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## G = C52⋊D6order 300 = 22·3·52

### The semidirect product of C52 and D6 acting faithfully

Aliases: C52⋊D6, C5⋊D5⋊S3, C52⋊S3⋊C2, C52⋊C6⋊C2, C52⋊C3⋊C22, SmallGroup(300,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C3 — C52⋊D6
 Chief series C1 — C52 — C52⋊C3 — C52⋊S3 — C52⋊D6
 Lower central C52⋊C3 — C52⋊D6
 Upper central C1

Generators and relations for C52⋊D6
G = < a,b,c,d | a5=b5=c6=d2=1, ab=ba, cac-1=dad=a2b3, cbc-1=a-1b-1, dbd=a-1b3, dcd=c-1 >

15C2
15C2
25C2
25C3
3C5
3C5
75C22
25C6
25S3
25S3
3D5
3D5
15D5
15C10
15D5
15C10
25D6
15D10
15D10
3D52

Character table of C52⋊D6

 class 1 2A 2B 2C 3 5A 5B 5C 5D 6 10A 10B 10C 10D size 1 15 15 25 50 6 6 6 6 50 30 30 30 30 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 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 -1 1 -1 1 1 1 1 1 -1 1 1 -1 -1 linear of order 2 ρ5 2 0 0 -2 -1 2 2 2 2 1 0 0 0 0 orthogonal lifted from D6 ρ6 2 0 0 2 -1 2 2 2 2 -1 0 0 0 0 orthogonal lifted from S3 ρ7 6 0 2 0 0 -3+√5/2 1+√5 1-√5 -3-√5/2 0 -1-√5/2 -1+√5/2 0 0 orthogonal faithful ρ8 6 0 -2 0 0 -3-√5/2 1-√5 1+√5 -3+√5/2 0 1-√5/2 1+√5/2 0 0 orthogonal faithful ρ9 6 2 0 0 0 1+√5 -3-√5/2 -3+√5/2 1-√5 0 0 0 -1+√5/2 -1-√5/2 orthogonal faithful ρ10 6 0 -2 0 0 -3+√5/2 1+√5 1-√5 -3-√5/2 0 1+√5/2 1-√5/2 0 0 orthogonal faithful ρ11 6 -2 0 0 0 1-√5 -3+√5/2 -3-√5/2 1+√5 0 0 0 1+√5/2 1-√5/2 orthogonal faithful ρ12 6 0 2 0 0 -3-√5/2 1-√5 1+√5 -3+√5/2 0 -1+√5/2 -1-√5/2 0 0 orthogonal faithful ρ13 6 2 0 0 0 1-√5 -3+√5/2 -3-√5/2 1+√5 0 0 0 -1-√5/2 -1+√5/2 orthogonal faithful ρ14 6 -2 0 0 0 1+√5 -3-√5/2 -3+√5/2 1-√5 0 0 0 1-√5/2 1+√5/2 orthogonal faithful

Permutation representations of C52⋊D6
On 15 points - transitive group 15T18
Generators in S15
```(1 6 12 15 9)(2 13 4 7 10)(3 8 14 11 5)
(2 4 10 13 7)(3 5 11 14 8)
(1 2 3)(4 5 6 7 8 9)(10 11 12 13 14 15)
(1 2)(4 9)(5 8)(6 7)(10 15)(11 14)(12 13)```

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

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

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

`G:=TransitiveGroup(15,18);`

On 25 points: primitive - transitive group 25T27
Generators in S25
```(1 4 8 11 7)(2 3 19 22 17)(5 14 25 16 6)(9 23 21 10 15)(12 18 13 24 20)
(1 24 14 17 21)(2 10 4 20 25)(3 15 8 12 16)(5 22 23 7 13)(6 19 9 11 18)
(2 3 4 5 6 7)(8 9 10 11 12 13)(14 15 16 17 18 19)(20 21 22 23 24 25)
(2 5)(3 4)(6 7)(8 13)(9 12)(10 11)(14 18)(15 17)(21 25)(22 24)```

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

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

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

`G:=TransitiveGroup(25,27);`

On 30 points - transitive group 30T66
Generators in S30
```(2 14 26 20 7)(3 15 27 21 8)(5 10 23 29 17)(6 11 24 30 18)
(1 25 12 13 19)(2 20 14 7 26)(3 15 27 21 8)(4 22 16 9 28)(5 29 10 17 23)(6 11 24 30 18)
(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)
(1 3)(4 6)(8 12)(9 11)(13 15)(16 18)(19 21)(22 24)(25 27)(28 30)```

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

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

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

`G:=TransitiveGroup(30,66);`

On 30 points - transitive group 30T72
Generators in S30
```(2 7 16 26 19)(3 8 17 27 20)(5 22 29 13 10)(6 23 30 14 11)
(1 15 24 12 25)(2 26 7 19 16)(3 8 17 27 20)(4 28 9 21 18)(5 13 22 10 29)(6 23 30 14 11)
(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)
(1 6)(2 5)(3 4)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)(13 26)(14 25)(15 30)(16 29)(17 28)(18 27)```

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

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

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

`G:=TransitiveGroup(30,72);`

On 30 points - transitive group 30T80
Generators in S30
```(1 9 21 24 12)(3 8 20 23 11)(5 17 30 27 14)(6 18 25 28 15)
(1 9 21 24 12)(2 19 10 7 22)(3 23 8 11 20)(4 29 13 16 26)(5 27 17 14 30)(6 18 25 28 15)
(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)
(1 4)(2 6)(3 5)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(19 25)(20 30)(21 29)(22 28)(23 27)(24 26)```

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

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

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

`G:=TransitiveGroup(30,80);`

Polynomial with Galois group C52⋊D6 over ℚ
actionf(x)Disc(f)
15T18x15-15x13-15x12+55x11+149x10+150x9-165x8-740x7-1105x6-1153x5-875x4-460x3-160x2-25x+1-518·172·235·532·1812·416517412

Matrix representation of C52⋊D6 in GL6(𝔽31)

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

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

C52⋊D6 in GAP, Magma, Sage, TeX

`C_5^2\rtimes D_6`
`% in TeX`

`G:=Group("C5^2:D6");`
`// GroupNames label`

`G:=SmallGroup(300,25);`
`// by ID`

`G=gap.SmallGroup(300,25);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-5,5,122,67,963,793,1804,3609,464]);`
`// Polycyclic`

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

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