metabelian, supersoluble, monomial
Aliases: C92⋊1C6, C9⋊D9⋊1C3, C92⋊C3⋊C2, He3⋊C3.1S3, C32.14(C32⋊C6), C3.2(He3.2S3), (C3×C9).27(C3×S3), SmallGroup(486,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C92 — C92⋊C6 |
Generators and relations for C92⋊C6
G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a-1b-1, cbc-1=a3b-1 >
Character table of C92⋊C6
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | 9K | 9L | 9M | 9N | |
| size | 1 | 81 | 2 | 6 | 27 | 27 | 81 | 81 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 54 | 54 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | linear of order 6 |
| ρ5 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | ζ6 | ζ65 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | ζ65 | ζ6 | complex lifted from C3×S3 |
| ρ10 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
| ρ11 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -ζ97-ζ92-1 | 2ζ95+2ζ94-1 | -ζ95-ζ94+2 | -ζ95-ζ94-1 | 2ζ98+2ζ9-1 | -ζ98-ζ9+2 | -ζ98-ζ9-1 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ97+2ζ92-1 | -ζ97-ζ92+2 | 0 | 0 | orthogonal faithful |
| ρ12 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 2ζ95+2ζ94-1 | -ζ98-ζ9+2 | -ζ98-ζ9-1 | 2ζ98+2ζ9-1 | -ζ97-ζ92+2 | -ζ97-ζ92-1 | 2ζ97+2ζ92-1 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ95-ζ94+2 | -ζ95-ζ94-1 | 0 | 0 | orthogonal faithful |
| ρ13 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -ζ95-ζ94+2 | -ζ98-ζ9-1 | 2ζ98+2ζ9-1 | -ζ98-ζ9+2 | -ζ97-ζ92-1 | 2ζ97+2ζ92-1 | -ζ97-ζ92+2 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ95-ζ94-1 | 2ζ95+2ζ94-1 | 0 | 0 | orthogonal faithful |
| ρ14 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 2ζ98+2ζ9-1 | -ζ97-ζ92+2 | -ζ97-ζ92-1 | 2ζ97+2ζ92-1 | -ζ95-ζ94+2 | -ζ95-ζ94-1 | 2ζ95+2ζ94-1 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98-ζ9+2 | -ζ98-ζ9-1 | 0 | 0 | orthogonal faithful |
| ρ15 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | orthogonal lifted from He3.2S3 |
| ρ16 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | orthogonal lifted from He3.2S3 |
| ρ17 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 2ζ97+2ζ92-1 | -ζ95-ζ94+2 | -ζ95-ζ94-1 | 2ζ95+2ζ94-1 | -ζ98-ζ9+2 | -ζ98-ζ9-1 | 2ζ98+2ζ9-1 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ97-ζ92+2 | -ζ97-ζ92-1 | 0 | 0 | orthogonal faithful |
| ρ18 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -ζ95-ζ94-1 | 2ζ98+2ζ9-1 | -ζ98-ζ9+2 | -ζ98-ζ9-1 | 2ζ97+2ζ92-1 | -ζ97-ζ92+2 | -ζ97-ζ92-1 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ95+2ζ94-1 | -ζ95-ζ94+2 | 0 | 0 | orthogonal faithful |
| ρ19 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -ζ98-ζ9+2 | -ζ97-ζ92-1 | 2ζ97+2ζ92-1 | -ζ97-ζ92+2 | -ζ95-ζ94-1 | 2ζ95+2ζ94-1 | -ζ95-ζ94+2 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98-ζ9-1 | 2ζ98+2ζ9-1 | 0 | 0 | orthogonal faithful |
| ρ20 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -ζ98-ζ9-1 | 2ζ97+2ζ92-1 | -ζ97-ζ92+2 | -ζ97-ζ92-1 | 2ζ95+2ζ94-1 | -ζ95-ζ94+2 | -ζ95-ζ94-1 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98+2ζ9-1 | -ζ98-ζ9+2 | 0 | 0 | orthogonal faithful |
| ρ21 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | orthogonal lifted from He3.2S3 |
| ρ22 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -ζ97-ζ92+2 | -ζ95-ζ94-1 | 2ζ95+2ζ94-1 | -ζ95-ζ94+2 | -ζ98-ζ9-1 | 2ζ98+2ζ9-1 | -ζ98-ζ9+2 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ97-ζ92-1 | 2ζ97+2ζ92-1 | 0 | 0 | orthogonal faithful |
►On 27 points - transitive group 27T172
Generators in S27
(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 7 6 2 8 4 3 9 5)(10 17 15 13 11 18 16 14 12)(19 24 20 25 21 26 22 27 23)
(1 10 22)(2 16 19 3 13 25)(4 11 23 8 18 21)(5 17 20 7 12 24)(6 14 26 9 15 27)
G:=sub<Sym(27)| (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), (1,7,6,2,8,4,3,9,5)(10,17,15,13,11,18,16,14,12)(19,24,20,25,21,26,22,27,23), (1,10,22)(2,16,19,3,13,25)(4,11,23,8,18,21)(5,17,20,7,12,24)(6,14,26,9,15,27)>;
G:=Group( (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), (1,7,6,2,8,4,3,9,5)(10,17,15,13,11,18,16,14,12)(19,24,20,25,21,26,22,27,23), (1,10,22)(2,16,19,3,13,25)(4,11,23,8,18,21)(5,17,20,7,12,24)(6,14,26,9,15,27) );
G=PermutationGroup([[(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)], [(1,7,6,2,8,4,3,9,5),(10,17,15,13,11,18,16,14,12),(19,24,20,25,21,26,22,27,23)], [(1,10,22),(2,16,19,3,13,25),(4,11,23,8,18,21),(5,17,20,7,12,24),(6,14,26,9,15,27)]])
G:=TransitiveGroup(27,172);
Matrix representation of C92⋊C6 ►in GL6(𝔽19)
| 0 | 18 | 0 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 14 | 0 | 0 |
| 0 | 0 | 5 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 5 |
| 0 | 0 | 0 | 0 | 14 | 7 |
| 17 | 7 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 7 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 17 |
| 0 | 0 | 0 | 0 | 2 | 12 |
| 0 | 0 | 0 | 0 | 17 | 14 |
| 0 | 0 | 0 | 0 | 12 | 2 |
| 17 | 14 | 0 | 0 | 0 | 0 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 14 | 0 | 0 |
| 0 | 0 | 12 | 2 | 0 | 0 |
G:=sub<GL(6,GF(19))| [0,1,0,0,0,0,18,18,0,0,0,0,0,0,7,5,0,0,0,0,14,2,0,0,0,0,0,0,2,14,0,0,0,0,5,7],[17,12,0,0,0,0,7,5,0,0,0,0,0,0,17,12,0,0,0,0,7,5,0,0,0,0,0,0,14,2,0,0,0,0,17,12],[0,0,17,12,0,0,0,0,14,2,0,0,0,0,0,0,17,12,0,0,0,0,14,2,17,12,0,0,0,0,14,2,0,0,0,0] >;
C92⋊C6 in GAP, Magma, Sage, TeX
C_9^2\rtimes C_6
% in TeX
G:=Group("C9^2:C6"); // GroupNames label
G:=SmallGroup(486,35);
// by ID
G=gap.SmallGroup(486,35);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,224,338,4755,873,453,3244,3250,11669]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations
Export
Subgroup lattice of C92⋊C6 in TeX
Character table of C92⋊C6 in TeX