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## G = C52⋊D10order 500 = 22·53

### The semidirect product of C52 and D10 acting faithfully

Aliases: C52⋊D10, He5⋊C22, C5⋊D5⋊D5, C5.2D52, C52⋊C10⋊C2, He5⋊C2⋊C2, SmallGroup(500,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — He5 — C52⋊D10
 Chief series C1 — C5 — C52 — He5 — C52⋊C10 — C52⋊D10
 Lower central He5 — C52⋊D10
 Upper central C1

Generators and relations for C52⋊D10
G = < a,b,c,d | a5=b5=c10=d2=1, ab=ba, cac-1=dad=a-1b3, cbc-1=b-1, bd=db, dcd=c-1 >

25C2
25C2
25C2
5C5
5C5
10C5
10C5
125C22
5D5
5D5
5D5
5D5
10D5
10D5
25C10
25D5
25C10
25D5
25C10
2C52
2C52
25D10
25D10
25D10
10C5×D5
10C5×D5
5D52
5D52

Character table of C52⋊D10

 class 1 2A 2B 2C 5A 5B 5C 5D 5E 5F 5G 5H 5I 5J 10A 10B 10C 10D 10E 10F size 1 25 25 25 2 2 10 10 10 10 20 20 20 20 50 50 50 50 50 50 ρ1 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 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 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 linear of order 2 ρ5 2 0 2 0 2 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 -1-√5/2 0 0 orthogonal lifted from D5 ρ6 2 0 2 0 2 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 -1+√5/2 0 0 orthogonal lifted from D5 ρ7 2 0 0 2 2 2 2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 0 -1-√5/2 -1+√5/2 0 0 0 orthogonal lifted from D5 ρ8 2 0 0 2 2 2 2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 0 -1+√5/2 -1-√5/2 0 0 0 orthogonal lifted from D5 ρ9 2 0 0 -2 2 2 2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 0 1-√5/2 1+√5/2 0 0 0 orthogonal lifted from D10 ρ10 2 0 0 -2 2 2 2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 0 1+√5/2 1-√5/2 0 0 0 orthogonal lifted from D10 ρ11 2 0 -2 0 2 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 1-√5/2 0 0 orthogonal lifted from D10 ρ12 2 0 -2 0 2 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 1+√5/2 0 0 orthogonal lifted from D10 ρ13 4 0 0 0 4 4 -1+√5 -1+√5 -1-√5 -1-√5 -1 -1 3+√5/2 3-√5/2 0 0 0 0 0 0 orthogonal lifted from D52 ρ14 4 0 0 0 4 4 -1+√5 -1-√5 -1+√5 -1-√5 3-√5/2 3+√5/2 -1 -1 0 0 0 0 0 0 orthogonal lifted from D52 ρ15 4 0 0 0 4 4 -1-√5 -1+√5 -1-√5 -1+√5 3+√5/2 3-√5/2 -1 -1 0 0 0 0 0 0 orthogonal lifted from D52 ρ16 4 0 0 0 4 4 -1-√5 -1-√5 -1+√5 -1+√5 -1 -1 3-√5/2 3+√5/2 0 0 0 0 0 0 orthogonal lifted from D52 ρ17 10 -2 0 0 -5-5√5/2 -5+5√5/2 0 0 0 0 0 0 0 0 0 0 0 0 1-√5/2 1+√5/2 orthogonal faithful ρ18 10 2 0 0 -5+5√5/2 -5-5√5/2 0 0 0 0 0 0 0 0 0 0 0 0 -1-√5/2 -1+√5/2 orthogonal faithful ρ19 10 2 0 0 -5-5√5/2 -5+5√5/2 0 0 0 0 0 0 0 0 0 0 0 0 -1+√5/2 -1-√5/2 orthogonal faithful ρ20 10 -2 0 0 -5+5√5/2 -5-5√5/2 0 0 0 0 0 0 0 0 0 0 0 0 1+√5/2 1-√5/2 orthogonal faithful

Permutation representations of C52⋊D10
On 25 points - transitive group 25T38
Generators in S25
```(1 20 8 13 25)(2 18 12 9 17)(3 22 14 7 23)(4 16 6 15 19)(5 24 10 11 21)
(1 3 5 4 2)(6 12 8 14 10)(7 11 15 9 13)(16 18 20 22 24)(17 25 23 21 19)
(2 3)(4 5)(6 7 8 9 10 11 12 13 14 15)(16 17 18 19 20 21 22 23 24 25)
(6 9)(7 8)(10 15)(11 14)(12 13)(16 23)(17 22)(18 21)(19 20)(24 25)```

`G:=sub<Sym(25)| (1,20,8,13,25)(2,18,12,9,17)(3,22,14,7,23)(4,16,6,15,19)(5,24,10,11,21), (1,3,5,4,2)(6,12,8,14,10)(7,11,15,9,13)(16,18,20,22,24)(17,25,23,21,19), (2,3)(4,5)(6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25), (6,9)(7,8)(10,15)(11,14)(12,13)(16,23)(17,22)(18,21)(19,20)(24,25)>;`

`G:=Group( (1,20,8,13,25)(2,18,12,9,17)(3,22,14,7,23)(4,16,6,15,19)(5,24,10,11,21), (1,3,5,4,2)(6,12,8,14,10)(7,11,15,9,13)(16,18,20,22,24)(17,25,23,21,19), (2,3)(4,5)(6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25), (6,9)(7,8)(10,15)(11,14)(12,13)(16,23)(17,22)(18,21)(19,20)(24,25) );`

`G=PermutationGroup([[(1,20,8,13,25),(2,18,12,9,17),(3,22,14,7,23),(4,16,6,15,19),(5,24,10,11,21)], [(1,3,5,4,2),(6,12,8,14,10),(7,11,15,9,13),(16,18,20,22,24),(17,25,23,21,19)], [(2,3),(4,5),(6,7,8,9,10,11,12,13,14,15),(16,17,18,19,20,21,22,23,24,25)], [(6,9),(7,8),(10,15),(11,14),(12,13),(16,23),(17,22),(18,21),(19,20),(24,25)]])`

`G:=TransitiveGroup(25,38);`

Matrix representation of C52⋊D10 in GL10(𝔽11)

 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
,
 0 1 0 0 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 10 3
,
 1 0 0 0 0 0 0 0 0 0 3 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 8 0 0 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 10 0 0 0 0 0 0 0 0 8 8 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 8 0 0 0 0 0 0 0 0 3 10 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 10 3 0 0 0 0 0 0 3 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 10 3 0 0 0 0 0 0 0 0 8 8 0 0 0 0 0 0

`G:=sub<GL(10,GF(11))| [0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,10,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,3],[1,3,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,3,8,0,0,0,0,0,0,0,0,10,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,10,0,0,0,0,0,0,0,0,8,3,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,8,0,0,0,0,0,0,0,0,3,8,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,8,3,0,0,0,0,0,0,0,0,8,10,0,0,0,0,0,0] >;`

C52⋊D10 in GAP, Magma, Sage, TeX

`C_5^2\rtimes D_{10}`
`% in TeX`

`G:=Group("C5^2:D10");`
`// GroupNames label`

`G:=SmallGroup(500,27);`
`// by ID`

`G=gap.SmallGroup(500,27);`
`# by ID`

`G:=PCGroup([5,-2,-2,-5,-5,-5,127,1603,613,10004,5009]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^5=c^10=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^3,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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