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## G = C52⋊C3order 75 = 3·52

### The semidirect product of C52 and C3 acting faithfully

Aliases: C52⋊C3, SmallGroup(75,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C3
 Chief series C1 — C52 — C52⋊C3
 Lower central C52 — C52⋊C3
 Upper central C1

Generators and relations for C52⋊C3
G = < a,b,c | a5=b5=c3=1, ab=ba, cac-1=a3b3, cbc-1=a-1b >

Character table of C52⋊C3

 class 1 3A 3B 5A 5B 5C 5D 5E 5F 5G 5H size 1 25 25 3 3 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ3 1 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ4 3 0 0 1+√5/2 1-√5/2 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 1-√5/2 1+√5/2 complex faithful ρ5 3 0 0 ζ53+2ζ5 ζ54+2ζ53 1-√5/2 1+√5/2 1-√5/2 1+√5/2 2ζ52+ζ5 2ζ54+ζ52 complex faithful ρ6 3 0 0 2ζ52+ζ5 ζ53+2ζ5 1+√5/2 1-√5/2 1+√5/2 1-√5/2 2ζ54+ζ52 ζ54+2ζ53 complex faithful ρ7 3 0 0 1+√5/2 1-√5/2 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 1-√5/2 1+√5/2 complex faithful ρ8 3 0 0 1-√5/2 1+√5/2 ζ54+2ζ53 ζ53+2ζ5 2ζ52+ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 complex faithful ρ9 3 0 0 2ζ54+ζ52 2ζ52+ζ5 1-√5/2 1+√5/2 1-√5/2 1+√5/2 ζ54+2ζ53 ζ53+2ζ5 complex faithful ρ10 3 0 0 ζ54+2ζ53 2ζ54+ζ52 1+√5/2 1-√5/2 1+√5/2 1-√5/2 ζ53+2ζ5 2ζ52+ζ5 complex faithful ρ11 3 0 0 1-√5/2 1+√5/2 2ζ52+ζ5 2ζ54+ζ52 ζ54+2ζ53 ζ53+2ζ5 1+√5/2 1-√5/2 complex faithful

Permutation representations of C52⋊C3
On 15 points - transitive group 15T9
Generators in S15
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(6 7 8 9 10)(11 14 12 15 13)
(1 13 9)(2 15 10)(3 12 6)(4 14 7)(5 11 8)```

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

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

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

`G:=TransitiveGroup(15,9);`

On 25 points: primitive - transitive group 25T6
Generators in S25
```(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)
(1 25 19 14 9)(2 21 20 15 10)(3 22 16 11 6)(4 23 17 12 7)(5 24 18 13 8)
(2 20 12)(3 6 21)(4 23 8)(5 13 16)(7 11 14)(9 18 10)(15 24 25)(17 19 22)```

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

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

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

`G:=TransitiveGroup(25,6);`

C52⋊C3 is a maximal subgroup of   C52⋊S3  C52⋊C6  C52⋊A4
C52⋊C3 is a maximal quotient of   C52⋊C9  C52⋊A4  He5⋊C3

Polynomial with Galois group C52⋊C3 over ℚ
actionf(x)Disc(f)
15T9x15-470x13-305x12+71840x11+85357x10-4292700x9-3714805x8+119761820x7+25284495x6-1542190154x5+717324725x4+7178878600x3-5452953875x2-7998223215x+4461221029224·36·524·750·4118·2932·15672·29272·16079412

Matrix representation of C52⋊C3 in GL3(𝔽11) generated by

 1 0 0 0 9 0 0 0 5
,
 4 0 0 0 4 0 0 0 9
,
 0 0 1 1 0 0 0 1 0
`G:=sub<GL(3,GF(11))| [1,0,0,0,9,0,0,0,5],[4,0,0,0,4,0,0,0,9],[0,1,0,0,0,1,1,0,0] >;`

C52⋊C3 in GAP, Magma, Sage, TeX

`C_5^2\rtimes C_3`
`% in TeX`

`G:=Group("C5^2:C3");`
`// GroupNames label`

`G:=SmallGroup(75,2);`
`// by ID`

`G=gap.SmallGroup(75,2);`
`# by ID`

`G:=PCGroup([3,-3,-5,5,199,434]);`
`// Polycyclic`

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

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