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## G = D10⋊C8order 160 = 25·5

### 2nd semidirect product of D10 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D10⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — D10⋊C8
 Lower central C5 — C10 — D10⋊C8
 Upper central C1 — C22 — C2×C4

Generators and relations for D10⋊C8
G = < a,b,c | a10=b2=c8=1, bab=a-1, cac-1=a3, cbc-1=a7b >

Character table of D10⋊C8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 20A 20B 20C 20D size 1 1 1 1 10 10 2 2 5 5 5 5 4 10 10 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 -1 -1 -1 -1 1 -i -i i i i i -i -i 1 1 1 -1 -1 -1 -1 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 -1 -1 -1 -1 linear of order 4 ρ7 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 -i i -i -i i i -i i 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 i -i i i -i -i i -i 1 1 1 1 1 1 1 linear of order 4 ρ9 1 -1 1 -1 -1 1 -i i i -i i -i 1 ζ87 ζ87 ζ8 ζ85 ζ85 ζ8 ζ83 ζ83 -1 -1 1 i -i -i i linear of order 8 ρ10 1 -1 1 -1 -1 1 -i i i -i i -i 1 ζ83 ζ83 ζ85 ζ8 ζ8 ζ85 ζ87 ζ87 -1 -1 1 i -i -i i linear of order 8 ρ11 1 -1 1 -1 -1 1 i -i -i i -i i 1 ζ8 ζ8 ζ87 ζ83 ζ83 ζ87 ζ85 ζ85 -1 -1 1 -i i i -i linear of order 8 ρ12 1 -1 1 -1 -1 1 i -i -i i -i i 1 ζ85 ζ85 ζ83 ζ87 ζ87 ζ83 ζ8 ζ8 -1 -1 1 -i i i -i linear of order 8 ρ13 1 -1 1 -1 1 -1 i -i i -i i -i 1 ζ83 ζ87 ζ8 ζ85 ζ8 ζ85 ζ87 ζ83 -1 -1 1 -i i i -i linear of order 8 ρ14 1 -1 1 -1 1 -1 -i i -i i -i i 1 ζ8 ζ85 ζ83 ζ87 ζ83 ζ87 ζ85 ζ8 -1 -1 1 i -i -i i linear of order 8 ρ15 1 -1 1 -1 1 -1 -i i -i i -i i 1 ζ85 ζ8 ζ87 ζ83 ζ87 ζ83 ζ8 ζ85 -1 -1 1 i -i -i i linear of order 8 ρ16 1 -1 1 -1 1 -1 i -i i -i i -i 1 ζ87 ζ83 ζ85 ζ8 ζ85 ζ8 ζ83 ζ87 -1 -1 1 -i i i -i linear of order 8 ρ17 2 2 -2 -2 0 0 0 0 -2 2 2 -2 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 0 0 0 0 2 -2 -2 2 2 0 0 0 0 0 0 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 -2 2 0 0 0 0 2i 2i -2i -2i 2 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 complex lifted from M4(2) ρ20 2 -2 -2 2 0 0 0 0 -2i -2i 2i 2i 2 0 0 0 0 0 0 0 0 -2 2 -2 0 0 0 0 complex lifted from M4(2) ρ21 4 4 4 4 0 0 4 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ22 4 4 4 4 0 0 -4 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 -√5 √5 -√5 √5 orthogonal lifted from C22⋊F5 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 √5 -√5 √5 -√5 orthogonal lifted from C22⋊F5 ρ25 4 -4 4 -4 0 0 4i -4i 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 i -i -i i complex lifted from D5⋊C8, Schur index 2 ρ26 4 -4 4 -4 0 0 -4i 4i 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 -i i i -i complex lifted from D5⋊C8, Schur index 2 ρ27 4 -4 -4 4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 √-5 √-5 -√-5 -√-5 complex lifted from C4.F5 ρ28 4 -4 -4 4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 -√-5 -√-5 √-5 √-5 complex lifted from C4.F5

Smallest permutation representation of D10⋊C8
On 80 points
Generators in S80
```(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)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 80)(19 79)(20 78)(31 44)(32 43)(33 42)(34 41)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 70)(60 69)
(1 14 46 65 21 80 40 60)(2 11 45 68 22 77 39 53)(3 18 44 61 23 74 38 56)(4 15 43 64 24 71 37 59)(5 12 42 67 25 78 36 52)(6 19 41 70 26 75 35 55)(7 16 50 63 27 72 34 58)(8 13 49 66 28 79 33 51)(9 20 48 69 29 76 32 54)(10 17 47 62 30 73 31 57)```

`G:=sub<Sym(80)| (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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,80)(19,79)(20,78)(31,44)(32,43)(33,42)(34,41)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,70)(60,69), (1,14,46,65,21,80,40,60)(2,11,45,68,22,77,39,53)(3,18,44,61,23,74,38,56)(4,15,43,64,24,71,37,59)(5,12,42,67,25,78,36,52)(6,19,41,70,26,75,35,55)(7,16,50,63,27,72,34,58)(8,13,49,66,28,79,33,51)(9,20,48,69,29,76,32,54)(10,17,47,62,30,73,31,57)>;`

`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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,80)(19,79)(20,78)(31,44)(32,43)(33,42)(34,41)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,70)(60,69), (1,14,46,65,21,80,40,60)(2,11,45,68,22,77,39,53)(3,18,44,61,23,74,38,56)(4,15,43,64,24,71,37,59)(5,12,42,67,25,78,36,52)(6,19,41,70,26,75,35,55)(7,16,50,63,27,72,34,58)(8,13,49,66,28,79,33,51)(9,20,48,69,29,76,32,54)(10,17,47,62,30,73,31,57) );`

`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),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,77),(12,76),(13,75),(14,74),(15,73),(16,72),(17,71),(18,80),(19,79),(20,78),(31,44),(32,43),(33,42),(34,41),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(51,68),(52,67),(53,66),(54,65),(55,64),(56,63),(57,62),(58,61),(59,70),(60,69)], [(1,14,46,65,21,80,40,60),(2,11,45,68,22,77,39,53),(3,18,44,61,23,74,38,56),(4,15,43,64,24,71,37,59),(5,12,42,67,25,78,36,52),(6,19,41,70,26,75,35,55),(7,16,50,63,27,72,34,58),(8,13,49,66,28,79,33,51),(9,20,48,69,29,76,32,54),(10,17,47,62,30,73,31,57)]])`

Matrix representation of D10⋊C8 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 1 0 0 0 0 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40 0
,
 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 40 0 0 0 0 0 0 0 0 0 1 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 39 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 14 27 23 0 0 0 0 0 37 27 0 14 0 0 0 0 14 0 27 37 0 0 0 0 0 23 27 14

`G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,5,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,14,37,14,0,0,0,0,0,27,27,0,23,0,0,0,0,23,0,27,27,0,0,0,0,0,14,37,14] >;`

D10⋊C8 in GAP, Magma, Sage, TeX

`D_{10}\rtimes C_8`
`% in TeX`

`G:=Group("D10:C8");`
`// GroupNames label`

`G:=SmallGroup(160,78);`
`// by ID`

`G=gap.SmallGroup(160,78);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,86,2309,1169]);`
`// Polycyclic`

`G:=Group<a,b,c|a^10=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^7*b>;`
`// generators/relations`

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