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## G = D10⋊2Q8order 160 = 25·5

### 2nd semidirect product of D10 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D10⋊2Q8
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊2Q8
 Lower central C5 — C2×C10 — D10⋊2Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D102Q8
G = < a,b,c,d | a10=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 240 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, D5 [×2], C10 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C22⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, D102Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, D20 [×2], C22×D5, C2×D20, D42D5, Q8×D5, D102Q8

Smallest permutation representation of D102Q8
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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 73)(12 72)(13 71)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 50)(39 49)(40 48)(51 62)(52 61)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)
(1 39 30 45)(2 38 21 44)(3 37 22 43)(4 36 23 42)(5 35 24 41)(6 34 25 50)(7 33 26 49)(8 32 27 48)(9 31 28 47)(10 40 29 46)(11 70 79 54)(12 69 80 53)(13 68 71 52)(14 67 72 51)(15 66 73 60)(16 65 74 59)(17 64 75 58)(18 63 76 57)(19 62 77 56)(20 61 78 55)
(1 70 30 54)(2 61 21 55)(3 62 22 56)(4 63 23 57)(5 64 24 58)(6 65 25 59)(7 66 26 60)(8 67 27 51)(9 68 28 52)(10 69 29 53)(11 45 79 39)(12 46 80 40)(13 47 71 31)(14 48 72 32)(15 49 73 33)(16 50 74 34)(17 41 75 35)(18 42 76 36)(19 43 77 37)(20 44 78 38)```

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

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

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

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 D20 D4⋊2D5 Q8×D5 kernel D10⋊2Q8 C4⋊Dic5 D10⋊C4 C5×C4⋊C4 C2×Dic10 C2×C4×D5 C20 D10 C4⋊C4 C10 C2×C4 C4 C2 C2 # reps 1 2 2 1 1 1 2 2 2 2 6 8 2 2

Matrix representation of D102Q8 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 7 0 0 0 0 35 6
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 32 40 0 0 0 0 0 0 35 7 0 0 0 0 36 6
,
 9 0 0 0 0 0 9 32 0 0 0 0 0 0 32 39 0 0 0 0 0 9 0 0 0 0 0 0 6 1 0 0 0 0 6 35
,
 1 39 0 0 0 0 1 40 0 0 0 0 0 0 9 0 0 0 0 0 1 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,35,0,0,0,0,7,6],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,32,0,0,0,0,0,40,0,0,0,0,0,0,35,36,0,0,0,0,7,6],[9,9,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,39,9,0,0,0,0,0,0,6,6,0,0,0,0,1,35],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;`

D102Q8 in GAP, Magma, Sage, TeX

`D_{10}\rtimes_2Q_8`
`% in TeX`

`G:=Group("D10:2Q8");`
`// GroupNames label`

`G:=SmallGroup(160,118);`
`// by ID`

`G=gap.SmallGroup(160,118);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,122,4613]);`
`// 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=a^-1,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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