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## G = C52⋊5D10order 500 = 22·53

### 2nd semidirect product of C52 and D10 acting via D10/C5=C22

Aliases: C525D10, C533C22, C52D52, C5⋊D52D5, (C5×C5⋊D5)⋊3C2, SmallGroup(500,52)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C53 — C52⋊5D10
 Chief series C1 — C5 — C52 — C53 — C5×C5⋊D5 — C52⋊5D10
 Lower central C53 — C52⋊5D10
 Upper central C1

Generators and relations for C525D10
G = < a,b,c,d | a5=b5=c10=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 616 in 72 conjugacy classes, 15 normal (3 characteristic)
C1, C2, C22, C5, C5, D5, C10, D10, C52, C52, C5×D5, C5⋊D5, D52, C53, C5×C5⋊D5, C525D10
Quotients: C1, C2, C22, D5, D10, D52, C525D10

Permutation representations of C525D10
On 20 points - transitive group 20T128
Generators in S20
```(1 7 3 9 5)(2 6 10 4 8)(11 17 13 19 15)(12 16 20 14 18)
(1 9 7 5 3)(2 4 6 8 10)(11 13 15 17 19)(12 20 18 16 14)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 20)(8 19)(9 18)(10 17)```

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

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

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

`G:=TransitiveGroup(20,128);`

44 conjugacy classes

 class 1 2A 2B 2C 5A ··· 5F 5G ··· 5AH 10A ··· 10F order 1 2 2 2 5 ··· 5 5 ··· 5 10 ··· 10 size 1 25 25 25 2 ··· 2 4 ··· 4 50 ··· 50

44 irreducible representations

 dim 1 1 2 2 4 4 type + + + + + image C1 C2 D5 D10 D52 C52⋊5D10 kernel C52⋊5D10 C5×C5⋊D5 C5⋊D5 C52 C5 C1 # reps 1 3 6 6 12 16

Matrix representation of C525D10 in GL6(𝔽11)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 10 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 10 0 0 0 0 1 0
,
 7 4 0 0 0 0 7 1 0 0 0 0 0 0 1 0 0 0 0 0 7 10 0 0 0 0 0 0 1 0 0 0 0 0 3 10
,
 7 1 0 0 0 0 7 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 3 10

`G:=sub<GL(6,GF(11))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,1,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,1,0,0,0,0,10,0],[7,7,0,0,0,0,4,1,0,0,0,0,0,0,1,7,0,0,0,0,0,10,0,0,0,0,0,0,1,3,0,0,0,0,0,10],[7,7,0,0,0,0,1,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,10] >;`

C525D10 in GAP, Magma, Sage, TeX

`C_5^2\rtimes_5D_{10}`
`% in TeX`

`G:=Group("C5^2:5D10");`
`// GroupNames label`

`G:=SmallGroup(500,52);`
`// by ID`

`G=gap.SmallGroup(500,52);`
`# by ID`

`G:=PCGroup([5,-2,-2,-5,-5,-5,242,127,808,10004]);`
`// Polycyclic`

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

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