non-abelian, supersoluble, monomial
Aliases: C3≀S3, He3⋊C6, C33⋊1S3, C3≀C3⋊2C2, He3⋊C2⋊1C3, C32.1(C3×S3), C3.6(C32⋊C6), SmallGroup(162,10)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C3≀S3 |
Generators and relations for C3≀S3
G = < a,b,c,d | a3=b3=c3=d6=1, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a-1c-1 >
Character table of C3≀S3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 9A | 9B | |
| size | 1 | 9 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 18 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 18 | 18 | |
| 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 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ6 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | -1 | ζ32 | ζ3 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ65 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | -1 | ζ3 | ζ32 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | 2 | -1-√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | 2 | -1+√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | complex lifted from C3×S3 |
| ρ10 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | -√-3 | -3-√-3/2 | 3+√-3/2 | -3+√-3/2 | 3-√-3/2 | √-3 | 0 | 0 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | 0 | 0 | complex faithful |
| ρ11 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | -3-√-3/2 | 3+√-3/2 | √-3 | 3-√-3/2 | -√-3 | -3+√-3/2 | 0 | 0 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | 0 | 0 | complex faithful |
| ρ12 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | √-3 | -3+√-3/2 | 3-√-3/2 | -3-√-3/2 | 3+√-3/2 | -√-3 | 0 | 0 | -1 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | ζ6 | 0 | 0 | complex faithful |
| ρ13 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | √-3 | -3+√-3/2 | 3-√-3/2 | -3-√-3/2 | 3+√-3/2 | -√-3 | 0 | 0 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | 0 | 0 | complex faithful |
| ρ14 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | -3+√-3/2 | 3-√-3/2 | -√-3 | 3+√-3/2 | √-3 | -3-√-3/2 | 0 | 0 | ζ6 | ζ6 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ65 | 0 | 0 | complex faithful |
| ρ15 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | -3-√-3/2 | 3+√-3/2 | √-3 | 3-√-3/2 | -√-3 | -3+√-3/2 | 0 | 0 | ζ65 | ζ65 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ6 | 0 | 0 | complex faithful |
| ρ16 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | -√-3 | -3-√-3/2 | 3+√-3/2 | -3+√-3/2 | 3-√-3/2 | √-3 | 0 | 0 | -1 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | ζ65 | 0 | 0 | complex faithful |
| ρ17 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 3-√-3/2 | -√-3 | -3-√-3/2 | √-3 | -3+√-3/2 | 3+√-3/2 | 0 | 0 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | -1 | ζ6 | 0 | 0 | complex faithful |
| ρ18 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 3-√-3/2 | -√-3 | -3-√-3/2 | √-3 | -3+√-3/2 | 3+√-3/2 | 0 | 0 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | 0 | 0 | complex faithful |
| ρ19 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | -3+√-3/2 | 3-√-3/2 | -√-3 | 3+√-3/2 | √-3 | -3-√-3/2 | 0 | 0 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | 0 | 0 | complex faithful |
| ρ20 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 3+√-3/2 | √-3 | -3+√-3/2 | -√-3 | -3-√-3/2 | 3-√-3/2 | 0 | 0 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | 0 | 0 | complex faithful |
| ρ21 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 3+√-3/2 | √-3 | -3+√-3/2 | -√-3 | -3-√-3/2 | 3-√-3/2 | 0 | 0 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | -1 | ζ65 | 0 | 0 | complex faithful |
| ρ22 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
►On 9 points - transitive group 9T20
Generators in S9
(2 3 6)(7 9 8)
(1 4 5)(2 3 6)(7 8 9)
(1 7 2)(3 4 8)(5 9 6)
(1 2)(3 4)(5 6)(7 8 9)
G:=sub<Sym(9)| (2,3,6)(7,9,8), (1,4,5)(2,3,6)(7,8,9), (1,7,2)(3,4,8)(5,9,6), (1,2)(3,4)(5,6)(7,8,9)>;
G:=Group( (2,3,6)(7,9,8), (1,4,5)(2,3,6)(7,8,9), (1,7,2)(3,4,8)(5,9,6), (1,2)(3,4)(5,6)(7,8,9) );
G=PermutationGroup([[(2,3,6),(7,9,8)], [(1,4,5),(2,3,6),(7,8,9)], [(1,7,2),(3,4,8),(5,9,6)], [(1,2),(3,4),(5,6),(7,8,9)]])
G:=TransitiveGroup(9,20);
►On 18 points - transitive group 18T86
Generators in S18
(2 6 3)(7 10 11)(13 15 17)(14 16 18)
(1 5 4)(2 6 3)(7 10 11)(8 9 12)(13 17 15)(14 18 16)
(1 15 10)(2 9 18)(3 8 14)(4 17 7)(5 13 11)(6 12 16)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14 15 16 17 18)
G:=sub<Sym(18)| (2,6,3)(7,10,11)(13,15,17)(14,16,18), (1,5,4)(2,6,3)(7,10,11)(8,9,12)(13,17,15)(14,18,16), (1,15,10)(2,9,18)(3,8,14)(4,17,7)(5,13,11)(6,12,16), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15,16,17,18)>;
G:=Group( (2,6,3)(7,10,11)(13,15,17)(14,16,18), (1,5,4)(2,6,3)(7,10,11)(8,9,12)(13,17,15)(14,18,16), (1,15,10)(2,9,18)(3,8,14)(4,17,7)(5,13,11)(6,12,16), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15,16,17,18) );
G=PermutationGroup([[(2,6,3),(7,10,11),(13,15,17),(14,16,18)], [(1,5,4),(2,6,3),(7,10,11),(8,9,12),(13,17,15),(14,18,16)], [(1,15,10),(2,9,18),(3,8,14),(4,17,7),(5,13,11),(6,12,16)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14,15,16,17,18)]])
G:=TransitiveGroup(18,86);
►On 27 points - transitive group 27T37
Generators in S27
(1 4 7)(2 5 8)(3 6 9)(11 27 19)(13 23 21)(15 25 17)
(1 7 4)(2 8 5)(3 9 6)(10 26 18)(11 27 19)(12 22 20)(13 23 21)(14 24 16)(15 25 17)
(1 25 22)(2 21 18)(3 11 14)(4 15 12)(5 23 26)(6 19 16)(7 17 20)(8 13 10)(9 27 24)
(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)| (1,4,7)(2,5,8)(3,6,9)(11,27,19)(13,23,21)(15,25,17), (1,7,4)(2,8,5)(3,9,6)(10,26,18)(11,27,19)(12,22,20)(13,23,21)(14,24,16)(15,25,17), (1,25,22)(2,21,18)(3,11,14)(4,15,12)(5,23,26)(6,19,16)(7,17,20)(8,13,10)(9,27,24), (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( (1,4,7)(2,5,8)(3,6,9)(11,27,19)(13,23,21)(15,25,17), (1,7,4)(2,8,5)(3,9,6)(10,26,18)(11,27,19)(12,22,20)(13,23,21)(14,24,16)(15,25,17), (1,25,22)(2,21,18)(3,11,14)(4,15,12)(5,23,26)(6,19,16)(7,17,20)(8,13,10)(9,27,24), (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([[(1,4,7),(2,5,8),(3,6,9),(11,27,19),(13,23,21),(15,25,17)], [(1,7,4),(2,8,5),(3,9,6),(10,26,18),(11,27,19),(12,22,20),(13,23,21),(14,24,16),(15,25,17)], [(1,25,22),(2,21,18),(3,11,14),(4,15,12),(5,23,26),(6,19,16),(7,17,20),(8,13,10),(9,27,24)], [(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(27,37);
►On 27 points - transitive group 27T50
Generators in S27
(1 5 4)(2 9 8)(3 7 6)(10 14 12)(11 16 27)(13 18 23)(15 20 25)(17 21 19)(22 26 24)
(1 2 3)(4 8 6)(5 9 7)(10 19 22)(11 20 23)(12 21 24)(13 16 25)(14 17 26)(15 18 27)
(1 23 26)(2 11 14)(3 20 17)(4 15 10)(5 16 21)(6 27 22)(7 13 12)(8 18 19)(9 25 24)
(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)| (1,5,4)(2,9,8)(3,7,6)(10,14,12)(11,16,27)(13,18,23)(15,20,25)(17,21,19)(22,26,24), (1,2,3)(4,8,6)(5,9,7)(10,19,22)(11,20,23)(12,21,24)(13,16,25)(14,17,26)(15,18,27), (1,23,26)(2,11,14)(3,20,17)(4,15,10)(5,16,21)(6,27,22)(7,13,12)(8,18,19)(9,25,24), (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( (1,5,4)(2,9,8)(3,7,6)(10,14,12)(11,16,27)(13,18,23)(15,20,25)(17,21,19)(22,26,24), (1,2,3)(4,8,6)(5,9,7)(10,19,22)(11,20,23)(12,21,24)(13,16,25)(14,17,26)(15,18,27), (1,23,26)(2,11,14)(3,20,17)(4,15,10)(5,16,21)(6,27,22)(7,13,12)(8,18,19)(9,25,24), (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([[(1,5,4),(2,9,8),(3,7,6),(10,14,12),(11,16,27),(13,18,23),(15,20,25),(17,21,19),(22,26,24)], [(1,2,3),(4,8,6),(5,9,7),(10,19,22),(11,20,23),(12,21,24),(13,16,25),(14,17,26),(15,18,27)], [(1,23,26),(2,11,14),(3,20,17),(4,15,10),(5,16,21),(6,27,22),(7,13,12),(8,18,19),(9,25,24)], [(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(27,50);
►On 27 points - transitive group 27T70
Generators in S27
(1 15 20)(2 13 18)(3 11 16)(4 21 25)(5 19 23)(6 17 27)(7 24 12)(8 22 10)(9 26 14)
(1 7 6)(2 8 4)(3 9 5)(10 25 18)(11 26 19)(12 27 20)(13 22 21)(14 23 16)(15 24 17)
(1 15 12)(2 18 21)(4 25 22)(6 17 20)(7 24 27)(8 10 13)(11 26 19)(14 16 23)
(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)| (1,15,20)(2,13,18)(3,11,16)(4,21,25)(5,19,23)(6,17,27)(7,24,12)(8,22,10)(9,26,14), (1,7,6)(2,8,4)(3,9,5)(10,25,18)(11,26,19)(12,27,20)(13,22,21)(14,23,16)(15,24,17), (1,15,12)(2,18,21)(4,25,22)(6,17,20)(7,24,27)(8,10,13)(11,26,19)(14,16,23), (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( (1,15,20)(2,13,18)(3,11,16)(4,21,25)(5,19,23)(6,17,27)(7,24,12)(8,22,10)(9,26,14), (1,7,6)(2,8,4)(3,9,5)(10,25,18)(11,26,19)(12,27,20)(13,22,21)(14,23,16)(15,24,17), (1,15,12)(2,18,21)(4,25,22)(6,17,20)(7,24,27)(8,10,13)(11,26,19)(14,16,23), (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([[(1,15,20),(2,13,18),(3,11,16),(4,21,25),(5,19,23),(6,17,27),(7,24,12),(8,22,10),(9,26,14)], [(1,7,6),(2,8,4),(3,9,5),(10,25,18),(11,26,19),(12,27,20),(13,22,21),(14,23,16),(15,24,17)], [(1,15,12),(2,18,21),(4,25,22),(6,17,20),(7,24,27),(8,10,13),(11,26,19),(14,16,23)], [(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(27,70);
C3≀S3 is a maximal subgroup of
He3⋊D6 C3≀S3⋊3C3 C3≀C3⋊C6 C3≀C3.C6 C34⋊5S3
C3≀S3 is a maximal quotient of He3⋊C12 C3.C3≀S3 C32⋊C9⋊C6 C3.3C3≀S3 C33⋊1D9 He3⋊C18 C34⋊5S3
| action | f(x) | Disc(f) |
|---|---|---|
| 9T20 | x9-4x8-4x7+22x6-x5-31x4+4x3+15x2-1 | 372·2293 |
Matrix representation of C3≀S3 ►in GL3(𝔽7) generated by
| 0 | 6 | 5 |
| 6 | 0 | 5 |
| 5 | 6 | 0 |
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 0 | 6 | 5 |
| 1 | 6 | 4 |
| 0 | 0 | 1 |
| 1 | 4 | 0 |
| 0 | 4 | 1 |
| 0 | 5 | 0 |
G:=sub<GL(3,GF(7))| [0,6,5,6,0,6,5,5,0],[2,0,0,0,2,0,0,0,2],[0,1,0,6,6,0,5,4,1],[1,0,0,4,4,5,0,1,0] >;
C3≀S3 in GAP, Magma, Sage, TeX
C_3\wr S_3
% in TeX
G:=Group("C3wrS3"); // GroupNames label
G:=SmallGroup(162,10);
// by ID
G=gap.SmallGroup(162,10);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,182,187,1803,253]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^6=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*c^-1>;
// generators/relations
Export
Subgroup lattice of C3≀S3 in TeX
Character table of C3≀S3 in TeX