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## G = C7⋊F5order 140 = 22·5·7

### The semidirect product of C7 and F5 acting via F5/D5=C2

Aliases: C7⋊F5, C5⋊Dic7, C351C4, D5.D7, (C7×D5).1C2, SmallGroup(140,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — C7⋊F5
 Chief series C1 — C7 — C35 — C7×D5 — C7⋊F5
 Lower central C35 — C7⋊F5
 Upper central C1

Generators and relations for C7⋊F5
G = < a,b,c | a7=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >

Character table of C7⋊F5

 class 1 2 4A 4B 5 7A 7B 7C 14A 14B 14C 35A 35B 35C 35D 35E 35F size 1 5 35 35 4 2 2 2 10 10 10 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 1 -1 i -i 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 linear of order 4 ρ4 1 -1 -i i 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 linear of order 4 ρ5 2 2 0 0 2 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ6 2 2 0 0 2 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ7 2 2 0 0 2 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ8 2 -2 0 0 2 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 symplectic lifted from Dic7, Schur index 2 ρ9 2 -2 0 0 2 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 symplectic lifted from Dic7, Schur index 2 ρ10 2 -2 0 0 2 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 symplectic lifted from Dic7, Schur index 2 ρ11 4 0 0 0 -1 4 4 4 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ12 4 0 0 0 -1 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 0 0 0 ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 complex faithful ρ13 4 0 0 0 -1 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 0 0 0 -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 complex faithful ρ14 4 0 0 0 -1 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 0 0 0 ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 complex faithful ρ15 4 0 0 0 -1 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 0 0 0 -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 complex faithful ρ16 4 0 0 0 -1 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 0 0 0 ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 complex faithful ρ17 4 0 0 0 -1 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 0 0 0 ζ54ζ75-ζ54ζ72+ζ5ζ75-ζ5ζ72-ζ72 ζ53ζ76-ζ53ζ7+ζ52ζ76-ζ52ζ7-ζ7 -ζ53ζ74+ζ53ζ73-ζ52ζ74+ζ52ζ73-ζ74 ζ53ζ74-ζ53ζ73+ζ52ζ74-ζ52ζ73-ζ73 ζ54ζ76-ζ54ζ7+ζ5ζ76-ζ5ζ7-ζ7 -ζ54ζ75+ζ54ζ72-ζ5ζ75+ζ5ζ72-ζ75 complex faithful

Smallest permutation representation of C7⋊F5
On 35 points
Generators in S35
```(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)
(1 13 20 27 34)(2 14 21 28 35)(3 8 15 22 29)(4 9 16 23 30)(5 10 17 24 31)(6 11 18 25 32)(7 12 19 26 33)
(2 7)(3 6)(4 5)(8 18 29 25)(9 17 30 24)(10 16 31 23)(11 15 32 22)(12 21 33 28)(13 20 34 27)(14 19 35 26)```

`G:=sub<Sym(35)| (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), (1,13,20,27,34)(2,14,21,28,35)(3,8,15,22,29)(4,9,16,23,30)(5,10,17,24,31)(6,11,18,25,32)(7,12,19,26,33), (2,7)(3,6)(4,5)(8,18,29,25)(9,17,30,24)(10,16,31,23)(11,15,32,22)(12,21,33,28)(13,20,34,27)(14,19,35,26)>;`

`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), (1,13,20,27,34)(2,14,21,28,35)(3,8,15,22,29)(4,9,16,23,30)(5,10,17,24,31)(6,11,18,25,32)(7,12,19,26,33), (2,7)(3,6)(4,5)(8,18,29,25)(9,17,30,24)(10,16,31,23)(11,15,32,22)(12,21,33,28)(13,20,34,27)(14,19,35,26) );`

`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)], [(1,13,20,27,34),(2,14,21,28,35),(3,8,15,22,29),(4,9,16,23,30),(5,10,17,24,31),(6,11,18,25,32),(7,12,19,26,33)], [(2,7),(3,6),(4,5),(8,18,29,25),(9,17,30,24),(10,16,31,23),(11,15,32,22),(12,21,33,28),(13,20,34,27),(14,19,35,26)]])`

C7⋊F5 is a maximal subgroup of   D7×F5  C35⋊C12  C5⋊Dic21
C7⋊F5 is a maximal quotient of   C35⋊C8  C5⋊Dic21

Matrix representation of C7⋊F5 in GL4(𝔽281) generated by

 280 1 0 0 39 241 0 0 0 0 280 1 0 0 39 241
,
 267 201 280 0 252 15 0 280 268 201 280 0 252 16 0 280
,
 27 94 254 187 58 254 223 27 40 1 0 0 87 241 0 0
`G:=sub<GL(4,GF(281))| [280,39,0,0,1,241,0,0,0,0,280,39,0,0,1,241],[267,252,268,252,201,15,201,16,280,0,280,0,0,280,0,280],[27,58,40,87,94,254,1,241,254,223,0,0,187,27,0,0] >;`

C7⋊F5 in GAP, Magma, Sage, TeX

`C_7\rtimes F_5`
`% in TeX`

`G:=Group("C7:F5");`
`// GroupNames label`

`G:=SmallGroup(140,6);`
`// by ID`

`G=gap.SmallGroup(140,6);`
`# by ID`

`G:=PCGroup([4,-2,-2,-5,-7,8,146,102,1923]);`
`// Polycyclic`

`G:=Group<a,b,c|a^7=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;`
`// generators/relations`

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