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## G = D5.D16order 320 = 26·5

### The non-split extension by D5 of D16 acting via D16/D8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D5.D16
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C8×D5 — D5.D8 — D5.D16
 Lower central C5 — C10 — C20 — C40 — D5.D16
 Upper central C1 — C2 — C4 — C8 — D8

Generators and relations for D5.D16
G = < a,b,c,d | a5=b2=c16=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=dbd-1=a2b, dcd-1=a-1bc-1 >

Subgroups: 458 in 66 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, C23, D5, D5, C10, C10, C16, C4⋊C4, C2×C8, D8, D8, C2×D4, Dic5, C20, F5, D10, D10, C2×C10, C2.D8, C2×C16, C2×D8, C52C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C22×D5, C2.D16, C5⋊C16, C8×D5, D40, D4⋊D5, C5×D8, C4⋊F5, D4×D5, D5⋊C16, D5.D8, D5×D8, D5.D16
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, F5, D4⋊C4, D16, SD32, C2×F5, C2.D16, C22⋊F5, D20⋊C4, D5.D16

Character table of D5.D16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5 8A 8B 8C 8D 10A 10B 10C 16A 16B 16C 16D 16E 16F 16G 16H 20 40A 40B size 1 1 5 5 8 40 2 10 40 40 4 2 2 10 10 4 16 16 10 10 10 10 10 10 10 10 8 8 8 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 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 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 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 1 linear of order 2 ρ5 1 1 -1 -1 1 -1 1 -1 -i i 1 1 1 -1 -1 1 1 1 -i i i i i -i -i -i 1 1 1 linear of order 4 ρ6 1 1 -1 -1 1 -1 1 -1 i -i 1 1 1 -1 -1 1 1 1 i -i -i -i -i i i i 1 1 1 linear of order 4 ρ7 1 1 -1 -1 -1 1 1 -1 -i i 1 1 1 -1 -1 1 -1 -1 i -i -i -i -i i i i 1 1 1 linear of order 4 ρ8 1 1 -1 -1 -1 1 1 -1 i -i 1 1 1 -1 -1 1 -1 -1 -i i i i i -i -i -i 1 1 1 linear of order 4 ρ9 2 2 -2 -2 0 0 2 -2 0 0 2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 2 -2 -2 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 2 2 0 0 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 2 -2 -2 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -2 -2 0 0 2 0 0 0 0 2 0 0 -√2 √2 √2 -√2 -√2 √2 √2 -√2 -2 0 0 orthogonal lifted from D8 ρ12 2 2 2 2 0 0 -2 -2 0 0 2 0 0 0 0 2 0 0 √2 -√2 -√2 √2 √2 -√2 -√2 √2 -2 0 0 orthogonal lifted from D8 ρ13 2 -2 2 -2 0 0 0 0 0 0 2 -√2 √2 -√2 √2 -2 0 0 ζ165-ζ163 -ζ167+ζ16 ζ167-ζ16 -ζ165+ζ163 ζ165-ζ163 -ζ167+ζ16 ζ167-ζ16 -ζ165+ζ163 0 -√2 √2 orthogonal lifted from D16 ρ14 2 -2 2 -2 0 0 0 0 0 0 2 √2 -√2 √2 -√2 -2 0 0 ζ167-ζ16 ζ165-ζ163 -ζ165+ζ163 -ζ167+ζ16 ζ167-ζ16 ζ165-ζ163 -ζ165+ζ163 -ζ167+ζ16 0 √2 -√2 orthogonal lifted from D16 ρ15 2 -2 2 -2 0 0 0 0 0 0 2 -√2 √2 -√2 √2 -2 0 0 -ζ165+ζ163 ζ167-ζ16 -ζ167+ζ16 ζ165-ζ163 -ζ165+ζ163 ζ167-ζ16 -ζ167+ζ16 ζ165-ζ163 0 -√2 √2 orthogonal lifted from D16 ρ16 2 -2 2 -2 0 0 0 0 0 0 2 √2 -√2 √2 -√2 -2 0 0 -ζ167+ζ16 -ζ165+ζ163 ζ165-ζ163 ζ167-ζ16 -ζ167+ζ16 -ζ165+ζ163 ζ165-ζ163 ζ167-ζ16 0 √2 -√2 orthogonal lifted from D16 ρ17 2 2 -2 -2 0 0 -2 2 0 0 2 0 0 0 0 2 0 0 -√-2 -√-2 -√-2 √-2 √-2 √-2 √-2 -√-2 -2 0 0 complex lifted from SD16 ρ18 2 -2 -2 2 0 0 0 0 0 0 2 -√2 √2 √2 -√2 -2 0 0 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 ζ1615+ζ169 0 -√2 √2 complex lifted from SD32 ρ19 2 2 -2 -2 0 0 -2 2 0 0 2 0 0 0 0 2 0 0 √-2 √-2 √-2 -√-2 -√-2 -√-2 -√-2 √-2 -2 0 0 complex lifted from SD16 ρ20 2 -2 -2 2 0 0 0 0 0 0 2 √2 -√2 -√2 √2 -2 0 0 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 0 √2 -√2 complex lifted from SD32 ρ21 2 -2 -2 2 0 0 0 0 0 0 2 √2 -√2 -√2 √2 -2 0 0 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 ζ165+ζ163 0 √2 -√2 complex lifted from SD32 ρ22 2 -2 -2 2 0 0 0 0 0 0 2 -√2 √2 √2 -√2 -2 0 0 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 0 -√2 √2 complex lifted from SD32 ρ23 4 4 0 0 -4 0 4 0 0 0 -1 4 4 0 0 -1 1 1 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from C2×F5 ρ24 4 4 0 0 4 0 4 0 0 0 -1 4 4 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from F5 ρ25 4 4 0 0 0 0 4 0 0 0 -1 -4 -4 0 0 -1 -√5 √5 0 0 0 0 0 0 0 0 -1 1 1 orthogonal lifted from C22⋊F5 ρ26 4 4 0 0 0 0 4 0 0 0 -1 -4 -4 0 0 -1 √5 -√5 0 0 0 0 0 0 0 0 -1 1 1 orthogonal lifted from C22⋊F5 ρ27 8 8 0 0 0 0 -8 0 0 0 -2 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 0 0 orthogonal lifted from D20⋊C4, Schur index 2 ρ28 8 -8 0 0 0 0 0 0 0 0 -2 -4√2 4√2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 orthogonal faithful, Schur index 2 ρ29 8 -8 0 0 0 0 0 0 0 0 -2 4√2 -4√2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 orthogonal faithful, Schur index 2

Smallest permutation representation of D5.D16
On 80 points
Generators in S80
```(1 24 40 62 73)(2 63 25 74 41)(3 75 64 42 26)(4 43 76 27 49)(5 28 44 50 77)(6 51 29 78 45)(7 79 52 46 30)(8 47 80 31 53)(9 32 48 54 65)(10 55 17 66 33)(11 67 56 34 18)(12 35 68 19 57)(13 20 36 58 69)(14 59 21 70 37)(15 71 60 38 22)(16 39 72 23 61)
(1 73)(2 41)(3 26)(4 49)(5 77)(6 45)(7 30)(8 53)(9 65)(10 33)(11 18)(12 57)(13 69)(14 37)(15 22)(16 61)(19 35)(20 58)(23 39)(24 62)(27 43)(28 50)(31 47)(32 54)(34 67)(38 71)(42 75)(46 79)(51 78)(55 66)(59 70)(63 74)
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 30 66 79)(18 78 67 29)(19 28 68 77)(20 76 69 27)(21 26 70 75)(22 74 71 25)(23 24 72 73)(31 32 80 65)(33 46 55 52)(34 51 56 45)(35 44 57 50)(36 49 58 43)(37 42 59 64)(38 63 60 41)(39 40 61 62)(47 48 53 54)```

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

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

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

Matrix representation of D5.D16 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 240 189 0 0 0 0 0 0 240 189 0 0 0 0 52 52
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 240 189 0 0 0 0 0 1
,
 144 161 0 0 0 0 36 62 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 0 0 0 0 0 52 1 0 0
,
 144 161 0 0 0 0 208 97 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 189 240 0 0

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,1,189,0,0,0,0,0,0,240,52,0,0,0,0,189,52],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,189,1],[144,36,0,0,0,0,161,62,0,0,0,0,0,0,0,0,240,52,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[144,208,0,0,0,0,161,97,0,0,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0] >;`

D5.D16 in GAP, Magma, Sage, TeX

`D_5.D_{16}`
`% in TeX`

`G:=Group("D5.D16");`
`// GroupNames label`

`G:=SmallGroup(320,242);`
`// by ID`

`G=gap.SmallGroup(320,242);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,675,346,192,1684,851,102,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^2=c^16=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^-1*b*c^-1>;`
`// generators/relations`

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