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## G = D8⋊15D10order 320 = 26·5

### 4th semidirect product of D8 and D10 acting through Inn(D8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D8⋊15D10
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C4○D20 — D4⋊8D10 — D8⋊15D10
 Lower central C5 — C10 — C20 — D8⋊15D10
 Upper central C1 — C2 — C2×C4 — C4○D8

Generators and relations for D815D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a4b, dbd=a2b, dcd=c-1 >

Subgroups: 1238 in 268 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×4], C22, C22 [×14], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4 [×2], D4 [×19], Q8 [×2], Q8, C23 [×6], D5 [×6], C10, C10 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8, D8 [×8], SD16 [×2], SD16 [×4], Q16, C2×D4 [×12], C4○D4 [×2], C4○D4 [×7], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×10], C2×C10, C2×C10 [×2], C8○D4, C2×D8 [×3], C4○D8, C4○D8 [×2], C8⋊C22 [×6], 2+ 1+4 [×2], C52C8 [×2], C40 [×2], Dic10, C4×D5 [×2], C4×D5 [×4], D20, D20 [×4], D20 [×6], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×6], D4○D8, C8×D5 [×2], C8⋊D5 [×2], D40 [×4], C4.Dic5, D4⋊D5 [×4], Q8⋊D5 [×4], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C2×D20 [×2], C2×D20 [×2], C4○D20, C4○D20 [×2], D4×D5 [×4], D4×D5 [×4], Q82D5 [×4], C5×C4○D4 [×2], D20.3C4, C2×D40, D5×D8 [×2], D40⋊C2 [×4], Q8.D10 [×2], D4⋊D10 [×2], C5×C4○D8, D48D10 [×2], D815D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D815D10

Smallest permutation representation of D815D10
On 80 points
Generators in S80
```(1 28 51 46 63 73 40 12)(2 29 52 47 64 74 31 13)(3 30 53 48 65 75 32 14)(4 21 54 49 66 76 33 15)(5 22 55 50 67 77 34 16)(6 23 56 41 68 78 35 17)(7 24 57 42 69 79 36 18)(8 25 58 43 70 80 37 19)(9 26 59 44 61 71 38 20)(10 27 60 45 62 72 39 11)
(2 64)(4 66)(6 68)(8 70)(10 62)(11 72)(12 28)(13 74)(14 30)(15 76)(16 22)(17 78)(18 24)(19 80)(20 26)(21 49)(23 41)(25 43)(27 45)(29 47)(32 53)(34 55)(36 57)(38 59)(40 51)(42 79)(44 71)(46 73)(48 75)(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 30)(9 29)(10 28)(11 51)(12 60)(13 59)(14 58)(15 57)(16 56)(17 55)(18 54)(19 53)(20 52)(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)| (1,28,51,46,63,73,40,12)(2,29,52,47,64,74,31,13)(3,30,53,48,65,75,32,14)(4,21,54,49,66,76,33,15)(5,22,55,50,67,77,34,16)(6,23,56,41,68,78,35,17)(7,24,57,42,69,79,36,18)(8,25,58,43,70,80,37,19)(9,26,59,44,61,71,38,20)(10,27,60,45,62,72,39,11), (2,64)(4,66)(6,68)(8,70)(10,62)(11,72)(12,28)(13,74)(14,30)(15,76)(16,22)(17,78)(18,24)(19,80)(20,26)(21,49)(23,41)(25,43)(27,45)(29,47)(32,53)(34,55)(36,57)(38,59)(40,51)(42,79)(44,71)(46,73)(48,75)(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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,30)(9,29)(10,28)(11,51)(12,60)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(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( (1,28,51,46,63,73,40,12)(2,29,52,47,64,74,31,13)(3,30,53,48,65,75,32,14)(4,21,54,49,66,76,33,15)(5,22,55,50,67,77,34,16)(6,23,56,41,68,78,35,17)(7,24,57,42,69,79,36,18)(8,25,58,43,70,80,37,19)(9,26,59,44,61,71,38,20)(10,27,60,45,62,72,39,11), (2,64)(4,66)(6,68)(8,70)(10,62)(11,72)(12,28)(13,74)(14,30)(15,76)(16,22)(17,78)(18,24)(19,80)(20,26)(21,49)(23,41)(25,43)(27,45)(29,47)(32,53)(34,55)(36,57)(38,59)(40,51)(42,79)(44,71)(46,73)(48,75)(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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,30)(9,29)(10,28)(11,51)(12,60)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(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([(1,28,51,46,63,73,40,12),(2,29,52,47,64,74,31,13),(3,30,53,48,65,75,32,14),(4,21,54,49,66,76,33,15),(5,22,55,50,67,77,34,16),(6,23,56,41,68,78,35,17),(7,24,57,42,69,79,36,18),(8,25,58,43,70,80,37,19),(9,26,59,44,61,71,38,20),(10,27,60,45,62,72,39,11)], [(2,64),(4,66),(6,68),(8,70),(10,62),(11,72),(12,28),(13,74),(14,30),(15,76),(16,22),(17,78),(18,24),(19,80),(20,26),(21,49),(23,41),(25,43),(27,45),(29,47),(32,53),(34,55),(36,57),(38,59),(40,51),(42,79),(44,71),(46,73),(48,75),(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,30),(9,29),(10,28),(11,51),(12,60),(13,59),(14,58),(15,57),(16,56),(17,55),(18,54),(19,53),(20,52),(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)])`

50 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 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 2 2 4 4 4 4 4 4 5 5 8 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 20 20 2 2 4 4 10 10 2 2 2 2 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 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 D10 D4○D8 D4×D5 D4×D5 D8⋊15D10 kernel D8⋊15D10 D20.3C4 C2×D40 D5×D8 D40⋊C2 Q8.D10 D4⋊D10 C5×C4○D8 D4⋊8D10 Dic10 D20 C5⋊D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C5 C4 C22 C1 # reps 1 1 1 2 4 2 2 1 2 1 1 2 2 2 2 4 2 4 2 2 2 8

Matrix representation of D815D10 in GL4(𝔽41) generated by

 29 0 29 0 0 29 0 29 12 0 29 0 0 12 0 29
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 0 0 30 27 0 0 14 14 11 14 0 0 27 27 0 0
,
 9 15 32 26 37 32 4 9 32 26 32 26 4 9 4 9
`G:=sub<GL(4,GF(41))| [29,0,12,0,0,29,0,12,29,0,29,0,0,29,0,29],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,0,11,27,0,0,14,27,30,14,0,0,27,14,0,0],[9,37,32,4,15,32,26,9,32,4,32,4,26,9,26,9] >;`

D815D10 in GAP, Magma, Sage, TeX

`D_8\rtimes_{15}D_{10}`
`% in TeX`

`G:=Group("D8:15D10");`
`// GroupNames label`

`G:=SmallGroup(320,1441);`
`// by ID`

`G=gap.SmallGroup(320,1441);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,570,185,438,235,102,12550]);`
`// Polycyclic`

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

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