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## G = Q16⋊D5order 160 = 25·5

### 2nd semidirect product of Q16 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Q16⋊D5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — Q8×D5 — Q16⋊D5
 Lower central C5 — C10 — C20 — Q16⋊D5
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q16⋊D5
G = < a,b,c,d | a8=c5=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 208 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, Q8, D5, C10, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C8.C22, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C5×Q8, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q82D5, Q16⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8.C22, C22×D5, D4×D5, Q16⋊D5

Character table of Q16⋊D5

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 8A 8B 10A 10B 20A 20B 20C 20D 20E 20F 40A 40B 40C 40D size 1 1 10 20 2 4 4 10 20 2 2 4 20 2 2 4 4 8 8 8 8 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 2 0 -2 0 0 -2 0 2 2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 -2 0 0 2 0 2 2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 0 0 2 -2 -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 ρ12 2 2 0 0 2 2 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 ρ13 2 2 0 0 2 2 -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 0 0 2 -2 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 0 0 2 -2 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 0 0 2 2 -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 0 0 2 -2 -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 0 0 2 2 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 -4 0 0 0 0 -1-√5 -1+√5 0 0 -1+√5 -1-√5 1+√5 1-√5 0 0 0 0 0 0 0 0 orthogonal lifted from D4×D5 ρ20 4 4 0 0 -4 0 0 0 0 -1+√5 -1-√5 0 0 -1-√5 -1+√5 1-√5 1+√5 0 0 0 0 0 0 0 0 orthogonal lifted from D4×D5 ρ21 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 symplectic lifted from C8.C22, Schur index 2 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 ζ83ζ54-ζ83ζ5-ζ8ζ54+ζ8ζ5 -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 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 ζ87ζ54-ζ87ζ5-ζ85ζ54+ζ85ζ5 ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 -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 -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 ζ87ζ54-ζ87ζ5-ζ85ζ54+ζ85ζ5 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 ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 ζ83ζ54-ζ83ζ5-ζ8ζ54+ζ8ζ5 ζ87ζ54-ζ87ζ5-ζ85ζ54+ζ85ζ5 complex faithful

Smallest permutation representation of Q16⋊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 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(17 45 21 41)(18 44 22 48)(19 43 23 47)(20 42 24 46)(25 75 29 79)(26 74 30 78)(27 73 31 77)(28 80 32 76)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
(1 66 12 41 75)(2 67 13 42 76)(3 68 14 43 77)(4 69 15 44 78)(5 70 16 45 79)(6 71 9 46 80)(7 72 10 47 73)(8 65 11 48 74)(17 29 60 50 40)(18 30 61 51 33)(19 31 62 52 34)(20 32 63 53 35)(21 25 64 54 36)(22 26 57 55 37)(23 27 58 56 38)(24 28 59 49 39)
(1 25)(2 30)(3 27)(4 32)(5 29)(6 26)(7 31)(8 28)(9 37)(10 34)(11 39)(12 36)(13 33)(14 38)(15 35)(16 40)(17 70)(18 67)(19 72)(20 69)(21 66)(22 71)(23 68)(24 65)(41 54)(42 51)(43 56)(44 53)(45 50)(46 55)(47 52)(48 49)(57 80)(58 77)(59 74)(60 79)(61 76)(62 73)(63 78)(64 75)```

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

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

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

Matrix representation of Q16⋊D5 in GL4(𝔽41) generated by

 0 0 39 17 0 0 24 2 1 12 39 17 29 40 24 2
,
 17 11 2 0 30 22 0 2 18 11 24 30 30 23 11 19
,
 0 1 0 0 40 34 0 0 0 0 0 1 0 0 40 34
,
 30 22 0 2 17 11 2 0 0 40 30 24 40 0 19 11
`G:=sub<GL(4,GF(41))| [0,0,1,29,0,0,12,40,39,24,39,24,17,2,17,2],[17,30,18,30,11,22,11,23,2,0,24,11,0,2,30,19],[0,40,0,0,1,34,0,0,0,0,0,40,0,0,1,34],[30,17,0,40,22,11,40,0,0,2,30,19,2,0,24,11] >;`

Q16⋊D5 in GAP, Magma, Sage, TeX

`Q_{16}\rtimes D_5`
`% in TeX`

`G:=Group("Q16:D5");`
`// GroupNames label`

`G:=SmallGroup(160,139);`
`// by ID`

`G=gap.SmallGroup(160,139);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,362,116,86,297,159,69,4613]);`
`// Polycyclic`

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

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