direct product, non-abelian, soluble, monomial
Aliases: S3×C3.S4, C62.17D6, (C2×C6)⋊D18, (C3×S3).S4, C3.3(S3×S4), C3.A4⋊1D6, (C22×S3)⋊D9, C22⋊2(S3×D9), C32.3S4⋊C2, C32.2(C2×S4), (S3×C2×C6).S3, (C3×C3.S4)⋊C2, (S3×C3.A4)⋊C2, (C2×C6).3S32, C3⋊1(C2×C3.S4), (C3×C3.A4)⋊C22, SmallGroup(432,522)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C3.A4 — S3×C3.S4 |
Generators and relations for S3×C3.S4
G = < a,b,c,d,e,f,g | a3=b2=c3=d2=e2=g2=1, f3=c, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg=c-1, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=c-1f2 >
Subgroups: 1120 in 123 conjugacy classes, 19 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, S3, C6, C2×C4, D4, C23, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×D4, D9, C18, C3×S3, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C22×S3, C22×C6, C3×C9, C3.A4, C3.A4, D18, C3×Dic3, C3⋊Dic3, S32, S3×C6, C2×C3⋊S3, C62, S3×D4, C2×C3⋊D4, C3×D9, S3×C9, C9⋊S3, C3.S4, C3.S4, C2×C3.A4, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C32⋊7D4, C2×S32, S3×C2×C6, S3×D9, C3×C3.A4, C2×C3.S4, S3×C3⋊D4, C3×C3.S4, C32.3S4, S3×C3.A4, S3×C3.S4
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, S32, C2×S4, C3.S4, S3×D9, C2×C3.S4, S3×S4, S3×C3.S4
Character table of S3×C3.S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 9A | 9B | 9C | 9D | 9E | 9F | 12 | 18A | 18B | 18C | |
| size | 1 | 3 | 3 | 9 | 18 | 54 | 2 | 2 | 4 | 18 | 54 | 6 | 6 | 6 | 12 | 18 | 36 | 8 | 8 | 8 | 16 | 16 | 16 | 36 | 24 | 24 | 24 | |
| 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 | -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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | -2 | 0 | -1 | 2 | -1 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 1 | 2 | 2 | 2 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 0 | 0 | 2 | 0 | -1 | 2 | -1 | 2 | 0 | 0 | -1 | 2 | -1 | 0 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 1 | 2 | -1 | -1 | 1 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 1 | 2 | -1 | -1 | 1 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 1 | 2 | -1 | -1 | 1 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ15 | 3 | -1 | 3 | -1 | -1 | -1 | 3 | 3 | 3 | 1 | 1 | 3 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ16 | 3 | -1 | 3 | -1 | 1 | 1 | 3 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ17 | 3 | -1 | -3 | 1 | -1 | 1 | 3 | 3 | 3 | 1 | -1 | -3 | -1 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ18 | 3 | -1 | -3 | 1 | 1 | -1 | 3 | 3 | 3 | -1 | 1 | -3 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ19 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ20 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D9 |
| ρ21 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D9 |
| ρ22 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D9 |
| ρ23 | 6 | -2 | 0 | 0 | -2 | 0 | -3 | 6 | -3 | 2 | 0 | 0 | 1 | -2 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | orthogonal lifted from S3×S4 |
| ρ24 | 6 | -2 | 6 | -2 | 0 | 0 | 6 | -3 | -3 | 0 | 0 | -3 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ25 | 6 | -2 | 0 | 0 | 2 | 0 | -3 | 6 | -3 | -2 | 0 | 0 | 1 | -2 | 1 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | orthogonal lifted from S3×S4 |
| ρ26 | 6 | -2 | -6 | 2 | 0 | 0 | 6 | -3 | -3 | 0 | 0 | 3 | -2 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C3.S4 |
| ρ27 | 12 | -4 | 0 | 0 | 0 | 0 | -6 | -6 | 3 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 36 points
Generators in S36
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 28)(10 27)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)
(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)(2 19)(4 21)(5 22)(7 24)(8 25)(10 29)(11 30)(13 32)(14 33)(16 35)(17 36)
(2 19)(3 20)(5 22)(6 23)(8 25)(9 26)(11 30)(12 31)(14 33)(15 34)(17 36)(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)| (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,28)(10,27)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (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)(2,19)(4,21)(5,22)(7,24)(8,25)(10,29)(11,30)(13,32)(14,33)(16,35)(17,36), (2,19)(3,20)(5,22)(6,23)(8,25)(9,26)(11,30)(12,31)(14,33)(15,34)(17,36)(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( (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,28)(10,27)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (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)(2,19)(4,21)(5,22)(7,24)(8,25)(10,29)(11,30)(13,32)(14,33)(16,35)(17,36), (2,19)(3,20)(5,22)(6,23)(8,25)(9,26)(11,30)(12,31)(14,33)(15,34)(17,36)(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([[(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,28),(10,27),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26)], [(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),(2,19),(4,21),(5,22),(7,24),(8,25),(10,29),(11,30),(13,32),(14,33),(16,35),(17,36)], [(2,19),(3,20),(5,22),(6,23),(8,25),(9,26),(11,30),(12,31),(14,33),(15,34),(17,36),(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 S3×C3.S4 ►in GL7(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 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 | 19 | 0 | 0 | 0 | 0 | 0 |
| 35 | 36 | 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 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 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 | 36 | 0 | 36 |
| 6 | 13 | 0 | 0 | 0 | 0 | 0 |
| 22 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 36 | 35 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 36 | 35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(7,GF(37))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0,1,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,35,0,0,0,0,0,19,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,1,0,0,0,0,0,36,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,36,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[6,22,0,0,0,0,0,13,17,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,1,0,0,0,0,0,36,0,0,0,0,0,0,35,0,1],[1,35,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,36,0,0,0,0,0,0,35,1] >;
S3×C3.S4 in GAP, Magma, Sage, TeX
S_3\times C_3.S_4
% in TeX
G:=Group("S3xC3.S4"); // GroupNames label
G:=SmallGroup(432,522);
// by ID
G=gap.SmallGroup(432,522);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,471,394,675,1271,4548,2287,2659,3989]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^3=d^2=e^2=g^2=1,f^3=c,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,g*c*g=c^-1,f*d*f^-1=g*d*g=d*e=e*d,f*e*f^-1=d,e*g=g*e,g*f*g=c^-1*f^2>;
// generators/relations
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