direct product, non-abelian, soluble, monomial
Aliases: C6xS4, (C2xA4):C6, A4:(C2xC6), (C2xC6):2D6, C22:(S3xC6), C23:(C3xS3), (C6xA4):1C2, (C22xC6):1S3, (C3xA4):2C22, SmallGroup(144,188)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C6xS4 |
Generators and relations for C6xS4
G = < a,b,c,d,e | a6=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >
Subgroups: 216 in 70 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C32, C12, A4, A4, D6, C2xC6, C2xC6, C2xD4, C3xS3, C3xC6, C2xC12, C3xD4, S4, C2xA4, C2xA4, C22xC6, C22xC6, C3xA4, S3xC6, C6xD4, C2xS4, C3xS4, C6xA4, C6xS4
Quotients: C1, C2, C3, C22, S3, C6, D6, C2xC6, C3xS3, S4, S3xC6, C2xS4, C3xS4, C6xS4
Character table of C6xS4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 3 | 3 | 6 | 6 | 1 | 1 | 8 | 8 | 8 | 6 | 6 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 6 | 6 | 6 | 6 | |
| 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 | -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 |
| ρ4 | 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 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | 1 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | -1 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | ζ6 | ζ65 | ζ3 | ζ32 | ζ6 | -1 | ζ65 | ζ6 | ζ3 | ζ65 | ζ32 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | 1 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | -1 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | ζ65 | ζ6 | ζ32 | ζ3 | ζ65 | -1 | ζ6 | ζ65 | ζ32 | ζ6 | ζ3 | linear of order 6 |
| ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | 1 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ3 | ζ6 | ζ32 | ζ65 | linear of order 6 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | 1 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ32 | ζ65 | ζ3 | ζ6 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 0 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 0 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 1-√-3 | 1+√-3 | 1+√-3 | 1-√-3 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | ζ3 | 1 | ζ32 | 0 | 0 | 0 | 0 | complex lifted from S3xC6 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 1+√-3 | 1-√-3 | 1-√-3 | 1+√-3 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | ζ32 | 1 | ζ3 | 0 | 0 | 0 | 0 | complex lifted from S3xC6 |
| ρ19 | 3 | 3 | -1 | -1 | 1 | 1 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | 3 | 3 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 1 | 1 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ21 | 3 | -3 | -1 | 1 | -1 | 1 | 3 | 3 | 0 | 0 | 0 | -1 | 1 | -3 | -3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from C2xS4 |
| ρ22 | 3 | -3 | -1 | 1 | 1 | -1 | 3 | 3 | 0 | 0 | 0 | 1 | -1 | -3 | -3 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from C2xS4 |
| ρ23 | 3 | -3 | -1 | 1 | 1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1 | -1 | 3-3√-3/2 | 3+3√-3/2 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | ζ32 | ζ3 | 0 | 0 | 0 | ζ3 | ζ6 | ζ32 | ζ65 | complex faithful |
| ρ24 | 3 | -3 | -1 | 1 | 1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1 | -1 | 3+3√-3/2 | 3-3√-3/2 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | ζ3 | ζ32 | 0 | 0 | 0 | ζ32 | ζ65 | ζ3 | ζ6 | complex faithful |
| ρ25 | 3 | 3 | -1 | -1 | 1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | ζ6 | ζ65 | ζ65 | ζ6 | ζ3 | ζ32 | ζ32 | ζ3 | 0 | 0 | 0 | ζ65 | ζ6 | ζ6 | ζ65 | complex lifted from C3xS4 |
| ρ26 | 3 | 3 | -1 | -1 | 1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | ζ65 | ζ6 | ζ6 | ζ65 | ζ32 | ζ3 | ζ3 | ζ32 | 0 | 0 | 0 | ζ6 | ζ65 | ζ65 | ζ6 | complex lifted from C3xS4 |
| ρ27 | 3 | -3 | -1 | 1 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | 1 | 3-3√-3/2 | 3+3√-3/2 | ζ32 | ζ3 | ζ65 | ζ6 | ζ3 | ζ32 | ζ6 | ζ65 | 0 | 0 | 0 | ζ65 | ζ32 | ζ6 | ζ3 | complex faithful |
| ρ28 | 3 | 3 | -1 | -1 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1 | 1 | -3-3√-3/2 | -3+3√-3/2 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | 0 | 0 | ζ32 | ζ3 | ζ3 | ζ32 | complex lifted from C3xS4 |
| ρ29 | 3 | -3 | -1 | 1 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | 1 | 3+3√-3/2 | 3-3√-3/2 | ζ3 | ζ32 | ζ6 | ζ65 | ζ32 | ζ3 | ζ65 | ζ6 | 0 | 0 | 0 | ζ6 | ζ3 | ζ65 | ζ32 | complex faithful |
| ρ30 | 3 | 3 | -1 | -1 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1 | 1 | -3+3√-3/2 | -3-3√-3/2 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | 0 | 0 | ζ3 | ζ32 | ζ32 | ζ3 | complex lifted from C3xS4 |
►On 18 points - transitive group 18T61
Generators in S18
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 15 9)(2 16 10)(3 17 11)(4 18 12)(5 13 7)(6 14 8)
(1 4)(2 5)(3 6)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)
G:=sub<Sym(18)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18), (1,15,9)(2,16,10)(3,17,11)(4,18,12)(5,13,7)(6,14,8), (1,4)(2,5)(3,6)(7,16)(8,17)(9,18)(10,13)(11,14)(12,15)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18), (1,15,9)(2,16,10)(3,17,11)(4,18,12)(5,13,7)(6,14,8), (1,4)(2,5)(3,6)(7,16)(8,17)(9,18)(10,13)(11,14)(12,15) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,4),(2,5),(3,6),(13,16),(14,17),(15,18)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18)], [(1,15,9),(2,16,10),(3,17,11),(4,18,12),(5,13,7),(6,14,8)], [(1,4),(2,5),(3,6),(7,16),(8,17),(9,18),(10,13),(11,14),(12,15)]])
G:=TransitiveGroup(18,61);
►On 24 points - transitive group 24T254
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(7 17 19)(8 18 20)(9 13 21)(10 14 22)(11 15 23)(12 16 24)
(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
G:=sub<Sym(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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (7,17,19)(8,18,20)(9,13,21)(10,14,22)(11,15,23)(12,16,24), (7,19)(8,20)(9,21)(10,22)(11,23)(12,24)>;
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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (7,17,19)(8,18,20)(9,13,21)(10,14,22)(11,15,23)(12,16,24), (7,19)(8,20)(9,21)(10,22)(11,23)(12,24) );
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)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,7),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20)], [(1,18),(2,13),(3,14),(4,15),(5,16),(6,17),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(7,17,19),(8,18,20),(9,13,21),(10,14,22),(11,15,23),(12,16,24)], [(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)]])
G:=TransitiveGroup(24,254);
C6xS4 is a maximal subgroup of
Dic3:S4 D6:S4
Matrix representation of C6xS4 ►in GL3(F7) generated by
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 0 | 3 | 2 |
| 5 | 0 | 3 |
| 0 | 0 | 6 |
| 2 | 5 | 4 |
| 4 | 1 | 3 |
| 3 | 5 | 3 |
| 2 | 5 | 4 |
| 0 | 3 | 4 |
| 0 | 4 | 2 |
| 2 | 5 | 4 |
| 3 | 2 | 5 |
| 6 | 2 | 4 |
G:=sub<GL(3,GF(7))| [3,0,0,0,3,0,0,0,3],[0,5,0,3,0,0,2,3,6],[2,4,3,5,1,5,4,3,3],[2,0,0,5,3,4,4,4,2],[2,3,6,5,2,2,4,5,4] >;
C6xS4 in GAP, Magma, Sage, TeX
C_6\times S_4
% in TeX
G:=Group("C6xS4"); // GroupNames label
G:=SmallGroup(144,188);
// by ID
G=gap.SmallGroup(144,188);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,2,579,2164,202,1301,347]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations
Export