direct product, non-abelian, soluble, monomial
Aliases: C6×S4, (C2×A4)⋊C6, A4⋊(C2×C6), (C2×C6)⋊2D6, C22⋊(S3×C6), C23⋊(C3×S3), (C6×A4)⋊1C2, (C22×C6)⋊1S3, (C3×A4)⋊2C22, SmallGroup(144,188)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C6×S4 |
Generators and relations for C6×S4
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 [×4], C3, C3 [×2], C4 [×2], C22, C22 [×6], S3 [×2], C6, C6 [×6], C2×C4, D4 [×4], C23, C23, C32, C12 [×2], A4, A4, D6, C2×C6, C2×C6 [×6], C2×D4, C3×S3 [×2], C3×C6, C2×C12, C3×D4 [×4], S4 [×2], C2×A4, C2×A4, C22×C6, C22×C6, C3×A4, S3×C6, C6×D4, C2×S4, C3×S4 [×2], C6×A4, C6×S4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C3×S4, C6×S4
Character table of C6×S4
| 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 C3×S3 |
| 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 C3×S3 |
| 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 S3×C6 |
| 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 S3×C6 |
| 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 C2×S4 |
| 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 C2×S4 |
| 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 C3×S4 |
| 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 C3×S4 |
| 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 C3×S4 |
| 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 C3×S4 |
►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 16 9)(2 17 10)(3 18 11)(4 13 12)(5 14 7)(6 15 8)
(1 4)(2 5)(3 6)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)
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), (1,16,9)(2,17,10)(3,18,11)(4,13,12)(5,14,7)(6,15,8), (1,4)(2,5)(3,6)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16)>;
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), (1,16,9)(2,17,10)(3,18,11)(4,13,12)(5,14,7)(6,15,8), (1,4)(2,5)(3,6)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16) );
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)], [(1,16,9),(2,17,10),(3,18,11),(4,13,12),(5,14,7),(6,15,8)], [(1,4),(2,5),(3,6),(7,17),(8,18),(9,13),(10,14),(11,15),(12,16)])
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 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)
(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 19 17)(8 20 18)(9 21 13)(10 22 14)(11 23 15)(12 24 16)
(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), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16), (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (7,19,17)(8,20,18)(9,21,13)(10,22,14)(11,23,15)(12,24,16), (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), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16), (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (7,19,17)(8,20,18)(9,21,13)(10,22,14)(11,23,15)(12,24,16), (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)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,19),(7,17),(8,18),(9,13),(10,14),(11,15),(12,16)], [(1,18),(2,13),(3,14),(4,15),(5,16),(6,17),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(7,19,17),(8,20,18),(9,21,13),(10,22,14),(11,23,15),(12,24,16)], [(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)])
G:=TransitiveGroup(24,254);
C6×S4 is a maximal subgroup of
Dic3⋊S4 D6⋊S4
Matrix representation of C6×S4 ►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] >;
C6×S4 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
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