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## G = D10⋊Q8order 160 = 25·5

### 1st semidirect product of D10 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D10⋊Q8
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊Q8
 Lower central C5 — C2×C10 — D10⋊Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D10⋊Q8
G = < a,b,c,d | a10=b2=c4=1, d2=c2, bab=cac-1=dad-1=a-1, cbc-1=a3b, dbd-1=a8b, dcd-1=c-1 >

Subgroups: 240 in 74 conjugacy classes, 33 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×7], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×5], Q8 [×2], C23, D5 [×2], C10 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, Dic5 [×2], Dic5 [×2], C20 [×3], D10 [×2], D10 [×2], C2×C10, C22⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C10.D4 [×2], D10⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, D10⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C22×D5, C4○D20, D4×D5, Q8×D5, D10⋊Q8

Smallest permutation representation of D10⋊Q8
On 80 points
Generators in S80
```(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 22)(2 21)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 72)(12 71)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 50)(38 49)(39 48)(40 47)(51 61)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)
(1 50 28 38)(2 49 29 37)(3 48 30 36)(4 47 21 35)(5 46 22 34)(6 45 23 33)(7 44 24 32)(8 43 25 31)(9 42 26 40)(10 41 27 39)(11 65 73 53)(12 64 74 52)(13 63 75 51)(14 62 76 60)(15 61 77 59)(16 70 78 58)(17 69 79 57)(18 68 80 56)(19 67 71 55)(20 66 72 54)
(1 70 28 58)(2 69 29 57)(3 68 30 56)(4 67 21 55)(5 66 22 54)(6 65 23 53)(7 64 24 52)(8 63 25 51)(9 62 26 60)(10 61 27 59)(11 33 73 45)(12 32 74 44)(13 31 75 43)(14 40 76 42)(15 39 77 41)(16 38 78 50)(17 37 79 49)(18 36 80 48)(19 35 71 47)(20 34 72 46)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 10A ··· 10F 20A ··· 20L order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 10 10 2 2 4 4 10 10 20 20 2 2 2 ··· 2 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 D4 Q8 D5 C4○D4 D10 C4○D20 D4×D5 Q8×D5 kernel D10⋊Q8 C10.D4 D10⋊C4 C5×C4⋊C4 C2×Dic10 C2×C4×D5 Dic5 D10 C4⋊C4 C10 C2×C4 C2 C2 C2 # reps 1 2 2 1 1 1 2 2 2 2 6 8 2 2

Matrix representation of D10⋊Q8 in GL4(𝔽41) generated by

 1 6 0 0 35 6 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 35 40 0 0 0 0 1 0 0 0 1 40
,
 20 20 0 0 23 21 0 0 0 0 40 2 0 0 0 1
,
 13 28 0 0 32 28 0 0 0 0 40 0 0 0 0 40
`G:=sub<GL(4,GF(41))| [1,35,0,0,6,6,0,0,0,0,40,0,0,0,0,40],[1,35,0,0,0,40,0,0,0,0,1,1,0,0,0,40],[20,23,0,0,20,21,0,0,0,0,40,0,0,0,2,1],[13,32,0,0,28,28,0,0,0,0,40,0,0,0,0,40] >;`

D10⋊Q8 in GAP, Magma, Sage, TeX

`D_{10}\rtimes Q_8`
`% in TeX`

`G:=Group("D10:Q8");`
`// GroupNames label`

`G:=SmallGroup(160,117);`
`// by ID`

`G=gap.SmallGroup(160,117);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,55,506,188,86,4613]);`
`// Polycyclic`

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

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