metacyclic, supersoluble, monomial
Aliases: C9⋊C6, D9⋊C3, C32.S3, 3- 1+2⋊C2, C3.3(C3×S3), Aut(D9), Hol(C9), SmallGroup(54,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C9⋊C6 |
Generators and relations for C9⋊C6
G = < a,b | a9=b6=1, bab-1=a2 >
Character table of C9⋊C6
| class | 1 | 2 | 3A | 3B | 3C | 6A | 6B | 9A | 9B | 9C | |
| size | 1 | 9 | 2 | 3 | 3 | 9 | 9 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ32 | 1 | ζ3 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ3 | 1 | ζ32 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | ζ6 | -1 | ζ65 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | ζ65 | -1 | ζ6 | complex lifted from C3×S3 |
| ρ10 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 9 points - transitive group 9T10
Generators in S9
(1 2 3 4 5 6 7 8 9)
(2 6 8 9 5 3)(4 7)
G:=sub<Sym(9)| (1,2,3,4,5,6,7,8,9), (2,6,8,9,5,3)(4,7)>;
G:=Group( (1,2,3,4,5,6,7,8,9), (2,6,8,9,5,3)(4,7) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9)], [(2,6,8,9,5,3),(4,7)]])
G:=TransitiveGroup(9,10);
►On 18 points - transitive group 18T18
Generators in S18
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 16)(2 12 8 15 5 18)(3 17 6 14 9 11)(4 13)(7 10)
G:=sub<Sym(18)| (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,16)(2,12,8,15,5,18)(3,17,6,14,9,11)(4,13)(7,10)>;
G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,16)(2,12,8,15,5,18)(3,17,6,14,9,11)(4,13)(7,10) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,16),(2,12,8,15,5,18),(3,17,6,14,9,11),(4,13),(7,10)]])
G:=TransitiveGroup(18,18);
►On 27 points - transitive group 27T14
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 17 23)(2 13 21 9 12 25)(3 18 19 8 16 27)(4 14 26 7 11 20)(5 10 24 6 15 22)
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), (1,17,23)(2,13,21,9,12,25)(3,18,19,8,16,27)(4,14,26,7,11,20)(5,10,24,6,15,22)>;
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), (1,17,23)(2,13,21,9,12,25)(3,18,19,8,16,27)(4,14,26,7,11,20)(5,10,24,6,15,22) );
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)], [(1,17,23),(2,13,21,9,12,25),(3,18,19,8,16,27),(4,14,26,7,11,20),(5,10,24,6,15,22)]])
G:=TransitiveGroup(27,14);
C9⋊C6 is a maximal subgroup of
C33⋊S3 He3.3S3 3- 1+2.S3 C33.S3 He3.4S3 C32.S4 D9⋊A4 D45⋊C3 C63⋊C6 C63⋊6C6 C9⋊F7 C9⋊2F7 D63⋊C3
C9⋊C6 is a maximal quotient of
C9⋊C12 C32⋊D9 C9⋊C18 C33.S3 C32.S4 D9⋊A4 D45⋊C3 C63⋊C6 C63⋊6C6 C9⋊F7 C9⋊2F7 D63⋊C3
| action | f(x) | Disc(f) |
|---|---|---|
| 9T10 | x9-4x8-14x7+44x6+62x5-120x4-92x3+48x2+12x-4 | 216·194·374·42192 |
Matrix representation of C9⋊C6 ►in GL6(ℤ)
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 |
| 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,Integers())| [0,0,0,0,1,0,0,0,0,0,0,1,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,0,1,-1,0,0,0,0,0,-1,0,0] >;
C9⋊C6 in GAP, Magma, Sage, TeX
C_9\rtimes C_6
% in TeX
G:=Group("C9:C6"); // GroupNames label
G:=SmallGroup(54,6);
// by ID
G=gap.SmallGroup(54,6);
# by ID
G:=PCGroup([4,-2,-3,-3,-3,362,150,82,579]);
// Polycyclic
G:=Group<a,b|a^9=b^6=1,b*a*b^-1=a^2>;
// generators/relations
Export
Subgroup lattice of C9⋊C6 in TeX
Character table of C9⋊C6 in TeX