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## G = C52⋊F5order 500 = 22·53

### 3rd semidirect product of C52 and F5 acting faithfully

Aliases: He53C4, C523F5, He5⋊C2.2C2, C5.2(C5⋊F5), SmallGroup(500,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — He5 — C52⋊F5
 Chief series C1 — C5 — C52 — He5 — He5⋊C2 — C52⋊F5
 Lower central He5 — C52⋊F5
 Upper central C1

Generators and relations for C52⋊F5
G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, cac-1=ab-1, dad-1=a3b-1, bc=cb, dbd-1=b-1, dcd-1=c3 >

25C2
5C5
5C5
5C5
5C5
5C5
5C5
125C4
5D5
5D5
5D5
5D5
5D5
5D5
25C10
25F5
25F5
25F5
25F5
25F5
25F5
25Dic5

Character table of C52⋊F5

 class 1 2 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B size 1 25 125 125 2 2 20 20 20 20 20 20 50 50 ρ1 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 linear of order 2 ρ3 1 -1 i -i 1 1 1 1 1 1 1 1 -1 -1 linear of order 4 ρ4 1 -1 -i i 1 1 1 1 1 1 1 1 -1 -1 linear of order 4 ρ5 4 0 0 0 4 4 -1 -1 -1 4 -1 -1 0 0 orthogonal lifted from F5 ρ6 4 0 0 0 4 4 -1 -1 -1 -1 -1 4 0 0 orthogonal lifted from F5 ρ7 4 0 0 0 4 4 -1 -1 4 -1 -1 -1 0 0 orthogonal lifted from F5 ρ8 4 0 0 0 4 4 -1 -1 -1 -1 4 -1 0 0 orthogonal lifted from F5 ρ9 4 0 0 0 4 4 -1 4 -1 -1 -1 -1 0 0 orthogonal lifted from F5 ρ10 4 0 0 0 4 4 4 -1 -1 -1 -1 -1 0 0 orthogonal lifted from F5 ρ11 10 2 0 0 -5+5√5/2 -5-5√5/2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 orthogonal faithful ρ12 10 2 0 0 -5-5√5/2 -5+5√5/2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 orthogonal faithful ρ13 10 -2 0 0 -5-5√5/2 -5+5√5/2 0 0 0 0 0 0 1-√5/2 1+√5/2 symplectic faithful, Schur index 2 ρ14 10 -2 0 0 -5+5√5/2 -5-5√5/2 0 0 0 0 0 0 1+√5/2 1-√5/2 symplectic faithful, Schur index 2

Permutation representations of C52⋊F5
On 25 points - transitive group 25T33
Generators in S25
```(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)
(1 4 5 2 3)(6 8 10 7 9)(11 12 13 14 15)(16 20 19 18 17)(21 24 22 25 23)
(1 20 6 25 14)(2 17 7 24 12)(3 16 9 22 13)(4 19 8 23 15)(5 18 10 21 11)
(1 12 6 24)(2 14 7 25)(3 13 9 22)(4 11 8 21)(5 15 10 23)(17 20)(18 19)```

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

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

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

`G:=TransitiveGroup(25,33);`

Matrix representation of C52⋊F5 in GL10(𝔽41)

 34 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 7 7 0 0 0 0 0 0 0 0 34 40 0 0 0 0 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 7 7 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 34
,
 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
,
 1 0 0 0 0 0 0 0 0 0 34 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 34 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 34 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 34 40 0 0 0 0 1 0 0 0 0 0 0 0 0 0 34 40 0 0 0 0

`G:=sub<GL(10,GF(41))| [34,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,7,34,0,0,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34],[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],[1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0] >;`

C52⋊F5 in GAP, Magma, Sage, TeX

`C_5^2\rtimes F_5`
`% in TeX`

`G:=Group("C5^2:F5");`
`// GroupNames label`

`G:=SmallGroup(500,23);`
`// by ID`

`G=gap.SmallGroup(500,23);`
`# by ID`

`G:=PCGroup([5,-2,-2,-5,-5,-5,10,122,127,803,808,613,10004]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^3*b^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;`
`// generators/relations`

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