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## G = D8⋊D5order 160 = 25·5

### 2nd semidirect product of D8 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D8⋊D5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D4×D5 — D8⋊D5
 Lower central C5 — C10 — C20 — D8⋊D5
 Upper central C1 — C2 — C4 — D8

Generators and relations for D8⋊D5
G = < a,b,c,d | a8=b2=c5=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 272 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, M4(2), D8, D8, SD16, C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C22×D5, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, D8⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8⋊C22, C22×D5, D4×D5, D8⋊D5

Character table of D8⋊D5

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 5A 5B 8A 8B 10A 10B 10C 10D 10E 10F 20A 20B 40A 40B 40C 40D size 1 1 4 4 10 20 2 10 20 2 2 4 20 2 2 8 8 8 8 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 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 0 0 2 0 -2 -2 0 2 2 0 0 2 2 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 0 -2 0 -2 2 0 2 2 0 0 2 2 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 2 0 0 -1+√5/2 -1-√5/2 2 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 orthogonal lifted from D5 ρ12 2 2 -2 -2 0 0 2 0 0 -1+√5/2 -1-√5/2 2 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 orthogonal lifted from D10 ρ13 2 2 -2 -2 0 0 2 0 0 -1-√5/2 -1+√5/2 2 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 orthogonal lifted from D10 ρ14 2 2 2 -2 0 0 2 0 0 -1-√5/2 -1+√5/2 -2 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 orthogonal lifted from D10 ρ15 2 2 2 -2 0 0 2 0 0 -1+√5/2 -1-√5/2 -2 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 orthogonal lifted from D10 ρ16 2 2 -2 2 0 0 2 0 0 -1+√5/2 -1-√5/2 -2 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 orthogonal lifted from D10 ρ17 2 2 -2 2 0 0 2 0 0 -1-√5/2 -1+√5/2 -2 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 orthogonal lifted from D10 ρ18 2 2 2 2 0 0 2 0 0 -1-√5/2 -1+√5/2 2 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 orthogonal lifted from D5 ρ19 4 -4 0 0 0 0 0 0 0 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 4 0 0 0 0 -4 0 0 -1-√5 -1+√5 0 0 -1-√5 -1+√5 0 0 0 0 1+√5 1-√5 0 0 0 0 orthogonal lifted from D4×D5 ρ21 4 4 0 0 0 0 -4 0 0 -1+√5 -1-√5 0 0 -1+√5 -1-√5 0 0 0 0 1-√5 1+√5 0 0 0 0 orthogonal lifted from D4×D5 ρ22 4 -4 0 0 0 0 0 0 0 -1-√5 -1+√5 0 0 1+√5 1-√5 0 0 0 0 0 0 -ζ87ζ54+ζ87ζ5+ζ85ζ54-ζ85ζ5 -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 -ζ83ζ54+ζ83ζ5+ζ8ζ54-ζ8ζ5 ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 complex faithful ρ23 4 -4 0 0 0 0 0 0 0 -1-√5 -1+√5 0 0 1+√5 1-√5 0 0 0 0 0 0 -ζ83ζ54+ζ83ζ5+ζ8ζ54-ζ8ζ5 ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 -ζ87ζ54+ζ87ζ5+ζ85ζ54-ζ85ζ5 -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 complex faithful ρ24 4 -4 0 0 0 0 0 0 0 -1+√5 -1-√5 0 0 1-√5 1+√5 0 0 0 0 0 0 ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 -ζ87ζ54+ζ87ζ5+ζ85ζ54-ζ85ζ5 -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 -ζ83ζ54+ζ83ζ5+ζ8ζ54-ζ8ζ5 complex faithful ρ25 4 -4 0 0 0 0 0 0 0 -1+√5 -1-√5 0 0 1-√5 1+√5 0 0 0 0 0 0 -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 -ζ83ζ54+ζ83ζ5+ζ8ζ54-ζ8ζ5 ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 -ζ87ζ54+ζ87ζ5+ζ85ζ54-ζ85ζ5 complex faithful

Smallest permutation representation of D8⋊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)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 19)(20 24)(21 23)(25 31)(26 30)(27 29)(33 37)(34 36)(38 40)
(1 18 10 39 32)(2 19 11 40 25)(3 20 12 33 26)(4 21 13 34 27)(5 22 14 35 28)(6 23 15 36 29)(7 24 16 37 30)(8 17 9 38 31)
(1 32)(2 29)(3 26)(4 31)(5 28)(6 25)(7 30)(8 27)(9 13)(11 15)(17 34)(18 39)(19 36)(20 33)(21 38)(22 35)(23 40)(24 37)```

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

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

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

Matrix representation of D8⋊D5 in GL4(𝔽41) generated by

 0 0 11 31 0 0 10 30 19 20 22 21 21 22 20 19
,
 1 0 40 0 0 1 0 40 0 0 40 0 0 0 0 40
,
 6 1 0 0 40 0 0 0 0 0 6 1 0 0 40 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(41))| [0,0,19,21,0,0,20,22,11,10,22,20,31,30,21,19],[1,0,0,0,0,1,0,0,40,0,40,0,0,40,0,40],[6,40,0,0,1,0,0,0,0,0,6,40,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;`

D8⋊D5 in GAP, Magma, Sage, TeX

`D_8\rtimes D_5`
`% in TeX`

`G:=Group("D8:D5");`
`// GroupNames label`

`G:=SmallGroup(160,132);`
`// by ID`

`G=gap.SmallGroup(160,132);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,362,116,297,159,69,4613]);`
`// Polycyclic`

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

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