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## G = C8⋊D5order 80 = 24·5

### 3rd semidirect product of C8 and D5 acting via D5/C5=C2

Aliases: C83D5, C404C2, C53M4(2), D10.1C4, C4.13D10, Dic5.1C4, C20.13C22, C52C84C2, C2.3(C4×D5), C10.9(C2×C4), (C4×D5).2C2, SmallGroup(80,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C8⋊D5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C8⋊D5
 Lower central C5 — C10 — C8⋊D5
 Upper central C1 — C4 — C8

Generators and relations for C8⋊D5
G = < a,b,c | a8=b5=c2=1, ab=ba, cac=a5, cbc=b-1 >

Character table of C8⋊D5

 class 1 2A 2B 4A 4B 4C 5A 5B 8A 8B 8C 8D 10A 10B 20A 20B 20C 20D 40A 40B 40C 40D 40E 40F 40G 40H size 1 1 10 1 1 10 2 2 2 2 10 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 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 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 linear of order 2 ρ5 1 1 1 -1 -1 -1 1 1 i -i -i i 1 1 -1 -1 -1 -1 -i -i -i -i i i i i linear of order 4 ρ6 1 1 -1 -1 -1 1 1 1 i -i i -i 1 1 -1 -1 -1 -1 -i -i -i -i i i i i linear of order 4 ρ7 1 1 1 -1 -1 -1 1 1 -i i i -i 1 1 -1 -1 -1 -1 i i i i -i -i -i -i linear of order 4 ρ8 1 1 -1 -1 -1 1 1 1 -i i -i i 1 1 -1 -1 -1 -1 i i i i -i -i -i -i linear of order 4 ρ9 2 2 0 2 2 0 -1-√5/2 -1+√5/2 -2 -2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ10 2 2 0 2 2 0 -1+√5/2 -1-√5/2 -2 -2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ11 2 2 0 2 2 0 -1-√5/2 -1+√5/2 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ12 2 2 0 2 2 0 -1+√5/2 -1-√5/2 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ13 2 -2 0 -2i 2i 0 2 2 0 0 0 0 -2 -2 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ14 2 -2 0 2i -2i 0 2 2 0 0 0 0 -2 -2 2i -2i 2i -2i 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ15 2 2 0 -2 -2 0 -1+√5/2 -1-√5/2 -2i 2i 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ54+ζ43ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ43ζ53+ζ43ζ52 complex lifted from C4×D5 ρ16 2 2 0 -2 -2 0 -1-√5/2 -1+√5/2 2i -2i 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ53+ζ4ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ4ζ54+ζ4ζ5 complex lifted from C4×D5 ρ17 2 2 0 -2 -2 0 -1+√5/2 -1-√5/2 2i -2i 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ54+ζ4ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ4ζ53+ζ4ζ52 complex lifted from C4×D5 ρ18 2 2 0 -2 -2 0 -1-√5/2 -1+√5/2 -2i 2i 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ53+ζ43ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ43ζ54+ζ43ζ5 complex lifted from C4×D5 ρ19 2 -2 0 -2i 2i 0 -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 1+√5/2 ζ86ζ53+ζ86ζ52 ζ82ζ54+ζ82ζ5 ζ86ζ54+ζ86ζ5 ζ82ζ53+ζ82ζ52 ζ83ζ54-ζ83ζ5 ζ83ζ53-ζ83ζ52 -ζ83ζ53+ζ83ζ52 ζ87ζ54-ζ87ζ5 ζ8ζ53-ζ8ζ52 -ζ8ζ53+ζ8ζ52 ζ85ζ54-ζ85ζ5 ζ8ζ54-ζ8ζ5 complex faithful ρ20 2 -2 0 -2i 2i 0 -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 1+√5/2 ζ86ζ53+ζ86ζ52 ζ82ζ54+ζ82ζ5 ζ86ζ54+ζ86ζ5 ζ82ζ53+ζ82ζ52 ζ87ζ54-ζ87ζ5 -ζ83ζ53+ζ83ζ52 ζ83ζ53-ζ83ζ52 ζ83ζ54-ζ83ζ5 -ζ8ζ53+ζ8ζ52 ζ8ζ53-ζ8ζ52 ζ8ζ54-ζ8ζ5 ζ85ζ54-ζ85ζ5 complex faithful ρ21 2 -2 0 -2i 2i 0 -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 1-√5/2 ζ86ζ54+ζ86ζ5 ζ82ζ53+ζ82ζ52 ζ86ζ53+ζ86ζ52 ζ82ζ54+ζ82ζ5 -ζ83ζ53+ζ83ζ52 ζ83ζ54-ζ83ζ5 ζ87ζ54-ζ87ζ5 ζ83ζ53-ζ83ζ52 ζ8ζ54-ζ8ζ5 ζ85ζ54-ζ85ζ5 ζ8ζ53-ζ8ζ52 -ζ8ζ53+ζ8ζ52 complex faithful ρ22 2 -2 0 2i -2i 0 -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 1-√5/2 ζ82ζ54+ζ82ζ5 ζ86ζ53+ζ86ζ52 ζ82ζ53+ζ82ζ52 ζ86ζ54+ζ86ζ5 ζ8ζ53-ζ8ζ52 ζ85ζ54-ζ85ζ5 ζ8ζ54-ζ8ζ5 -ζ8ζ53+ζ8ζ52 ζ87ζ54-ζ87ζ5 ζ83ζ54-ζ83ζ5 -ζ83ζ53+ζ83ζ52 ζ83ζ53-ζ83ζ52 complex faithful ρ23 2 -2 0 -2i 2i 0 -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 1-√5/2 ζ86ζ54+ζ86ζ5 ζ82ζ53+ζ82ζ52 ζ86ζ53+ζ86ζ52 ζ82ζ54+ζ82ζ5 ζ83ζ53-ζ83ζ52 ζ87ζ54-ζ87ζ5 ζ83ζ54-ζ83ζ5 -ζ83ζ53+ζ83ζ52 ζ85ζ54-ζ85ζ5 ζ8ζ54-ζ8ζ5 -ζ8ζ53+ζ8ζ52 ζ8ζ53-ζ8ζ52 complex faithful ρ24 2 -2 0 2i -2i 0 -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 1+√5/2 ζ82ζ53+ζ82ζ52 ζ86ζ54+ζ86ζ5 ζ82ζ54+ζ82ζ5 ζ86ζ53+ζ86ζ52 ζ85ζ54-ζ85ζ5 -ζ8ζ53+ζ8ζ52 ζ8ζ53-ζ8ζ52 ζ8ζ54-ζ8ζ5 -ζ83ζ53+ζ83ζ52 ζ83ζ53-ζ83ζ52 ζ83ζ54-ζ83ζ5 ζ87ζ54-ζ87ζ5 complex faithful ρ25 2 -2 0 2i -2i 0 -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 1-√5/2 ζ82ζ54+ζ82ζ5 ζ86ζ53+ζ86ζ52 ζ82ζ53+ζ82ζ52 ζ86ζ54+ζ86ζ5 -ζ8ζ53+ζ8ζ52 ζ8ζ54-ζ8ζ5 ζ85ζ54-ζ85ζ5 ζ8ζ53-ζ8ζ52 ζ83ζ54-ζ83ζ5 ζ87ζ54-ζ87ζ5 ζ83ζ53-ζ83ζ52 -ζ83ζ53+ζ83ζ52 complex faithful ρ26 2 -2 0 2i -2i 0 -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 1+√5/2 ζ82ζ53+ζ82ζ52 ζ86ζ54+ζ86ζ5 ζ82ζ54+ζ82ζ5 ζ86ζ53+ζ86ζ52 ζ8ζ54-ζ8ζ5 ζ8ζ53-ζ8ζ52 -ζ8ζ53+ζ8ζ52 ζ85ζ54-ζ85ζ5 ζ83ζ53-ζ83ζ52 -ζ83ζ53+ζ83ζ52 ζ87ζ54-ζ87ζ5 ζ83ζ54-ζ83ζ5 complex faithful

Smallest permutation representation of C8⋊D5
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 35 32 24 11)(2 36 25 17 12)(3 37 26 18 13)(4 38 27 19 14)(5 39 28 20 15)(6 40 29 21 16)(7 33 30 22 9)(8 34 31 23 10)
(1 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(25 29)(27 31)```

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

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

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

C8⋊D5 is a maximal subgroup of
D20.3C4  D5×M4(2)  D20.2C4  D8⋊D5  D40⋊C2  SD16⋊D5  Q16⋊D5  C20.32D6  D30.5C4  C40⋊S3  C8⋊D25  C20.30D10  C20.31D10  C40⋊D5  D10.F5  Dic5.F5
C8⋊D5 is a maximal quotient of
C20.8Q8  C408C4  D101C8  C20.32D6  D30.5C4  C40⋊S3  C8⋊D25  C20.30D10  C20.31D10  C40⋊D5  D10.F5  Dic5.F5

Matrix representation of C8⋊D5 in GL2(𝔽29) generated by

 11 1 12 18
,
 12 18 13 22
,
 22 26 16 7
`G:=sub<GL(2,GF(29))| [11,12,1,18],[12,13,18,22],[22,16,26,7] >;`

C8⋊D5 in GAP, Magma, Sage, TeX

`C_8\rtimes D_5`
`% in TeX`

`G:=Group("C8:D5");`
`// GroupNames label`

`G:=SmallGroup(80,5);`
`// by ID`

`G=gap.SmallGroup(80,5);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-5,101,26,42,1604]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^5=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;`
`// generators/relations`

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