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## G = D8⋊5D10order 320 = 26·5

### 5th semidirect product of D8 and D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D8⋊5D10
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C4○D20 — D4⋊6D10 — D8⋊5D10
 Lower central C5 — C10 — C20 — D8⋊5D10
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D85D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=a-1, cac-1=a5, dad=a3, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 1190 in 268 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C2×D4, C2×D4, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C8○D4, C2×D8, C4○D8, C8⋊C22, C8⋊C22, 2+ 1+4, C52C8, C40, Dic10, C4×D5, C4×D5, D20, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×D4, C5×Q8, C22×D5, C22×C10, D4○D8, C8×D5, C8⋊D5, C40⋊C2, D40, C2×C52C8, D4⋊D5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C5×M4(2), C5×D8, C5×SD16, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q82D5, C2×C5⋊D4, D4×C10, C5×C4○D4, D20.2C4, C8⋊D10, D5×D8, D8⋊D5, D40⋊C2, SD163D5, C2×D4⋊D5, D4.8D10, C5×C8⋊C22, D46D10, D48D10, D85D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, D4○D8, D4×D5, C23×D5, C2×D4×D5, D85D10

Smallest permutation representation of D85D10
On 80 points
Generators in S80
```(1 30 53 46 63 73 40 16)(2 74 54 17 64 21 31 47)(3 22 55 48 65 75 32 18)(4 76 56 19 66 23 33 49)(5 24 57 50 67 77 34 20)(6 78 58 11 68 25 35 41)(7 26 59 42 69 79 36 12)(8 80 60 13 70 27 37 43)(9 28 51 44 61 71 38 14)(10 72 52 15 62 29 39 45)
(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 21)(18 22)(19 23)(20 24)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 51)(39 52)(40 53)(41 78)(42 79)(43 80)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)
(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 57)(12 56)(13 55)(14 54)(15 53)(16 52)(17 51)(18 60)(19 59)(20 58)(31 44)(32 43)(33 42)(34 41)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(61 74)(62 73)(63 72)(64 71)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 D10 D4○D8 D4×D5 D4×D5 D8⋊5D10 kernel D8⋊5D10 D20.2C4 C8⋊D10 D5×D8 D8⋊D5 D40⋊C2 SD16⋊3D5 C2×D4⋊D5 D4.8D10 C5×C8⋊C22 D4⋊6D10 D4⋊8D10 Dic10 D20 C5⋊D4 C8⋊C22 M4(2) D8 SD16 C2×D4 C4○D4 C5 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 2 2 2 4 4 2 2 2 2 2 2

Matrix representation of D85D10 in GL8(𝔽41)

 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 29 29 0 0 0 0 0 0 12 29 0 0 0 0 0 0 18 21 0 34 0 0 0 0 3 35 6 24
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 25 25 40 37 0 0 0 0 0 0 0 1
,
 6 6 0 0 0 0 0 0 35 1 0 0 0 0 0 0 0 0 6 6 0 0 0 0 0 0 35 1 0 0 0 0 0 0 0 0 7 37 9 0 0 0 0 0 16 27 32 5 0 0 0 0 20 6 30 25 0 0 0 0 1 1 0 18
,
 0 0 6 6 0 0 0 0 0 0 1 35 0 0 0 0 6 6 0 0 0 0 0 0 1 35 0 0 0 0 0 0 0 0 0 0 11 11 0 22 0 0 0 0 26 3 30 19 0 0 0 0 29 37 27 5 0 0 0 0 30 34 26 0

`G:=sub<GL(8,GF(41))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,29,12,18,3,0,0,0,0,29,29,21,35,0,0,0,0,0,0,0,6,0,0,0,0,0,0,34,24],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,25,0,0,0,0,0,1,0,25,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,37,1],[6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,7,16,20,1,0,0,0,0,37,27,6,1,0,0,0,0,9,32,30,0,0,0,0,0,0,5,25,18],[0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,0,0,0,0,0,0,11,26,29,30,0,0,0,0,11,3,37,34,0,0,0,0,0,30,27,26,0,0,0,0,22,19,5,0] >;`

D85D10 in GAP, Magma, Sage, TeX

`D_8\rtimes_5D_{10}`
`% in TeX`

`G:=Group("D8:5D10");`
`// GroupNames label`

`G:=SmallGroup(320,1446);`
`// by ID`

`G=gap.SmallGroup(320,1446);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,570,185,438,235,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,d*a*d=a^3,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;`
`// generators/relations`

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