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## G = D20order 40 = 23·5

### Dihedral group

Aliases: D20, C4⋊D5, C51D4, C201C2, D101C2, C2.4D10, C10.3C22, sometimes denoted D40 or Dih20 or Dih40, SmallGroup(40,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D20
 Chief series C1 — C5 — C10 — D10 — D20
 Lower central C5 — C10 — D20
 Upper central C1 — C2 — C4

Generators and relations for D20
G = < a,b | a20=b2=1, bab=a-1 >

Character table of D20

 class 1 2A 2B 2C 4 5A 5B 10A 10B 20A 20B 20C 20D size 1 1 10 10 2 2 2 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 -2 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 0 0 -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 -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 ρ8 2 2 0 0 -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 ρ9 2 2 0 0 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 ρ10 2 -2 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 orthogonal faithful ρ11 2 -2 0 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 orthogonal faithful ρ12 2 -2 0 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 orthogonal faithful ρ13 2 -2 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 orthogonal faithful

Permutation representations of D20
On 20 points - transitive group 20T10
Generators in S20
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)```

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

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

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

`G:=TransitiveGroup(20,10);`

D20 is a maximal subgroup of
C40⋊C2  D40  D4⋊D5  Q8⋊D5  C4○D20  D4×D5  Q82D5  C3⋊D20  D60  D100  C5⋊D20  C20⋊D5  C7⋊D20  D140  C32⋊D20  C11⋊D20  D220
D20 is a maximal quotient of
C40⋊C2  D40  Dic20  C4⋊Dic5  D10⋊C4  C3⋊D20  D60  D100  C5⋊D20  C20⋊D5  C7⋊D20  D140  C32⋊D20  C11⋊D20  D220

Matrix representation of D20 in GL2(𝔽19) generated by

 0 18 1 11
,
 11 6 18 8
`G:=sub<GL(2,GF(19))| [0,1,18,11],[11,18,6,8] >;`

D20 in GAP, Magma, Sage, TeX

`D_{20}`
`% in TeX`

`G:=Group("D20");`
`// GroupNames label`

`G:=SmallGroup(40,6);`
`// by ID`

`G=gap.SmallGroup(40,6);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-5,49,21,515]);`
`// Polycyclic`

`G:=Group<a,b|a^20=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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