metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic21, C6.D7, C3⋊Dic7, C7⋊Dic3, C21⋊1C4, C14.S3, C2.D21, C42.1C2, SmallGroup(84,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — Dic21 |
Generators and relations for Dic21
G = < a,b | a42=1, b2=a21, bab-1=a-1 >
Character table of Dic21
| class | 1 | 2 | 3 | 4A | 4B | 6 | 7A | 7B | 7C | 14A | 14B | 14C | 21A | 21B | 21C | 21D | 21E | 21F | 42A | 42B | 42C | 42D | 42E | 42F | |
| size | 1 | 1 | 2 | 21 | 21 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | -i | i | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | i | -i | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | -1 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -1 | 0 | 0 | -1 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ32ζ74-ζ32ζ73-ζ73 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ32ζ75+ζ32ζ72-ζ75 | orthogonal lifted from D21 |
| ρ7 | 2 | 2 | 2 | 0 | 0 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ8 | 2 | 2 | 2 | 0 | 0 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ10 | 2 | 2 | -1 | 0 | 0 | -1 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ32ζ74-ζ32ζ73-ζ73 | orthogonal lifted from D21 |
| ρ11 | 2 | 2 | -1 | 0 | 0 | -1 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | orthogonal lifted from D21 |
| ρ12 | 2 | 2 | -1 | 0 | 0 | -1 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ32ζ74-ζ32ζ73-ζ73 | ζ3ζ76-ζ3ζ7-ζ7 | orthogonal lifted from D21 |
| ρ13 | 2 | 2 | -1 | 0 | 0 | -1 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ3ζ75+ζ3ζ72-ζ75 | orthogonal lifted from D21 |
| ρ14 | 2 | 2 | -1 | 0 | 0 | -1 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ3ζ76-ζ3ζ7-ζ7 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ3ζ76-ζ3ζ7-ζ7 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ76+ζ3ζ7-ζ76 | orthogonal lifted from D21 |
| ρ15 | 2 | -2 | -1 | 0 | 0 | 1 | 2 | 2 | 2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ16 | 2 | -2 | 2 | 0 | 0 | -2 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | symplectic lifted from Dic7, Schur index 2 |
| ρ17 | 2 | -2 | 2 | 0 | 0 | -2 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | symplectic lifted from Dic7, Schur index 2 |
| ρ18 | 2 | -2 | -1 | 0 | 0 | 1 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ3ζ76-ζ3ζ7-ζ7 | ζ3ζ75-ζ3ζ72+ζ75 | -ζ3ζ75+ζ3ζ72+ζ72 | -ζ32ζ76+ζ32ζ7+ζ7 | ζ32ζ74-ζ32ζ73+ζ74 | ζ3ζ74-ζ3ζ73+ζ74 | ζ32ζ76-ζ32ζ7+ζ76 | symplectic faithful, Schur index 2 |
| ρ19 | 2 | -2 | -1 | 0 | 0 | 1 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ3ζ74-ζ3ζ73+ζ74 | ζ32ζ74-ζ32ζ73+ζ74 | -ζ3ζ75+ζ3ζ72+ζ72 | ζ32ζ76-ζ32ζ7+ζ76 | -ζ32ζ76+ζ32ζ7+ζ7 | ζ3ζ75-ζ3ζ72+ζ75 | symplectic faithful, Schur index 2 |
| ρ20 | 2 | -2 | -1 | 0 | 0 | 1 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ32ζ76-ζ32ζ7+ζ76 | -ζ32ζ76+ζ32ζ7+ζ7 | ζ3ζ74-ζ3ζ73+ζ74 | -ζ3ζ75+ζ3ζ72+ζ72 | ζ3ζ75-ζ3ζ72+ζ75 | ζ32ζ74-ζ32ζ73+ζ74 | symplectic faithful, Schur index 2 |
| ρ21 | 2 | -2 | -1 | 0 | 0 | 1 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ32ζ74-ζ32ζ73-ζ73 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ32ζ75+ζ32ζ72-ζ75 | ζ32ζ74-ζ32ζ73+ζ74 | ζ3ζ74-ζ3ζ73+ζ74 | ζ3ζ75-ζ3ζ72+ζ75 | -ζ32ζ76+ζ32ζ7+ζ7 | ζ32ζ76-ζ32ζ7+ζ76 | -ζ3ζ75+ζ3ζ72+ζ72 | symplectic faithful, Schur index 2 |
| ρ22 | 2 | -2 | -1 | 0 | 0 | 1 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ3ζ76+ζ3ζ7-ζ76 | ζ3ζ76-ζ3ζ7-ζ7 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | -ζ3ζ75+ζ3ζ72-ζ75 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ32ζ76+ζ32ζ7+ζ7 | ζ32ζ76-ζ32ζ7+ζ76 | ζ32ζ74-ζ32ζ73+ζ74 | ζ3ζ75-ζ3ζ72+ζ75 | -ζ3ζ75+ζ3ζ72+ζ72 | ζ3ζ74-ζ3ζ73+ζ74 | symplectic faithful, Schur index 2 |
| ρ23 | 2 | -2 | 2 | 0 | 0 | -2 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | symplectic lifted from Dic7, Schur index 2 |
| ρ24 | 2 | -2 | -1 | 0 | 0 | 1 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ32ζ75+ζ32ζ72-ζ75 | -ζ3ζ75+ζ3ζ72-ζ75 | -ζ32ζ74+ζ32ζ73-ζ74 | ζ3ζ76-ζ3ζ7-ζ7 | ζ32ζ74-ζ32ζ73-ζ73 | -ζ3ζ76+ζ3ζ7-ζ76 | -ζ3ζ75+ζ3ζ72+ζ72 | ζ3ζ75-ζ3ζ72+ζ75 | ζ32ζ76-ζ32ζ7+ζ76 | ζ3ζ74-ζ3ζ73+ζ74 | ζ32ζ74-ζ32ζ73+ζ74 | -ζ32ζ76+ζ32ζ7+ζ7 | symplectic faithful, Schur index 2 |
►Regular action on 84 points
Generators in S84
(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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)
(1 44 22 65)(2 43 23 64)(3 84 24 63)(4 83 25 62)(5 82 26 61)(6 81 27 60)(7 80 28 59)(8 79 29 58)(9 78 30 57)(10 77 31 56)(11 76 32 55)(12 75 33 54)(13 74 34 53)(14 73 35 52)(15 72 36 51)(16 71 37 50)(17 70 38 49)(18 69 39 48)(19 68 40 47)(20 67 41 46)(21 66 42 45)
G:=sub<Sym(84)| (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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,44,22,65)(2,43,23,64)(3,84,24,63)(4,83,25,62)(5,82,26,61)(6,81,27,60)(7,80,28,59)(8,79,29,58)(9,78,30,57)(10,77,31,56)(11,76,32,55)(12,75,33,54)(13,74,34,53)(14,73,35,52)(15,72,36,51)(16,71,37,50)(17,70,38,49)(18,69,39,48)(19,68,40,47)(20,67,41,46)(21,66,42,45)>;
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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,44,22,65)(2,43,23,64)(3,84,24,63)(4,83,25,62)(5,82,26,61)(6,81,27,60)(7,80,28,59)(8,79,29,58)(9,78,30,57)(10,77,31,56)(11,76,32,55)(12,75,33,54)(13,74,34,53)(14,73,35,52)(15,72,36,51)(16,71,37,50)(17,70,38,49)(18,69,39,48)(19,68,40,47)(20,67,41,46)(21,66,42,45) );
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),(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)], [(1,44,22,65),(2,43,23,64),(3,84,24,63),(4,83,25,62),(5,82,26,61),(6,81,27,60),(7,80,28,59),(8,79,29,58),(9,78,30,57),(10,77,31,56),(11,76,32,55),(12,75,33,54),(13,74,34,53),(14,73,35,52),(15,72,36,51),(16,71,37,50),(17,70,38,49),(18,69,39,48),(19,68,40,47),(20,67,41,46),(21,66,42,45)]])
Dic21 is a maximal subgroup of
Dic3×D7 S3×Dic7 C21⋊D4 C21⋊Q8 Dic42 C4×D21 C21⋊7D4 Dic63 C6.F7 C3⋊Dic21 Q8.D21 A4⋊Dic7 Dic105 C5⋊Dic21
Dic21 is a maximal quotient of
C21⋊C8 Dic63 C3⋊Dic21 A4⋊Dic7 Dic105 C5⋊Dic21
Matrix representation of Dic21 ►in GL2(𝔽41) generated by
| 26 | 2 |
| 2 | 27 |
| 9 | 25 |
| 0 | 32 |
G:=sub<GL(2,GF(41))| [26,2,2,27],[9,0,25,32] >;
Dic21 in GAP, Magma, Sage, TeX
{\rm Dic}_{21} % in TeX
G:=Group("Dic21"); // GroupNames label
G:=SmallGroup(84,5);
// by ID
G=gap.SmallGroup(84,5);
# by ID
G:=PCGroup([4,-2,-2,-3,-7,8,98,1155]);
// Polycyclic
G:=Group<a,b|a^42=1,b^2=a^21,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic21 in TeX
Character table of Dic21 in TeX