metacyclic, supersoluble, monomial
Aliases: C9:C18, D9:C9, C9:C9:C2, (C3xC9).C6, (C3xD9).C3, C3.3(S3xC9), (C3xC9).1S3, C3.2(C9:C6), C32.14(C3xS3), SmallGroup(162,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C9:C18 |
Generators and relations for C9:C18
G = < a,b | a9=b18=1, bab-1=a2 >
Quotients: C1, C2, C3, S3, C6, C9, C18, C3xS3, S3xC9, C9:C6, C9:C18
Character table of C9:C18
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | 9K | 9L | 9M | 9N | 9O | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 9 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ94 | ζ9 | ζ97 | ζ98 | ζ95 | ζ92 | 1 | ζ94 | ζ92 | ζ9 | ζ32 | ζ97 | ζ98 | ζ3 | ζ95 | ζ92 | ζ98 | ζ95 | ζ94 | ζ9 | ζ97 | linear of order 9 |
| ρ8 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ94 | ζ9 | ζ97 | ζ98 | ζ95 | ζ92 | 1 | ζ94 | ζ92 | ζ9 | ζ32 | ζ97 | ζ98 | ζ3 | ζ95 | -ζ92 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | -ζ97 | linear of order 18 |
| ρ9 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ9 | ζ97 | ζ94 | ζ92 | ζ98 | ζ95 | 1 | ζ9 | ζ95 | ζ97 | ζ32 | ζ94 | ζ92 | ζ3 | ζ98 | ζ95 | ζ92 | ζ98 | ζ9 | ζ97 | ζ94 | linear of order 9 |
| ρ10 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ92 | ζ95 | ζ98 | ζ94 | ζ97 | ζ9 | 1 | ζ92 | ζ9 | ζ95 | ζ3 | ζ98 | ζ94 | ζ32 | ζ97 | ζ9 | ζ94 | ζ97 | ζ92 | ζ95 | ζ98 | linear of order 9 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ92 | ζ95 | ζ98 | ζ94 | ζ97 | ζ9 | 1 | ζ92 | ζ9 | ζ95 | ζ3 | ζ98 | ζ94 | ζ32 | ζ97 | -ζ9 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | -ζ98 | linear of order 18 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ95 | ζ98 | ζ92 | ζ9 | ζ94 | ζ97 | 1 | ζ95 | ζ97 | ζ98 | ζ3 | ζ92 | ζ9 | ζ32 | ζ94 | -ζ97 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | -ζ92 | linear of order 18 |
| ρ13 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ9 | ζ97 | ζ94 | ζ92 | ζ98 | ζ95 | 1 | ζ9 | ζ95 | ζ97 | ζ32 | ζ94 | ζ92 | ζ3 | ζ98 | -ζ95 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | -ζ94 | linear of order 18 |
| ρ14 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ98 | ζ92 | ζ95 | ζ97 | ζ9 | ζ94 | 1 | ζ98 | ζ94 | ζ92 | ζ3 | ζ95 | ζ97 | ζ32 | ζ9 | -ζ94 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | -ζ95 | linear of order 18 |
| ρ15 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ97 | ζ94 | ζ9 | ζ95 | ζ92 | ζ98 | 1 | ζ97 | ζ98 | ζ94 | ζ32 | ζ9 | ζ95 | ζ3 | ζ92 | -ζ98 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | -ζ9 | linear of order 18 |
| ρ16 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ97 | ζ94 | ζ9 | ζ95 | ζ92 | ζ98 | 1 | ζ97 | ζ98 | ζ94 | ζ32 | ζ9 | ζ95 | ζ3 | ζ92 | ζ98 | ζ95 | ζ92 | ζ97 | ζ94 | ζ9 | linear of order 9 |
| ρ17 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ95 | ζ98 | ζ92 | ζ9 | ζ94 | ζ97 | 1 | ζ95 | ζ97 | ζ98 | ζ3 | ζ92 | ζ9 | ζ32 | ζ94 | ζ97 | ζ9 | ζ94 | ζ95 | ζ98 | ζ92 | linear of order 9 |
| ρ18 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ98 | ζ92 | ζ95 | ζ97 | ζ9 | ζ94 | 1 | ζ98 | ζ94 | ζ92 | ζ3 | ζ95 | ζ97 | ζ32 | ζ9 | ζ94 | ζ97 | ζ9 | ζ98 | ζ92 | ζ95 | linear of order 9 |
| ρ19 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ20 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1+√-3 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1-√-3 | -1 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | ζ6 | -1 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ21 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1-√-3 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1+√-3 | -1 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | ζ65 | -1 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ22 | 2 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 2 | 0 | 0 | 2ζ95 | 2ζ98 | 2ζ92 | 2ζ9 | 2ζ94 | 2ζ97 | -1 | -ζ95 | -ζ97 | -ζ98 | ζ65 | -ζ92 | -ζ9 | ζ6 | -ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from S3xC9 |
| ρ23 | 2 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 2 | 0 | 0 | 2ζ97 | 2ζ94 | 2ζ9 | 2ζ95 | 2ζ92 | 2ζ98 | -1 | -ζ97 | -ζ98 | -ζ94 | ζ6 | -ζ9 | -ζ95 | ζ65 | -ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from S3xC9 |
| ρ24 | 2 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 2 | 0 | 0 | 2ζ98 | 2ζ92 | 2ζ95 | 2ζ97 | 2ζ9 | 2ζ94 | -1 | -ζ98 | -ζ94 | -ζ92 | ζ65 | -ζ95 | -ζ97 | ζ6 | -ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from S3xC9 |
| ρ25 | 2 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 2 | 0 | 0 | 2ζ92 | 2ζ95 | 2ζ98 | 2ζ94 | 2ζ97 | 2ζ9 | -1 | -ζ92 | -ζ9 | -ζ95 | ζ65 | -ζ98 | -ζ94 | ζ6 | -ζ97 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from S3xC9 |
| ρ26 | 2 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 2 | 0 | 0 | 2ζ94 | 2ζ9 | 2ζ97 | 2ζ98 | 2ζ95 | 2ζ92 | -1 | -ζ94 | -ζ92 | -ζ9 | ζ6 | -ζ97 | -ζ98 | ζ65 | -ζ95 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from S3xC9 |
| ρ27 | 2 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 2 | 0 | 0 | 2ζ9 | 2ζ97 | 2ζ94 | 2ζ92 | 2ζ98 | 2ζ95 | -1 | -ζ9 | -ζ95 | -ζ97 | ζ6 | -ζ94 | -ζ92 | ζ65 | -ζ98 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from S3xC9 |
| ρ28 | 6 | 0 | 6 | 6 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9:C6 |
| ρ29 | 6 | 0 | -3+3√-3 | -3-3√-3 | 3-3√-3/2 | 3+3√-3/2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 6 | 0 | -3-3√-3 | -3+3√-3 | 3+3√-3/2 | 3-3√-3/2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 18 points - transitive group 18T80
Generators in S18
(1 5 15 13 17 9 7 11 3)(2 16 18 8 4 6 14 10 12)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)
G:=sub<Sym(18)| (1,5,15,13,17,9,7,11,3)(2,16,18,8,4,6,14,10,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)>;
G:=Group( (1,5,15,13,17,9,7,11,3)(2,16,18,8,4,6,14,10,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18) );
G=PermutationGroup([[(1,5,15,13,17,9,7,11,3),(2,16,18,8,4,6,14,10,12)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)]])
G:=TransitiveGroup(18,80);
C9:C18 is a maximal subgroup of
C27:C18 C9:C9.S3 C9:C9.3S3 C9:C9:S3 C9xC9:C6 D9:3- 1+2 C92:7C6 C92:8C6 C9:(S3xC9) C92:3S3 C9:C9:2S3 C92:6S3 C92:5S3
C9:C18 is a maximal quotient of
C9:C36 C9:S3:C9 C9:C54 C27:C18 C9:(S3xC9)
Matrix representation of C9:C18 ►in GL6(F19)
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(19))| [0,0,1,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,1,0,0],[0,0,0,0,0,11,0,0,0,1,0,0,0,0,0,0,11,0,0,0,11,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0] >;
C9:C18 in GAP, Magma, Sage, TeX
C_9\rtimes C_{18} % in TeX
G:=Group("C9:C18"); // GroupNames label
G:=SmallGroup(162,6);
// by ID
G=gap.SmallGroup(162,6);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,36,1803,728,138,2704]);
// Polycyclic
G:=Group<a,b|a^9=b^18=1,b*a*b^-1=a^2>;
// generators/relations
Export
Subgroup lattice of C9:C18 in TeX
Character table of C9:C18 in TeX