metacyclic, supersoluble, monomial
Aliases: C9⋊2F7, C63⋊4C6, D63⋊4C3, C7⋊2(C9⋊C6), C63⋊C3⋊1C2, C21.2(C3×S3), C3.2(C3⋊F7), (C3×C7⋊C3).2S3, SmallGroup(378,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C63 — C9⋊2F7 |
Generators and relations for C9⋊2F7
G = < a,b,c | a9=b7=c6=1, ab=ba, cac-1=a5, cbc-1=b5 >
Character table of C9⋊2F7
| class | 1 | 2 | 3A | 3B | 3C | 6A | 6B | 7 | 9A | 9B | 9C | 21A | 21B | 63A | 63B | 63C | 63D | 63E | 63F | |
| size | 1 | 63 | 2 | 21 | 21 | 63 | 63 | 6 | 6 | 42 | 42 | 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 | trivial |
| ρ2 | 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 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | -1 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | 2 | -1 | ζ65 | ζ6 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | 2 | -1 | ζ6 | ζ65 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | -1 | 6 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ11 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9⋊C6 |
| ρ12 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | -1 | -3 | 0 | 0 | -1 | -1 | 1-√21/2 | 1+√21/2 | 1+√21/2 | 1-√21/2 | 1+√21/2 | 1-√21/2 | orthogonal lifted from C3⋊F7 |
| ρ13 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | -1 | -3 | 0 | 0 | -1 | -1 | 1+√21/2 | 1-√21/2 | 1-√21/2 | 1+√21/2 | 1-√21/2 | 1+√21/2 | orthogonal lifted from C3⋊F7 |
| ρ14 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1+√21/2 | 1-√21/2 | -ζ95ζ76+ζ95ζ75-ζ94ζ76-ζ94ζ75-2ζ94ζ74-ζ94ζ73-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ7 | -ζ95ζ74+ζ95ζ7-2ζ94ζ75-ζ94ζ74-ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ74+ζ92ζ72-ζ9ζ75+ζ9ζ73 | ζ98ζ72-ζ98ζ7+ζ97ζ76-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ9ζ76-ζ9ζ74-2ζ9ζ73-ζ9ζ72-ζ9ζ7-ζ9 | -ζ98ζ75+ζ98ζ73-ζ97ζ74+ζ97ζ72-2ζ95ζ75-ζ95ζ74-ζ95ζ73-ζ95ζ72-ζ95ζ7-ζ95-ζ94ζ74+ζ94ζ7 | -ζ98ζ72+ζ98ζ7-2ζ97ζ76-ζ97ζ75-ζ97ζ74-ζ97ζ72-ζ97ζ7-ζ97-ζ94ζ76+ζ94ζ75+ζ92ζ74-ζ92ζ72 | -ζ98ζ76-ζ98ζ74-2ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ74-ζ94ζ7+ζ92ζ76-ζ92ζ73+ζ9ζ72-ζ9ζ7 | orthogonal faithful |
| ρ15 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1+√21/2 | 1-√21/2 | -ζ98ζ76-ζ98ζ74-2ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ74-ζ94ζ7+ζ92ζ76-ζ92ζ73+ζ9ζ72-ζ9ζ7 | -ζ98ζ72+ζ98ζ7-2ζ97ζ76-ζ97ζ75-ζ97ζ74-ζ97ζ72-ζ97ζ7-ζ97-ζ94ζ76+ζ94ζ75+ζ92ζ74-ζ92ζ72 | -ζ95ζ74+ζ95ζ7-2ζ94ζ75-ζ94ζ74-ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ74+ζ92ζ72-ζ9ζ75+ζ9ζ73 | -ζ95ζ76+ζ95ζ75-ζ94ζ76-ζ94ζ75-2ζ94ζ74-ζ94ζ73-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ7 | ζ98ζ72-ζ98ζ7+ζ97ζ76-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ9ζ76-ζ9ζ74-2ζ9ζ73-ζ9ζ72-ζ9ζ7-ζ9 | -ζ98ζ75+ζ98ζ73-ζ97ζ74+ζ97ζ72-2ζ95ζ75-ζ95ζ74-ζ95ζ73-ζ95ζ72-ζ95ζ7-ζ95-ζ94ζ74+ζ94ζ7 | orthogonal faithful |
| ρ16 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1-√21/2 | 1+√21/2 | -ζ95ζ74+ζ95ζ7-2ζ94ζ75-ζ94ζ74-ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ74+ζ92ζ72-ζ9ζ75+ζ9ζ73 | -ζ98ζ76-ζ98ζ74-2ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ74-ζ94ζ7+ζ92ζ76-ζ92ζ73+ζ9ζ72-ζ9ζ7 | -ζ95ζ76+ζ95ζ75-ζ94ζ76-ζ94ζ75-2ζ94ζ74-ζ94ζ73-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ7 | ζ98ζ72-ζ98ζ7+ζ97ζ76-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ9ζ76-ζ9ζ74-2ζ9ζ73-ζ9ζ72-ζ9ζ7-ζ9 | -ζ98ζ75+ζ98ζ73-ζ97ζ74+ζ97ζ72-2ζ95ζ75-ζ95ζ74-ζ95ζ73-ζ95ζ72-ζ95ζ7-ζ95-ζ94ζ74+ζ94ζ7 | -ζ98ζ72+ζ98ζ7-2ζ97ζ76-ζ97ζ75-ζ97ζ74-ζ97ζ72-ζ97ζ7-ζ97-ζ94ζ76+ζ94ζ75+ζ92ζ74-ζ92ζ72 | orthogonal faithful |
| ρ17 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1-√21/2 | 1+√21/2 | -ζ98ζ72+ζ98ζ7-2ζ97ζ76-ζ97ζ75-ζ97ζ74-ζ97ζ72-ζ97ζ7-ζ97-ζ94ζ76+ζ94ζ75+ζ92ζ74-ζ92ζ72 | -ζ98ζ75+ζ98ζ73-ζ97ζ74+ζ97ζ72-2ζ95ζ75-ζ95ζ74-ζ95ζ73-ζ95ζ72-ζ95ζ7-ζ95-ζ94ζ74+ζ94ζ7 | -ζ98ζ76-ζ98ζ74-2ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ74-ζ94ζ7+ζ92ζ76-ζ92ζ73+ζ9ζ72-ζ9ζ7 | -ζ95ζ74+ζ95ζ7-2ζ94ζ75-ζ94ζ74-ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ74+ζ92ζ72-ζ9ζ75+ζ9ζ73 | -ζ95ζ76+ζ95ζ75-ζ94ζ76-ζ94ζ75-2ζ94ζ74-ζ94ζ73-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ7 | ζ98ζ72-ζ98ζ7+ζ97ζ76-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ9ζ76-ζ9ζ74-2ζ9ζ73-ζ9ζ72-ζ9ζ7-ζ9 | orthogonal faithful |
| ρ18 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1-√21/2 | 1+√21/2 | ζ98ζ72-ζ98ζ7+ζ97ζ76-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ9ζ76-ζ9ζ74-2ζ9ζ73-ζ9ζ72-ζ9ζ7-ζ9 | -ζ95ζ76+ζ95ζ75-ζ94ζ76-ζ94ζ75-2ζ94ζ74-ζ94ζ73-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ7 | -ζ98ζ75+ζ98ζ73-ζ97ζ74+ζ97ζ72-2ζ95ζ75-ζ95ζ74-ζ95ζ73-ζ95ζ72-ζ95ζ7-ζ95-ζ94ζ74+ζ94ζ7 | -ζ98ζ72+ζ98ζ7-2ζ97ζ76-ζ97ζ75-ζ97ζ74-ζ97ζ72-ζ97ζ7-ζ97-ζ94ζ76+ζ94ζ75+ζ92ζ74-ζ92ζ72 | -ζ98ζ76-ζ98ζ74-2ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ74-ζ94ζ7+ζ92ζ76-ζ92ζ73+ζ9ζ72-ζ9ζ7 | -ζ95ζ74+ζ95ζ7-2ζ94ζ75-ζ94ζ74-ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ74+ζ92ζ72-ζ9ζ75+ζ9ζ73 | orthogonal faithful |
| ρ19 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1+√21/2 | 1-√21/2 | -ζ98ζ75+ζ98ζ73-ζ97ζ74+ζ97ζ72-2ζ95ζ75-ζ95ζ74-ζ95ζ73-ζ95ζ72-ζ95ζ7-ζ95-ζ94ζ74+ζ94ζ7 | ζ98ζ72-ζ98ζ7+ζ97ζ76-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ9ζ76-ζ9ζ74-2ζ9ζ73-ζ9ζ72-ζ9ζ7-ζ9 | -ζ98ζ72+ζ98ζ7-2ζ97ζ76-ζ97ζ75-ζ97ζ74-ζ97ζ72-ζ97ζ7-ζ97-ζ94ζ76+ζ94ζ75+ζ92ζ74-ζ92ζ72 | -ζ98ζ76-ζ98ζ74-2ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ74-ζ94ζ7+ζ92ζ76-ζ92ζ73+ζ9ζ72-ζ9ζ7 | -ζ95ζ74+ζ95ζ7-2ζ94ζ75-ζ94ζ74-ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ74+ζ92ζ72-ζ9ζ75+ζ9ζ73 | -ζ95ζ76+ζ95ζ75-ζ94ζ76-ζ94ζ75-2ζ94ζ74-ζ94ζ73-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ7 | orthogonal faithful |
►On 63 points
Generators in S63
(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)
(1 32 12 27 62 53 39)(2 33 13 19 63 54 40)(3 34 14 20 55 46 41)(4 35 15 21 56 47 42)(5 36 16 22 57 48 43)(6 28 17 23 58 49 44)(7 29 18 24 59 50 45)(8 30 10 25 60 51 37)(9 31 11 26 61 52 38)
(2 3 5 9 8 6)(4 7)(10 44 63 46 36 26)(11 37 58 54 34 22)(12 39 62 53 32 27)(13 41 57 52 30 23)(14 43 61 51 28 19)(15 45 56 50 35 24)(16 38 60 49 33 20)(17 40 55 48 31 25)(18 42 59 47 29 21)
G:=sub<Sym(63)| (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), (1,32,12,27,62,53,39)(2,33,13,19,63,54,40)(3,34,14,20,55,46,41)(4,35,15,21,56,47,42)(5,36,16,22,57,48,43)(6,28,17,23,58,49,44)(7,29,18,24,59,50,45)(8,30,10,25,60,51,37)(9,31,11,26,61,52,38), (2,3,5,9,8,6)(4,7)(10,44,63,46,36,26)(11,37,58,54,34,22)(12,39,62,53,32,27)(13,41,57,52,30,23)(14,43,61,51,28,19)(15,45,56,50,35,24)(16,38,60,49,33,20)(17,40,55,48,31,25)(18,42,59,47,29,21)>;
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), (1,32,12,27,62,53,39)(2,33,13,19,63,54,40)(3,34,14,20,55,46,41)(4,35,15,21,56,47,42)(5,36,16,22,57,48,43)(6,28,17,23,58,49,44)(7,29,18,24,59,50,45)(8,30,10,25,60,51,37)(9,31,11,26,61,52,38), (2,3,5,9,8,6)(4,7)(10,44,63,46,36,26)(11,37,58,54,34,22)(12,39,62,53,32,27)(13,41,57,52,30,23)(14,43,61,51,28,19)(15,45,56,50,35,24)(16,38,60,49,33,20)(17,40,55,48,31,25)(18,42,59,47,29,21) );
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)], [(1,32,12,27,62,53,39),(2,33,13,19,63,54,40),(3,34,14,20,55,46,41),(4,35,15,21,56,47,42),(5,36,16,22,57,48,43),(6,28,17,23,58,49,44),(7,29,18,24,59,50,45),(8,30,10,25,60,51,37),(9,31,11,26,61,52,38)], [(2,3,5,9,8,6),(4,7),(10,44,63,46,36,26),(11,37,58,54,34,22),(12,39,62,53,32,27),(13,41,57,52,30,23),(14,43,61,51,28,19),(15,45,56,50,35,24),(16,38,60,49,33,20),(17,40,55,48,31,25),(18,42,59,47,29,21)]])
Matrix representation of C9⋊2F7 ►in GL6(𝔽127)
| 124 | 2 | 121 | 19 | 49 | 109 |
| 125 | 122 | 108 | 102 | 18 | 67 |
| 67 | 49 | 35 | 87 | 18 | 67 |
| 78 | 18 | 40 | 75 | 60 | 78 |
| 92 | 55 | 55 | 90 | 95 | 38 |
| 72 | 37 | 37 | 92 | 89 | 57 |
| 1 | 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 | 0 | 1 |
| 39 | 77 | 38 | 77 | 38 | 77 |
| 50 | 89 | 50 | 88 | 50 | 88 |
| 126 | 0 | 0 | 0 | 126 | 0 |
| 0 | 126 | 0 | 0 | 0 | 126 |
| 39 | 77 | 0 | 0 | 38 | 77 |
| 38 | 88 | 0 | 0 | 39 | 89 |
| 0 | 0 | 126 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 89 | 50 | 1 | 0 | 1 | 0 |
| 88 | 38 | 126 | 126 | 126 | 126 |
G:=sub<GL(6,GF(127))| [124,125,67,78,92,72,2,122,49,18,55,37,121,108,35,40,55,37,19,102,87,75,90,92,49,18,18,60,95,89,109,67,67,78,38,57],[1,0,39,50,126,0,0,1,77,89,0,126,1,0,38,50,0,0,0,1,77,88,0,0,1,0,38,50,126,0,0,1,77,88,0,126],[39,38,0,0,89,88,77,88,0,0,50,38,0,0,126,1,1,126,0,0,0,1,0,126,38,39,0,0,1,126,77,89,0,0,0,126] >;
C9⋊2F7 in GAP, Magma, Sage, TeX
C_9\rtimes_2F_7
% in TeX
G:=Group("C9:2F7"); // GroupNames label
G:=SmallGroup(378,19);
// by ID
G=gap.SmallGroup(378,19);
# by ID
G:=PCGroup([5,-2,-3,-3,-7,-3,3962,997,327,2163,368,6304]);
// Polycyclic
G:=Group<a,b,c|a^9=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^5>;
// generators/relations
Export
Subgroup lattice of C9⋊2F7 in TeX
Character table of C9⋊2F7 in TeX