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## G = D9order 18 = 2·32

### Dihedral group

Aliases: D9, C9⋊C2, C3.S3, sometimes denoted D18 or Dih9 or Dih18, SmallGroup(18,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — D9
 Chief series C1 — C3 — C9 — D9
 Lower central C9 — D9
 Upper central C1

Generators and relations for D9
G = < a,b | a9=b2=1, bab=a-1 >

Character table of D9

 class 1 2 3 9A 9B 9C size 1 9 2 2 2 2 ρ1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 linear of order 2 ρ3 2 0 2 -1 -1 -1 orthogonal lifted from S3 ρ4 2 0 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 orthogonal faithful ρ5 2 0 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 orthogonal faithful ρ6 2 0 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 orthogonal faithful

Permutation representations of D9
On 9 points - transitive group 9T3
Generators in S9
```(1 2 3 4 5 6 7 8 9)
(1 9)(2 8)(3 7)(4 6)```

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

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

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

`G:=TransitiveGroup(9,3);`

Regular action on 18 points - transitive group 18T5
Generators in S18
```(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)```

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

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

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

`G:=TransitiveGroup(18,5);`

D9 is a maximal subgroup of
C9⋊C6  C9⋊S3  C3.S4  C52⋊D9
D9p: D27  D45  D63  D99  D117  D153  D171  D207 ...
D9 is a maximal quotient of
C9⋊S3  C3.S4  C52⋊D9
C3p.S3: Dic9  D27  D45  D63  D99  D117  D153  D171 ...

Polynomial with Galois group D9 over ℚ
actionf(x)Disc(f)
9T3x9+3x8-67x7-226x6+699x5+1211x4-3137x3+940x2+904x-392222·74·132·196·292·2294·5472

Matrix representation of D9 in GL2(𝔽17) generated by

 16 10 7 14
,
 14 6 10 3
`G:=sub<GL(2,GF(17))| [16,7,10,14],[14,10,6,3] >;`

D9 in GAP, Magma, Sage, TeX

`D_9`
`% in TeX`

`G:=Group("D9");`
`// GroupNames label`

`G:=SmallGroup(18,1);`
`// by ID`

`G=gap.SmallGroup(18,1);`
`# by ID`

`G:=PCGroup([3,-2,-3,-3,97,22,110]);`
`// Polycyclic`

`G:=Group<a,b|a^9=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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