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## G = C5⋊2C8order 40 = 23·5

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

Aliases: C52C8, C4.2D5, C2.Dic5, C10.2C4, C20.2C2, SmallGroup(40,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8
 Lower central C5 — C5⋊2C8
 Upper central C1 — C4

Generators and relations for C52C8
G = < a,b | a5=b8=1, bab-1=a-1 >

Character table of C52C8

 class 1 2 4A 4B 5A 5B 8A 8B 8C 8D 10A 10B 20A 20B 20C 20D size 1 1 1 1 2 2 5 5 5 5 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 1 1 -1 -1 1 1 -i i -i i 1 1 -1 -1 -1 -1 linear of order 4 ρ4 1 1 -1 -1 1 1 i -i i -i 1 1 -1 -1 -1 -1 linear of order 4 ρ5 1 -1 i -i 1 1 ζ87 ζ85 ζ83 ζ8 -1 -1 i -i i -i linear of order 8 ρ6 1 -1 i -i 1 1 ζ83 ζ8 ζ87 ζ85 -1 -1 i -i i -i linear of order 8 ρ7 1 -1 -i i 1 1 ζ8 ζ83 ζ85 ζ87 -1 -1 -i i -i i linear of order 8 ρ8 1 -1 -i i 1 1 ζ85 ζ87 ζ8 ζ83 -1 -1 -i i -i i linear of order 8 ρ9 2 2 2 2 -1-√5/2 -1+√5/2 0 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ10 2 2 2 2 -1+√5/2 -1-√5/2 0 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ11 2 2 -2 -2 -1+√5/2 -1-√5/2 0 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ12 2 2 -2 -2 -1-√5/2 -1+√5/2 0 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ13 2 -2 -2i 2i -1+√5/2 -1-√5/2 0 0 0 0 1-√5/2 1+√5/2 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 complex faithful, Schur index 2 ρ14 2 -2 2i -2i -1-√5/2 -1+√5/2 0 0 0 0 1+√5/2 1-√5/2 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 complex faithful, Schur index 2 ρ15 2 -2 -2i 2i -1-√5/2 -1+√5/2 0 0 0 0 1+√5/2 1-√5/2 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 complex faithful, Schur index 2 ρ16 2 -2 2i -2i -1+√5/2 -1-√5/2 0 0 0 0 1-√5/2 1+√5/2 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 complex faithful, Schur index 2

Smallest permutation representation of C52C8
Regular action on 40 points
Generators in S40
```(1 35 9 32 24)(2 17 25 10 36)(3 37 11 26 18)(4 19 27 12 38)(5 39 13 28 20)(6 21 29 14 40)(7 33 15 30 22)(8 23 31 16 34)
(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)| (1,35,9,32,24)(2,17,25,10,36)(3,37,11,26,18)(4,19,27,12,38)(5,39,13,28,20)(6,21,29,14,40)(7,33,15,30,22)(8,23,31,16,34), (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( (1,35,9,32,24)(2,17,25,10,36)(3,37,11,26,18)(4,19,27,12,38)(5,39,13,28,20)(6,21,29,14,40)(7,33,15,30,22)(8,23,31,16,34), (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([[(1,35,9,32,24),(2,17,25,10,36),(3,37,11,26,18),(4,19,27,12,38),(5,39,13,28,20),(6,21,29,14,40),(7,33,15,30,22),(8,23,31,16,34)], [(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)]])`

C52C8 is a maximal subgroup of
C5⋊C16  C8×D5  C8⋊D5  C4.Dic5  D4⋊D5  D4.D5  Q8⋊D5  C5⋊Q16  C153C8  C252C8  C527C8  C523C8  C353C8  (C3×C15)⋊9C8  C52F9  C553C8
C52C8 is a maximal quotient of
C52C16  C153C8  C252C8  C527C8  C523C8  C353C8  (C3×C15)⋊9C8  C52F9  C553C8

Matrix representation of C52C8 in GL2(𝔽29) generated by

 11 21 9 12
,
 0 17 1 0
`G:=sub<GL(2,GF(29))| [11,9,21,12],[0,1,17,0] >;`

C52C8 in GAP, Magma, Sage, TeX

`C_5\rtimes_2C_8`
`% in TeX`

`G:=Group("C5:2C8");`
`// GroupNames label`

`G:=SmallGroup(40,1);`
`// by ID`

`G=gap.SmallGroup(40,1);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-5,8,21,515]);`
`// Polycyclic`

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

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