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## G = D10⋊D4order 160 = 25·5

### 1st semidirect product of D10 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D10⋊D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊D4
 Lower central C5 — C2×C10 — D10⋊D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for D10⋊D4
G = < a,b,c,d | a10=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a8b, dbd=a3b, dcd=c-1 >

Subgroups: 352 in 94 conjugacy classes, 33 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×5], C22, C22 [×10], C5, C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], D5 [×3], C10 [×3], C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], Dic5, C20 [×2], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], C4⋊D4, C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C22×D5 [×2], C22×C10, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4 [×2], D10⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C22×D5, C4○D20, D4×D5 [×2], D10⋊D4

Smallest permutation representation of D10⋊D4
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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 80)(9 79)(10 78)(11 22)(12 21)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 60)(41 63)(42 62)(43 61)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)
(1 38 28 50)(2 37 29 49)(3 36 30 48)(4 35 21 47)(5 34 22 46)(6 33 23 45)(7 32 24 44)(8 31 25 43)(9 40 26 42)(10 39 27 41)(11 70 73 58)(12 69 74 57)(13 68 75 56)(14 67 76 55)(15 66 77 54)(16 65 78 53)(17 64 79 52)(18 63 80 51)(19 62 71 60)(20 61 72 59)
(1 23)(2 22)(3 21)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 73)(12 72)(13 71)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(31 40)(32 39)(33 38)(34 37)(35 36)(41 44)(42 43)(45 50)(46 49)(47 48)(51 55)(52 54)(56 60)(57 59)(61 69)(62 68)(63 67)(64 66)```

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

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

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

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 C4○D20 D4×D5 kernel D10⋊D4 C10.D4 D10⋊C4 C5×C22⋊C4 C2×C4×D5 C2×D20 C2×C5⋊D4 Dic5 D10 C22⋊C4 C10 C2×C4 C23 C2 C2 # reps 1 1 1 1 1 1 2 2 2 2 2 4 2 8 4

Matrix representation of D10⋊D4 in GL4(𝔽41) generated by

 34 34 0 0 7 1 0 0 0 0 40 0 0 0 0 40
,
 14 27 0 0 11 27 0 0 0 0 32 9 0 0 23 9
,
 17 40 0 0 3 24 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 34 40 0 0 0 0 40 0 0 0 39 1
`G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[14,11,0,0,27,27,0,0,0,0,32,23,0,0,9,9],[17,3,0,0,40,24,0,0,0,0,40,0,0,0,0,40],[1,34,0,0,0,40,0,0,0,0,40,39,0,0,0,1] >;`

D10⋊D4 in GAP, Magma, Sage, TeX

`D_{10}\rtimes D_4`
`% in TeX`

`G:=Group("D10:D4");`
`// GroupNames label`

`G:=SmallGroup(160,105);`
`// by ID`

`G=gap.SmallGroup(160,105);`
`# by ID`

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

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

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