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
| C9 — D18 |
Generators and relations for D18
G = < a,b | a18=b2=1, bab=a-1 >
Character table of D18
| class | 1 | 2A | 2B | 2C | 3 | 6 | 9A | 9B | 9C | 18A | 18B | 18C | |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | 2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | -2 | 0 | 0 | -1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal faithful |
| ρ8 | 2 | 2 | 0 | 0 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ9 | 2 | -2 | 0 | 0 | -1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal faithful |
| ρ10 | 2 | 2 | 0 | 0 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ11 | 2 | -2 | 0 | 0 | -1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal faithful |
| ρ12 | 2 | 2 | 0 | 0 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
►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);
D18 is a maximal subgroup of
D36 C9⋊D4 Q8⋊D9
D18 is a maximal quotient of Dic18 D36 C9⋊D4
Matrix representation of D18 ►in GL2(𝔽17) generated by
| 3 | 1 |
| 16 | 0 |
| 0 | 1 |
| 1 | 0 |
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
Export
Subgroup lattice of D18 in TeX
Character table of D18 in TeX