metabelian, supersoluble, monomial
Aliases: C32⋊C6, He3⋊1C2, C32⋊1S3, C3⋊S3⋊C3, C3.2(C3×S3), SmallGroup(54,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C32⋊C6 |
Generators and relations for C32⋊C6
G = < a,b,c | a3=b3=c6=1, ab=ba, cac-1=a-1b-1, cbc-1=b-1 >
Character table of C32⋊C6
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | |
| size | 1 | 9 | 2 | 3 | 3 | 6 | 6 | 6 | 9 | 9 | |
| 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 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | -1 | ζ6 | ζ65 | 0 | 0 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | -1 | ζ65 | ζ6 | 0 | 0 | complex lifted from C3×S3 |
| ρ10 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 9 points - transitive group 9T11
Generators in S9
(1 7 4)(2 5 6)(3 9 8)
(1 3 2)(4 8 6)(5 7 9)
(2 3)(4 5 6 7 8 9)
G:=sub<Sym(9)| (1,7,4)(2,5,6)(3,9,8), (1,3,2)(4,8,6)(5,7,9), (2,3)(4,5,6,7,8,9)>;
G:=Group( (1,7,4)(2,5,6)(3,9,8), (1,3,2)(4,8,6)(5,7,9), (2,3)(4,5,6,7,8,9) );
G=PermutationGroup([[(1,7,4),(2,5,6),(3,9,8)], [(1,3,2),(4,8,6),(5,7,9)], [(2,3),(4,5,6,7,8,9)]])
G:=TransitiveGroup(9,11);
►On 9 points - transitive group 9T13
Generators in S9
(2 5 8)(3 6 9)
(1 7 4)(2 5 8)(3 9 6)
(1 2 3)(4 5 6 7 8 9)
G:=sub<Sym(9)| (2,5,8)(3,6,9), (1,7,4)(2,5,8)(3,9,6), (1,2,3)(4,5,6,7,8,9)>;
G:=Group( (2,5,8)(3,6,9), (1,7,4)(2,5,8)(3,9,6), (1,2,3)(4,5,6,7,8,9) );
G=PermutationGroup([[(2,5,8),(3,6,9)], [(1,7,4),(2,5,8),(3,9,6)], [(1,2,3),(4,5,6,7,8,9)]])
G:=TransitiveGroup(9,13);
►On 18 points - transitive group 18T20
Generators in S18
(2 12 15)(3 7 16)(5 18 9)(6 13 10)
(1 14 11)(2 12 15)(3 16 7)(4 8 17)(5 18 9)(6 10 13)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
G:=sub<Sym(18)| (2,12,15)(3,7,16)(5,18,9)(6,13,10), (1,14,11)(2,12,15)(3,16,7)(4,8,17)(5,18,9)(6,10,13), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;
G:=Group( (2,12,15)(3,7,16)(5,18,9)(6,13,10), (1,14,11)(2,12,15)(3,16,7)(4,8,17)(5,18,9)(6,10,13), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );
G=PermutationGroup([[(2,12,15),(3,7,16),(5,18,9),(6,13,10)], [(1,14,11),(2,12,15),(3,16,7),(4,8,17),(5,18,9),(6,10,13)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)]])
G:=TransitiveGroup(18,20);
►On 18 points - transitive group 18T21
Generators in S18
(1 12 14)(2 17 9)(3 15 11)(4 8 18)(5 13 7)(6 10 16)
(1 4 6)(2 5 3)(7 11 9)(8 10 12)(13 15 17)(14 18 16)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
G:=sub<Sym(18)| (1,12,14)(2,17,9)(3,15,11)(4,8,18)(5,13,7)(6,10,16), (1,4,6)(2,5,3)(7,11,9)(8,10,12)(13,15,17)(14,18,16), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;
G:=Group( (1,12,14)(2,17,9)(3,15,11)(4,8,18)(5,13,7)(6,10,16), (1,4,6)(2,5,3)(7,11,9)(8,10,12)(13,15,17)(14,18,16), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );
G=PermutationGroup([[(1,12,14),(2,17,9),(3,15,11),(4,8,18),(5,13,7),(6,10,16)], [(1,4,6),(2,5,3),(7,11,9),(8,10,12),(13,15,17),(14,18,16)], [(1,2),(3,4),(5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)]])
G:=TransitiveGroup(18,21);
►On 18 points - transitive group 18T22
Generators in S18
(1 11 3)(2 13 15)(4 6 8)(5 18 16)(7 14 9)(10 12 17)
(1 7 16)(2 17 8)(3 9 18)(4 13 10)(5 11 14)(6 15 12)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
G:=sub<Sym(18)| (1,11,3)(2,13,15)(4,6,8)(5,18,16)(7,14,9)(10,12,17), (1,7,16)(2,17,8)(3,9,18)(4,13,10)(5,11,14)(6,15,12), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;
G:=Group( (1,11,3)(2,13,15)(4,6,8)(5,18,16)(7,14,9)(10,12,17), (1,7,16)(2,17,8)(3,9,18)(4,13,10)(5,11,14)(6,15,12), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );
G=PermutationGroup([[(1,11,3),(2,13,15),(4,6,8),(5,18,16),(7,14,9),(10,12,17)], [(1,7,16),(2,17,8),(3,9,18),(4,13,10),(5,11,14),(6,15,12)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)]])
G:=TransitiveGroup(18,22);
►On 27 points - transitive group 27T11
Generators in S27
(1 24 27)(2 11 14)(3 20 17)(4 15 23)(5 13 21)(6 22 19)(7 26 12)(8 18 10)(9 16 25)
(1 5 8)(2 9 6)(3 7 4)(10 27 21)(11 16 22)(12 23 17)(13 18 24)(14 25 19)(15 20 26)
(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)
G:=sub<Sym(27)| (1,24,27)(2,11,14)(3,20,17)(4,15,23)(5,13,21)(6,22,19)(7,26,12)(8,18,10)(9,16,25), (1,5,8)(2,9,6)(3,7,4)(10,27,21)(11,16,22)(12,23,17)(13,18,24)(14,25,19)(15,20,26), (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)>;
G:=Group( (1,24,27)(2,11,14)(3,20,17)(4,15,23)(5,13,21)(6,22,19)(7,26,12)(8,18,10)(9,16,25), (1,5,8)(2,9,6)(3,7,4)(10,27,21)(11,16,22)(12,23,17)(13,18,24)(14,25,19)(15,20,26), (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) );
G=PermutationGroup([[(1,24,27),(2,11,14),(3,20,17),(4,15,23),(5,13,21),(6,22,19),(7,26,12),(8,18,10),(9,16,25)], [(1,5,8),(2,9,6),(3,7,4),(10,27,21),(11,16,22),(12,23,17),(13,18,24),(14,25,19),(15,20,26)], [(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)]])
G:=TransitiveGroup(27,11);
C32⋊C6 is a maximal subgroup of
C32⋊D6 C33⋊C6 He3.S3 He3.2S3 C33⋊S3 He3.3S3 He3⋊S3 He3⋊4S3 He3.4S3 C62⋊S3 C62⋊C6 ASL2(𝔽3) He3⋊D5 C7⋊He3⋊C2 C32⋊F7 He3⋊D7
C32⋊C6 is a maximal quotient of
C32⋊C12 C32⋊C18 C32⋊D9 C3≀S3 C33⋊C6 He3.C6 He3.S3 He3.2C6 He3.2S3 He3⋊4S3 C62⋊S3 C62⋊C6 He3⋊D5 C7⋊He3⋊C2 C32⋊F7 He3⋊D7
| action | f(x) | Disc(f) |
|---|---|---|
| 9T11 | x9-27x7+30x6+189x5-378x4-21x3+378x2-126x-42 | 214·318·76·30232 |
| 9T13 | x9-4x8-30x7+142x6+79x5-680x4-247x3+998x2+716x+104 | 218·33·76·414·126132 |
Matrix representation of C32⋊C6 ►in GL6(ℤ)
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| -1 | -1 | 0 | 0 | 0 | 0 |
| 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 |
| -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,Integers())| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,-1,1,0,0] >;
C32⋊C6 in GAP, Magma, Sage, TeX
C_3^2\rtimes C_6
% in TeX
G:=Group("C3^2:C6"); // GroupNames label
G:=SmallGroup(54,5);
// by ID
G=gap.SmallGroup(54,5);
# by ID
G:=PCGroup([4,-2,-3,-3,-3,146,150,579]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^6=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C32⋊C6 in TeX
Character table of C32⋊C6 in TeX