direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C17⋊C4, D51⋊C4, D17.1D6, C17⋊(C4×S3), C51⋊(C2×C4), C51⋊C4⋊C2, (S3×C17)⋊C4, (S3×D17).C2, (C3×D17).C22, (C3×C17⋊C4)⋊C2, C3⋊1(C2×C17⋊C4), SmallGroup(408,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — S3×C17⋊C4 |
Generators and relations for S3×C17⋊C4
G = < a,b,c,d | a3=b2=c17=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Character table of S3×C17⋊C4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | 51A | 51B | 51C | 51D | |
| size | 1 | 3 | 17 | 51 | 2 | 17 | 17 | 51 | 51 | 34 | 34 | 34 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -1 | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | i | -i | i | -i | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | i | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 0 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | 1 | 1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 0 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 0 | -2 | 0 | -1 | -2i | 2i | 0 | 0 | 1 | i | -i | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | complex lifted from C4×S3 |
| ρ12 | 2 | 0 | -2 | 0 | -1 | 2i | -2i | 0 | 0 | 1 | -i | i | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | complex lifted from C4×S3 |
| ρ13 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | orthogonal lifted from C17⋊C4 |
| ρ14 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | orthogonal lifted from C17⋊C4 |
| ρ15 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | orthogonal lifted from C2×C17⋊C4 |
| ρ16 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | orthogonal lifted from C17⋊C4 |
| ρ17 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | orthogonal lifted from C2×C17⋊C4 |
| ρ18 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | orthogonal lifted from C17⋊C4 |
| ρ19 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | orthogonal lifted from C2×C17⋊C4 |
| ρ20 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | orthogonal lifted from C2×C17⋊C4 |
| ρ21 | 8 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1711+2ζ1710+2ζ177+2ζ176 | 2ζ1715+2ζ179+2ζ178+2ζ172 | 2ζ1714+2ζ1712+2ζ175+2ζ173 | 2ζ1716+2ζ1713+2ζ174+2ζ17 | 0 | 0 | 0 | 0 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | orthogonal faithful |
| ρ22 | 8 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1715+2ζ179+2ζ178+2ζ172 | 2ζ1714+2ζ1712+2ζ175+2ζ173 | 2ζ1716+2ζ1713+2ζ174+2ζ17 | 2ζ1711+2ζ1710+2ζ177+2ζ176 | 0 | 0 | 0 | 0 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | orthogonal faithful |
| ρ23 | 8 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1714+2ζ1712+2ζ175+2ζ173 | 2ζ1716+2ζ1713+2ζ174+2ζ17 | 2ζ1711+2ζ1710+2ζ177+2ζ176 | 2ζ1715+2ζ179+2ζ178+2ζ172 | 0 | 0 | 0 | 0 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | orthogonal faithful |
| ρ24 | 8 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ1716+2ζ1713+2ζ174+2ζ17 | 2ζ1711+2ζ1710+2ζ177+2ζ176 | 2ζ1715+2ζ179+2ζ178+2ζ172 | 2ζ1714+2ζ1712+2ζ175+2ζ173 | 0 | 0 | 0 | 0 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | orthogonal faithful |
►On 51 points
Generators in S51
(1 18 35)(2 19 36)(3 20 37)(4 21 38)(5 22 39)(6 23 40)(7 24 41)(8 25 42)(9 26 43)(10 27 44)(11 28 45)(12 29 46)(13 30 47)(14 31 48)(15 32 49)(16 33 50)(17 34 51)
(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)
(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)
(2 14 17 5)(3 10 16 9)(4 6 15 13)(7 11 12 8)(19 31 34 22)(20 27 33 26)(21 23 32 30)(24 28 29 25)(36 48 51 39)(37 44 50 43)(38 40 49 47)(41 45 46 42)
G:=sub<Sym(51)| (1,18,35)(2,19,36)(3,20,37)(4,21,38)(5,22,39)(6,23,40)(7,24,41)(8,25,42)(9,26,43)(10,27,44)(11,28,45)(12,29,46)(13,30,47)(14,31,48)(15,32,49)(16,33,50)(17,34,51), (18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51), (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), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)(36,48,51,39)(37,44,50,43)(38,40,49,47)(41,45,46,42)>;
G:=Group( (1,18,35)(2,19,36)(3,20,37)(4,21,38)(5,22,39)(6,23,40)(7,24,41)(8,25,42)(9,26,43)(10,27,44)(11,28,45)(12,29,46)(13,30,47)(14,31,48)(15,32,49)(16,33,50)(17,34,51), (18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51), (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), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)(36,48,51,39)(37,44,50,43)(38,40,49,47)(41,45,46,42) );
G=PermutationGroup([[(1,18,35),(2,19,36),(3,20,37),(4,21,38),(5,22,39),(6,23,40),(7,24,41),(8,25,42),(9,26,43),(10,27,44),(11,28,45),(12,29,46),(13,30,47),(14,31,48),(15,32,49),(16,33,50),(17,34,51)], [(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51)], [(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)], [(2,14,17,5),(3,10,16,9),(4,6,15,13),(7,11,12,8),(19,31,34,22),(20,27,33,26),(21,23,32,30),(24,28,29,25),(36,48,51,39),(37,44,50,43),(38,40,49,47),(41,45,46,42)]])
Matrix representation of S3×C17⋊C4 ►in GL6(𝔽409)
| 407 | 329 | 0 | 0 | 0 | 0 |
| 271 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 138 | 408 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 72 | 17 | 408 |
| 0 | 0 | 330 | 89 | 288 | 368 |
| 0 | 0 | 365 | 234 | 322 | 366 |
| 0 | 0 | 383 | 305 | 340 | 365 |
| 143 | 0 | 0 | 0 | 0 | 0 |
| 0 | 143 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 4 | 53 | 162 |
| 0 | 0 | 0 | 208 | 388 | 129 |
| 0 | 0 | 31 | 62 | 373 | 113 |
| 0 | 0 | 63 | 22 | 153 | 219 |
G:=sub<GL(6,GF(409))| [407,271,0,0,0,0,329,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,138,0,0,0,0,0,408,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,330,365,383,0,0,72,89,234,305,0,0,17,288,322,340,0,0,408,368,366,365],[143,0,0,0,0,0,0,143,0,0,0,0,0,0,18,0,31,63,0,0,4,208,62,22,0,0,53,388,373,153,0,0,162,129,113,219] >;
S3×C17⋊C4 in GAP, Magma, Sage, TeX
S_3\times C_{17}\rtimes C_4 % in TeX
G:=Group("S3xC17:C4"); // GroupNames label
G:=SmallGroup(408,35);
// by ID
G=gap.SmallGroup(408,35);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,20,168,7804,2414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^17=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations
Export
Subgroup lattice of S3×C17⋊C4 in TeX
Character table of S3×C17⋊C4 in TeX