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## G = D8.D5order 160 = 25·5

### The non-split extension by D8 of D5 acting via D5/C5=C2

Aliases: D8.D5, C52SD32, C20.4D4, C10.9D8, C8.5D10, Dic203C2, C40.3C22, C52C162C2, (C5×D8).1C2, C2.5(D4⋊D5), C4.2(C5⋊D4), SmallGroup(160,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D8.D5
 Chief series C1 — C5 — C10 — C20 — C40 — Dic20 — D8.D5
 Lower central C5 — C10 — C20 — C40 — D8.D5
 Upper central C1 — C2 — C4 — C8 — D8

Generators and relations for D8.D5
G = < a,b,c,d | a8=b2=c5=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Character table of D8.D5

 class 1 2A 2B 4A 4B 5A 5B 8A 8B 10A 10B 10C 10D 10E 10F 16A 16B 16C 16D 20A 20B 40A 40B 40C 40D size 1 1 8 2 40 2 2 2 2 2 2 8 8 8 8 10 10 10 10 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 2 2 0 2 0 2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 2 2 -2 -2 -2 -2 orthogonal lifted from D4 ρ6 2 2 0 -2 0 2 2 0 0 2 2 0 0 0 0 -√2 √2 √2 -√2 -2 -2 0 0 0 0 orthogonal lifted from D8 ρ7 2 2 2 2 0 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -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 ρ8 2 2 -2 2 0 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 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 D10 ρ9 2 2 2 2 0 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -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 0 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 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 D10 ρ11 2 2 0 -2 0 2 2 0 0 2 2 0 0 0 0 √2 -√2 -√2 √2 -2 -2 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 0 2 0 -1-√5/2 -1+√5/2 -2 -2 -1-√5/2 -1+√5/2 ζ53-ζ52 ζ54-ζ5 -ζ53+ζ52 -ζ54+ζ5 0 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 complex lifted from C5⋊D4 ρ13 2 2 0 2 0 -1+√5/2 -1-√5/2 -2 -2 -1+√5/2 -1-√5/2 ζ54-ζ5 -ζ53+ζ52 -ζ54+ζ5 ζ53-ζ52 0 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 complex lifted from C5⋊D4 ρ14 2 2 0 2 0 -1-√5/2 -1+√5/2 -2 -2 -1-√5/2 -1+√5/2 -ζ53+ζ52 -ζ54+ζ5 ζ53-ζ52 ζ54-ζ5 0 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 complex lifted from C5⋊D4 ρ15 2 2 0 2 0 -1+√5/2 -1-√5/2 -2 -2 -1+√5/2 -1-√5/2 -ζ54+ζ5 ζ53-ζ52 ζ54-ζ5 -ζ53+ζ52 0 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 complex lifted from C5⋊D4 ρ16 2 -2 0 0 0 2 2 -√2 √2 -2 -2 0 0 0 0 ζ167+ζ16 ζ1613+ζ1611 ζ165+ζ163 ζ1615+ζ169 0 0 √2 -√2 -√2 √2 complex lifted from SD32 ρ17 2 -2 0 0 0 2 2 √2 -√2 -2 -2 0 0 0 0 ζ165+ζ163 ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 0 0 -√2 √2 √2 -√2 complex lifted from SD32 ρ18 2 -2 0 0 0 2 2 √2 -√2 -2 -2 0 0 0 0 ζ1613+ζ1611 ζ1615+ζ169 ζ167+ζ16 ζ165+ζ163 0 0 -√2 √2 √2 -√2 complex lifted from SD32 ρ19 2 -2 0 0 0 2 2 -√2 √2 -2 -2 0 0 0 0 ζ1615+ζ169 ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 0 0 √2 -√2 -√2 √2 complex lifted from SD32 ρ20 4 4 0 -4 0 -1+√5 -1-√5 0 0 -1+√5 -1-√5 0 0 0 0 0 0 0 0 1-√5 1+√5 0 0 0 0 orthogonal lifted from D4⋊D5, Schur index 2 ρ21 4 4 0 -4 0 -1-√5 -1+√5 0 0 -1-√5 -1+√5 0 0 0 0 0 0 0 0 1+√5 1-√5 0 0 0 0 orthogonal lifted from D4⋊D5, Schur index 2 ρ22 4 -4 0 0 0 -1-√5 -1+√5 -2√2 2√2 1+√5 1-√5 0 0 0 0 0 0 0 0 0 0 ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 symplectic faithful, Schur index 2 ρ23 4 -4 0 0 0 -1-√5 -1+√5 2√2 -2√2 1+√5 1-√5 0 0 0 0 0 0 0 0 0 0 ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 symplectic faithful, Schur index 2 ρ24 4 -4 0 0 0 -1+√5 -1-√5 -2√2 2√2 1-√5 1+√5 0 0 0 0 0 0 0 0 0 0 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 symplectic faithful, Schur index 2 ρ25 4 -4 0 0 0 -1+√5 -1-√5 2√2 -2√2 1-√5 1+√5 0 0 0 0 0 0 0 0 0 0 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 symplectic faithful, Schur index 2

Smallest permutation representation of D8.D5
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 8)(2 7)(3 6)(4 5)(10 16)(11 15)(12 14)(17 19)(20 24)(21 23)(25 29)(26 28)(30 32)(33 39)(34 38)(35 37)(41 46)(42 45)(43 44)(47 48)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 64)(62 63)(65 68)(66 67)(69 72)(70 71)(73 77)(74 76)(78 80)
(1 53 71 63 44)(2 54 72 64 45)(3 55 65 57 46)(4 56 66 58 47)(5 49 67 59 48)(6 50 68 60 41)(7 51 69 61 42)(8 52 70 62 43)(9 27 18 36 75)(10 28 19 37 76)(11 29 20 38 77)(12 30 21 39 78)(13 31 22 40 79)(14 32 23 33 80)(15 25 24 34 73)(16 26 17 35 74)
(1 32 5 28)(2 31 6 27)(3 30 7 26)(4 29 8 25)(9 54 13 50)(10 53 14 49)(11 52 15 56)(12 51 16 55)(17 46 21 42)(18 45 22 41)(19 44 23 48)(20 43 24 47)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)(65 78 69 74)(66 77 70 73)(67 76 71 80)(68 75 72 79)```

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

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

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

D8.D5 is a maximal subgroup of
D16⋊D5  D163D5  D5×SD32  SD32⋊D5  D8.D10  C40.30C23  C40.31C23  C15⋊SD32  D24.D5  D8.D15
D8.D5 is a maximal quotient of
C10.SD32  C10.Q32  C10.D16  C15⋊SD32  D24.D5  D8.D15

Matrix representation of D8.D5 in GL4(𝔽241) generated by

 11 230 0 0 11 11 0 0 0 0 240 0 0 0 0 240
,
 11 230 0 0 230 230 0 0 0 0 240 0 0 0 195 1
,
 1 0 0 0 0 1 0 0 0 0 98 0 0 0 161 91
,
 200 103 0 0 103 41 0 0 0 0 142 151 0 0 216 99
`G:=sub<GL(4,GF(241))| [11,11,0,0,230,11,0,0,0,0,240,0,0,0,0,240],[11,230,0,0,230,230,0,0,0,0,240,195,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,98,161,0,0,0,91],[200,103,0,0,103,41,0,0,0,0,142,216,0,0,151,99] >;`

D8.D5 in GAP, Magma, Sage, TeX

`D_8.D_5`
`% in TeX`

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

`G:=SmallGroup(160,34);`
`// by ID`

`G=gap.SmallGroup(160,34);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,73,218,116,122,579,297,69,4613]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^5=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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