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## G = C8⋊F7order 336 = 24·3·7

### 3rd semidirect product of C8 and F7 acting via F7/C7⋊C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C8⋊F7
 Chief series C1 — C7 — C14 — C28 — C4×C7⋊C3 — C4×F7 — C8⋊F7
 Lower central C7 — C14 — C8⋊F7
 Upper central C1 — C4 — C8

Generators and relations for C8⋊F7
G = < a,b,c | a8=b7=c6=1, ab=ba, cac-1=a5, cbc-1=b5 >

Smallest permutation representation of C8⋊F7
On 56 points
Generators in S56
```(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)
(1 31 34 16 19 46 49)(2 32 35 9 20 47 50)(3 25 36 10 21 48 51)(4 26 37 11 22 41 52)(5 27 38 12 23 42 53)(6 28 39 13 24 43 54)(7 29 40 14 17 44 55)(8 30 33 15 18 45 56)
(2 6)(4 8)(9 39 50 24 47 28)(10 36 51 21 48 25)(11 33 52 18 41 30)(12 38 53 23 42 27)(13 35 54 20 43 32)(14 40 55 17 44 29)(15 37 56 22 45 26)(16 34 49 19 46 31)```

`G:=sub<Sym(56)| (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), (1,31,34,16,19,46,49)(2,32,35,9,20,47,50)(3,25,36,10,21,48,51)(4,26,37,11,22,41,52)(5,27,38,12,23,42,53)(6,28,39,13,24,43,54)(7,29,40,14,17,44,55)(8,30,33,15,18,45,56), (2,6)(4,8)(9,39,50,24,47,28)(10,36,51,21,48,25)(11,33,52,18,41,30)(12,38,53,23,42,27)(13,35,54,20,43,32)(14,40,55,17,44,29)(15,37,56,22,45,26)(16,34,49,19,46,31)>;`

`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), (1,31,34,16,19,46,49)(2,32,35,9,20,47,50)(3,25,36,10,21,48,51)(4,26,37,11,22,41,52)(5,27,38,12,23,42,53)(6,28,39,13,24,43,54)(7,29,40,14,17,44,55)(8,30,33,15,18,45,56), (2,6)(4,8)(9,39,50,24,47,28)(10,36,51,21,48,25)(11,33,52,18,41,30)(12,38,53,23,42,27)(13,35,54,20,43,32)(14,40,55,17,44,29)(15,37,56,22,45,26)(16,34,49,19,46,31) );`

`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)], [(1,31,34,16,19,46,49),(2,32,35,9,20,47,50),(3,25,36,10,21,48,51),(4,26,37,11,22,41,52),(5,27,38,12,23,42,53),(6,28,39,13,24,43,54),(7,29,40,14,17,44,55),(8,30,33,15,18,45,56)], [(2,6),(4,8),(9,39,50,24,47,28),(10,36,51,21,48,25),(11,33,52,18,41,30),(12,38,53,23,42,27),(13,35,54,20,43,32),(14,40,55,17,44,29),(15,37,56,22,45,26),(16,34,49,19,46,31)]])`

38 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 6C 6D 7 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 14 24A ··· 24H 28A 28B 56A 56B 56C 56D order 1 2 2 3 3 4 4 4 6 6 6 6 7 8 8 8 8 12 12 12 12 12 12 14 24 ··· 24 28 28 56 56 56 56 size 1 1 14 7 7 1 1 14 7 7 14 14 6 2 2 14 14 7 7 7 7 14 14 6 14 ··· 14 6 6 6 6 6 6

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 6 6 6 6 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 M4(2) C3×M4(2) F7 C2×F7 C4×F7 C8⋊F7 kernel C8⋊F7 C7⋊C24 C8×C7⋊C3 C4×F7 C8⋊D7 C7⋊C12 C2×F7 C7⋊C8 C56 C4×D7 Dic7 D14 C7⋊C3 C7 C8 C4 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 2 4 1 1 2 4

Matrix representation of C8⋊F7 in GL6(𝔽337)

 97 0 143 194 143 0 194 97 143 0 0 143 194 194 240 0 143 0 0 194 0 97 143 143 194 0 0 194 240 143 0 194 143 194 0 240
,
 0 0 0 0 0 336 1 0 0 0 0 336 0 1 0 0 0 336 0 0 1 0 0 336 0 0 0 1 0 336 0 0 0 0 1 336
,
 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0

`G:=sub<GL(6,GF(337))| [97,194,194,0,194,0,0,97,194,194,0,194,143,143,240,0,0,143,194,0,0,97,194,194,143,0,143,143,240,0,0,143,0,143,143,240],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,336,336,336,336,336,336],[0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C8⋊F7 in GAP, Magma, Sage, TeX

`C_8\rtimes F_7`
`% in TeX`

`G:=Group("C8:F7");`
`// GroupNames label`

`G:=SmallGroup(336,8);`
`// by ID`

`G=gap.SmallGroup(336,8);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-2,-2,-7,313,79,69,10373,1745]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^5>;`
`// generators/relations`

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