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## G = D19order 38 = 2·19

### Dihedral group

Aliases: D19, C19⋊C2, sometimes denoted D38 or Dih19 or Dih38, SmallGroup(38,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — D19
 Chief series C1 — C19 — D19
 Lower central C19 — D19
 Upper central C1

Generators and relations for D19
G = < a,b | a19=b2=1, bab=a-1 >

Character table of D19

 class 1 2 19A 19B 19C 19D 19E 19F 19G 19H 19I size 1 19 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 ζ1915+ζ194 ζ1913+ζ196 ζ1911+ζ198 ζ1910+ζ199 ζ1912+ζ197 ζ1914+ζ195 ζ1916+ζ193 ζ1918+ζ19 ζ1917+ζ192 orthogonal faithful ρ4 2 0 ζ1918+ζ19 ζ1911+ζ198 ζ1917+ζ192 ζ1912+ζ197 ζ1916+ζ193 ζ1913+ζ196 ζ1915+ζ194 ζ1914+ζ195 ζ1910+ζ199 orthogonal faithful ρ5 2 0 ζ1916+ζ193 ζ1914+ζ195 ζ1913+ζ196 ζ1917+ζ192 ζ1910+ζ199 ζ1918+ζ19 ζ1912+ζ197 ζ1915+ζ194 ζ1911+ζ198 orthogonal faithful ρ6 2 0 ζ1912+ζ197 ζ1918+ζ19 ζ1914+ζ195 ζ1911+ζ198 ζ1917+ζ192 ζ1915+ζ194 ζ1910+ζ199 ζ1916+ζ193 ζ1913+ζ196 orthogonal faithful ρ7 2 0 ζ1913+ζ196 ζ1910+ζ199 ζ1912+ζ197 ζ1915+ζ194 ζ1918+ζ19 ζ1917+ζ192 ζ1914+ζ195 ζ1911+ζ198 ζ1916+ζ193 orthogonal faithful ρ8 2 0 ζ1917+ζ192 ζ1916+ζ193 ζ1915+ζ194 ζ1914+ζ195 ζ1913+ζ196 ζ1912+ζ197 ζ1911+ζ198 ζ1910+ζ199 ζ1918+ζ19 orthogonal faithful ρ9 2 0 ζ1910+ζ199 ζ1915+ζ194 ζ1918+ζ19 ζ1913+ζ196 ζ1911+ζ198 ζ1916+ζ193 ζ1917+ζ192 ζ1912+ζ197 ζ1914+ζ195 orthogonal faithful ρ10 2 0 ζ1914+ζ195 ζ1917+ζ192 ζ1910+ζ199 ζ1916+ζ193 ζ1915+ζ194 ζ1911+ζ198 ζ1918+ζ19 ζ1913+ζ196 ζ1912+ζ197 orthogonal faithful ρ11 2 0 ζ1911+ζ198 ζ1912+ζ197 ζ1916+ζ193 ζ1918+ζ19 ζ1914+ζ195 ζ1910+ζ199 ζ1913+ζ196 ζ1917+ζ192 ζ1915+ζ194 orthogonal faithful

Permutation representations of D19
On 19 points: primitive - transitive group 19T2
Generators in S19
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)```

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

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

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

`G:=TransitiveGroup(19,2);`

D19 is a maximal subgroup of
C19⋊C6
D19p: D57  D95  D133  D209  D247 ...
D19 is a maximal quotient of
Dic19
D19p: D57  D95  D133  D209  D247 ...

Matrix representation of D19 in GL2(𝔽37) generated by

 9 26 11 36
,
 36 0 26 1
`G:=sub<GL(2,GF(37))| [9,11,26,36],[36,26,0,1] >;`

D19 in GAP, Magma, Sage, TeX

`D_{19}`
`% in TeX`

`G:=Group("D19");`
`// GroupNames label`

`G:=SmallGroup(38,1);`
`// by ID`

`G=gap.SmallGroup(38,1);`
`# by ID`

`G:=PCGroup([2,-2,-19,145]);`
`// Polycyclic`

`G:=Group<a,b|a^19=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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