direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D9, C9⋊3C6, C32.2S3, (C3×C9)⋊2C2, C3.1(C3×S3), SmallGroup(54,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C3×D9 |
Generators and relations for C3×D9
G = < a,b,c | a3=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C3×D9
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | |
| size | 1 | 9 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ9 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ10 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ11 | 2 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 2 | 0 | 0 | ζ65 | -1 | -1 | ζ6 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | complex lifted from C3×S3 |
| ρ12 | 2 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 2 | 0 | 0 | ζ6 | -1 | -1 | ζ65 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | complex lifted from C3×S3 |
| ρ13 | 2 | 0 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | 0 | ζ98+ζ97 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ95 | ζ98+ζ94 | ζ92+ζ9 | ζ97+ζ92 | ζ94+ζ92 | ζ95+ζ9 | complex faithful |
| ρ14 | 2 | 0 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | 0 | ζ98+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ97 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ9 | ζ92+ζ9 | ζ97+ζ95 | complex faithful |
| ρ15 | 2 | 0 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | 0 | ζ95+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ92+ζ9 | ζ97+ζ95 | ζ98+ζ94 | ζ98+ζ9 | ζ98+ζ97 | ζ94+ζ92 | complex faithful |
| ρ16 | 2 | 0 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | 0 | ζ97+ζ95 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ9 | ζ98+ζ97 | ζ94+ζ92 | ζ95+ζ94 | ζ98+ζ94 | ζ92+ζ9 | complex faithful |
| ρ17 | 2 | 0 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | 0 | ζ94+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ94 | ζ92+ζ9 | ζ97+ζ95 | ζ95+ζ94 | ζ95+ζ9 | ζ98+ζ97 | complex faithful |
| ρ18 | 2 | 0 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | 0 | ζ92+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ97 | ζ97+ζ92 | ζ97+ζ95 | ζ98+ζ94 | complex faithful |
►On 18 points - transitive group 18T19
Generators in S18
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 12)(2 11)(3 10)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)
G:=sub<Sym(18)| (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,12)(2,11)(3,10)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)>;
G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,12)(2,11)(3,10)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13) );
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,12),(2,11),(3,10),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13)]])
G:=TransitiveGroup(18,19);
►On 27 points - transitive group 27T9
Generators in S27
(1 23 14)(2 24 15)(3 25 16)(4 26 17)(5 27 18)(6 19 10)(7 20 11)(8 21 12)(9 22 13)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)
(1 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)
G:=sub<Sym(27)| (1,23,14)(2,24,15)(3,25,16)(4,26,17)(5,27,18)(6,19,10)(7,20,11)(8,21,12)(9,22,13), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)>;
G:=Group( (1,23,14)(2,24,15)(3,25,16)(4,26,17)(5,27,18)(6,19,10)(7,20,11)(8,21,12)(9,22,13), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23) );
G=PermutationGroup([[(1,23,14),(2,24,15),(3,25,16),(4,26,17),(5,27,18),(6,19,10),(7,20,11),(8,21,12),(9,22,13)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23)]])
G:=TransitiveGroup(27,9);
C3×D9 is a maximal subgroup of
C9⋊C18 C27⋊C6 C32⋊2D9 He3.3S3 He3⋊S3 3- 1+2.S3 He3.4S3 C9⋊5F7
C3×D9 is a maximal quotient of
C32⋊D9 C27⋊C6 C9⋊5F7
Matrix representation of C3×D9 ►in GL2(𝔽19) generated by
| 11 | 0 |
| 0 | 11 |
| 5 | 0 |
| 0 | 4 |
| 0 | 4 |
| 5 | 0 |
G:=sub<GL(2,GF(19))| [11,0,0,11],[5,0,0,4],[0,5,4,0] >;
C3×D9 in GAP, Magma, Sage, TeX
C_3\times D_9
% in TeX
G:=Group("C3xD9"); // GroupNames label
G:=SmallGroup(54,3);
// by ID
G=gap.SmallGroup(54,3);
# by ID
G:=PCGroup([4,-2,-3,-3,-3,362,82,579]);
// Polycyclic
G:=Group<a,b,c|a^3=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C3×D9 in TeX
Character table of C3×D9 in TeX