metacyclic, supersoluble, monomial, Z-group
Aliases: D65⋊C3, C65⋊1C6, C13⋊C3⋊D5, C5⋊(C13⋊C6), C13⋊(C3×D5), (C5×C13⋊C3)⋊1C2, SmallGroup(390,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C65 — D65⋊C3 |
Generators and relations for D65⋊C3
G = < a,b,c | a65=b2=c3=1, bab=a-1, cac-1=a61, cbc-1=a60b >
Character table of D65⋊C3
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 13A | 13B | 15A | 15B | 15C | 15D | 65A | 65B | 65C | 65D | 65E | 65F | 65G | 65H | |
| size | 1 | 65 | 13 | 13 | 2 | 2 | 65 | 65 | 6 | 6 | 26 | 26 | 26 | 26 | 6 | 6 | 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 | trivial |
| ρ2 | 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 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ65 | ζ6 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ6 | ζ65 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ8 | 2 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ9 | 2 | 0 | -1-√-3 | -1+√-3 | -1+√5/2 | -1-√5/2 | 0 | 0 | 2 | 2 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | complex lifted from C3×D5 |
| ρ10 | 2 | 0 | -1-√-3 | -1+√-3 | -1-√5/2 | -1+√5/2 | 0 | 0 | 2 | 2 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | complex lifted from C3×D5 |
| ρ11 | 2 | 0 | -1+√-3 | -1-√-3 | -1+√5/2 | -1-√5/2 | 0 | 0 | 2 | 2 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | complex lifted from C3×D5 |
| ρ12 | 2 | 0 | -1+√-3 | -1-√-3 | -1-√5/2 | -1+√5/2 | 0 | 0 | 2 | 2 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | complex lifted from C3×D5 |
| ρ13 | 6 | 0 | 0 | 0 | 6 | 6 | 0 | 0 | -1+√13/2 | -1-√13/2 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | -1-√13/2 | -1+√13/2 | -1-√13/2 | -1+√13/2 | -1-√13/2 | -1+√13/2 | orthogonal lifted from C13⋊C6 |
| ρ14 | 6 | 0 | 0 | 0 | 6 | 6 | 0 | 0 | -1-√13/2 | -1+√13/2 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | -1+√13/2 | -1-√13/2 | -1+√13/2 | -1-√13/2 | -1+√13/2 | -1-√13/2 | orthogonal lifted from C13⋊C6 |
| ρ15 | 6 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | -1-√13/2 | -1+√13/2 | 0 | 0 | 0 | 0 | ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 | ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 | ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 | ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 | ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 | ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 | ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 | ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 | orthogonal faithful |
| ρ16 | 6 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | -1+√13/2 | -1-√13/2 | 0 | 0 | 0 | 0 | ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 | ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 | ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 | ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 | ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 | ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 | ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 | ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 | orthogonal faithful |
| ρ17 | 6 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | -1+√13/2 | -1-√13/2 | 0 | 0 | 0 | 0 | ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 | ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 | ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 | ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 | ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 | ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 | ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 | ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 | orthogonal faithful |
| ρ18 | 6 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | -1-√13/2 | -1+√13/2 | 0 | 0 | 0 | 0 | ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 | ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 | ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 | ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 | ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 | ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 | ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 | ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 | orthogonal faithful |
| ρ19 | 6 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | -1-√13/2 | -1+√13/2 | 0 | 0 | 0 | 0 | ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 | ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 | ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 | ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 | ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 | ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 | ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 | ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 | orthogonal faithful |
| ρ20 | 6 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | -1+√13/2 | -1-√13/2 | 0 | 0 | 0 | 0 | ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 | ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 | ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 | ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 | ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 | ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 | ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 | ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 | orthogonal faithful |
| ρ21 | 6 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | -1-√13/2 | -1+√13/2 | 0 | 0 | 0 | 0 | ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 | ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 | ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 | ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 | ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 | ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 | ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 | ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 | orthogonal faithful |
| ρ22 | 6 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | -1+√13/2 | -1-√13/2 | 0 | 0 | 0 | 0 | ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 | ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 | ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 | ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 | ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 | ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 | ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 | ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 | orthogonal faithful |
►On 65 points
Generators in S65
(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 57 58 59 60 61 62 63 64 65)
(1 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)
(2 17 62)(3 33 58)(4 49 54)(5 65 50)(6 16 46)(7 32 42)(8 48 38)(9 64 34)(10 15 30)(11 31 26)(12 47 22)(13 63 18)(19 29 59)(20 45 55)(21 61 51)(23 28 43)(24 44 39)(25 60 35)(36 41 56)(37 57 52)
G:=sub<Sym(65)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34), (2,17,62)(3,33,58)(4,49,54)(5,65,50)(6,16,46)(7,32,42)(8,48,38)(9,64,34)(10,15,30)(11,31,26)(12,47,22)(13,63,18)(19,29,59)(20,45,55)(21,61,51)(23,28,43)(24,44,39)(25,60,35)(36,41,56)(37,57,52)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34), (2,17,62)(3,33,58)(4,49,54)(5,65,50)(6,16,46)(7,32,42)(8,48,38)(9,64,34)(10,15,30)(11,31,26)(12,47,22)(13,63,18)(19,29,59)(20,45,55)(21,61,51)(23,28,43)(24,44,39)(25,60,35)(36,41,56)(37,57,52) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65)], [(1,65),(2,64),(3,63),(4,62),(5,61),(6,60),(7,59),(8,58),(9,57),(10,56),(11,55),(12,54),(13,53),(14,52),(15,51),(16,50),(17,49),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42),(25,41),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34)], [(2,17,62),(3,33,58),(4,49,54),(5,65,50),(6,16,46),(7,32,42),(8,48,38),(9,64,34),(10,15,30),(11,31,26),(12,47,22),(13,63,18),(19,29,59),(20,45,55),(21,61,51),(23,28,43),(24,44,39),(25,60,35),(36,41,56),(37,57,52)]])
Matrix representation of D65⋊C3 ►in GL6(𝔽1171)
| 152 | 39 | 239 | 844 | 391 | 126 |
| 1045 | 592 | 958 | 805 | 592 | 831 |
| 340 | 880 | 101 | 453 | 314 | 427 |
| 744 | 530 | 26 | 718 | 770 | 504 |
| 667 | 162 | 693 | 1119 | 881 | 188 |
| 983 | 896 | 957 | 1110 | 743 | 1110 |
| 152 | 39 | 239 | 844 | 391 | 126 |
| 1146 | 1106 | 932 | 87 | 62 | 1019 |
| 870 | 982 | 997 | 112 | 783 | 25 |
| 61 | 428 | 61 | 214 | 275 | 301 |
| 957 | 1110 | 804 | 153 | 830 | 1110 |
| 214 | 61 | 214 | 87 | 214 | 214 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 742 | 311 | 741 | 741 | 311 | 742 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 429 | 861 | 1169 | 431 | 428 | 430 |
| 1170 | 431 | 1169 | 432 | 1169 | 431 |
G:=sub<GL(6,GF(1171))| [152,1045,340,744,667,983,39,592,880,530,162,896,239,958,101,26,693,957,844,805,453,718,1119,1110,391,592,314,770,881,743,126,831,427,504,188,1110],[152,1146,870,61,957,214,39,1106,982,428,1110,61,239,932,997,61,804,214,844,87,112,214,153,87,391,62,783,275,830,214,126,1019,25,301,1110,214],[1,742,0,0,429,1170,0,311,0,1,861,431,0,741,0,0,1169,1169,0,741,0,0,431,432,0,311,0,0,428,1169,0,742,1,0,430,431] >;
D65⋊C3 in GAP, Magma, Sage, TeX
D_{65}\rtimes C_3 % in TeX
G:=Group("D65:C3"); // GroupNames label
G:=SmallGroup(390,3);
// by ID
G=gap.SmallGroup(390,3);
# by ID
G:=PCGroup([4,-2,-3,-5,-13,290,5763,727]);
// Polycyclic
G:=Group<a,b,c|a^65=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^61,c*b*c^-1=a^60*b>;
// generators/relations
Export
Subgroup lattice of D65⋊C3 in TeX
Character table of D65⋊C3 in TeX