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G = D9order 18 = 2·32

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C9 — D9
C1C3C9 — D9
C9 — D9
C1

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

9C2
3S3

Character table of D9

 class 1239A9B9C
 size 192222
ρ1111111    trivial
ρ21-11111    linear of order 2
ρ3202-1-1-1    orthogonal lifted from S3
ρ420-1ζ9594ζ9792ζ989    orthogonal faithful
ρ520-1ζ989ζ9594ζ9792    orthogonal faithful
ρ620-1ζ9792ζ989ζ9594    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);

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

1610
714
,
146
103
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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