metabelian, supersoluble, monomial
Aliases: C92⋊2C6, C9⋊D9⋊2C3, C92⋊2C3⋊2C2, He3⋊C3.2S3, C32.15(C32⋊C6), C3.3(He3.2S3), (C3×C9).28(C3×S3), SmallGroup(486,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C92 — C92⋊2C6 |
Generators and relations for C92⋊2C6
G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a-1b-1, cbc-1=a3b2 >
Character table of C92⋊2C6
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | 9K | 9L | |
| size | 1 | 81 | 2 | 6 | 27 | 27 | 54 | 54 | 81 | 81 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| 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 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | -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 | ζ65 | ζ6 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | 2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 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 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -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 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | orthogonal lifted from He3.2S3 |
| ρ12 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95+2ζ94+2 | -ζ95-ζ94-1 | -ζ98-ζ9-1 | 2ζ98+2ζ9+2 | -ζ98-ζ9-1 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ97-ζ92-1 | 2ζ97+2ζ92+2 | -ζ97-ζ92-1 | -ζ95-ζ94-1 | orthogonal faithful |
| ρ13 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98+2ζ9+2 | -ζ98-ζ9-1 | -ζ97-ζ92-1 | 2ζ97+2ζ92+2 | -ζ97-ζ92-1 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ95-ζ94-1 | 2ζ95+2ζ94+2 | -ζ95-ζ94-1 | -ζ98-ζ9-1 | orthogonal faithful |
| ρ14 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ95-ζ94-1 | -ζ95-ζ94-1 | 2ζ98+2ζ9+2 | -ζ98-ζ9-1 | -ζ98-ζ9-1 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ97+2ζ92+2 | -ζ97-ζ92-1 | -ζ97-ζ92-1 | 2ζ95+2ζ94+2 | orthogonal faithful |
| ρ15 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | orthogonal lifted from He3.2S3 |
| ρ16 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ97-ζ92-1 | -ζ97-ζ92-1 | 2ζ95+2ζ94+2 | -ζ95-ζ94-1 | -ζ95-ζ94-1 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98+2ζ9+2 | -ζ98-ζ9-1 | -ζ98-ζ9-1 | 2ζ97+2ζ92+2 | orthogonal faithful |
| ρ17 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98-ζ9-1 | 2ζ98+2ζ9+2 | -ζ97-ζ92-1 | -ζ97-ζ92-1 | 2ζ97+2ζ92+2 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ95-ζ94-1 | -ζ95-ζ94-1 | 2ζ95+2ζ94+2 | -ζ98-ζ9-1 | orthogonal faithful |
| ρ18 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | orthogonal lifted from He3.2S3 |
| ρ19 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ95-ζ94-1 | 2ζ95+2ζ94+2 | -ζ98-ζ9-1 | -ζ98-ζ9-1 | 2ζ98+2ζ9+2 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ97-ζ92-1 | -ζ97-ζ92-1 | 2ζ97+2ζ92+2 | -ζ95-ζ94-1 | orthogonal faithful |
| ρ20 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ97-ζ92-1 | 2ζ97+2ζ92+2 | -ζ95-ζ94-1 | -ζ95-ζ94-1 | 2ζ95+2ζ94+2 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98-ζ9-1 | -ζ98-ζ9-1 | 2ζ98+2ζ9+2 | -ζ97-ζ92-1 | orthogonal faithful |
| ρ21 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98-ζ9-1 | -ζ98-ζ9-1 | 2ζ97+2ζ92+2 | -ζ97-ζ92-1 | -ζ97-ζ92-1 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ95+2ζ94+2 | -ζ95-ζ94-1 | -ζ95-ζ94-1 | 2ζ98+2ζ9+2 | orthogonal faithful |
| ρ22 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ97+2ζ92+2 | -ζ97-ζ92-1 | -ζ95-ζ94-1 | 2ζ95+2ζ94+2 | -ζ95-ζ94-1 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98-ζ9-1 | 2ζ98+2ζ9+2 | -ζ98-ζ9-1 | -ζ97-ζ92-1 | orthogonal faithful |
►On 27 points - transitive group 27T158
Generators in S27
(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 27 26 25 24 23 22 21 20)
(1 10 27)(2 16 24 3 13 21)(4 14 22 8 15 23)(5 11 19 7 18 26)(6 17 25 9 12 20)
G:=sub<Sym(27)| (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)(19,27,26,25,24,23,22,21,20), (1,10,27)(2,16,24,3,13,21)(4,14,22,8,15,23)(5,11,19,7,18,26)(6,17,25,9,12,20)>;
G:=Group( (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)(19,27,26,25,24,23,22,21,20), (1,10,27)(2,16,24,3,13,21)(4,14,22,8,15,23)(5,11,19,7,18,26)(6,17,25,9,12,20) );
G=PermutationGroup([[(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),(19,27,26,25,24,23,22,21,20)], [(1,10,27),(2,16,24,3,13,21),(4,14,22,8,15,23),(5,11,19,7,18,26),(6,17,25,9,12,20)]])
G:=TransitiveGroup(27,158);
Matrix representation of C92⋊2C6 ►in GL6(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 7 | 14 | 0 | 0 | 0 | 0 |
| 5 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 7 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 14 |
| 0 | 0 | 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 0 | 17 | 7 |
| 0 | 0 | 0 | 0 | 5 | 2 |
| 17 | 7 | 0 | 0 | 0 | 0 |
| 5 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 7 | 0 | 0 |
| 0 | 0 | 5 | 2 | 0 | 0 |
G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,5,0,0,0,0,14,2,0,0,0,0,0,0,2,14,0,0,0,0,5,7],[7,5,0,0,0,0,14,2,0,0,0,0,0,0,17,12,0,0,0,0,7,5,0,0,0,0,0,0,7,5,0,0,0,0,14,2],[0,0,17,5,0,0,0,0,7,2,0,0,0,0,0,0,17,5,0,0,0,0,7,2,17,5,0,0,0,0,7,2,0,0,0,0] >;
C92⋊2C6 in GAP, Magma, Sage, TeX
C_9^2\rtimes_2C_6
% in TeX
G:=Group("C9^2:2C6"); // GroupNames label
G:=SmallGroup(486,37);
// by ID
G=gap.SmallGroup(486,37);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,224,338,4755,2817,453,3244,3250,11669]);
// 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^2>;
// generators/relations
Export
Subgroup lattice of C92⋊2C6 in TeX
Character table of C92⋊2C6 in TeX