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## G = C52⋊C6order 150 = 2·3·52

### The semidirect product of C52 and C6 acting faithfully

Aliases: C52⋊C6, C5⋊D5⋊C3, C52⋊C32C2, SmallGroup(150,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C6
 Chief series C1 — C52 — C52⋊C3 — C52⋊C6
 Lower central C52 — C52⋊C6
 Upper central C1

Generators and relations for C52⋊C6
G = < a,b,c | a5=b5=c6=1, ab=ba, cac-1=a2b3, cbc-1=a-1b-1 >

25C2
25C3
3C5
3C5
25C6
15D5
15D5

Character table of C52⋊C6

 class 1 2 3A 3B 5A 5B 5C 5D 6A 6B size 1 25 25 25 6 6 6 6 25 25 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ3 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 linear of order 3 ρ4 1 -1 ζ32 ζ3 1 1 1 1 ζ6 ζ65 linear of order 6 ρ5 1 -1 ζ3 ζ32 1 1 1 1 ζ65 ζ6 linear of order 6 ρ6 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 linear of order 3 ρ7 6 0 0 0 -3-√5/2 1-√5 1+√5 -3+√5/2 0 0 orthogonal faithful ρ8 6 0 0 0 1+√5 -3-√5/2 -3+√5/2 1-√5 0 0 orthogonal faithful ρ9 6 0 0 0 -3+√5/2 1+√5 1-√5 -3-√5/2 0 0 orthogonal faithful ρ10 6 0 0 0 1-√5 -3+√5/2 -3-√5/2 1+√5 0 0 orthogonal faithful

Permutation representations of C52⋊C6
On 15 points - transitive group 15T12
Generators in S15
```(1 8 11 14 5)(2 12 6 9 15)(3 4 13 10 7)
(2 6 15 12 9)(3 7 10 13 4)
(1 2 3)(4 5 6 7 8 9)(10 11 12 13 14 15)```

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

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

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

`G:=TransitiveGroup(15,12);`

On 25 points: primitive - transitive group 25T15
Generators in S25
```(1 21 5 2 24)(3 14 4 11 13)(6 10 8 7 17)(9 19 25 20 15)(12 18 23 22 16)
(1 11 16 19 8)(2 14 23 15 6)(3 18 20 17 5)(4 22 9 10 24)(7 21 13 12 25)
(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)| (1,21,5,2,24)(3,14,4,11,13)(6,10,8,7,17)(9,19,25,20,15)(12,18,23,22,16), (1,11,16,19,8)(2,14,23,15,6)(3,18,20,17,5)(4,22,9,10,24)(7,21,13,12,25), (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( (1,21,5,2,24)(3,14,4,11,13)(6,10,8,7,17)(9,19,25,20,15)(12,18,23,22,16), (1,11,16,19,8)(2,14,23,15,6)(3,18,20,17,5)(4,22,9,10,24)(7,21,13,12,25), (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([(1,21,5,2,24),(3,14,4,11,13),(6,10,8,7,17),(9,19,25,20,15),(12,18,23,22,16)], [(1,11,16,19,8),(2,14,23,15,6),(3,18,20,17,5),(4,22,9,10,24),(7,21,13,12,25)], [(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(25,15);`

On 30 points - transitive group 30T35
Generators in S30
```(1 22 25 8 14)(2 26 15 23 9)(3 24 27 10 16)(4 17 11 28 19)(5 12 20 18 29)(6 13 7 30 21)
(2 15 9 26 23)(3 16 10 27 24)(5 20 29 12 18)(6 21 30 7 13)
(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)| (1,22,25,8,14)(2,26,15,23,9)(3,24,27,10,16)(4,17,11,28,19)(5,12,20,18,29)(6,13,7,30,21), (2,15,9,26,23)(3,16,10,27,24)(5,20,29,12,18)(6,21,30,7,13), (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( (1,22,25,8,14)(2,26,15,23,9)(3,24,27,10,16)(4,17,11,28,19)(5,12,20,18,29)(6,13,7,30,21), (2,15,9,26,23)(3,16,10,27,24)(5,20,29,12,18)(6,21,30,7,13), (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([(1,22,25,8,14),(2,26,15,23,9),(3,24,27,10,16),(4,17,11,28,19),(5,12,20,18,29),(6,13,7,30,21)], [(2,15,9,26,23),(3,16,10,27,24),(5,20,29,12,18),(6,21,30,7,13)], [(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(30,35);`

C52⋊C6 is a maximal subgroup of   C52⋊Dic3  C52⋊C12  C52⋊D6  C5⋊D15⋊C3
C52⋊C6 is a maximal quotient of   C522C12  C52⋊C18  C5⋊D15⋊C3

Polynomial with Galois group C52⋊C6 over ℚ
actionf(x)Disc(f)
15T12x15-235x13+70x12+15930x11-14493x10-325950x9+112750x8+2876560x7+708890x6-11702794x5-8278925x4+19171805x3+20087950x2-6362585x-9179941312·518·722·132·4118·22672·48712·5483031859935751932

Matrix representation of C52⋊C6 in GL6(𝔽31)

 1 0 0 0 0 0 0 1 0 0 0 0 13 1 30 18 0 0 30 19 13 13 0 0 30 19 0 0 13 13 13 1 0 0 18 30
,
 30 1 0 0 0 0 11 19 0 0 0 0 30 0 0 1 0 0 12 1 30 18 0 0 29 19 0 0 13 13 12 1 0 0 18 30
,
 13 1 0 0 17 30 30 19 0 0 12 13 0 0 0 0 30 0 12 1 0 0 30 0 0 0 1 0 30 0 13 1 18 30 30 0

`G:=sub<GL(6,GF(31))| [1,0,13,30,30,13,0,1,1,19,19,1,0,0,30,13,0,0,0,0,18,13,0,0,0,0,0,0,13,18,0,0,0,0,13,30],[30,11,30,12,29,12,1,19,0,1,19,1,0,0,0,30,0,0,0,0,1,18,0,0,0,0,0,0,13,18,0,0,0,0,13,30],[13,30,0,12,0,13,1,19,0,1,0,1,0,0,0,0,1,18,0,0,0,0,0,30,17,12,30,30,30,30,30,13,0,0,0,0] >;`

C52⋊C6 in GAP, Magma, Sage, TeX

`C_5^2\rtimes C_6`
`% in TeX`

`G:=Group("C5^2:C6");`
`// GroupNames label`

`G:=SmallGroup(150,6);`
`// by ID`

`G=gap.SmallGroup(150,6);`
`# by ID`

`G:=PCGroup([4,-2,-3,-5,5,290,474,1923,295]);`
`// Polycyclic`

`G:=Group<a,b,c|a^5=b^5=c^6=1,a*b=b*a,c*a*c^-1=a^2*b^3,c*b*c^-1=a^-1*b^-1>;`
`// generators/relations`

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