metabelian, supersoluble, monomial, A-group, rational
Aliases: C3⋊S3, C32⋊2C2, SmallGroup(18,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3⋊S3 |
Generators and relations for C3⋊S3
G = < a,b,c | a3=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Character table of C3⋊S3
| class | 1 | 2 | 3A | 3B | 3C | 3D | |
| size | 1 | 9 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 2 | 0 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ4 | 2 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ5 | 2 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 0 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
►On 9 points - transitive group 9T5
Generators in S9
(1 2 3)(4 5 6)(7 8 9)
(1 5 8)(2 6 9)(3 4 7)
(2 3)(4 9)(5 8)(6 7)
G:=sub<Sym(9)| (1,2,3)(4,5,6)(7,8,9), (1,5,8)(2,6,9)(3,4,7), (2,3)(4,9)(5,8)(6,7)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9), (1,5,8)(2,6,9)(3,4,7), (2,3)(4,9)(5,8)(6,7) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9)], [(1,5,8),(2,6,9),(3,4,7)], [(2,3),(4,9),(5,8),(6,7)]])
G:=TransitiveGroup(9,5);
►Regular action on 18 points - transitive group 18T4
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 8)(2 6 9)(3 4 7)(10 16 13)(11 17 14)(12 18 15)
(1 10)(2 12)(3 11)(4 14)(5 13)(6 15)(7 17)(8 16)(9 18)
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,8)(2,6,9)(3,4,7)(10,16,13)(11,17,14)(12,18,15), (1,10)(2,12)(3,11)(4,14)(5,13)(6,15)(7,17)(8,16)(9,18)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,5,8)(2,6,9)(3,4,7)(10,16,13)(11,17,14)(12,18,15), (1,10)(2,12)(3,11)(4,14)(5,13)(6,15)(7,17)(8,16)(9,18) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,5,8),(2,6,9),(3,4,7),(10,16,13),(11,17,14),(12,18,15)], [(1,10),(2,12),(3,11),(4,14),(5,13),(6,15),(7,17),(8,16),(9,18)]])
G:=TransitiveGroup(18,4);
C3⋊S3 is a maximal subgroup of
C32⋊C4 C32⋊C6 C33⋊C2 C3⋊S4 C52⋊(C3⋊S3)
C3⋊D3p: S32 C9⋊S3 C3⋊D15 C3⋊D21 C3⋊D33 C3⋊D39 C3⋊D51 C3⋊D57 ...
C3⋊S3 is a maximal quotient of
C3⋊Dic3 He3⋊C2 C33⋊C2 C3⋊S4 C52⋊(C3⋊S3)
C3⋊D3p: C9⋊S3 C3⋊D15 C3⋊D21 C3⋊D33 C3⋊D39 C3⋊D51 C3⋊D57 C3⋊D69 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 9T5 | x9-54x7-12x6+756x5+180x4-2652x3-864x2+288x+64 | 232·318·72·238·432·1512 |
Matrix representation of C3⋊S3 ►in GL4(ℤ) generated by
| 0 | 1 | 0 | 0 |
| -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | -1 |
| 1 | 0 | 0 | 0 |
| -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | -1 | 0 |
G:=sub<GL(4,Integers())| [0,-1,0,0,1,-1,0,0,0,0,0,-1,0,0,1,-1],[1,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,-1],[1,-1,0,0,0,-1,0,0,0,0,0,-1,0,0,-1,0] >;
C3⋊S3 in GAP, Magma, Sage, TeX
C_3\rtimes S_3
% in TeX
G:=Group("C3:S3"); // GroupNames label
G:=SmallGroup(18,4);
// by ID
G=gap.SmallGroup(18,4);
# by ID
G:=PCGroup([3,-2,-3,-3,25,110]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C3⋊S3 in TeX
Character table of C3⋊S3 in TeX