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## G = D10order 20 = 22·5

### Dihedral group

Aliases: D10, C2×D5, C10⋊C2, C5⋊C22, sometimes denoted D20 or Dih10 or Dih20, SmallGroup(20,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D10
 Chief series C1 — C5 — D5 — D10
 Lower central C5 — D10
 Upper central C1 — C2

Generators and relations for D10
G = < a,b | a10=b2=1, bab=a-1 >

Character table of D10

 class 1 2A 2B 2C 5A 5B 10A 10B size 1 1 5 5 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ4 1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ5 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 orthogonal faithful ρ6 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ7 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ8 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 orthogonal faithful

Permutation representations of D10
On 10 points - transitive group 10T3
Generators in S10
```(1 2 3 4 5 6 7 8 9 10)
(1 5)(2 4)(6 10)(7 9)```

`G:=sub<Sym(10)| (1,2,3,4,5,6,7,8,9,10), (1,5)(2,4)(6,10)(7,9)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10), (1,5)(2,4)(6,10)(7,9) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10)], [(1,5),(2,4),(6,10),(7,9)]])`

`G:=TransitiveGroup(10,3);`

Regular action on 20 points - transitive group 20T4
Generators in S20
```(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 12)(2 11)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)```

`G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,12),(2,11),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13)]])`

`G:=TransitiveGroup(20,4);`

D10 is a maximal subgroup of   D20  C5⋊D4  2- 1+4⋊D5
D10 is a maximal quotient of   Dic10  D20  C5⋊D4

Polynomial with Galois group D10 over ℚ
actionf(x)Disc(f)
10T3x10-x9-16x8+11x7+58x6-19x5-68x4+8x3+21x2-3x-136·55·132·4014

Matrix representation of D10 in GL2(𝔽11) generated by

 4 1 10 0
,
 0 1 1 0
`G:=sub<GL(2,GF(11))| [4,10,1,0],[0,1,1,0] >;`

D10 in GAP, Magma, Sage, TeX

`D_{10}`
`% in TeX`

`G:=Group("D10");`
`// GroupNames label`

`G:=SmallGroup(20,4);`
`// by ID`

`G=gap.SmallGroup(20,4);`
`# by ID`

`G:=PCGroup([3,-2,-2,-5,146]);`
`// Polycyclic`

`G:=Group<a,b|a^10=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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