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## G = D10.Q16order 320 = 26·5

### 2nd non-split extension by D10 of Q16 acting via Q16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D10.Q16
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.3Q8 — D10.Q16
 Lower central C5 — C2×C10 — C2×C20 — D10.Q16
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for D10.Q16
G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, dbd-1=a5b, dcd-1=a4bc-1 >

Subgroups: 410 in 80 conjugacy classes, 24 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, Dic5, C20, F5, D10, D10, C2×C10, C2.C42, C22⋊C8, C22⋊Q8, C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C2×F5, C22×D5, C23.31D4, C10.D4, C4⋊Dic5, D10⋊C4, C2×C5⋊C8, C2×C4×D5, Q8×C10, C22×F5, D10⋊C8, D10.3Q8, D103Q8, D10.Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, SD16, Q16, F5, C23⋊C4, Q8⋊C4, C4≀C2, C2×F5, C23.31D4, C22⋊F5, Q8⋊F5, Q82F5, C23⋊F5, D10.Q16

Character table of D10.Q16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 5 8A 8B 8C 8D 10A 10B 10C 20A 20B 20C 20D 20E 20F size 1 1 1 1 10 10 4 8 10 10 20 20 20 20 40 4 20 20 20 20 4 4 4 8 8 8 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 -1 -1 -i i i -i -1 1 -i i i -i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 1 -1 -1 1 1 -1 -1 i -i -i i -1 1 i -i -i i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 1 -1 -1 1 -1 -1 -1 i -i -i i 1 1 -i i i -i 1 1 1 -1 1 1 -1 -1 -1 linear of order 4 ρ8 1 1 1 1 -1 -1 1 -1 -1 -1 -i i i -i 1 1 i -i -i i 1 1 1 -1 1 1 -1 -1 -1 linear of order 4 ρ9 2 2 2 2 2 2 -2 0 -2 -2 0 0 0 0 0 2 0 0 0 0 2 2 2 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 -2 0 2 2 0 0 0 0 0 2 0 0 0 0 2 2 2 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 2 -√2 -√2 √2 √2 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ12 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 2 √2 √2 -√2 -√2 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ13 2 2 -2 -2 0 0 0 0 -2i 2i -1-i 1-i -1+i 1+i 0 2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4≀C2 ρ14 2 2 -2 -2 0 0 0 0 2i -2i -1+i 1+i -1-i 1-i 0 2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4≀C2 ρ15 2 2 -2 -2 0 0 0 0 -2i 2i 1+i -1+i 1-i -1-i 0 2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4≀C2 ρ16 2 2 -2 -2 0 0 0 0 2i -2i 1-i -1-i 1+i -1+i 0 2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4≀C2 ρ17 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 2 √-2 -√-2 √-2 -√-2 2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ18 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 2 -√-2 √-2 -√-2 √-2 2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ19 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 -4 4 -4 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ20 4 4 4 4 0 0 4 4 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ21 4 4 4 4 0 0 4 -4 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 1 -1 -1 1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 4 4 0 0 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 √5 1 1 √5 -√5 -√5 orthogonal lifted from C22⋊F5 ρ23 4 4 4 4 0 0 -4 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 -√5 1 1 -√5 √5 √5 orthogonal lifted from C22⋊F5 ρ24 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 1 2ζ53+2ζ5+1 √5 -√5 2ζ54+2ζ52+1 2ζ54+2ζ53+1 2ζ52+2ζ5+1 complex lifted from C23⋊F5 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 1 2ζ54+2ζ52+1 √5 -√5 2ζ53+2ζ5+1 2ζ52+2ζ5+1 2ζ54+2ζ53+1 complex lifted from C23⋊F5 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 1 2ζ52+2ζ5+1 -√5 √5 2ζ54+2ζ53+1 2ζ53+2ζ5+1 2ζ54+2ζ52+1 complex lifted from C23⋊F5 ρ27 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 1 2ζ54+2ζ53+1 -√5 √5 2ζ52+2ζ5+1 2ζ54+2ζ52+1 2ζ53+2ζ5+1 complex lifted from C23⋊F5 ρ28 8 8 -8 -8 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 2 2 -2 0 0 0 0 0 0 orthogonal lifted from Q8⋊2F5 ρ29 8 -8 8 -8 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from Q8⋊F5, Schur index 2

Smallest permutation representation of D10.Q16
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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 75)(12 74)(13 73)(14 72)(15 71)(16 80)(17 79)(18 78)(19 77)(20 76)(31 44)(32 43)(33 42)(34 41)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 70)
(1 70 50 11 30 56 36 71)(2 67 49 14 21 53 35 74)(3 64 48 17 22 60 34 77)(4 61 47 20 23 57 33 80)(5 68 46 13 24 54 32 73)(6 65 45 16 25 51 31 76)(7 62 44 19 26 58 40 79)(8 69 43 12 27 55 39 72)(9 66 42 15 28 52 38 75)(10 63 41 18 29 59 37 78)
(1 70 30 56)(2 61 21 57)(3 62 22 58)(4 63 23 59)(5 64 24 60)(6 65 25 51)(7 66 26 52)(8 67 27 53)(9 68 28 54)(10 69 29 55)(11 45 71 31)(12 46 72 32)(13 47 73 33)(14 48 74 34)(15 49 75 35)(16 50 76 36)(17 41 77 37)(18 42 78 38)(19 43 79 39)(20 44 80 40)```

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

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

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

Matrix representation of D10.Q16 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 40 40 40 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 0 39 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
,
 4 4 0 0 0 0 0 0 4 37 0 0 0 0 0 0 0 0 12 29 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 22 0 3 3 0 0 0 0 3 3 0 22 0 0 0 0 38 19 38 0 0 0 0 0 19 22 22 19
,
 5 5 0 0 0 0 0 0 3 36 0 0 0 0 0 0 0 0 15 15 0 0 0 0 0 0 15 26 0 0 0 0 0 0 0 0 19 0 38 38 0 0 0 0 3 22 3 0 0 0 0 0 0 3 22 3 0 0 0 0 38 38 0 19

`G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0],[1,39,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[4,4,0,0,0,0,0,0,4,37,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,29,12,0,0,0,0,0,0,0,0,22,3,38,19,0,0,0,0,0,3,19,22,0,0,0,0,3,0,38,22,0,0,0,0,3,22,0,19],[5,3,0,0,0,0,0,0,5,36,0,0,0,0,0,0,0,0,15,15,0,0,0,0,0,0,15,26,0,0,0,0,0,0,0,0,19,3,0,38,0,0,0,0,0,22,3,38,0,0,0,0,38,3,22,0,0,0,0,0,38,0,3,19] >;`

D10.Q16 in GAP, Magma, Sage, TeX

`D_{10}.Q_{16}`
`% in TeX`

`G:=Group("D10.Q16");`
`// GroupNames label`

`G:=SmallGroup(320,264);`
`// by ID`

`G=gap.SmallGroup(320,264);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,1571,570,136,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^7*b,d*b*d^-1=a^5*b,d*c*d^-1=a^4*b*c^-1>;`
`// generators/relations`

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