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G = C52⋊C10order 250 = 2·53

The semidirect product of C52 and C10 acting faithfully

Aliases: C52⋊C10, He51C2, C521D5, C5⋊D5⋊C5, C5.2(C5×D5), SmallGroup(250,5)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C52⋊C10
G = < a,b,c | a5=b5=c10=1, ab=ba, cac-1=a-1b3, cbc-1=b-1 >

25C2
5C5
5C5
10C5
10C5
5D5
25C10
25D5
2C52
2C52

Character table of C52⋊C10

 class 1 2 5A 5B 5C 5D 5E 5F 5G 5H 5I 5J 5K 5L 5M 5N 5O 5P 10A 10B 10C 10D size 1 25 2 2 5 5 5 5 10 10 10 10 10 10 10 10 10 10 25 25 25 25 ρ1 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 linear of order 2 ρ3 1 -1 1 1 ζ5 ζ52 ζ53 ζ54 1 ζ54 ζ54 1 ζ5 ζ5 ζ52 ζ53 ζ53 ζ52 -ζ52 -ζ54 -ζ5 -ζ53 linear of order 10 ρ4 1 1 1 1 ζ5 ζ52 ζ53 ζ54 1 ζ54 ζ54 1 ζ5 ζ5 ζ52 ζ53 ζ53 ζ52 ζ52 ζ54 ζ5 ζ53 linear of order 5 ρ5 1 1 1 1 ζ54 ζ53 ζ52 ζ5 1 ζ5 ζ5 1 ζ54 ζ54 ζ53 ζ52 ζ52 ζ53 ζ53 ζ5 ζ54 ζ52 linear of order 5 ρ6 1 -1 1 1 ζ52 ζ54 ζ5 ζ53 1 ζ53 ζ53 1 ζ52 ζ52 ζ54 ζ5 ζ5 ζ54 -ζ54 -ζ53 -ζ52 -ζ5 linear of order 10 ρ7 1 -1 1 1 ζ54 ζ53 ζ52 ζ5 1 ζ5 ζ5 1 ζ54 ζ54 ζ53 ζ52 ζ52 ζ53 -ζ53 -ζ5 -ζ54 -ζ52 linear of order 10 ρ8 1 1 1 1 ζ52 ζ54 ζ5 ζ53 1 ζ53 ζ53 1 ζ52 ζ52 ζ54 ζ5 ζ5 ζ54 ζ54 ζ53 ζ52 ζ5 linear of order 5 ρ9 1 1 1 1 ζ53 ζ5 ζ54 ζ52 1 ζ52 ζ52 1 ζ53 ζ53 ζ5 ζ54 ζ54 ζ5 ζ5 ζ52 ζ53 ζ54 linear of order 5 ρ10 1 -1 1 1 ζ53 ζ5 ζ54 ζ52 1 ζ52 ζ52 1 ζ53 ζ53 ζ5 ζ54 ζ54 ζ5 -ζ5 -ζ52 -ζ53 -ζ54 linear of order 10 ρ11 2 0 2 2 2 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 -1-√5/2 -1-√5/2 0 0 0 0 orthogonal lifted from D5 ρ12 2 0 2 2 2 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 -1+√5/2 -1+√5/2 0 0 0 0 orthogonal lifted from D5 ρ13 2 0 2 2 2ζ52 2ζ54 2ζ5 2ζ53 -1+√5/2 ζ5+1 ζ54+ζ52 -1-√5/2 ζ54+1 ζ53+ζ5 ζ52+ζ5 ζ54+ζ53 ζ52+1 ζ53+1 0 0 0 0 complex lifted from C5×D5 ρ14 2 0 2 2 2ζ52 2ζ54 2ζ5 2ζ53 -1-√5/2 ζ54+ζ52 ζ5+1 -1+√5/2 ζ53+ζ5 ζ54+1 ζ53+1 ζ52+1 ζ54+ζ53 ζ52+ζ5 0 0 0 0 complex lifted from C5×D5 ρ15 2 0 2 2 2ζ5 2ζ52 2ζ53 2ζ54 -1-√5/2 ζ53+1 ζ52+ζ5 -1+√5/2 ζ52+1 ζ54+ζ53 ζ53+ζ5 ζ54+ζ52 ζ5+1 ζ54+1 0 0 0 0 complex lifted from C5×D5 ρ16 2 0 2 2 2ζ54 2ζ53 2ζ52 2ζ5 -1+√5/2 ζ54+ζ53 ζ52+1 -1-√5/2 ζ52+ζ5 ζ53+1 ζ5+1 ζ54+1 ζ53+ζ5 ζ54+ζ52 0 0 0 0 complex lifted from C5×D5 ρ17 2 0 2 2 2ζ53 2ζ5 2ζ54 2ζ52 -1+√5/2 ζ54+1 ζ53+ζ5 -1-√5/2 ζ5+1 ζ54+ζ52 ζ54+ζ53 ζ52+ζ5 ζ53+1 ζ52+1 0 0 0 0 complex lifted from C5×D5 ρ18 2 0 2 2 2ζ5 2ζ52 2ζ53 2ζ54 -1+√5/2 ζ52+ζ5 ζ53+1 -1-√5/2 ζ54+ζ53 ζ52+1 ζ54+1 ζ5+1 ζ54+ζ52 ζ53+ζ5 0 0 0 0 complex lifted from C5×D5 ρ19 2 0 2 2 2ζ53 2ζ5 2ζ54 2ζ52 -1-√5/2 ζ53+ζ5 ζ54+1 -1+√5/2 ζ54+ζ52 ζ5+1 ζ52+1 ζ53+1 ζ52+ζ5 ζ54+ζ53 0 0 0 0 complex lifted from C5×D5 ρ20 2 0 2 2 2ζ54 2ζ53 2ζ52 2ζ5 -1-√5/2 ζ52+1 ζ54+ζ53 -1+√5/2 ζ53+1 ζ52+ζ5 ζ54+ζ52 ζ53+ζ5 ζ54+1 ζ5+1 0 0 0 0 complex lifted from C5×D5 ρ21 10 0 -5+5√5/2 -5-5√5/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ22 10 0 -5-5√5/2 -5+5√5/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C52⋊C10
On 25 points - transitive group 25T23
Generators in S25
```(1 16 6 11 21)(2 24 10 7 23)(3 18 12 15 19)(4 22 14 13 25)(5 20 8 9 17)
(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)```

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

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

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

`G:=TransitiveGroup(25,23);`

On 25 points - transitive group 25T24
Generators in S25
```(1 15 25 20 10)(2 16 11 6 21)(4 23 8 13 18)(5 14 24 19 9)
(1 15 25 20 10)(2 11 21 16 6)(3 7 17 22 12)(4 13 23 18 8)(5 9 19 24 14)
(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)```

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

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

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

`G:=TransitiveGroup(25,24);`

C52⋊C10 is a maximal subgroup of   C52⋊C20  He5⋊C4  C52⋊D10
C52⋊C10 is a maximal quotient of   He55C4

Matrix representation of C52⋊C10 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

`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] >;`

C52⋊C10 in GAP, Magma, Sage, TeX

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

`G:=Group("C5^2:C10");`
`// GroupNames label`

`G:=SmallGroup(250,5);`
`// by ID`

`G=gap.SmallGroup(250,5);`
`# by ID`

`G:=PCGroup([4,-2,-5,-5,-5,482,366,3203]);`
`// Polycyclic`

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

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