direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×S3, C3≀C2, U2(𝔽2), AΣL1(𝔽9), CU2(𝔽2), C3⋊C6, C32⋊1C2, SmallGroup(18,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C3×S3 |
Generators and relations for C3×S3
G = < a,b,c | a3=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of C3×S3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ4 | 1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | -1+√-3 | -1-√-3 | -1 | ζ65 | ζ6 | 0 | 0 | complex faithful |
| ρ9 | 2 | 0 | -1-√-3 | -1+√-3 | -1 | ζ6 | ζ65 | 0 | 0 | complex faithful |
►On 6 points - transitive group 6T5
Generators in S6
(1 2 3)(4 5 6)
(1 2 3)(4 6 5)
(1 4)(2 5)(3 6)
G:=sub<Sym(6)| (1,2,3)(4,5,6), (1,2,3)(4,6,5), (1,4)(2,5)(3,6)>;
G:=Group( (1,2,3)(4,5,6), (1,2,3)(4,6,5), (1,4)(2,5)(3,6) );
G=PermutationGroup([[(1,2,3),(4,5,6)], [(1,2,3),(4,6,5)], [(1,4),(2,5),(3,6)]])
G:=TransitiveGroup(6,5);
►On 9 points - transitive group 9T4
Generators in S9
(1 2 3)(4 5 6)(7 8 9)
(1 5 9)(2 6 7)(3 4 8)
(4 8)(5 9)(6 7)
G:=sub<Sym(9)| (1,2,3)(4,5,6)(7,8,9), (1,5,9)(2,6,7)(3,4,8), (4,8)(5,9)(6,7)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9), (1,5,9)(2,6,7)(3,4,8), (4,8)(5,9)(6,7) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9)], [(1,5,9),(2,6,7),(3,4,8)], [(4,8),(5,9),(6,7)]])
G:=TransitiveGroup(9,4);
►Regular action on 18 points - transitive group 18T3
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 10 14)(2 11 15)(3 12 13)(4 18 8)(5 16 9)(6 17 7)
(1 16)(2 17)(3 18)(4 12)(5 10)(6 11)(7 15)(8 13)(9 14)
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,10,14)(2,11,15)(3,12,13)(4,18,8)(5,16,9)(6,17,7), (1,16)(2,17)(3,18)(4,12)(5,10)(6,11)(7,15)(8,13)(9,14)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,10,14)(2,11,15)(3,12,13)(4,18,8)(5,16,9)(6,17,7), (1,16)(2,17)(3,18)(4,12)(5,10)(6,11)(7,15)(8,13)(9,14) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,10,14),(2,11,15),(3,12,13),(4,18,8),(5,16,9),(6,17,7)], [(1,16),(2,17),(3,18),(4,12),(5,10),(6,11),(7,15),(8,13),(9,14)]])
G:=TransitiveGroup(18,3);
C3×S3 is a maximal subgroup of
C32⋊C6 C9⋊C6 He3⋊C2 C3⋊F7 D39⋊C3 A4≀C2 D57⋊C3 C5⋊D15⋊C3
C3×S3 is a maximal quotient of
C32⋊C6 C9⋊C6 C3⋊F7 D39⋊C3 A4≀C2 D57⋊C3 C5⋊D15⋊C3
| action | f(x) | Disc(f) |
|---|---|---|
| 6T5 | x6-2x5+2x3-x2+1 | -26·132 |
| 9T4 | x9-3x8-8x7+13x6+22x5-13x4-20x3+x2+5x+1 | 76·533 |
Matrix representation of C3×S3 ►in GL2(𝔽7) generated by
| 2 | 0 |
| 0 | 2 |
| 2 | 0 |
| 0 | 4 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(7))| [2,0,0,2],[2,0,0,4],[0,1,1,0] >;
C3×S3 in GAP, Magma, Sage, TeX
C_3\times S_3
% in TeX
G:=Group("C3xS3"); // GroupNames label
G:=SmallGroup(18,3);
// by ID
G=gap.SmallGroup(18,3);
# by ID
G:=PCGroup([3,-2,-3,-3,110]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C3×S3 in TeX
Character table of C3×S3 in TeX