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## G = D8⋊3D5order 160 = 25·5

### The semidirect product of D8 and D5 acting through Inn(D8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D8⋊3D5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D4⋊2D5 — D8⋊3D5
 Lower central C5 — C10 — C20 — D8⋊3D5
 Upper central C1 — C2 — C4 — D8

Generators and relations for D83D5
G = < a,b,c,d | a8=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 200 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, D4, Q8, D5, C10, C10, C2×C8, D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C4○D8, C52C8, C40, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C8×D5, Dic20, D4.D5, C5×D8, D42D5, D83D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4○D8, C22×D5, D4×D5, D83D5

Character table of D83D5

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 20A 20B 40A 40B 40C 40D size 1 1 4 4 10 2 5 5 20 20 2 2 2 2 10 10 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 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 -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 -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 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 1 1 1 linear of order 2 ρ9 2 2 0 0 2 -2 -2 -2 0 0 2 2 0 0 0 0 2 2 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 0 -2 -2 2 2 0 0 2 2 0 0 0 0 2 2 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 0 2 0 0 0 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 orthogonal lifted from D10 ρ12 2 2 -2 2 0 2 0 0 0 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 orthogonal lifted from D10 ρ13 2 2 -2 -2 0 2 0 0 0 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 orthogonal lifted from D10 ρ14 2 2 2 2 0 2 0 0 0 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 orthogonal lifted from D5 ρ15 2 2 2 2 0 2 0 0 0 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 orthogonal lifted from D5 ρ16 2 2 -2 -2 0 2 0 0 0 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 orthogonal lifted from D10 ρ17 2 2 2 -2 0 2 0 0 0 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 orthogonal lifted from D10 ρ18 2 2 2 -2 0 2 0 0 0 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 orthogonal lifted from D10 ρ19 2 -2 0 0 0 0 -2i 2i 0 0 2 2 -√2 √2 √-2 -√-2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 complex lifted from C4○D8 ρ20 2 -2 0 0 0 0 2i -2i 0 0 2 2 √2 -√2 √-2 -√-2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 complex lifted from C4○D8 ρ21 2 -2 0 0 0 0 2i -2i 0 0 2 2 -√2 √2 -√-2 √-2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 complex lifted from C4○D8 ρ22 2 -2 0 0 0 0 -2i 2i 0 0 2 2 √2 -√2 -√-2 √-2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 complex lifted from C4○D8 ρ23 4 4 0 0 0 -4 0 0 0 0 -1+√5 -1-√5 0 0 0 0 -1-√5 -1+√5 0 0 0 0 1+√5 1-√5 0 0 0 0 orthogonal lifted from D4×D5 ρ24 4 4 0 0 0 -4 0 0 0 0 -1-√5 -1+√5 0 0 0 0 -1+√5 -1-√5 0 0 0 0 1-√5 1+√5 0 0 0 0 orthogonal lifted from D4×D5 ρ25 4 -4 0 0 0 0 0 0 0 0 -1+√5 -1-√5 -2√2 2√2 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 symplectic faithful, Schur index 2 ρ26 4 -4 0 0 0 0 0 0 0 0 -1-√5 -1+√5 2√2 -2√2 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 symplectic faithful, Schur index 2 ρ27 4 -4 0 0 0 0 0 0 0 0 -1-√5 -1+√5 -2√2 2√2 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 symplectic faithful, Schur index 2 ρ28 4 -4 0 0 0 0 0 0 0 0 -1+√5 -1-√5 2√2 -2√2 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 symplectic faithful, Schur index 2

Smallest permutation representation of D83D5
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 74)(2 73)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)(33 64)(34 63)(35 62)(36 61)(37 60)(38 59)(39 58)(40 57)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 72)(56 71)
(1 66 12 28 60)(2 67 13 29 61)(3 68 14 30 62)(4 69 15 31 63)(5 70 16 32 64)(6 71 9 25 57)(7 72 10 26 58)(8 65 11 27 59)(17 38 75 54 46)(18 39 76 55 47)(19 40 77 56 48)(20 33 78 49 41)(21 34 79 50 42)(22 35 80 51 43)(23 36 73 52 44)(24 37 74 53 45)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(25 71)(26 72)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(41 45)(42 46)(43 47)(44 48)```

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

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

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

Matrix representation of D83D5 in GL4(𝔽41) generated by

 27 0 0 0 0 38 0 0 0 0 40 0 0 0 0 40
,
 0 38 0 0 27 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 6 1 0 0 40 0
,
 1 0 0 0 0 40 0 0 0 0 1 6 0 0 0 40
`G:=sub<GL(4,GF(41))| [27,0,0,0,0,38,0,0,0,0,40,0,0,0,0,40],[0,27,0,0,38,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,40,0,0,1,0],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,6,40] >;`

D83D5 in GAP, Magma, Sage, TeX

`D_8\rtimes_3D_5`
`% in TeX`

`G:=Group("D8:3D5");`
`// GroupNames label`

`G:=SmallGroup(160,133);`
`// by ID`

`G=gap.SmallGroup(160,133);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,362,116,297,159,69,4613]);`
`// 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,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

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