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## G = D10⋊3Q8order 160 = 25·5

### 3rd semidirect product of D10 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D10⋊3Q8
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊3Q8
 Lower central C5 — C2×C10 — D10⋊3Q8
 Upper central C1 — C22 — C2×Q8

Generators and relations for D103Q8
G = < a,b,c,d | a10=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 224 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 [×3], C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C22⋊Q8, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C2×C4×D5, Q8×C10, D103Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C5⋊D4 [×2], C22×D5, Q8×D5, Q82D5, C2×C5⋊D4, D103Q8

Smallest permutation representation of D103Q8
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 75)(12 74)(13 73)(14 72)(15 71)(16 80)(17 79)(18 78)(19 77)(20 76)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 50)(51 64)(52 63)(53 62)(54 61)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)
(1 50 30 36)(2 41 21 37)(3 42 22 38)(4 43 23 39)(5 44 24 40)(6 45 25 31)(7 46 26 32)(8 47 27 33)(9 48 28 34)(10 49 29 35)(11 51 71 65)(12 52 72 66)(13 53 73 67)(14 54 74 68)(15 55 75 69)(16 56 76 70)(17 57 77 61)(18 58 78 62)(19 59 79 63)(20 60 80 64)
(1 70 30 56)(2 61 21 57)(3 62 22 58)(4 63 23 59)(5 64 24 60)(6 65 25 51)(7 66 26 52)(8 67 27 53)(9 68 28 54)(10 69 29 55)(11 45 71 31)(12 46 72 32)(13 47 73 33)(14 48 74 34)(15 49 75 35)(16 50 76 36)(17 41 77 37)(18 42 78 38)(19 43 79 39)(20 44 80 40)```

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

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

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

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 C5⋊D4 Q8×D5 Q8⋊2D5 kernel D10⋊3Q8 C10.D4 C4⋊Dic5 D10⋊C4 C2×C4×D5 Q8×C10 C20 D10 C2×Q8 C10 C2×C4 C4 C2 C2 # reps 1 2 1 2 1 1 2 2 2 2 6 8 2 2

Matrix representation of D103Q8 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 0 6 0 0 34 7
,
 1 0 0 0 0 40 0 0 0 0 34 6 0 0 33 7
,
 0 1 0 0 40 0 0 0 0 0 24 35 0 0 7 17
,
 32 0 0 0 0 9 0 0 0 0 40 0 0 0 0 40
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,0,34,0,0,6,7],[1,0,0,0,0,40,0,0,0,0,34,33,0,0,6,7],[0,40,0,0,1,0,0,0,0,0,24,7,0,0,35,17],[32,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40] >;`

D103Q8 in GAP, Magma, Sage, TeX

`D_{10}\rtimes_3Q_8`
`% in TeX`

`G:=Group("D10:3Q8");`
`// GroupNames label`

`G:=SmallGroup(160,167);`
`// by ID`

`G=gap.SmallGroup(160,167);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,86,4613]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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