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## G = C5⋊D20order 200 = 23·52

### The semidirect product of C5 and D20 acting via D20/D10=C2

Aliases: C52D20, Dic5⋊D5, C523D4, D102D5, C10.4D10, C2.4D52, (D5×C10)⋊2C2, C51(C5⋊D4), (C5×Dic5)⋊1C2, (C5×C10).4C22, (C2×C5⋊D5)⋊1C2, SmallGroup(200,25)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5⋊D20
G = < a,b,c | a5=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >

10C2
50C2
2C5
2C5
5C4
5C22
25C22
2D5
2C10
2C10
10D5
10D5
10D5
10D5
10C10
10D5
10D5
25D4
5D10
5C20
5D10
10D10
10D10
5D20

Character table of C5⋊D20

 class 1 2A 2B 2C 4 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A 20B 20C 20D size 1 1 10 50 10 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 10 10 10 10 10 10 10 10 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ5 2 -2 0 0 0 2 2 2 2 2 2 2 2 -2 -2 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 0 0 2 2 -1+√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ7 2 2 0 0 2 2 -1-√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ8 2 2 -2 0 0 -1-√5/2 2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 0 0 0 0 orthogonal lifted from D10 ρ9 2 2 0 0 -2 2 -1-√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from D10 ρ10 2 -2 0 0 0 2 -1-√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -2 -2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 0 0 0 0 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 orthogonal lifted from D20 ρ11 2 2 -2 0 0 -1+√5/2 2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 0 0 0 0 orthogonal lifted from D10 ρ12 2 2 2 0 0 -1-√5/2 2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 0 0 0 0 orthogonal lifted from D5 ρ13 2 -2 0 0 0 2 -1+√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -2 -2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 0 0 0 0 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 orthogonal lifted from D20 ρ14 2 2 0 0 -2 2 -1+√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from D10 ρ15 2 2 2 0 0 -1+√5/2 2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 0 0 0 0 orthogonal lifted from D5 ρ16 2 -2 0 0 0 2 -1+√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -2 -2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 0 0 0 0 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 orthogonal lifted from D20 ρ17 2 -2 0 0 0 2 -1-√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -2 -2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 0 0 0 0 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 orthogonal lifted from D20 ρ18 2 -2 0 0 0 -1+√5/2 2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 -2 -2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 ζ53-ζ52 -ζ53+ζ52 ζ54-ζ5 -ζ54+ζ5 0 0 0 0 complex lifted from C5⋊D4 ρ19 2 -2 0 0 0 -1-√5/2 2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 -2 -2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 -ζ54+ζ5 ζ54-ζ5 ζ53-ζ52 -ζ53+ζ52 0 0 0 0 complex lifted from C5⋊D4 ρ20 2 -2 0 0 0 -1+√5/2 2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 -2 -2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 -ζ53+ζ52 ζ53-ζ52 -ζ54+ζ5 ζ54-ζ5 0 0 0 0 complex lifted from C5⋊D4 ρ21 2 -2 0 0 0 -1-√5/2 2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 -2 -2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 ζ54-ζ5 -ζ54+ζ5 -ζ53+ζ52 ζ53-ζ52 0 0 0 0 complex lifted from C5⋊D4 ρ22 4 4 0 0 0 -1+√5 -1+√5 -1-√5 -1-√5 -1 3-√5/2 -1 3+√5/2 -1-√5 -1+√5 -1+√5 -1-√5 -1 3+√5/2 3-√5/2 -1 0 0 0 0 0 0 0 0 orthogonal lifted from D52 ρ23 4 -4 0 0 0 -1-√5 -1-√5 -1+√5 -1+√5 -1 3+√5/2 -1 3-√5/2 1-√5 1+√5 1+√5 1-√5 1 -3+√5/2 -3-√5/2 1 0 0 0 0 0 0 0 0 orthogonal faithful ρ24 4 -4 0 0 0 -1+√5 -1+√5 -1-√5 -1-√5 -1 3-√5/2 -1 3+√5/2 1+√5 1-√5 1-√5 1+√5 1 -3-√5/2 -3+√5/2 1 0 0 0 0 0 0 0 0 orthogonal faithful ρ25 4 -4 0 0 0 -1-√5 -1+√5 -1+√5 -1-√5 3-√5/2 -1 3+√5/2 -1 1-√5 1+√5 1-√5 1+√5 -3+√5/2 1 1 -3-√5/2 0 0 0 0 0 0 0 0 orthogonal faithful ρ26 4 4 0 0 0 -1-√5 -1-√5 -1+√5 -1+√5 -1 3+√5/2 -1 3-√5/2 -1+√5 -1-√5 -1-√5 -1+√5 -1 3-√5/2 3+√5/2 -1 0 0 0 0 0 0 0 0 orthogonal lifted from D52 ρ27 4 4 0 0 0 -1+√5 -1-√5 -1-√5 -1+√5 3+√5/2 -1 3-√5/2 -1 -1-√5 -1+√5 -1-√5 -1+√5 3+√5/2 -1 -1 3-√5/2 0 0 0 0 0 0 0 0 orthogonal lifted from D52 ρ28 4 -4 0 0 0 -1+√5 -1-√5 -1-√5 -1+√5 3+√5/2 -1 3-√5/2 -1 1+√5 1-√5 1+√5 1-√5 -3-√5/2 1 1 -3+√5/2 0 0 0 0 0 0 0 0 orthogonal faithful ρ29 4 4 0 0 0 -1-√5 -1+√5 -1+√5 -1-√5 3-√5/2 -1 3+√5/2 -1 -1+√5 -1-√5 -1+√5 -1-√5 3-√5/2 -1 -1 3+√5/2 0 0 0 0 0 0 0 0 orthogonal lifted from D52

Permutation representations of C5⋊D20
On 20 points - transitive group 20T60
Generators in S20
```(1 9 17 5 13)(2 14 6 18 10)(3 11 19 7 15)(4 16 8 20 12)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)```

`G:=sub<Sym(20)| (1,9,17,5,13)(2,14,6,18,10)(3,11,19,7,15)(4,16,8,20,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)>;`

`G:=Group( (1,9,17,5,13)(2,14,6,18,10)(3,11,19,7,15)(4,16,8,20,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19) );`

`G=PermutationGroup([[(1,9,17,5,13),(2,14,6,18,10),(3,11,19,7,15),(4,16,8,20,12)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19)]])`

`G:=TransitiveGroup(20,60);`

C5⋊D20 is a maximal subgroup of   D20⋊D5  D10.9D10  Dic105D5  D5×D20  Dic5.D10  D5×C5⋊D4  D10⋊D10
C5⋊D20 is a maximal quotient of   C5⋊D40  C523SD16  C524SD16  C523Q16  D10⋊Dic5  C10.D20  Dic5⋊Dic5

Matrix representation of C5⋊D20 in GL4(𝔽41) generated by

 6 40 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 18 35 0 0 20 23 0 0 0 0 0 40 0 0 1 6
,
 40 0 0 0 35 1 0 0 0 0 1 6 0 0 0 40
`G:=sub<GL(4,GF(41))| [6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[18,20,0,0,35,23,0,0,0,0,0,1,0,0,40,6],[40,35,0,0,0,1,0,0,0,0,1,0,0,0,6,40] >;`

C5⋊D20 in GAP, Magma, Sage, TeX

`C_5\rtimes D_{20}`
`% in TeX`

`G:=Group("C5:D20");`
`// GroupNames label`

`G:=SmallGroup(200,25);`
`// by ID`

`G=gap.SmallGroup(200,25);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-5,-5,61,26,328,4004]);`
`// Polycyclic`

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

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