direct product, non-abelian, supersoluble, monomial
Aliases: C2×3- 1+2.S3, 3- 1+2.D6, (C3×C9).2D6, (C3×C18).10S3, C3.He3⋊C22, C6.10(He3⋊C2), (C2×3- 1+2).4S3, (C3×C6).8(C3⋊S3), (C2×C3.He3)⋊C2, C32.4(C2×C3⋊S3), C3.5(C2×He3⋊C2), SmallGroup(324,80)
Series: Derived ►Chief ►Lower central ►Upper central
| C3.He3 — C2×3- 1+2.S3 |
Generators and relations for C2×3- 1+2.S3
G = < a,b,c,d,e | a2=b9=c3=e2=1, d3=b6, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b4, dbd-1=b7c, ebe=b5c-1, dcd-1=b3c, ce=ec, ede=b3d2 >
Subgroups: 337 in 65 conjugacy classes, 19 normal (11 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, D6, C2×C6, D9, C18, C3×S3, C3×C6, C3×C9, 3- 1+2, D18, S3×C6, C3×D9, C9⋊C6, C3×C18, C2×3- 1+2, C3.He3, C6×D9, C2×C9⋊C6, 3- 1+2.S3, C2×C3.He3, C2×3- 1+2.S3
Quotients: C1, C2, C22, S3, D6, C3⋊S3, C2×C3⋊S3, He3⋊C2, C2×He3⋊C2, 3- 1+2.S3, C2×3- 1+2.S3
Character table of C2×3- 1+2.S3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 9D | 9E | 9F | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 1 | 27 | 27 | 2 | 3 | 3 | 2 | 3 | 3 | 27 | 27 | 27 | 27 | 6 | 6 | 6 | 18 | 18 | 18 | 6 | 6 | 6 | 18 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | 1 | 1 | 1 | 1 | -2 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | 1 | 1 | -2 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -2 | -2 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 3 | -3 | -1 | 1 | 3 | -3-3√-3/2 | -3+3√-3/2 | -3 | 3-3√-3/2 | 3+3√-3/2 | ζ65 | ζ32 | ζ3 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×He3⋊C2 |
| ρ14 | 3 | -3 | -1 | 1 | 3 | -3+3√-3/2 | -3-3√-3/2 | -3 | 3+3√-3/2 | 3-3√-3/2 | ζ6 | ζ3 | ζ32 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×He3⋊C2 |
| ρ15 | 3 | 3 | 1 | 1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 3 | -3+3√-3/2 | -3-3√-3/2 | ζ3 | ζ32 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ16 | 3 | -3 | 1 | -1 | 3 | -3-3√-3/2 | -3+3√-3/2 | -3 | 3-3√-3/2 | 3+3√-3/2 | ζ3 | ζ6 | ζ65 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×He3⋊C2 |
| ρ17 | 3 | 3 | -1 | -1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 3 | -3+3√-3/2 | -3-3√-3/2 | ζ65 | ζ6 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ18 | 3 | 3 | -1 | -1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 3 | -3-3√-3/2 | -3+3√-3/2 | ζ6 | ζ65 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ19 | 3 | -3 | 1 | -1 | 3 | -3+3√-3/2 | -3-3√-3/2 | -3 | 3+3√-3/2 | 3-3√-3/2 | ζ32 | ζ65 | ζ6 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×He3⋊C2 |
| ρ20 | 3 | 3 | 1 | 1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 3 | -3-3√-3/2 | -3+3√-3/2 | ζ32 | ζ3 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ21 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | 0 | 0 | 0 | orthogonal faithful |
| ρ22 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ23 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ24 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | orthogonal lifted from 3- 1+2.S3 |
| ρ25 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 0 | 0 | 0 | 2ζ95+ζ94+ζ92-ζ9 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 0 | 0 | 0 | orthogonal faithful |
| ρ26 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 0 | 0 | 0 | ζ98+ζ94-ζ92+2ζ9 | ζ98+ζ97-ζ94+2ζ92 | 2ζ95+ζ94+ζ92-ζ9 | 0 | 0 | 0 | orthogonal faithful |
►On 54 points
Generators in S54
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 43)(20 44)(21 45)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 46)(35 47)(36 48)
(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)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 25 22)(20 23 26)(28 34 31)(29 32 35)(37 43 40)(38 41 44)(46 52 49)(47 50 53)
(1 38 31 7 44 28 4 41 34)(2 45 32 8 42 29 5 39 35)(3 37 30 9 43 36 6 40 33)(10 25 54 16 22 51 13 19 48)(11 20 49 17 26 46 14 23 52)(12 27 50 18 24 47 15 21 53)
(2 3)(4 7)(5 9)(6 8)(10 12)(11 17)(13 18)(15 16)(19 50)(20 49)(21 54)(22 47)(23 46)(24 51)(25 53)(26 52)(27 48)(28 44)(29 43)(30 39)(31 41)(32 40)(33 45)(34 38)(35 37)(36 42)
G:=sub<Sym(54)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,46)(35,47)(36,48), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,52,49)(47,50,53), (1,38,31,7,44,28,4,41,34)(2,45,32,8,42,29,5,39,35)(3,37,30,9,43,36,6,40,33)(10,25,54,16,22,51,13,19,48)(11,20,49,17,26,46,14,23,52)(12,27,50,18,24,47,15,21,53), (2,3)(4,7)(5,9)(6,8)(10,12)(11,17)(13,18)(15,16)(19,50)(20,49)(21,54)(22,47)(23,46)(24,51)(25,53)(26,52)(27,48)(28,44)(29,43)(30,39)(31,41)(32,40)(33,45)(34,38)(35,37)(36,42)>;
G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,46)(35,47)(36,48), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,52,49)(47,50,53), (1,38,31,7,44,28,4,41,34)(2,45,32,8,42,29,5,39,35)(3,37,30,9,43,36,6,40,33)(10,25,54,16,22,51,13,19,48)(11,20,49,17,26,46,14,23,52)(12,27,50,18,24,47,15,21,53), (2,3)(4,7)(5,9)(6,8)(10,12)(11,17)(13,18)(15,16)(19,50)(20,49)(21,54)(22,47)(23,46)(24,51)(25,53)(26,52)(27,48)(28,44)(29,43)(30,39)(31,41)(32,40)(33,45)(34,38)(35,37)(36,42) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13),(19,43),(20,44),(21,45),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,46),(35,47),(36,48)], [(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),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,25,22),(20,23,26),(28,34,31),(29,32,35),(37,43,40),(38,41,44),(46,52,49),(47,50,53)], [(1,38,31,7,44,28,4,41,34),(2,45,32,8,42,29,5,39,35),(3,37,30,9,43,36,6,40,33),(10,25,54,16,22,51,13,19,48),(11,20,49,17,26,46,14,23,52),(12,27,50,18,24,47,15,21,53)], [(2,3),(4,7),(5,9),(6,8),(10,12),(11,17),(13,18),(15,16),(19,50),(20,49),(21,54),(22,47),(23,46),(24,51),(25,53),(26,52),(27,48),(28,44),(29,43),(30,39),(31,41),(32,40),(33,45),(34,38),(35,37),(36,42)]])
Matrix representation of C2×3- 1+2.S3 ►in GL6(𝔽19)
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 7 | 17 | 0 | 0 |
| 14 | 14 | 14 | 14 | 7 | 16 |
| 12 | 12 | 12 | 12 | 10 | 7 |
| 0 | 7 | 0 | 7 | 7 | 5 |
| 5 | 0 | 0 | 7 | 7 | 5 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 1 | 18 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 18 | 18 | 18 | 0 | 18 | 18 |
| 18 | 18 | 18 | 18 | 18 | 17 |
| 0 | 0 | 0 | 0 | 1 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 18 |
| 18 | 18 | 18 | 18 | 18 | 17 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,0,14,12,0,5,0,0,14,12,7,0,5,7,14,12,0,0,12,17,14,12,7,7,0,0,7,10,7,7,0,0,16,7,5,5],[1,0,0,0,0,18,0,1,0,0,0,18,0,0,0,1,0,18,0,0,18,18,1,0,0,0,0,0,0,18,0,0,0,0,1,18],[18,0,1,0,0,0,18,0,0,1,0,0,18,0,0,0,0,1,18,0,0,0,0,0,18,1,0,0,0,0,17,18,0,0,1,1],[0,1,0,18,0,0,1,0,0,18,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,18,17,1,1] >;
C2×3- 1+2.S3 in GAP, Magma, Sage, TeX
C_2\times 3_-^{1+2}.S_3 % in TeX
G:=Group("C2xES-(3,1).S3"); // GroupNames label
G:=SmallGroup(324,80);
// by ID
G=gap.SmallGroup(324,80);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,2090,986,3171,303,453,7564,1096,7781]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^9=c^3=e^2=1,d^3=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^4,d*b*d^-1=b^7*c,e*b*e=b^5*c^-1,d*c*d^-1=b^3*c,c*e=e*c,e*d*e=b^3*d^2>;
// generators/relations
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