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## G = D20⋊D5order 400 = 24·52

### 3rd semidirect product of D20 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D20⋊D5
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — D5×Dic5 — D20⋊D5
 Lower central C52 — C5×C10 — D20⋊D5
 Upper central C1 — C2 — C4

Generators and relations for D20⋊D5
G = < a,b,c,d | a20=b2=c5=d2=1, bab=a-1, ac=ca, dad=a9, bc=cb, dbd=a18b, dcd=c-1 >

Subgroups: 604 in 96 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D4, Q8, D5, C10, C10, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C52, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C5×D5, C5⋊D5, C5×C10, D42D5, Q82D5, C5×Dic5, C526C4, C5×C20, D5×C10, C2×C5⋊D5, D5×Dic5, C5⋊D20, C5×Dic10, C5×D20, C4×C5⋊D5, D20⋊D5
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, D42D5, Q82D5, D52, C2×D52, D20⋊D5

Smallest permutation representation of D20⋊D5
On 40 points
Generators in S40
```(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)
(1 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 40)(20 39)
(1 17 13 9 5)(2 18 14 10 6)(3 19 15 11 7)(4 20 16 12 8)(21 25 29 33 37)(22 26 30 34 38)(23 27 31 35 39)(24 28 32 36 40)
(1 5)(2 14)(4 12)(6 10)(7 19)(9 17)(11 15)(16 20)(21 23)(22 32)(24 30)(25 39)(26 28)(27 37)(29 35)(31 33)(34 40)(36 38)```

`G:=sub<Sym(40)| (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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,40)(20,39), (1,17,13,9,5)(2,18,14,10,6)(3,19,15,11,7)(4,20,16,12,8)(21,25,29,33,37)(22,26,30,34,38)(23,27,31,35,39)(24,28,32,36,40), (1,5)(2,14)(4,12)(6,10)(7,19)(9,17)(11,15)(16,20)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38)>;`

`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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,40)(20,39), (1,17,13,9,5)(2,18,14,10,6)(3,19,15,11,7)(4,20,16,12,8)(21,25,29,33,37)(22,26,30,34,38)(23,27,31,35,39)(24,28,32,36,40), (1,5)(2,14)(4,12)(6,10)(7,19)(9,17)(11,15)(16,20)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38) );`

`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)], [(1,38),(2,37),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(16,23),(17,22),(18,21),(19,40),(20,39)], [(1,17,13,9,5),(2,18,14,10,6),(3,19,15,11,7),(4,20,16,12,8),(21,25,29,33,37),(22,26,30,34,38),(23,27,31,35,39),(24,28,32,36,40)], [(1,5),(2,14),(4,12),(6,10),(7,19),(9,17),(11,15),(16,20),(21,23),(22,32),(24,30),(25,39),(26,28),(27,37),(29,35),(31,33),(34,40),(36,38)]])`

46 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A ··· 20L 20M 20N 20O 20P order 1 2 2 2 2 4 4 4 4 4 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 10 10 50 2 10 10 25 25 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 20 20 20 20 4 ··· 4 20 20 20 20

46 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 D5 D5 C4○D4 D10 D10 D10 D4⋊2D5 Q8⋊2D5 D52 C2×D52 D20⋊D5 kernel D20⋊D5 D5×Dic5 C5⋊D20 C5×Dic10 C5×D20 C4×C5⋊D5 Dic10 D20 C52 Dic5 C20 D10 C5 C5 C4 C2 C1 # reps 1 2 2 1 1 1 2 2 2 4 4 4 2 2 4 4 8

Matrix representation of D20⋊D5 in GL6(𝔽41)

 32 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 35 1 0 0 0 0 40 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 1 0 0 0 0 6 6
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 1 34 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 34 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 6 40

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

D20⋊D5 in GAP, Magma, Sage, TeX

`D_{20}\rtimes D_5`
`% in TeX`

`G:=Group("D20:D5");`
`// GroupNames label`

`G:=SmallGroup(400,165);`
`// by ID`

`G=gap.SmallGroup(400,165);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,116,50,970,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^18*b,d*c*d=c^-1>;`
`// generators/relations`

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