direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C32, C33⋊1C2, C32⋊3C6, C3⋊(C3×C6), SmallGroup(54,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C32 |
Generators and relations for S3×C32
G = < a,b,c,d | a3=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Character table of S3×C32
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 3L | 3M | 3N | 3O | 3P | 3Q | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | |
| size | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 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 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | linear of order 3 |
| ρ4 | 1 | -1 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | -1 | ζ6 | linear of order 6 |
| ρ5 | 1 | -1 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | 1 | ζ32 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | -1 | linear of order 6 |
| ρ8 | 1 | -1 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | linear of order 6 |
| ρ9 | 1 | 1 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ10 | 1 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | linear of order 3 |
| ρ11 | 1 | -1 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | 1 | ζ3 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | -1 | linear of order 6 |
| ρ12 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ13 | 1 | -1 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | -1 | ζ65 | linear of order 6 |
| ρ14 | 1 | 1 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | linear of order 3 |
| ρ15 | 1 | 1 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ16 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | -1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ17 | 1 | 1 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | linear of order 3 |
| ρ18 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | -1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ19 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ20 | 2 | 0 | 2 | -1+√-3 | 2 | -1-√-3 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | ζ6 | ζ6 | ζ6 | ζ65 | -1 | ζ65 | -1 | -1 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ21 | 2 | 0 | -1-√-3 | -1-√-3 | -1+√-3 | 2 | -1+√-3 | -1-√-3 | 2 | -1+√-3 | ζ65 | -1 | ζ6 | ζ6 | -1 | -1 | ζ65 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ22 | 2 | 0 | -1+√-3 | -1+√-3 | -1-√-3 | 2 | -1-√-3 | -1+√-3 | 2 | -1-√-3 | ζ6 | -1 | ζ65 | ζ65 | -1 | -1 | ζ6 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ23 | 2 | 0 | -1-√-3 | 2 | -1+√-3 | -1-√-3 | 2 | -1+√-3 | -1+√-3 | -1-√-3 | -1 | ζ6 | ζ65 | -1 | -1 | ζ65 | ζ65 | ζ6 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ24 | 2 | 0 | 2 | -1-√-3 | 2 | -1+√-3 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | ζ65 | ζ65 | ζ65 | ζ6 | -1 | ζ6 | -1 | -1 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ25 | 2 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 2 | -1-√-3 | 2 | ζ6 | ζ65 | -1 | ζ65 | -1 | ζ6 | ζ65 | ζ6 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ26 | 2 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 2 | -1+√-3 | 2 | ζ65 | ζ6 | -1 | ζ6 | -1 | ζ65 | ζ6 | ζ65 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ27 | 2 | 0 | -1+√-3 | 2 | -1-√-3 | -1+√-3 | 2 | -1-√-3 | -1-√-3 | -1+√-3 | -1 | ζ65 | ζ6 | -1 | -1 | ζ6 | ζ6 | ζ65 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
►On 18 points - transitive group 18T17
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 8 18)(5 9 16)(6 7 17)
(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)(4,8,18)(5,9,16)(6,7,17), (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)(4,8,18)(5,9,16)(6,7,17), (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),(4,8,18),(5,9,16),(6,7,17)], [(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(18,17);
►On 27 points - transitive group 27T15
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 6 14)(2 4 15)(3 5 13)(7 24 16)(8 22 17)(9 23 18)(10 27 19)(11 25 20)(12 26 21)
(1 8 26)(2 9 27)(3 7 25)(4 23 19)(5 24 20)(6 22 21)(10 15 18)(11 13 16)(12 14 17)
(7 25)(8 26)(9 27)(10 18)(11 16)(12 17)(19 23)(20 24)(21 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), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,27,19)(11,25,20)(12,26,21), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (7,25)(8,26)(9,27)(10,18)(11,16)(12,17)(19,23)(20,24)(21,22)>;
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), (1,6,14)(2,4,15)(3,5,13)(7,24,16)(8,22,17)(9,23,18)(10,27,19)(11,25,20)(12,26,21), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (7,25)(8,26)(9,27)(10,18)(11,16)(12,17)(19,23)(20,24)(21,22) );
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)], [(1,6,14),(2,4,15),(3,5,13),(7,24,16),(8,22,17),(9,23,18),(10,27,19),(11,25,20),(12,26,21)], [(1,8,26),(2,9,27),(3,7,25),(4,23,19),(5,24,20),(6,22,21),(10,15,18),(11,13,16),(12,14,17)], [(7,25),(8,26),(9,27),(10,18),(11,16),(12,17),(19,23),(20,24),(21,22)]])
G:=TransitiveGroup(27,15);
S3×C32 is a maximal subgroup of
C3≀S3
Matrix representation of S3×C32 ►in GL3(𝔽7) generated by
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 4 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(7))| [2,0,0,0,2,0,0,0,2],[4,0,0,0,1,0,0,0,1],[1,0,0,0,2,0,0,0,4],[1,0,0,0,0,1,0,1,0] >;
S3×C32 in GAP, Magma, Sage, TeX
S_3\times C_3^2
% in TeX
G:=Group("S3xC3^2"); // GroupNames label
G:=SmallGroup(54,12);
// by ID
G=gap.SmallGroup(54,12);
# by ID
G:=PCGroup([4,-2,-3,-3,-3,579]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of S3×C32 in TeX
Character table of S3×C32 in TeX