dih50, SmallGroup(50,1)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D25
G = < a,b | a25=b2=1, bab=a-1 >

Character table of D25

 class 1 2 5A 5B 25A 25B 25C 25D 25E 25F 25G 25H 25I 25J size 1 25 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 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 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 orthogonal lifted from D5 ρ4 2 0 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 orthogonal lifted from D5 ρ5 2 0 -1-√5/2 -1+√5/2 ζ2521+ζ254 ζ2524+ζ25 ζ2519+ζ256 ζ2514+ζ2511 ζ2518+ζ257 ζ2513+ζ2512 ζ2517+ζ258 ζ2522+ζ253 ζ2523+ζ252 ζ2516+ζ259 orthogonal faithful ρ6 2 0 -1+√5/2 -1-√5/2 ζ2517+ζ258 ζ2523+ζ252 ζ2513+ζ2512 ζ2522+ζ253 ζ2514+ζ2511 ζ2524+ζ25 ζ2516+ζ259 ζ2519+ζ256 ζ2521+ζ254 ζ2518+ζ257 orthogonal faithful ρ7 2 0 -1-√5/2 -1+√5/2 ζ2524+ζ25 ζ2519+ζ256 ζ2514+ζ2511 ζ2516+ζ259 ζ2517+ζ258 ζ2522+ζ253 ζ2523+ζ252 ζ2518+ζ257 ζ2513+ζ2512 ζ2521+ζ254 orthogonal faithful ρ8 2 0 -1+√5/2 -1-√5/2 ζ2522+ζ253 ζ2518+ζ257 ζ2517+ζ258 ζ2523+ζ252 ζ2524+ζ25 ζ2516+ζ259 ζ2519+ζ256 ζ2521+ζ254 ζ2514+ζ2511 ζ2513+ζ2512 orthogonal faithful ρ9 2 0 -1+√5/2 -1-√5/2 ζ2518+ζ257 ζ2517+ζ258 ζ2523+ζ252 ζ2513+ζ2512 ζ2519+ζ256 ζ2521+ζ254 ζ2514+ζ2511 ζ2524+ζ25 ζ2516+ζ259 ζ2522+ζ253 orthogonal faithful ρ10 2 0 -1-√5/2 -1+√5/2 ζ2514+ζ2511 ζ2516+ζ259 ζ2521+ζ254 ζ2524+ζ25 ζ2513+ζ2512 ζ2517+ζ258 ζ2522+ζ253 ζ2523+ζ252 ζ2518+ζ257 ζ2519+ζ256 orthogonal faithful ρ11 2 0 -1+√5/2 -1-√5/2 ζ2523+ζ252 ζ2513+ζ2512 ζ2522+ζ253 ζ2518+ζ257 ζ2516+ζ259 ζ2519+ζ256 ζ2521+ζ254 ζ2514+ζ2511 ζ2524+ζ25 ζ2517+ζ258 orthogonal faithful ρ12 2 0 -1+√5/2 -1-√5/2 ζ2513+ζ2512 ζ2522+ζ253 ζ2518+ζ257 ζ2517+ζ258 ζ2521+ζ254 ζ2514+ζ2511 ζ2524+ζ25 ζ2516+ζ259 ζ2519+ζ256 ζ2523+ζ252 orthogonal faithful ρ13 2 0 -1-√5/2 -1+√5/2 ζ2519+ζ256 ζ2514+ζ2511 ζ2516+ζ259 ζ2521+ζ254 ζ2523+ζ252 ζ2518+ζ257 ζ2513+ζ2512 ζ2517+ζ258 ζ2522+ζ253 ζ2524+ζ25 orthogonal faithful ρ14 2 0 -1-√5/2 -1+√5/2 ζ2516+ζ259 ζ2521+ζ254 ζ2524+ζ25 ζ2519+ζ256 ζ2522+ζ253 ζ2523+ζ252 ζ2518+ζ257 ζ2513+ζ2512 ζ2517+ζ258 ζ2514+ζ2511 orthogonal faithful

Permutation representations of D25
On 25 points - transitive group 25T4
Generators in S25
```(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)
(1 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)```

`G:=sub<Sym(25)| (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)>;`

`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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14) );`

`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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14)]])`

`G:=TransitiveGroup(25,4);`

D25 is a maximal subgroup of   C25⋊C4  D75  D125  C25⋊C10  C25⋊D5  D175
D25 is a maximal quotient of   Dic25  D75  D125  C25⋊D5  D175

Matrix representation of D25 in GL2(𝔽101) generated by

 68 97 17 51
,
 99 64 11 2
`G:=sub<GL(2,GF(101))| [68,17,97,51],[99,11,64,2] >;`

D25 in GAP, Magma, Sage, TeX

`D_{25}`
`% in TeX`

`G:=Group("D25");`
`// GroupNames label`

`G:=SmallGroup(50,1);`
`// by ID`

`G=gap.SmallGroup(50,1);`
`# by ID`

`G:=PCGroup([3,-2,-5,-5,289,34,362]);`
`// Polycyclic`

`G:=Group<a,b|a^25=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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