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## G = C52⋊C12order 300 = 22·3·52

### The semidirect product of C52 and C12 acting faithfully

Aliases: C52⋊C12, C5⋊D5.C6, C5⋊F5⋊C3, C52⋊C32C4, C52⋊C6.1C2, SmallGroup(300,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C12
 Chief series C1 — C52 — C5⋊D5 — C52⋊C6 — C52⋊C12
 Lower central C52 — C52⋊C12
 Upper central C1

Generators and relations for C52⋊C12
G = < a,b,c | a5=b5=c12=1, ab=ba, cac-1=a-1b, cbc-1=a3b3 >

25C2
25C3
3C5
3C5
25C4
25C6
15D5
15D5
25C12
15F5
15F5

Character table of C52⋊C12

 class 1 2 3A 3B 4A 4B 5A 5B 6A 6B 12A 12B 12C 12D size 1 25 25 25 25 25 12 12 25 25 25 25 25 25 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 ζ32 ζ3 -1 -1 1 1 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ4 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ5 1 1 ζ3 ζ32 -1 -1 1 1 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ6 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ7 1 -1 1 1 -i i 1 1 -1 -1 i -i i -i linear of order 4 ρ8 1 -1 1 1 i -i 1 1 -1 -1 -i i -i i linear of order 4 ρ9 1 -1 ζ3 ζ32 -i i 1 1 ζ6 ζ65 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 linear of order 12 ρ10 1 -1 ζ32 ζ3 -i i 1 1 ζ65 ζ6 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 linear of order 12 ρ11 1 -1 ζ3 ζ32 i -i 1 1 ζ6 ζ65 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 linear of order 12 ρ12 1 -1 ζ32 ζ3 i -i 1 1 ζ65 ζ6 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 linear of order 12 ρ13 12 0 0 0 0 0 2 -3 0 0 0 0 0 0 orthogonal faithful ρ14 12 0 0 0 0 0 -3 2 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C52⋊C12
On 15 points - transitive group 15T19
Generators in S15
```(1 9 6 12 15)(2 10 7 13 4)(3 14 11 5 8)
(1 6 15 9 12)(2 4 13 7 10)
(1 2 3)(4 5 6 7 8 9 10 11 12 13 14 15)```

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

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

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

`G:=TransitiveGroup(15,19);`

On 25 points: primitive - transitive group 25T26
Generators in S25
```(1 11 8 2 5)(3 13 20 25 24)(4 16 6 23 15)(7 9 18 19 14)(10 21 17 12 22)
(1 21 18 24 15)(2 22 7 20 6)(3 4 11 17 19)(5 10 9 25 23)(8 12 14 13 16)
(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)| (1,11,8,2,5)(3,13,20,25,24)(4,16,6,23,15)(7,9,18,19,14)(10,21,17,12,22), (1,21,18,24,15)(2,22,7,20,6)(3,4,11,17,19)(5,10,9,25,23)(8,12,14,13,16), (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( (1,11,8,2,5)(3,13,20,25,24)(4,16,6,23,15)(7,9,18,19,14)(10,21,17,12,22), (1,21,18,24,15)(2,22,7,20,6)(3,4,11,17,19)(5,10,9,25,23)(8,12,14,13,16), (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([[(1,11,8,2,5),(3,13,20,25,24),(4,16,6,23,15),(7,9,18,19,14),(10,21,17,12,22)], [(1,21,18,24,15),(2,22,7,20,6),(3,4,11,17,19),(5,10,9,25,23),(8,12,14,13,16)], [(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(25,26);`

On 30 points - transitive group 30T78
Generators in S30
```(2 17 26 20 11)(3 27 12 18 21)(5 29 14 8 23)(6 15 24 30 9)
(1 16 25 19 10)(2 20 17 11 26)(3 27 12 18 21)(4 28 13 7 22)(5 8 29 23 14)(6 15 24 30 9)
(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 26 27 28 29 30)```

`G:=sub<Sym(30)| (2,17,26,20,11)(3,27,12,18,21)(5,29,14,8,23)(6,15,24,30,9), (1,16,25,19,10)(2,20,17,11,26)(3,27,12,18,21)(4,28,13,7,22)(5,8,29,23,14)(6,15,24,30,9), (1,2,3,4,5,6)(7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30)>;`

`G:=Group( (2,17,26,20,11)(3,27,12,18,21)(5,29,14,8,23)(6,15,24,30,9), (1,16,25,19,10)(2,20,17,11,26)(3,27,12,18,21)(4,28,13,7,22)(5,8,29,23,14)(6,15,24,30,9), (1,2,3,4,5,6)(7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30) );`

`G=PermutationGroup([[(2,17,26,20,11),(3,27,12,18,21),(5,29,14,8,23),(6,15,24,30,9)], [(1,16,25,19,10),(2,20,17,11,26),(3,27,12,18,21),(4,28,13,7,22),(5,8,29,23,14),(6,15,24,30,9)], [(1,2,3,4,5,6),(7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30)]])`

`G:=TransitiveGroup(30,78);`

Polynomial with Galois group C52⋊C12 over ℚ
actionf(x)Disc(f)
15T19x15+x10-2x5-1515·710

Matrix representation of C52⋊C12 in GL12(ℤ)

 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 -1 -1
,
 0 0 0 0 1 0 0 0 0 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 0 0 -1 -1 -1 -1 0 0 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 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 1 0 0 0 0 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 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0

`G:=sub<GL(12,Integers())| [0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0] >;`

C52⋊C12 in GAP, Magma, Sage, TeX

`C_5^2\rtimes C_{12}`
`% in TeX`

`G:=Group("C5^2:C12");`
`// GroupNames label`

`G:=SmallGroup(300,24);`
`// by ID`

`G=gap.SmallGroup(300,24);`
`# by ID`

`G:=PCGroup([5,-2,-3,-2,-5,5,30,483,1928,173,3004,2859,1014]);`
`// Polycyclic`

`G:=Group<a,b,c|a^5=b^5=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^3*b^3>;`
`// generators/relations`

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