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## G = D8⋊D10order 320 = 26·5

### 2nd semidirect product of D8 and D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D8⋊D10
 Chief series C1 — C5 — C10 — C20 — C40 — D40 — C2×D40 — D8⋊D10
 Lower central C5 — C10 — C20 — C40 — D8⋊D10
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4○D8

Generators and relations for D8⋊D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a4b, dbd=ab, dcd=c-1 >

Subgroups: 494 in 90 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, M5(2), D16, SD32, C2×D8, C4○D8, C40, D20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C16⋊C22, C52C16, D40, D40, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×D20, C5×C4○D4, C20.4C8, C5⋊D16, C5⋊SD32, C2×D40, C5×C4○D8, D8⋊D10
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C5⋊D4, C22×D5, C16⋊C22, D4⋊D5, C2×C5⋊D4, C2×D4⋊D5, D8⋊D10

Smallest permutation representation of D8⋊D10
On 80 points
Generators in S80
```(1 33 13 28 8 40 18 23)(2 34 14 29 9 36 19 24)(3 35 15 30 10 37 20 25)(4 31 11 26 6 38 16 21)(5 32 12 27 7 39 17 22)(41 63 58 78 46 68 53 73)(42 64 59 79 47 69 54 74)(43 65 60 80 48 70 55 75)(44 66 51 71 49 61 56 76)(45 67 52 72 50 62 57 77)
(1 73)(2 79)(3 75)(4 71)(5 77)(6 76)(7 72)(8 78)(9 74)(10 80)(11 66)(12 62)(13 68)(14 64)(15 70)(16 61)(17 67)(18 63)(19 69)(20 65)(21 49)(22 45)(23 41)(24 47)(25 43)(26 44)(27 50)(28 46)(29 42)(30 48)(31 51)(32 57)(33 53)(34 59)(35 55)(36 54)(37 60)(38 56)(39 52)(40 58)
(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 5)(2 4)(6 9)(7 8)(11 19)(12 18)(13 17)(14 16)(15 20)(21 34)(22 33)(23 32)(24 31)(25 35)(26 36)(27 40)(28 39)(29 38)(30 37)(41 62)(42 61)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 80)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 5A 5B 8A 8B 8C 10A 10B 10C 10D 10E 10F 10G 10H 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 40A ··· 40H order 1 2 2 2 2 2 4 4 4 5 5 8 8 8 10 10 10 10 10 10 10 10 16 16 16 16 20 20 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 2 8 40 40 2 2 8 2 2 2 2 4 2 2 4 4 8 8 8 8 20 20 20 20 2 2 2 2 4 4 8 8 8 8 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 D10 C5⋊D4 C5⋊D4 C16⋊C22 D4⋊D5 D4⋊D5 D8⋊D10 kernel D8⋊D10 C20.4C8 C5⋊D16 C5⋊SD32 C2×D40 C5×C4○D8 C40 C2×C20 C4○D8 C20 C2×C10 C2×C8 D8 Q16 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 2 2 1 1 1 1 2 2 2 2 2 2 4 4 2 2 2 8

Matrix representation of D8⋊D10 in GL4(𝔽241) generated by

 20 212 0 0 29 199 0 0 0 212 228 29 29 208 4 232
,
 29 29 198 170 20 20 22 192 0 29 13 212 20 33 194 179
,
 240 52 0 0 189 52 0 0 175 227 190 189 131 184 51 0
,
 1 0 0 0 52 240 0 0 9 8 214 20 15 204 60 27
`G:=sub<GL(4,GF(241))| [20,29,0,29,212,199,212,208,0,0,228,4,0,0,29,232],[29,20,0,20,29,20,29,33,198,22,13,194,170,192,212,179],[240,189,175,131,52,52,227,184,0,0,190,51,0,0,189,0],[1,52,9,15,0,240,8,204,0,0,214,60,0,0,20,27] >;`

D8⋊D10 in GAP, Magma, Sage, TeX

`D_8\rtimes D_{10}`
`% in TeX`

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

`G:=SmallGroup(320,820);`
`// by ID`

`G=gap.SmallGroup(320,820);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,387,675,185,192,1684,438,102,12550]);`
`// 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*b,d*c*d=c^-1>;`
`// generators/relations`

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