Copied to
clipboard

## G = C9⋊F7order 378 = 2·33·7

### 1st semidirect product of C9 and F7 acting via F7/C7=C6

Aliases: C91F7, C633C6, D633C3, C71(C9⋊C6), C633C31C2, C21.1(C3×S3), C3.1(C3⋊F7), (C3×C7⋊C3).1S3, SmallGroup(378,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C63 — C9⋊F7
 Chief series C1 — C3 — C21 — C63 — C63⋊3C3 — C9⋊F7
 Lower central C63 — C9⋊F7
 Upper central C1

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

Smallest permutation representation of C9⋊F7
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

׿
×
𝔽