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## G = C20⋊D10order 400 = 24·52

### 2nd semidirect product of C20 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C20⋊D10
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — C2×D52 — C20⋊D10
 Lower central C52 — C5×C10 — C20⋊D10
 Upper central C1 — C2 — C4

Generators and relations for C20⋊D10
G = < a,b,c | a20=b10=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >

Subgroups: 940 in 124 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D4, C23, D5, C10, C10, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C52, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C5×D5, C5⋊D5, C5×C10, D4×D5, C526C4, C5×C20, D52, D5×C10, C2×C5⋊D5, C522D4, C5×D20, C4×C5⋊D5, C2×D52, C20⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22×D5, D4×D5, D52, C2×D52, C20⋊D10

Smallest permutation representation of C20⋊D10
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 40 13 28 5 36 17 24 9 32)(2 39 14 27 6 35 18 23 10 31)(3 38 15 26 7 34 19 22 11 30)(4 37 16 25 8 33 20 21 12 29)
(1 5)(2 14)(4 12)(6 10)(7 19)(9 17)(11 15)(16 20)(21 37)(22 26)(23 35)(25 33)(27 31)(28 40)(30 38)(32 36)```

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

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

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

46 conjugacy classes

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

46 irreducible representations

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

Matrix representation of C20⋊D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 40 40 0 0 0 0 0 0 0 40 0 0 0 0 1 35
,
 1 35 0 0 0 0 6 6 0 0 0 0 0 0 1 2 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 35 1
,
 40 6 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 1 0 0 0 0 0 6 40

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40,0,0,0,0,0,0,0,1,0,0,0,0,40,35],[1,6,0,0,0,0,35,6,0,0,0,0,0,0,1,0,0,0,0,0,2,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1],[40,0,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;`

C20⋊D10 in GAP, Magma, Sage, TeX

`C_{20}\rtimes D_{10}`
`% in TeX`

`G:=Group("C20:D10");`
`// GroupNames label`

`G:=SmallGroup(400,171);`
`// by ID`

`G=gap.SmallGroup(400,171);`
`# by ID`

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

`G:=Group<a,b,c|a^20=b^10=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;`
`// generators/relations`

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