metacyclic, supersoluble, monomial
Aliases: C9⋊1F7, C63⋊3C6, D63⋊3C3, C7⋊1(C9⋊C6), C63⋊3C3⋊1C2, C21.1(C3×S3), C3.1(C3⋊F7), (C3×C7⋊C3).1S3, SmallGroup(378,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C63 — C9⋊F7 |
Generators and relations for C9⋊F7
G = < a,b,c | a9=b7=c6=1, ab=ba, cac-1=a2, cbc-1=b5 >
Character table of C9⋊F7
| 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 | -ζ95ζ72+ζ95ζ7-ζ94ζ75-ζ94ζ74-2ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94+ζ92ζ74-ζ92ζ72+ζ9ζ75-ζ9ζ73 | -ζ95ζ76+ζ95ζ73-ζ94ζ76-ζ94ζ75-ζ94ζ73-2ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ75-ζ9ζ72+ζ9ζ7 | -ζ98ζ76-2ζ98ζ75-ζ98ζ74-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ72-ζ94ζ7+ζ92ζ76-ζ92ζ75+ζ9ζ74-ζ9ζ7 | ζ97ζ75-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ92ζ76-ζ92ζ75-ζ92ζ74-ζ92ζ73-2ζ92ζ7-ζ92+ζ9ζ76-ζ9ζ73 | -2ζ98ζ76-ζ98ζ74-ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98-ζ94ζ74+ζ94ζ7-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ72 | ζ98ζ74-ζ98ζ72-ζ97ζ75-ζ97ζ74-2ζ97ζ73-ζ97ζ72-ζ97ζ7-ζ97+ζ94ζ75-ζ94ζ73-ζ92ζ72+ζ92ζ7 | orthogonal faithful |
| ρ15 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1+√21/2 | 1-√21/2 | ζ97ζ75-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ92ζ76-ζ92ζ75-ζ92ζ74-ζ92ζ73-2ζ92ζ7-ζ92+ζ9ζ76-ζ9ζ73 | -ζ98ζ76-2ζ98ζ75-ζ98ζ74-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ72-ζ94ζ7+ζ92ζ76-ζ92ζ75+ζ9ζ74-ζ9ζ7 | -2ζ98ζ76-ζ98ζ74-ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98-ζ94ζ74+ζ94ζ7-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ72 | ζ98ζ74-ζ98ζ72-ζ97ζ75-ζ97ζ74-2ζ97ζ73-ζ97ζ72-ζ97ζ7-ζ97+ζ94ζ75-ζ94ζ73-ζ92ζ72+ζ92ζ7 | -ζ95ζ76+ζ95ζ73-ζ94ζ76-ζ94ζ75-ζ94ζ73-2ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ75-ζ9ζ72+ζ9ζ7 | -ζ95ζ72+ζ95ζ7-ζ94ζ75-ζ94ζ74-2ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94+ζ92ζ74-ζ92ζ72+ζ9ζ75-ζ9ζ73 | orthogonal faithful |
| ρ16 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1-√21/2 | 1+√21/2 | -2ζ98ζ76-ζ98ζ74-ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98-ζ94ζ74+ζ94ζ7-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ72 | ζ97ζ75-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ92ζ76-ζ92ζ75-ζ92ζ74-ζ92ζ73-2ζ92ζ7-ζ92+ζ9ζ76-ζ9ζ73 | ζ98ζ74-ζ98ζ72-ζ97ζ75-ζ97ζ74-2ζ97ζ73-ζ97ζ72-ζ97ζ7-ζ97+ζ94ζ75-ζ94ζ73-ζ92ζ72+ζ92ζ7 | -ζ95ζ76+ζ95ζ73-ζ94ζ76-ζ94ζ75-ζ94ζ73-2ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ75-ζ9ζ72+ζ9ζ7 | -ζ95ζ72+ζ95ζ7-ζ94ζ75-ζ94ζ74-2ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94+ζ92ζ74-ζ92ζ72+ζ9ζ75-ζ9ζ73 | -ζ98ζ76-2ζ98ζ75-ζ98ζ74-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ72-ζ94ζ7+ζ92ζ76-ζ92ζ75+ζ9ζ74-ζ9ζ7 | orthogonal faithful |
| ρ17 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1+√21/2 | 1-√21/2 | ζ98ζ74-ζ98ζ72-ζ97ζ75-ζ97ζ74-2ζ97ζ73-ζ97ζ72-ζ97ζ7-ζ97+ζ94ζ75-ζ94ζ73-ζ92ζ72+ζ92ζ7 | -2ζ98ζ76-ζ98ζ74-ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98-ζ94ζ74+ζ94ζ7-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ72 | -ζ95ζ76+ζ95ζ73-ζ94ζ76-ζ94ζ75-ζ94ζ73-2ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ75-ζ9ζ72+ζ9ζ7 | -ζ95ζ72+ζ95ζ7-ζ94ζ75-ζ94ζ74-2ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94+ζ92ζ74-ζ92ζ72+ζ9ζ75-ζ9ζ73 | -ζ98ζ76-2ζ98ζ75-ζ98ζ74-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ72-ζ94ζ7+ζ92ζ76-ζ92ζ75+ζ9ζ74-ζ9ζ7 | ζ97ζ75-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ92ζ76-ζ92ζ75-ζ92ζ74-ζ92ζ73-2ζ92ζ7-ζ92+ζ9ζ76-ζ9ζ73 | orthogonal faithful |
| ρ18 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1-√21/2 | 1+√21/2 | -ζ95ζ76+ζ95ζ73-ζ94ζ76-ζ94ζ75-ζ94ζ73-2ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ75-ζ9ζ72+ζ9ζ7 | ζ98ζ74-ζ98ζ72-ζ97ζ75-ζ97ζ74-2ζ97ζ73-ζ97ζ72-ζ97ζ7-ζ97+ζ94ζ75-ζ94ζ73-ζ92ζ72+ζ92ζ7 | -ζ95ζ72+ζ95ζ7-ζ94ζ75-ζ94ζ74-2ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94+ζ92ζ74-ζ92ζ72+ζ9ζ75-ζ9ζ73 | -ζ98ζ76-2ζ98ζ75-ζ98ζ74-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ72-ζ94ζ7+ζ92ζ76-ζ92ζ75+ζ9ζ74-ζ9ζ7 | ζ97ζ75-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ92ζ76-ζ92ζ75-ζ92ζ74-ζ92ζ73-2ζ92ζ7-ζ92+ζ9ζ76-ζ9ζ73 | -2ζ98ζ76-ζ98ζ74-ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98-ζ94ζ74+ζ94ζ7-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ72 | orthogonal faithful |
| ρ19 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1-√21/2 | 1+√21/2 | -ζ98ζ76-2ζ98ζ75-ζ98ζ74-ζ98ζ72-ζ98ζ7-ζ98+ζ94ζ72-ζ94ζ7+ζ92ζ76-ζ92ζ75+ζ9ζ74-ζ9ζ7 | -ζ95ζ72+ζ95ζ7-ζ94ζ75-ζ94ζ74-2ζ94ζ73-ζ94ζ72-ζ94ζ7-ζ94+ζ92ζ74-ζ92ζ72+ζ9ζ75-ζ9ζ73 | ζ97ζ75-ζ97ζ73+ζ95ζ74-ζ95ζ7-ζ92ζ76-ζ92ζ75-ζ92ζ74-ζ92ζ73-2ζ92ζ7-ζ92+ζ9ζ76-ζ9ζ73 | -2ζ98ζ76-ζ98ζ74-ζ98ζ73-ζ98ζ72-ζ98ζ7-ζ98-ζ94ζ74+ζ94ζ7-ζ92ζ76+ζ92ζ73-ζ9ζ74+ζ9ζ72 | ζ98ζ74-ζ98ζ72-ζ97ζ75-ζ97ζ74-2ζ97ζ73-ζ97ζ72-ζ97ζ7-ζ97+ζ94ζ75-ζ94ζ73-ζ92ζ72+ζ92ζ7 | -ζ95ζ76+ζ95ζ73-ζ94ζ76-ζ94ζ75-ζ94ζ73-2ζ94ζ72-ζ94ζ7-ζ94-ζ92ζ76+ζ92ζ75-ζ9ζ72+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 30 14 26 56 52 45)(2 31 15 27 57 53 37)(3 32 16 19 58 54 38)(4 33 17 20 59 46 39)(5 34 18 21 60 47 40)(6 35 10 22 61 48 41)(7 36 11 23 62 49 42)(8 28 12 24 63 50 43)(9 29 13 25 55 51 44)
(2 6 8 9 5 3)(4 7)(10 43 55 47 32 27)(11 39 62 46 36 20)(12 44 60 54 31 22)(13 40 58 53 35 24)(14 45 56 52 30 26)(15 41 63 51 34 19)(16 37 61 50 29 21)(17 42 59 49 33 23)(18 38 57 48 28 25)
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), (1,30,14,26,56,52,45)(2,31,15,27,57,53,37)(3,32,16,19,58,54,38)(4,33,17,20,59,46,39)(5,34,18,21,60,47,40)(6,35,10,22,61,48,41)(7,36,11,23,62,49,42)(8,28,12,24,63,50,43)(9,29,13,25,55,51,44), (2,6,8,9,5,3)(4,7)(10,43,55,47,32,27)(11,39,62,46,36,20)(12,44,60,54,31,22)(13,40,58,53,35,24)(14,45,56,52,30,26)(15,41,63,51,34,19)(16,37,61,50,29,21)(17,42,59,49,33,23)(18,38,57,48,28,25)>;
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), (1,30,14,26,56,52,45)(2,31,15,27,57,53,37)(3,32,16,19,58,54,38)(4,33,17,20,59,46,39)(5,34,18,21,60,47,40)(6,35,10,22,61,48,41)(7,36,11,23,62,49,42)(8,28,12,24,63,50,43)(9,29,13,25,55,51,44), (2,6,8,9,5,3)(4,7)(10,43,55,47,32,27)(11,39,62,46,36,20)(12,44,60,54,31,22)(13,40,58,53,35,24)(14,45,56,52,30,26)(15,41,63,51,34,19)(16,37,61,50,29,21)(17,42,59,49,33,23)(18,38,57,48,28,25) );
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)], [(1,30,14,26,56,52,45),(2,31,15,27,57,53,37),(3,32,16,19,58,54,38),(4,33,17,20,59,46,39),(5,34,18,21,60,47,40),(6,35,10,22,61,48,41),(7,36,11,23,62,49,42),(8,28,12,24,63,50,43),(9,29,13,25,55,51,44)], [(2,6,8,9,5,3),(4,7),(10,43,55,47,32,27),(11,39,62,46,36,20),(12,44,60,54,31,22),(13,40,58,53,35,24),(14,45,56,52,30,26),(15,41,63,51,34,19),(16,37,61,50,29,21),(17,42,59,49,33,23),(18,38,57,48,28,25)]])
Matrix representation of C9⋊F7 ►in GL6(𝔽127)
| 22 | 57 | 45 | 97 | 15 | 82 |
| 70 | 79 | 30 | 15 | 45 | 97 |
| 15 | 82 | 63 | 104 | 71 | 62 |
| 45 | 97 | 23 | 40 | 65 | 6 |
| 71 | 62 | 65 | 6 | 42 | 93 |
| 65 | 6 | 121 | 71 | 34 | 8 |
| 88 | 77 | 89 | 77 | 1 | 0 |
| 50 | 38 | 50 | 39 | 0 | 1 |
| 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 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 126 | 0 | 126 | 50 | 38 |
| 126 | 0 | 126 | 0 | 88 | 77 |
| 0 | 1 | 77 | 89 | 77 | 88 |
| 1 | 0 | 39 | 50 | 38 | 50 |
G:=sub<GL(6,GF(127))| [22,70,15,45,71,65,57,79,82,97,62,6,45,30,63,23,65,121,97,15,104,40,6,71,15,45,71,65,42,34,82,97,62,6,93,8],[88,50,1,0,0,0,77,38,0,1,0,0,89,50,0,0,1,0,77,39,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0],[0,1,0,126,0,1,1,0,126,0,1,0,0,0,0,126,77,39,0,0,126,0,89,50,0,0,50,88,77,38,0,0,38,77,88,50] >;
C9⋊F7 in GAP, Magma, Sage, TeX
C_9\rtimes F_7
% in TeX
G:=Group("C9:F7"); // GroupNames label
G:=SmallGroup(378,18);
// by ID
G=gap.SmallGroup(378,18);
# by ID
G:=PCGroup([5,-2,-3,-3,-7,-3,2072,997,642,2163,368,6304]);
// Polycyclic
G:=Group<a,b,c|a^9=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^2,c*b*c^-1=b^5>;
// generators/relations
Export
Subgroup lattice of C9⋊F7 in TeX
Character table of C9⋊F7 in TeX