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## G = C5⋊D16order 160 = 25·5

### The semidirect product of C5 and D16 acting via D16/D8=C2

Aliases: C52D16, D81D5, D403C2, C20.3D4, C10.8D8, C8.4D10, C40.2C22, (C5×D8)⋊1C2, C52C161C2, C2.4(D4⋊D5), C4.1(C5⋊D4), SmallGroup(160,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C5⋊D16
 Chief series C1 — C5 — C10 — C20 — C40 — D40 — C5⋊D16
 Lower central C5 — C10 — C20 — C40 — C5⋊D16
 Upper central C1 — C2 — C4 — C8 — D8

Generators and relations for C5⋊D16
G = < a,b,c | a5=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >

Character table of C5⋊D16

 class 1 2A 2B 2C 4 5A 5B 8A 8B 10A 10B 10C 10D 10E 10F 16A 16B 16C 16D 20A 20B 40A 40B 40C 40D size 1 1 8 40 2 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 0 2 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 0 -2 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 0 2 -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 0 0 0 2 2 √2 -√2 -2 -2 0 0 0 0 -ζ167+ζ16 -ζ165+ζ163 ζ165-ζ163 ζ167-ζ16 0 0 -√2 √2 √2 -√2 orthogonal lifted from D16 ρ9 2 2 -2 0 2 -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 ρ10 2 -2 0 0 0 2 2 -√2 √2 -2 -2 0 0 0 0 ζ165-ζ163 -ζ167+ζ16 ζ167-ζ16 -ζ165+ζ163 0 0 √2 -√2 -√2 √2 orthogonal lifted from D16 ρ11 2 2 2 0 2 -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 ρ12 2 2 -2 0 2 -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 ρ13 2 2 0 0 -2 2 2 0 0 2 2 0 0 0 0 -√2 √2 √2 -√2 -2 -2 0 0 0 0 orthogonal lifted from D8 ρ14 2 -2 0 0 0 2 2 √2 -√2 -2 -2 0 0 0 0 ζ167-ζ16 ζ165-ζ163 -ζ165+ζ163 -ζ167+ζ16 0 0 -√2 √2 √2 -√2 orthogonal lifted from D16 ρ15 2 -2 0 0 0 2 2 -√2 √2 -2 -2 0 0 0 0 -ζ165+ζ163 ζ167-ζ16 -ζ167+ζ16 ζ165-ζ163 0 0 √2 -√2 -√2 √2 orthogonal lifted from D16 ρ16 2 2 0 0 2 -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 ρ17 2 2 0 0 2 -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 ρ18 2 2 0 0 2 -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 ρ19 2 2 0 0 2 -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 ρ20 4 4 0 0 -4 -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 0 -4 -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 orthogonal 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 orthogonal 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 orthogonal 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 orthogonal faithful, Schur index 2

Smallest permutation representation of C5⋊D16
On 80 points
Generators in S80
```(1 63 27 44 73)(2 74 45 28 64)(3 49 29 46 75)(4 76 47 30 50)(5 51 31 48 77)(6 78 33 32 52)(7 53 17 34 79)(8 80 35 18 54)(9 55 19 36 65)(10 66 37 20 56)(11 57 21 38 67)(12 68 39 22 58)(13 59 23 40 69)(14 70 41 24 60)(15 61 25 42 71)(16 72 43 26 62)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)```

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

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

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

C5⋊D16 is a maximal subgroup of
D5×D16  D16⋊D5  C16⋊D10  SD323D5  D8.D10  D8⋊D10  C40.30C23  C15⋊D16  C5⋊D48  C157D16
C5⋊D16 is a maximal quotient of
C40.2Q8  C40.5D4  C5⋊D32  D16.D5  C5⋊SD64  C5⋊Q64  C10.D16  C15⋊D16  C5⋊D48  C157D16

Matrix representation of C5⋊D16 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 240 1 0 0 50 190
,
 58 54 0 0 214 112 0 0 0 0 69 27 0 0 20 172
,
 1 0 0 0 1 240 0 0 0 0 51 1 0 0 51 190
`G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,50,0,0,1,190],[58,214,0,0,54,112,0,0,0,0,69,20,0,0,27,172],[1,1,0,0,0,240,0,0,0,0,51,51,0,0,1,190] >;`

C5⋊D16 in GAP, Magma, Sage, TeX

`C_5\rtimes D_{16}`
`% in TeX`

`G:=Group("C5:D16");`
`// GroupNames label`

`G:=SmallGroup(160,33);`
`// by ID`

`G=gap.SmallGroup(160,33);`
`# by ID`

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

`G:=Group<a,b,c|a^5=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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