Copied to
clipboard

G = D18order 36 = 22·32

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D18, C2×D9, C18⋊C2, C9⋊C22, C3.D6, C6.2S3, sometimes denoted D36 or Dih18 or Dih36, SmallGroup(36,4)

Series: Derived Chief Lower central Upper central

C1C9 — D18
C1C3C9D9 — D18
C9 — D18
C1C2

Generators and relations for D18
 G = < a,b | a18=b2=1, bab=a-1 >

9C2
9C2
9C22
3S3
3S3
3D6

Character table of D18

 class 12A2B2C369A9B9C18A18B18C
 size 119922222222
ρ1111111111111    trivial
ρ21-1-111-1111-1-1-1    linear of order 2
ρ31-11-11-1111-1-1-1    linear of order 2
ρ411-1-111111111    linear of order 2
ρ52-2002-2-1-1-1111    orthogonal lifted from D6
ρ6220022-1-1-1-1-1-1    orthogonal lifted from S3
ρ72-200-11ζ9792ζ9594ζ98997929594989    orthogonal faithful
ρ82200-1-1ζ9594ζ989ζ9792ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ92-200-11ζ989ζ9792ζ959498997929594    orthogonal faithful
ρ102200-1-1ζ9792ζ9594ζ989ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ112-200-11ζ9594ζ989ζ979295949899792    orthogonal faithful
ρ122200-1-1ζ989ζ9792ζ9594ζ989ζ9792ζ9594    orthogonal lifted from D9

Permutation representations of D18
On 18 points - transitive group 18T13
Generators in S18
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)
(1 9)(2 8)(3 7)(4 6)(10 18)(11 17)(12 16)(13 15)

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

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

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

G:=TransitiveGroup(18,13);

Matrix representation of D18 in GL2(𝔽17) generated by

31
160
,
01
10
G:=sub<GL(2,GF(17))| [3,16,1,0],[0,1,1,0] >;

D18 in GAP, Magma, Sage, TeX

D_{18}
% in TeX

G:=Group("D18");
// GroupNames label

G:=SmallGroup(36,4);
// by ID

G=gap.SmallGroup(36,4);
# by ID

G:=PCGroup([4,-2,-2,-3,-3,242,82,387]);
// Polycyclic

G:=Group<a,b|a^18=b^2=1,b*a*b=a^-1>;
// generators/relations

׿
×
𝔽