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## G = D8⋊6D10order 320 = 26·5

### 6th semidirect product of D8 and D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for D86D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=a-1, cac-1=dad=a3, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 998 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4, D4 [×2], D4 [×13], Q8, Q8 [×7], C23 [×3], D5 [×3], C10, C10 [×4], C2×C8 [×3], M4(2), M4(2) [×2], D8 [×2], D8, SD16 [×2], SD16 [×8], Q16 [×3], C2×D4, C2×D4 [×5], C2×Q8 [×4], C4○D4, C4○D4 [×10], Dic5 [×2], Dic5 [×3], C20 [×2], C20, D10 [×2], D10 [×3], C2×C10, C2×C10 [×4], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22, C8⋊C22 [×2], C8.C22 [×3], 2+ 1+4, 2- 1+4, C52C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×2], Dic10 [×3], C4×D5 [×2], C4×D5 [×3], D20 [×2], C2×Dic5 [×5], C5⋊D4 [×2], C5⋊D4 [×7], C2×C20, C2×C20, C5×D4, C5×D4 [×2], C5×D4 [×2], C5×Q8, C22×D5 [×2], C22×C10, D4○SD16, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], Dic20 [×2], C2×C52C8, D4⋊D5, D4.D5, D4.D5 [×4], Q8⋊D5, C5⋊Q16, C5×M4(2), C5×D8 [×2], C5×SD16 [×2], C2×Dic10, C2×Dic10, C4○D20 [×2], C4○D20, D4×D5 [×2], D4×D5, D42D5 [×4], D42D5 [×3], Q8×D5 [×2], C2×C5⋊D4 [×2], D4×C10, C5×C4○D4, D20.2C4, C8.D10, D8⋊D5 [×2], D83D5 [×2], D5×SD16 [×2], SD16⋊D5 [×2], C2×D4.D5, D4.8D10, C5×C8⋊C22, D46D10, D4.10D10, D86D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○SD16, D4×D5 [×2], C23×D5, C2×D4×D5, D86D10

Smallest permutation representation of D86D10
On 80 points
Generators in S80
```(1 41 40 78 63 20 59 21)(2 79 60 42 64 22 31 11)(3 43 32 80 65 12 51 23)(4 71 52 44 66 24 33 13)(5 45 34 72 67 14 53 25)(6 73 54 46 68 26 35 15)(7 47 36 74 69 16 55 27)(8 75 56 48 70 28 37 17)(9 49 38 76 61 18 57 29)(10 77 58 50 62 30 39 19)
(11 79)(12 80)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 60)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)
(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 53)(2 52)(3 51)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 13)(14 20)(15 19)(16 18)(21 72)(22 71)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 70)(38 69)(39 68)(40 67)(41 45)(42 44)(46 50)(47 49)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 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 4 4 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 2 2 4 10 10 20 20 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○SD16 D4×D5 D4×D5 D8⋊6D10 kernel D8⋊6D10 D20.2C4 C8.D10 D8⋊D5 D8⋊3D5 D5×SD16 SD16⋊D5 C2×D4.D5 D4.8D10 C5×C8⋊C22 D4⋊6D10 D4.10D10 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 D86D10 in GL8(𝔽41)

 1 0 13 28 0 0 0 0 0 1 13 0 0 0 0 0 0 3 40 0 0 0 0 0 38 3 0 40 0 0 0 0 0 0 0 0 15 26 26 0 0 0 0 0 15 26 15 26 0 0 0 0 0 30 0 15 0 0 0 0 11 30 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 40 0 0 0 0 0 38 3 0 40 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 40 0 0 0 0 0 0 0 40
,
 1 6 0 0 0 0 0 0 35 6 0 0 0 0 0 0 23 18 0 35 0 0 0 0 20 0 7 34 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 40 0 0 0 0 0 0 0 39 0 40
,
 1 0 0 0 0 0 0 0 35 40 0 0 0 0 0 0 23 38 7 35 0 0 0 0 20 38 8 34 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 39 1

`G:=sub<GL(8,GF(41))| [1,0,0,38,0,0,0,0,0,1,3,3,0,0,0,0,13,13,40,0,0,0,0,0,28,0,0,40,0,0,0,0,0,0,0,0,15,15,0,11,0,0,0,0,26,26,30,30,0,0,0,0,26,15,0,0,0,0,0,0,0,26,15,0],[1,0,0,38,0,0,0,0,0,1,3,3,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40],[1,35,23,20,0,0,0,0,6,6,18,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,40,39,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40],[1,35,23,20,0,0,0,0,0,40,38,38,0,0,0,0,0,0,7,8,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,40,39,0,0,0,0,0,0,0,1] >;`

D86D10 in GAP, Magma, Sage, TeX

`D_8\rtimes_6D_{10}`
`% in TeX`

`G:=Group("D8:6D10");`
`// GroupNames label`

`G:=SmallGroup(320,1447);`
`// by ID`

`G=gap.SmallGroup(320,1447);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,570,185,136,438,235,102,12550]);`
`// 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=d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

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