metacyclic, supersoluble, monomial
Aliases: D56⋊C3, C8⋊1F7, C56⋊1C6, D28⋊4C6, C7⋊C3⋊1D8, C7⋊1(C3×D8), C4⋊F7⋊4C2, C4.9(C2×F7), C28.9(C2×C6), C14.2(C3×D4), C2.4(C4⋊F7), (C8×C7⋊C3)⋊1C2, (C2×C7⋊C3).2D4, (C4×C7⋊C3).9C22, SmallGroup(336,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56⋊C3
G = < a,b,c | a56=b2=c3=1, bab=a-1, cac-1=a9, cbc-1=a8b >
28C2
28C2
7C3
14C22
14C22
7C6
28C6
28C6
4D7
4D7
7D4
7D4
7C12
14C2×C6
14C2×C6
2D14
2D14
4F7
4F7
7D8
7C24
Character table of D56⋊C3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 7 | 8A | 8B | 12A | 12B | 14 | 24A | 24B | 24C | 24D | 28A | 28B | 56A | 56B | 56C | 56D | |
| size | 1 | 1 | 28 | 28 | 7 | 7 | 2 | 7 | 7 | 28 | 28 | 28 | 28 | 6 | 2 | 2 | 14 | 14 | 6 | 14 | 14 | 14 | 14 | 6 | 6 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ65 | ζ32 | ζ6 | ζ3 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ6 | ζ32 | ζ65 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ7 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ8 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ6 | ζ3 | ζ65 | ζ32 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ9 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ65 | ζ3 | ζ6 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ12 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | 0 | 0 | -2 | √2 | -√2 | √2 | -√2 | 0 | 0 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | 0 | 0 | -2 | -√2 | √2 | -√2 | √2 | 0 | 0 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 1-√-3 | 1+√-3 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ17 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 1+√-3 | 1-√-3 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ18 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | 0 | 0 | -2 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ32+ζ85ζ32 | 0 | 0 | -√2 | √2 | √2 | -√2 | complex lifted from C3×D8 |
| ρ19 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | 0 | 0 | -2 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ32+ζ8ζ32 | 0 | 0 | √2 | -√2 | -√2 | √2 | complex lifted from C3×D8 |
| ρ20 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | 0 | 0 | -2 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ3+ζ8ζ3 | 0 | 0 | √2 | -√2 | -√2 | √2 | complex lifted from C3×D8 |
| ρ21 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | 0 | 0 | -2 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ3+ζ85ζ3 | 0 | 0 | -√2 | √2 | √2 | -√2 | complex lifted from C3×D8 |
| ρ22 | 6 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 6 | 6 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ23 | 6 | 6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -6 | -6 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F7 |
| ρ24 | 6 | 6 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | √7 | √7 | -√7 | -√7 | orthogonal lifted from C4⋊F7 |
| ρ25 | 6 | 6 | 0 | 0 | 0 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -√7 | -√7 | √7 | √7 | orthogonal lifted from C4⋊F7 |
| ρ26 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 3√2 | -3√2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | √7 | -√7 | ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ83+ζ8ζ74+ζ8ζ72+ζ8ζ7 | ζ83ζ76+ζ83ζ75+ζ83ζ73+ζ8ζ76+ζ8ζ75+ζ8ζ73+ζ8 | ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ87+ζ85ζ76+ζ85ζ75+ζ85ζ73 | ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ85ζ74+ζ85ζ72+ζ85ζ7+ζ85 | orthogonal faithful |
| ρ27 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 3√2 | -3√2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -√7 | √7 | ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ85ζ74+ζ85ζ72+ζ85ζ7+ζ85 | ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ87+ζ85ζ76+ζ85ζ75+ζ85ζ73 | ζ83ζ76+ζ83ζ75+ζ83ζ73+ζ8ζ76+ζ8ζ75+ζ8ζ73+ζ8 | ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ83+ζ8ζ74+ζ8ζ72+ζ8ζ7 | orthogonal faithful |
| ρ28 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -3√2 | 3√2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -√7 | √7 | ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ87+ζ85ζ76+ζ85ζ75+ζ85ζ73 | ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ85ζ74+ζ85ζ72+ζ85ζ7+ζ85 | ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ83+ζ8ζ74+ζ8ζ72+ζ8ζ7 | ζ83ζ76+ζ83ζ75+ζ83ζ73+ζ8ζ76+ζ8ζ75+ζ8ζ73+ζ8 | orthogonal faithful |
| ρ29 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -3√2 | 3√2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | √7 | -√7 | ζ83ζ76+ζ83ζ75+ζ83ζ73+ζ8ζ76+ζ8ζ75+ζ8ζ73+ζ8 | ζ83ζ74+ζ83ζ72+ζ83ζ7+ζ83+ζ8ζ74+ζ8ζ72+ζ8ζ7 | ζ87ζ74+ζ87ζ72+ζ87ζ7+ζ85ζ74+ζ85ζ72+ζ85ζ7+ζ85 | ζ87ζ76+ζ87ζ75+ζ87ζ73+ζ87+ζ85ζ76+ζ85ζ75+ζ85ζ73 | orthogonal faithful |
►On 56 points
Generators in S56
(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 55 56)
(1 7)(2 6)(3 5)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)
(2 26 10)(3 51 19)(4 20 28)(5 45 37)(6 14 46)(7 39 55)(9 33 17)(11 27 35)(12 52 44)(13 21 53)(16 40 24)(18 34 42)(23 47 31)(25 41 49)(30 54 38)(32 48 56)
G:=sub<Sym(56)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33), (2,26,10)(3,51,19)(4,20,28)(5,45,37)(6,14,46)(7,39,55)(9,33,17)(11,27,35)(12,52,44)(13,21,53)(16,40,24)(18,34,42)(23,47,31)(25,41,49)(30,54,38)(32,48,56)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33), (2,26,10)(3,51,19)(4,20,28)(5,45,37)(6,14,46)(7,39,55)(9,33,17)(11,27,35)(12,52,44)(13,21,53)(16,40,24)(18,34,42)(23,47,31)(25,41,49)(30,54,38)(32,48,56) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)], [(1,7),(2,6),(3,5),(8,56),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33)], [(2,26,10),(3,51,19),(4,20,28),(5,45,37),(6,14,46),(7,39,55),(9,33,17),(11,27,35),(12,52,44),(13,21,53),(16,40,24),(18,34,42),(23,47,31),(25,41,49),(30,54,38),(32,48,56)]])
Matrix representation of D56⋊C3 ►in GL8(𝔽337)
| 0 | 100 | 0 | 0 | 0 | 0 | 0 | 0 |
| 246 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 336 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 336 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 336 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 336 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 336 | 1 |
| 336 | 241 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 336 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 336 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 336 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 336 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 336 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 336 |
| 208 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 208 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(337))| [0,246,0,0,0,0,0,0,100,26,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,1,1,1,1,1,1],[336,0,0,0,0,0,0,0,241,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,336,336,336,336,336,336],[208,0,0,0,0,0,0,0,0,208,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0] >;
D56⋊C3 in GAP, Magma, Sage, TeX
D_{56}\rtimes C_3 % in TeX
G:=Group("D56:C3"); // GroupNames label
G:=SmallGroup(336,10);
// by ID
G=gap.SmallGroup(336,10);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,223,867,69,10373,1745]);
// Polycyclic
G:=Group<a,b,c|a^56=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^8*b>;
// generators/relations
Export
Subgroup lattice of D56⋊C3 in TeX
Character table of D56⋊C3 in TeX