metacyclic, supersoluble, monomial
Aliases: C27⋊C6, D27⋊C3, C32.D9, C27⋊C3⋊C2, C9.3(C3×S3), (C3×C9).3S3, C3.3(C3×D9), SmallGroup(162,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C27⋊C6 |
Generators and relations for C27⋊C6
G = < a,b | a27=b6=1, bab-1=a17 >
Character table of C27⋊C6
| class | 1 | 2 | 3A | 3B | 3C | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 27A | 27B | 27C | 27D | 27E | 27F | 27G | 27H | 27I | |
| size | 1 | 27 | 2 | 3 | 3 | 27 | 27 | 2 | 2 | 2 | 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 | trivial |
| ρ2 | 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 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | -1 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | 1 | linear of order 3 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ9 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ10 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ11 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | 2 | 2 | 2 | -1+√-3 | -1-√-3 | -1 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | -1 | -1 | complex lifted from C3×S3 |
| ρ12 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | 2 | 2 | 2 | -1-√-3 | -1+√-3 | -1 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | -1 | -1 | complex lifted from C3×S3 |
| ρ13 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -1 | -1 | -1 | ζ6 | ζ65 | ζ95+ζ94 | ζ97+ζ95 | ζ98+ζ94 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ97 | ζ92+ζ9 | ζ98+ζ9 | ζ97+ζ92 | complex lifted from C3×D9 |
| ρ14 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -1 | -1 | -1 | ζ6 | ζ65 | ζ97+ζ92 | ζ92+ζ9 | ζ97+ζ95 | ζ98+ζ97 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ94 | ζ95+ζ94 | ζ98+ζ9 | complex lifted from C3×D9 |
| ρ15 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -1 | -1 | -1 | ζ65 | ζ6 | ζ97+ζ92 | ζ98+ζ97 | ζ94+ζ92 | ζ92+ζ9 | ζ97+ζ95 | ζ98+ζ94 | ζ95+ζ9 | ζ95+ζ94 | ζ98+ζ9 | complex lifted from C3×D9 |
| ρ16 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -1 | -1 | -1 | ζ65 | ζ6 | ζ95+ζ94 | ζ94+ζ92 | ζ95+ζ9 | ζ97+ζ95 | ζ98+ζ94 | ζ92+ζ9 | ζ98+ζ97 | ζ98+ζ9 | ζ97+ζ92 | complex lifted from C3×D9 |
| ρ17 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -1 | -1 | -1 | ζ65 | ζ6 | ζ98+ζ9 | ζ95+ζ9 | ζ98+ζ97 | ζ98+ζ94 | ζ92+ζ9 | ζ97+ζ95 | ζ94+ζ92 | ζ97+ζ92 | ζ95+ζ94 | complex lifted from C3×D9 |
| ρ18 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -1 | -1 | -1 | ζ6 | ζ65 | ζ98+ζ9 | ζ98+ζ94 | ζ92+ζ9 | ζ95+ζ9 | ζ98+ζ97 | ζ94+ζ92 | ζ97+ζ95 | ζ97+ζ92 | ζ95+ζ94 | complex lifted from C3×D9 |
| ρ19 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 3ζ97+3ζ92 | 3ζ95+3ζ94 | 3ζ98+3ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ20 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 3ζ95+3ζ94 | 3ζ98+3ζ9 | 3ζ97+3ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ21 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 3ζ98+3ζ9 | 3ζ97+3ζ92 | 3ζ95+3ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 27 points - transitive group 27T55
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)
(2 9 11 27 20 18)(3 17 21 26 12 8)(4 25)(5 6 14 24 23 15)(7 22)(10 19)(13 16)
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), (2,9,11,27,20,18)(3,17,21,26,12,8)(4,25)(5,6,14,24,23,15)(7,22)(10,19)(13,16)>;
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), (2,9,11,27,20,18)(3,17,21,26,12,8)(4,25)(5,6,14,24,23,15)(7,22)(10,19)(13,16) );
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)], [(2,9,11,27,20,18),(3,17,21,26,12,8),(4,25),(5,6,14,24,23,15),(7,22),(10,19),(13,16)]])
G:=TransitiveGroup(27,55);
C27⋊C6 is a maximal subgroup of
C27⋊C18 C33.D9 He3.3D9 He3.4D9 C33.5D9 He3.5D9
C27⋊C6 is a maximal quotient of C27⋊C12 C27⋊3C18 C32⋊D27 C33.5D9
Matrix representation of C27⋊C6 ►in GL6(𝔽109)
| 0 | 0 | 59 | 82 | 0 | 0 |
| 0 | 0 | 27 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 82 |
| 0 | 0 | 0 | 0 | 27 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 50 |
| 0 | 0 | 0 | 0 | 82 | 77 |
| 0 | 0 | 50 | 27 | 0 | 0 |
| 0 | 0 | 77 | 59 | 0 | 0 |
G:=sub<GL(6,GF(109))| [0,0,0,0,1,0,0,0,0,0,0,1,59,27,0,0,0,0,82,32,0,0,0,0,0,0,59,27,0,0,0,0,82,32,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,50,77,0,0,0,0,27,59,0,0,32,82,0,0,0,0,50,77,0,0] >;
C27⋊C6 in GAP, Magma, Sage, TeX
C_{27}\rtimes C_6 % in TeX
G:=Group("C27:C6"); // GroupNames label
G:=SmallGroup(162,9);
// by ID
G=gap.SmallGroup(162,9);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,452,457,237,1803,138,2704]);
// Polycyclic
G:=Group<a,b|a^27=b^6=1,b*a*b^-1=a^17>;
// generators/relations
Export
Subgroup lattice of C27⋊C6 in TeX
Character table of C27⋊C6 in TeX