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## G = C8⋊F5order 160 = 25·5

### 3rd semidirect product of C8 and F5 acting via F5/D5=C2

Aliases: C83F5, C404C4, D5.M4(2), C10.2C42, C5⋊C81C4, C52C88C4, (C2×F5).C4, C51(C8⋊C4), C2.3(C4×F5), D5⋊C8.2C2, (C4×F5).2C2, C4.17(C2×F5), C20.16(C2×C4), (C8×D5).10C2, D10.6(C2×C4), Dic5.8(C2×C4), (C4×D5).32C22, SmallGroup(160,67)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C8⋊F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — C8⋊F5
 Lower central C5 — C10 — C8⋊F5
 Upper central C1 — C4 — C8

Generators and relations for C8⋊F5
G = < a,b,c | a8=b5=c4=1, ab=ba, cac-1=a5, cbc-1=b3 >

Character table of C8⋊F5

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 5 8A 8B 8C 8D 8E 8F 8G 8H 10 20A 20B 40A 40B 40C 40D size 1 1 5 5 1 1 5 5 10 10 10 10 4 2 2 10 10 10 10 10 10 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 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 1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 -1 -1 -1 -1 1 1 -i -i i i 1 -i i -1 -i -1 1 1 i 1 -1 -1 -i i i -i linear of order 4 ρ6 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 -i i -i i i -i i -i 1 -1 -1 -i i i -i linear of order 4 ρ7 1 1 -1 -1 -1 -1 1 1 i i -i -i 1 i -i -1 i -1 1 1 -i 1 -1 -1 i -i -i i linear of order 4 ρ8 1 1 -1 -1 -1 -1 1 1 -i -i i i 1 i -i 1 i 1 -1 -1 -i 1 -1 -1 i -i -i i linear of order 4 ρ9 1 1 -1 -1 -1 -1 1 1 i i -i -i 1 -i i 1 -i 1 -1 -1 i 1 -1 -1 -i i i -i linear of order 4 ρ10 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 i -i -i -i i -i i i 1 -1 -1 i -i -i i linear of order 4 ρ11 1 1 -1 -1 1 1 -1 -1 -i i i -i 1 1 1 -i -1 i i -i -1 1 1 1 1 1 1 1 linear of order 4 ρ12 1 1 -1 -1 1 1 -1 -1 i -i -i i 1 -1 -1 -i 1 i i -i 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ13 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 -i i i i -i i -i -i 1 -1 -1 -i i i -i linear of order 4 ρ14 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 i -i i -i -i i -i i 1 -1 -1 i -i -i i linear of order 4 ρ15 1 1 -1 -1 1 1 -1 -1 -i i i -i 1 -1 -1 i 1 -i -i i 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ16 1 1 -1 -1 1 1 -1 -1 i -i -i i 1 1 1 i -1 -i -i i -1 1 1 1 1 1 1 1 linear of order 4 ρ17 2 -2 -2 2 -2i 2i -2i 2i 0 0 0 0 2 0 0 0 0 0 0 0 0 -2 2i -2i 0 0 0 0 complex lifted from M4(2) ρ18 2 -2 2 -2 -2i 2i 2i -2i 0 0 0 0 2 0 0 0 0 0 0 0 0 -2 2i -2i 0 0 0 0 complex lifted from M4(2) ρ19 2 -2 -2 2 2i -2i 2i -2i 0 0 0 0 2 0 0 0 0 0 0 0 0 -2 -2i 2i 0 0 0 0 complex lifted from M4(2) ρ20 2 -2 2 -2 2i -2i -2i 2i 0 0 0 0 2 0 0 0 0 0 0 0 0 -2 -2i 2i 0 0 0 0 complex lifted from M4(2) ρ21 4 4 0 0 4 4 0 0 0 0 0 0 -1 -4 -4 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 0 0 4 4 0 0 0 0 0 0 -1 4 4 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ23 4 4 0 0 -4 -4 0 0 0 0 0 0 -1 -4i 4i 0 0 0 0 0 0 -1 1 1 i -i -i i complex lifted from C4×F5 ρ24 4 4 0 0 -4 -4 0 0 0 0 0 0 -1 4i -4i 0 0 0 0 0 0 -1 1 1 -i i i -i complex lifted from C4×F5 ρ25 4 -4 0 0 4i -4i 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 i -i 2ζ8ζ54+2ζ8ζ5+ζ8 2ζ83ζ53+2ζ83ζ52+ζ83 2ζ83ζ54+2ζ83ζ5+ζ83 2ζ85ζ54+2ζ85ζ5+ζ85 complex faithful ρ26 4 -4 0 0 -4i 4i 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -i i 2ζ83ζ53+2ζ83ζ52+ζ83 2ζ8ζ54+2ζ8ζ5+ζ8 2ζ85ζ54+2ζ85ζ5+ζ85 2ζ83ζ54+2ζ83ζ5+ζ83 complex faithful ρ27 4 -4 0 0 -4i 4i 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -i i 2ζ83ζ54+2ζ83ζ5+ζ83 2ζ85ζ54+2ζ85ζ5+ζ85 2ζ8ζ54+2ζ8ζ5+ζ8 2ζ83ζ53+2ζ83ζ52+ζ83 complex faithful ρ28 4 -4 0 0 4i -4i 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 i -i 2ζ85ζ54+2ζ85ζ5+ζ85 2ζ83ζ54+2ζ83ζ5+ζ83 2ζ83ζ53+2ζ83ζ52+ζ83 2ζ8ζ54+2ζ8ζ5+ζ8 complex faithful

Smallest permutation representation of C8⋊F5
On 40 points
Generators in S40
```(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)
(1 11 24 32 35)(2 12 17 25 36)(3 13 18 26 37)(4 14 19 27 38)(5 15 20 28 39)(6 16 21 29 40)(7 9 22 30 33)(8 10 23 31 34)
(1 7 5 3)(2 4 6 8)(9 20 37 32)(10 17 38 29)(11 22 39 26)(12 19 40 31)(13 24 33 28)(14 21 34 25)(15 18 35 30)(16 23 36 27)```

`G:=sub<Sym(40)| (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), (1,11,24,32,35)(2,12,17,25,36)(3,13,18,26,37)(4,14,19,27,38)(5,15,20,28,39)(6,16,21,29,40)(7,9,22,30,33)(8,10,23,31,34), (1,7,5,3)(2,4,6,8)(9,20,37,32)(10,17,38,29)(11,22,39,26)(12,19,40,31)(13,24,33,28)(14,21,34,25)(15,18,35,30)(16,23,36,27)>;`

`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), (1,11,24,32,35)(2,12,17,25,36)(3,13,18,26,37)(4,14,19,27,38)(5,15,20,28,39)(6,16,21,29,40)(7,9,22,30,33)(8,10,23,31,34), (1,7,5,3)(2,4,6,8)(9,20,37,32)(10,17,38,29)(11,22,39,26)(12,19,40,31)(13,24,33,28)(14,21,34,25)(15,18,35,30)(16,23,36,27) );`

`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)], [(1,11,24,32,35),(2,12,17,25,36),(3,13,18,26,37),(4,14,19,27,38),(5,15,20,28,39),(6,16,21,29,40),(7,9,22,30,33),(8,10,23,31,34)], [(1,7,5,3),(2,4,6,8),(9,20,37,32),(10,17,38,29),(11,22,39,26),(12,19,40,31),(13,24,33,28),(14,21,34,25),(15,18,35,30),(16,23,36,27)]])`

C8⋊F5 is a maximal subgroup of
C16⋊F5  C164F5  C20.12C42  M4(2)×F5  M4(2)⋊5F5  D40⋊C4  D8⋊F5  SD16⋊F5  SD162F5  Dic20⋊C4  Q16⋊F5  C30.3C42  C30.4C42  C24⋊F5
C8⋊F5 is a maximal quotient of
C16⋊F5  C164F5  C40⋊C8  C20.31M4(2)  D10.3M4(2)  C30.3C42  C30.4C42  C24⋊F5

Matrix representation of C8⋊F5 in GL6(𝔽41)

 0 9 0 0 0 0 40 0 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

`G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;`

C8⋊F5 in GAP, Magma, Sage, TeX

`C_8\rtimes F_5`
`% in TeX`

`G:=Group("C8:F5");`
`// GroupNames label`

`G:=SmallGroup(160,67);`
`// by ID`

`G=gap.SmallGroup(160,67);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,69,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^3>;`
`// generators/relations`

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