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## G = D5.D20order 400 = 24·52

### The non-split extension by D5 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — D5.D20
 Chief series C1 — C5 — C52 — C5×D5 — D5×C10 — C10×F5 — D5.D20
 Lower central C52 — C5×C10 — D5.D20
 Upper central C1 — C2

Generators and relations for D5.D20
G = < a,b,c,d | a5=b2=c20=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=dbd-1=a2b, dcd-1=a-1bc-1 >

Subgroups: 616 in 73 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, C23, D5, D5, C10, C10, C22⋊C4, Dic5, C20, F5, D10, D10, C2×C10, C52, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C5×D5, C5×D5, C5⋊D5, C5×C10, D10⋊C4, C22⋊F5, C5×F5, D5.D5, D52, D5×C10, C2×C5⋊D5, C10×F5, C2×D5.D5, C2×D52, D5.D20
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, F5, D10, C4×D5, D20, C5⋊D4, C2×F5, D10⋊C4, C22⋊F5, D5×F5, D5.D20

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

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 5E 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 20A ··· 20H order 1 2 2 2 2 2 4 4 4 4 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 20 ··· 20 size 1 1 5 5 10 50 10 10 50 50 2 2 4 8 8 2 2 4 8 8 10 10 10 10 20 20 10 ··· 10

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 8 8 type + + + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D5 D10 D20 C5⋊D4 C4×D5 F5 C2×F5 C22⋊F5 D5×F5 D5.D20 kernel D5.D20 C10×F5 C2×D5.D5 C2×D52 D5×C10 C2×C5⋊D5 C5×D5 C2×F5 D10 D5 D5 C10 D10 C10 C5 C2 C1 # reps 1 1 1 1 2 2 2 2 2 4 4 4 1 1 2 2 2

Matrix representation of D5.D20 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 40 40 40 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 1 0 0 40 40 40 40 0 0 0 0 1 0 0 0 1 0 0 0
,
 38 17 0 0 0 0 24 1 0 0 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 1 0 0 0 0 0 0 1 0
,
 38 17 0 0 0 0 38 3 0 0 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,1,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,1,0,0,0,40,0,0,0,0,0,40,1,0,0,0,1,40,0,0],[38,24,0,0,0,0,17,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0],[38,38,0,0,0,0,17,3,0,0,0,0,0,0,1,40,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0] >;`

D5.D20 in GAP, Magma, Sage, TeX

`D_5.D_{20}`
`% in TeX`

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

`G:=SmallGroup(400,118);`
`// by ID`

`G=gap.SmallGroup(400,118);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,970,5765,2897]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^2=c^20=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^-1*b*c^-1>;`
`// generators/relations`

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