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## G = C5⋊C8order 40 = 23·5

### The semidirect product of C5 and C8 acting via C8/C2=C4

Aliases: C5⋊C8, C2.F5, C10.C4, Dic5.2C2, SmallGroup(40,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8
 Lower central C5 — C5⋊C8
 Upper central C1 — C2

Generators and relations for C5⋊C8
G = < a,b | a5=b8=1, bab-1=a3 >

Character table of C5⋊C8

 class 1 2 4A 4B 5 8A 8B 8C 8D 10 size 1 1 5 5 4 5 5 5 5 4 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ3 1 1 -1 -1 1 i -i i -i 1 linear of order 4 ρ4 1 1 -1 -1 1 -i i -i i 1 linear of order 4 ρ5 1 -1 i -i 1 ζ85 ζ87 ζ8 ζ83 -1 linear of order 8 ρ6 1 -1 -i i 1 ζ83 ζ8 ζ87 ζ85 -1 linear of order 8 ρ7 1 -1 i -i 1 ζ8 ζ83 ζ85 ζ87 -1 linear of order 8 ρ8 1 -1 -i i 1 ζ87 ζ85 ζ83 ζ8 -1 linear of order 8 ρ9 4 4 0 0 -1 0 0 0 0 -1 orthogonal lifted from F5 ρ10 4 -4 0 0 -1 0 0 0 0 1 symplectic faithful, Schur index 2

Smallest permutation representation of C5⋊C8
Regular action on 40 points
Generators in S40
```(1 35 27 20 11)(2 21 36 12 28)(3 13 22 29 37)(4 30 14 38 23)(5 39 31 24 15)(6 17 40 16 32)(7 9 18 25 33)(8 26 10 34 19)
(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)```

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

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

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

C5⋊C8 is a maximal subgroup of
D5⋊C8  C4.F5  C22.F5  C15⋊C8  C25⋊C8  C523C8  C524C8  C525C8  CSU2(𝔽5)  C2.S5  C35⋊C8  (C3×C6).F5  C5⋊F9  C55⋊C8
C5⋊C8 is a maximal quotient of
C5⋊C16  C15⋊C8  C25⋊C8  C523C8  C524C8  C525C8  C35⋊C8  (C3×C6).F5  C5⋊F9  C55⋊C8

Matrix representation of C5⋊C8 in GL4(𝔽3) generated by

 0 2 1 2 1 1 1 1 1 2 2 0 0 2 0 2
,
 2 0 2 0 0 1 0 2 2 0 0 0 2 2 0 0
`G:=sub<GL(4,GF(3))| [0,1,1,0,2,1,2,2,1,1,2,0,2,1,0,2],[2,0,2,2,0,1,0,2,2,0,0,0,0,2,0,0] >;`

C5⋊C8 in GAP, Magma, Sage, TeX

`C_5\rtimes C_8`
`% in TeX`

`G:=Group("C5:C8");`
`// GroupNames label`

`G:=SmallGroup(40,3);`
`// by ID`

`G=gap.SmallGroup(40,3);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-5,8,21,259,263]);`
`// Polycyclic`

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

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