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## G = Q16⋊D10order 320 = 26·5

### 4th semidirect product of Q16 and D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Q16⋊D10
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — D5×C4○D4 — Q16⋊D10
 Lower central C5 — C10 — C20 — Q16⋊D10
 Upper central C1 — C4 — C2×C4 — C4○D8

Generators and relations for Q16⋊D10
G = < a,b,c,d | a8=c10=d2=1, b2=a4, bab-1=cac-1=a-1, dad=a3, cbc-1=a6b, dbd=a2b, dcd=c-1 >

Subgroups: 1022 in 262 conjugacy classes, 99 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), D8, D8, SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×M4(2), C4○D8, C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, D8⋊C22, C8⋊D5, C40⋊C2, D40, Dic20, C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, C2×C8⋊D5, D407C2, D8⋊D5, D40⋊C2, SD16⋊D5, Q16⋊D5, D4.8D10, C5×C4○D8, D5×C4○D4, Q16⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, D8⋊C22, D4×D5, C23×D5, C2×D4×D5, Q16⋊D10

Smallest permutation representation of Q16⋊D10
On 80 points
Generators in S80
```(1 50 13 38 33 18 45 6)(2 7 46 19 34 39 14 41)(3 42 15 40 35 20 47 8)(4 9 48 11 36 31 16 43)(5 44 17 32 37 12 49 10)(21 26 78 68 55 60 63 73)(22 74 64 51 56 69 79 27)(23 28 80 70 57 52 65 75)(24 76 66 53 58 61 71 29)(25 30 72 62 59 54 67 77)
(1 68 33 73)(2 51 34 27)(3 70 35 75)(4 53 36 29)(5 62 37 77)(6 55 38 21)(7 64 39 79)(8 57 40 23)(9 66 31 71)(10 59 32 25)(11 24 43 58)(12 67 44 72)(13 26 45 60)(14 69 46 74)(15 28 47 52)(16 61 48 76)(17 30 49 54)(18 63 50 78)(19 22 41 56)(20 65 42 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 72)(2 71)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 30)(19 29)(20 28)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)(37 63)(38 62)(39 61)(40 70)(41 53)(42 52)(43 51)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 40A ··· 40H order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 2 4 4 10 10 20 20 1 1 2 4 4 10 10 20 20 2 2 4 4 20 20 2 2 4 4 8 8 8 8 2 2 2 2 4 4 8 8 8 8 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 D10 D8⋊C22 D4×D5 D4×D5 Q16⋊D10 kernel Q16⋊D10 C2×C8⋊D5 D40⋊7C2 D8⋊D5 D40⋊C2 SD16⋊D5 Q16⋊D5 D4.8D10 C5×C4○D8 D5×C4○D4 C4×D5 C2×Dic5 C22×D5 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C5 C4 C22 C1 # reps 1 1 1 2 2 2 2 2 1 2 2 1 1 2 2 2 4 2 4 2 2 2 8

Matrix representation of Q16⋊D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 40 0 0 0 0 40 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 32 0 0 0 0 32 0 0 0 0 0 0 0 9 0 0 0 0 0 0 32
,
 40 35 0 0 0 0 6 35 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 35 40 0 0 0 0 0 0 0 0 0 32 0 0 0 0 9 0 0 0 0 32 0 0 0 0 9 0 0 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[40,6,0,0,0,0,35,35,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,9,0,0,0,0,32,0,0,0] >;`

Q16⋊D10 in GAP, Magma, Sage, TeX

`Q_{16}\rtimes D_{10}`
`% in TeX`

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

`G:=SmallGroup(320,1440);`
`// by ID`

`G=gap.SmallGroup(320,1440);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,570,185,438,235,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^10=d^2=1,b^2=a^4,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^3,c*b*c^-1=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

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