direct product, metabelian, supersoluble, monomial
Aliases: C3×C9⋊C6, D9⋊C32, C33.3S3, 3- 1+2⋊3C6, C9⋊(C3×C6), (C3×D9)⋊C3, (C3×C9)⋊4C6, C3.3(S3×C32), C32.4(C3×S3), (C3×3- 1+2)⋊1C2, SmallGroup(162,36)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C3×C9⋊C6 |
Generators and relations for C3×C9⋊C6
G = < a,b,c | a3=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >
Character table of C3×C9⋊C6
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | |
| size | 1 | 9 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 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 | 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 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | -1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | linear of order 6 |
| ρ7 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ8 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ9 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | 1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | -1 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ32 | ζ3 | linear of order 6 |
| ρ10 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | -1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | linear of order 3 |
| ρ12 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ13 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | -1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | 1 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | linear of order 6 |
| ρ14 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | linear of order 3 |
| ρ15 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | 1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | -1 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ3 | ζ32 | linear of order 6 |
| ρ16 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ17 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | linear of order 3 |
| ρ18 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | -1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | 1 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | linear of order 6 |
| ρ19 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ20 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | -1 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | -1 | complex lifted from C3×S3 |
| ρ21 | 2 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 2 | 2 | -1+√-3 | 2 | -1-√-3 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | -1 | -1 | -1 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3×S3 |
| ρ22 | 2 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 2 | -1-√-3 | 2 | -1+√-3 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | -1 | -1 | ζ6 | ζ65 | -1 | ζ65 | ζ65 | ζ6 | complex lifted from C3×S3 |
| ρ23 | 2 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 2 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | ζ65 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | -1 | ζ6 | complex lifted from C3×S3 |
| ρ24 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | -1 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | -1 | complex lifted from C3×S3 |
| ρ25 | 2 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 2 | -1+√-3 | 2 | -1-√-3 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | -1 | -1 | ζ65 | ζ6 | -1 | ζ6 | ζ6 | ζ65 | complex lifted from C3×S3 |
| ρ26 | 2 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 2 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | ζ6 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | -1 | ζ65 | complex lifted from C3×S3 |
| ρ27 | 2 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 2 | 2 | -1-√-3 | 2 | -1+√-3 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | -1 | -1 | -1 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3×S3 |
| 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 18T83
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 13 7 16 4 10)(2 18 5 15 8 12)(3 14)(6 11)(9 17)
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), (1,13,7,16,4,10)(2,18,5,15,8,12)(3,14)(6,11)(9,17)>;
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), (1,13,7,16,4,10)(2,18,5,15,8,12)(3,14)(6,11)(9,17) );
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)], [(1,13,7,16,4,10),(2,18,5,15,8,12),(3,14),(6,11),(9,17)]])
G:=TransitiveGroup(18,83);
►On 27 points - transitive group 27T57
Generators in S27
(1 20 11)(2 21 12)(3 22 13)(4 23 14)(5 24 15)(6 25 16)(7 26 17)(8 27 18)(9 19 10)
(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 11 20)(2 16 27 9 15 22)(3 12 25 8 10 24)(4 17 23 7 14 26)(5 13 21 6 18 19)
G:=sub<Sym(27)| (1,20,11)(2,21,12)(3,22,13)(4,23,14)(5,24,15)(6,25,16)(7,26,17)(8,27,18)(9,19,10), (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), (1,11,20)(2,16,27,9,15,22)(3,12,25,8,10,24)(4,17,23,7,14,26)(5,13,21,6,18,19)>;
G:=Group( (1,20,11)(2,21,12)(3,22,13)(4,23,14)(5,24,15)(6,25,16)(7,26,17)(8,27,18)(9,19,10), (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), (1,11,20)(2,16,27,9,15,22)(3,12,25,8,10,24)(4,17,23,7,14,26)(5,13,21,6,18,19) );
G=PermutationGroup([[(1,20,11),(2,21,12),(3,22,13),(4,23,14),(5,24,15),(6,25,16),(7,26,17),(8,27,18),(9,19,10)], [(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)], [(1,11,20),(2,16,27,9,15,22),(3,12,25,8,10,24),(4,17,23,7,14,26),(5,13,21,6,18,19)]])
G:=TransitiveGroup(27,57);
C3×C9⋊C6 is a maximal subgroup of
D9⋊He3 C92⋊7C6 C92⋊8C6 C34.7S3 (C32×C9)⋊S3 C33⋊(C3×S3) He3.C3⋊2C6 He3⋊(C3×S3) C3.He3⋊C6 3- 1+4⋊C2
C3×C9⋊C6 is a maximal quotient of
C34.S3 D9⋊He3 D9⋊3- 1+2 C92⋊7C6 C92⋊8C6
Matrix representation of C3×C9⋊C6 ►in GL6(𝔽19)
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0] >;
C3×C9⋊C6 in GAP, Magma, Sage, TeX
C_3\times C_9\rtimes C_6
% in TeX
G:=Group("C3xC9:C6"); // GroupNames label
G:=SmallGroup(162,36);
// by ID
G=gap.SmallGroup(162,36);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,1803,728,138,2704]);
// Polycyclic
G:=Group<a,b,c|a^3=b^9=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations
Export
Subgroup lattice of C3×C9⋊C6 in TeX
Character table of C3×C9⋊C6 in TeX