direct product, non-abelian, soluble, monomial
Aliases: C3xC42:S3, (C4xC12):2S3, C42:C3:2C6, (C2xC6).1S4, C22.(C3xS4), C42:1(C3xS3), (C3xC42:C3):6C2, SmallGroup(288,397)
Series: Derived ►Chief ►Lower central ►Upper central
| C42:C3 — C3xC42:S3 |
Generators and relations for C3xC42:S3
G = < a,b,c,d,e | a3=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=c, dcd-1=b-1c-1, ece=b, ede=d-1 >
Quotients: C1, C2, C3, S3, C6, C3xS3, S4, C3xS4, C42:S3, C3xC42:S3
3C2
12C2
16C3
32C3
3C4
3C4
6C22
6C4
3C6
12C6
16S3
16C32
3D4
3Q8
6C8
6D4
3C12
3C12
4A4
6C12
8A4
16C3xS3
4S4
6C24
Character table of C3xC42:S3
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | 24B | 24C | 24D | |
| size | 1 | 3 | 12 | 1 | 1 | 32 | 32 | 32 | 3 | 3 | 6 | 12 | 3 | 3 | 12 | 12 | 12 | 12 | 3 | 3 | 3 | 3 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | |
| 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 | 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 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 2 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 0 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 2 | 2 | 2 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ9 | 2 | 2 | 0 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 2 | 2 | 2 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ10 | 3 | 3 | 1 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 3 | 3 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ11 | 3 | 3 | -1 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 3 | 3 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ12 | 3 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | ζ65 | ζ6 | 1 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | complex lifted from C3xS4 |
| ρ13 | 3 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | ζ6 | ζ65 | 1 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | complex lifted from C3xS4 |
| ρ14 | 3 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | ζ3 | ζ32 | -1 | -1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | ζ3 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3xS4 |
| ρ15 | 3 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | ζ32 | ζ3 | -1 | -1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | ζ32 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3xS4 |
| ρ16 | 3 | -1 | 1 | 3 | 3 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | -1+2i | -1+2i | -1-2i | -1-2i | 1 | 1 | -1 | -1 | i | -i | i | -i | complex lifted from C42:S3 |
| ρ17 | 3 | -1 | 1 | 3 | 3 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -1-2i | -1-2i | -1+2i | -1+2i | 1 | 1 | -1 | -1 | -i | i | -i | i | complex lifted from C42:S3 |
| ρ18 | 3 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | -1+2i | -1+2i | -1-2i | -1-2i | 1 | 1 | 1 | 1 | -i | i | -i | i | complex lifted from C42:S3 |
| ρ19 | 3 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | -1-2i | -1-2i | -1+2i | -1+2i | 1 | 1 | 1 | 1 | i | -i | i | -i | complex lifted from C42:S3 |
| ρ20 | 3 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | -1 | ζ6 | ζ65 | ζ32 | ζ3 | i | -i | 2ζ43ζ3-ζ3 | 2ζ43ζ32-ζ32 | 2ζ4ζ3-ζ3 | 2ζ4ζ32-ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | complex faithful |
| ρ21 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | i | -i | 2ζ4ζ32-ζ32 | 2ζ4ζ3-ζ3 | 2ζ43ζ32-ζ32 | 2ζ43ζ3-ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | complex faithful |
| ρ22 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | i | -i | 2ζ4ζ3-ζ3 | 2ζ4ζ32-ζ32 | 2ζ43ζ3-ζ3 | 2ζ43ζ32-ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | complex faithful |
| ρ23 | 3 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | -1 | ζ65 | ζ6 | ζ3 | ζ32 | i | -i | 2ζ43ζ32-ζ32 | 2ζ43ζ3-ζ3 | 2ζ4ζ32-ζ32 | 2ζ4ζ3-ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | complex faithful |
| ρ24 | 3 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | -1 | ζ6 | ζ65 | ζ32 | ζ3 | -i | i | 2ζ4ζ3-ζ3 | 2ζ4ζ32-ζ32 | 2ζ43ζ3-ζ3 | 2ζ43ζ32-ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | complex faithful |
| ρ25 | 3 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | -1 | ζ65 | ζ6 | ζ3 | ζ32 | -i | i | 2ζ4ζ32-ζ32 | 2ζ4ζ3-ζ3 | 2ζ43ζ32-ζ32 | 2ζ43ζ3-ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | complex faithful |
| ρ26 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | -i | i | 2ζ43ζ32-ζ32 | 2ζ43ζ3-ζ3 | 2ζ4ζ32-ζ32 | 2ζ4ζ3-ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | complex faithful |
| ρ27 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | -i | i | 2ζ43ζ3-ζ3 | 2ζ43ζ32-ζ32 | 2ζ4ζ3-ζ3 | 2ζ4ζ32-ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | complex faithful |
| ρ28 | 6 | -2 | 0 | 6 | 6 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C42:S3 |
| ρ29 | 6 | -2 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 6 | -2 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 36 points
Generators in S36
(1 9 2)(3 6 11)(4 8 5)(7 10 12)(13 32 24)(14 29 21)(15 30 22)(16 31 23)(17 26 34)(18 27 35)(19 28 36)(20 25 33)
(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)
(1 6 7 5)(2 3 12 8)(4 9 11 10)(13 16 15 14)(21 24 23 22)(29 32 31 30)
(1 32 28)(2 13 19)(3 14 18)(4 23 33)(5 31 25)(6 29 27)(7 30 26)(8 16 20)(9 24 36)(10 22 34)(11 21 35)(12 15 17)
(1 25)(2 20)(3 17)(4 36)(5 28)(6 26)(7 27)(8 19)(9 33)(10 35)(11 34)(12 18)(13 16)(14 15)(21 22)(23 24)(29 30)(31 32)
G:=sub<Sym(36)| (1,9,2)(3,6,11)(4,8,5)(7,10,12)(13,32,24)(14,29,21)(15,30,22)(16,31,23)(17,26,34)(18,27,35)(19,28,36)(20,25,33), (13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,6,7,5)(2,3,12,8)(4,9,11,10)(13,16,15,14)(21,24,23,22)(29,32,31,30), (1,32,28)(2,13,19)(3,14,18)(4,23,33)(5,31,25)(6,29,27)(7,30,26)(8,16,20)(9,24,36)(10,22,34)(11,21,35)(12,15,17), (1,25)(2,20)(3,17)(4,36)(5,28)(6,26)(7,27)(8,19)(9,33)(10,35)(11,34)(12,18)(13,16)(14,15)(21,22)(23,24)(29,30)(31,32)>;
G:=Group( (1,9,2)(3,6,11)(4,8,5)(7,10,12)(13,32,24)(14,29,21)(15,30,22)(16,31,23)(17,26,34)(18,27,35)(19,28,36)(20,25,33), (13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,6,7,5)(2,3,12,8)(4,9,11,10)(13,16,15,14)(21,24,23,22)(29,32,31,30), (1,32,28)(2,13,19)(3,14,18)(4,23,33)(5,31,25)(6,29,27)(7,30,26)(8,16,20)(9,24,36)(10,22,34)(11,21,35)(12,15,17), (1,25)(2,20)(3,17)(4,36)(5,28)(6,26)(7,27)(8,19)(9,33)(10,35)(11,34)(12,18)(13,16)(14,15)(21,22)(23,24)(29,30)(31,32) );
G=PermutationGroup([[(1,9,2),(3,6,11),(4,8,5),(7,10,12),(13,32,24),(14,29,21),(15,30,22),(16,31,23),(17,26,34),(18,27,35),(19,28,36),(20,25,33)], [(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)], [(1,6,7,5),(2,3,12,8),(4,9,11,10),(13,16,15,14),(21,24,23,22),(29,32,31,30)], [(1,32,28),(2,13,19),(3,14,18),(4,23,33),(5,31,25),(6,29,27),(7,30,26),(8,16,20),(9,24,36),(10,22,34),(11,21,35),(12,15,17)], [(1,25),(2,20),(3,17),(4,36),(5,28),(6,26),(7,27),(8,19),(9,33),(10,35),(11,34),(12,18),(13,16),(14,15),(21,22),(23,24),(29,30),(31,32)]])
Matrix representation of C3xC42:S3 ►in GL3(F13) generated by
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 5 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 12 |
| 5 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 5 |
| 0 | 0 | 9 |
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 12 | 0 | 0 |
| 0 | 0 | 10 |
| 0 | 4 | 0 |
G:=sub<GL(3,GF(13))| [3,0,0,0,3,0,0,0,3],[5,0,0,0,5,0,0,0,12],[5,0,0,0,12,0,0,0,5],[0,9,0,0,0,9,9,0,0],[12,0,0,0,0,4,0,10,0] >;
C3xC42:S3 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes S_3
% in TeX
G:=Group("C3xC4^2:S3"); // GroupNames label
G:=SmallGroup(288,397);
// by ID
G=gap.SmallGroup(288,397);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,254,1011,185,360,634,1173,102,9077,1027,1784]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=c,d*c*d^-1=b^-1*c^-1,e*c*e=b,e*d*e=d^-1>;
// generators/relations
Export
Subgroup lattice of C3xC42:S3 in TeX
Character table of C3xC42:S3 in TeX