direct product, non-abelian, soluble, monomial
Aliases: C3×C3.S4, C32.2S4, C62.6S3, (C2×C6)⋊1D9, C22⋊(C3×D9), C3.A4⋊4C6, C3.2(C3×S4), (C3×C3.A4)⋊2C2, (C2×C6).3(C3×S3), SmallGroup(216,91)
Series: Derived ►Chief ►Lower central ►Upper central
| C3.A4 — C3×C3.S4 |
Generators and relations for C3×C3.S4
G = < a,b,c,d,e,f | a3=b3=c2=d2=f2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=b-1e2 >
Character table of C3×C3.S4
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 12A | 12B | |
| size | 1 | 3 | 18 | 1 | 1 | 2 | 2 | 2 | 18 | 3 | 3 | 6 | 6 | 6 | 18 | 18 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 18 | 18 | |
| 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 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | 1 | 1 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | 1 | 1 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ7 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | 0 | 0 | orthogonal lifted from D9 |
| ρ9 | 2 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | 0 | 0 | orthogonal lifted from D9 |
| ρ10 | 2 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | 0 | 0 | orthogonal lifted from D9 |
| ρ11 | 2 | 2 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 2 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 2 | 0 | 0 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | -1 | -1 | -1 | ζ6 | 0 | 0 | complex lifted from C3×S3 |
| ρ12 | 2 | 2 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 0 | ζ92+ζ9 | ζ98+ζ94 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ97 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ95 | 0 | 0 | complex lifted from C3×D9 |
| ρ13 | 2 | 2 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 0 | ζ95+ζ9 | ζ94+ζ92 | ζ92+ζ9 | ζ97+ζ95 | ζ98+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ97 | 0 | 0 | complex lifted from C3×D9 |
| ρ14 | 2 | 2 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 2 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 2 | 0 | 0 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | -1 | -1 | -1 | ζ65 | 0 | 0 | complex lifted from C3×S3 |
| ρ15 | 2 | 2 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 0 | ζ97+ζ95 | ζ92+ζ9 | ζ95+ζ9 | ζ98+ζ97 | ζ94+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ94 | 0 | 0 | complex lifted from C3×D9 |
| ρ16 | 2 | 2 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 0 | ζ98+ζ97 | ζ95+ζ9 | ζ97+ζ95 | ζ98+ζ94 | ζ92+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ94+ζ92 | 0 | 0 | complex lifted from C3×D9 |
| ρ17 | 2 | 2 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 0 | ζ94+ζ92 | ζ98+ζ97 | ζ98+ζ94 | ζ92+ζ9 | ζ97+ζ95 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ9 | 0 | 0 | complex lifted from C3×D9 |
| ρ18 | 2 | 2 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 0 | ζ98+ζ94 | ζ97+ζ95 | ζ98+ζ97 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ92+ζ9 | 0 | 0 | complex lifted from C3×D9 |
| ρ19 | 3 | -1 | 1 | 3 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S4 |
| ρ20 | 3 | -1 | -1 | 3 | 3 | 3 | 3 | 3 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S4 |
| ρ21 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | complex lifted from C3×S4 |
| ρ22 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | complex lifted from C3×S4 |
| ρ23 | 3 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | -1 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | complex lifted from C3×S4 |
| ρ24 | 3 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | -1 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | complex lifted from C3×S4 |
| ρ25 | 6 | -2 | 0 | 6 | 6 | -3 | -3 | -3 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ26 | 6 | -2 | 0 | -3+3√-3 | -3-3√-3 | 3+3√-3/2 | 3-3√-3/2 | -3 | 0 | 1+√-3 | 1-√-3 | ζ3 | ζ32 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 6 | -2 | 0 | -3-3√-3 | -3+3√-3 | 3-3√-3/2 | 3+3√-3/2 | -3 | 0 | 1-√-3 | 1+√-3 | ζ32 | ζ3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 36 points
Generators in S36
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(1 27)(3 20)(4 21)(6 23)(7 24)(9 26)(11 30)(12 31)(14 33)(15 34)(17 36)(18 28)
(1 27)(2 19)(4 21)(5 22)(7 24)(8 25)(10 29)(12 31)(13 32)(15 34)(16 35)(18 28)
(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)(28 29 30 31 32 33 34 35 36)
(1 29)(2 28)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)
G:=sub<Sym(36)| (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (1,27)(3,20)(4,21)(6,23)(7,24)(9,26)(11,30)(12,31)(14,33)(15,34)(17,36)(18,28), (1,27)(2,19)(4,21)(5,22)(7,24)(8,25)(10,29)(12,31)(13,32)(15,34)(16,35)(18,28), (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)(28,29,30,31,32,33,34,35,36), (1,29)(2,28)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;
G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (1,27)(3,20)(4,21)(6,23)(7,24)(9,26)(11,30)(12,31)(14,33)(15,34)(17,36)(18,28), (1,27)(2,19)(4,21)(5,22)(7,24)(8,25)(10,29)(12,31)(13,32)(15,34)(16,35)(18,28), (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)(28,29,30,31,32,33,34,35,36), (1,29)(2,28)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36)], [(1,27),(3,20),(4,21),(6,23),(7,24),(9,26),(11,30),(12,31),(14,33),(15,34),(17,36),(18,28)], [(1,27),(2,19),(4,21),(5,22),(7,24),(8,25),(10,29),(12,31),(13,32),(15,34),(16,35),(18,28)], [(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),(28,29,30,31,32,33,34,35,36)], [(1,29),(2,28),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)]])
Matrix representation of C3×C3.S4 ►in GL7(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 12 | 0 | 0 | 0 |
| 0 | 0 | 21 | 34 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 25 | 0 | 0 | 0 |
| 0 | 0 | 28 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(7,GF(37))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[26,10,0,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36],[16,1,0,0,0,0,0,0,7,0,0,0,0,0,0,0,2,21,0,0,0,0,0,12,34,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,28,36,0,0,0,0,0,0,0,35,28,0,0,0,0,0,25,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0] >;
C3×C3.S4 in GAP, Magma, Sage, TeX
C_3\times C_3.S_4
% in TeX
G:=Group("C3xC3.S4"); // GroupNames label
G:=SmallGroup(216,91);
// by ID
G=gap.SmallGroup(216,91);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,2,542,122,867,3244,556,1949,989]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^2=f^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=b^-1*e^2>;
// generators/relations
Export
Subgroup lattice of C3×C3.S4 in TeX
Character table of C3×C3.S4 in TeX