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## G = C52⋊S3order 150 = 2·3·52

### The semidirect product of C52 and S3 acting faithfully

Aliases: C52⋊S3, C52⋊C31C2, SmallGroup(150,5)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C52⋊S3
G = < a,b,c,d | a5=b5=c3=d2=1, ab=ba, cac-1=ab3, ad=da, cbc-1=a-1b3, dbd=a-1b-1, dcd=c-1 >

15C2
25C3
3C5
3C5
25S3
3D5
15C10

Character table of C52⋊S3

 class 1 2 3 5A 5B 5C 5D 5E 5F 10A 10B 10C 10D size 1 15 50 3 3 3 3 6 6 15 15 15 15 ρ1 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 linear of order 2 ρ3 2 0 -1 2 2 2 2 2 2 0 0 0 0 orthogonal lifted from S3 ρ4 3 1 0 2ζ52+ζ5 ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 1-√5/2 1+√5/2 ζ54 ζ53 ζ52 ζ5 complex faithful ρ5 3 1 0 ζ54+2ζ53 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 1-√5/2 1+√5/2 ζ5 ζ52 ζ53 ζ54 complex faithful ρ6 3 -1 0 2ζ52+ζ5 ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 1-√5/2 1+√5/2 -ζ54 -ζ53 -ζ52 -ζ5 complex faithful ρ7 3 1 0 ζ53+2ζ5 ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 ζ52 ζ54 ζ5 ζ53 complex faithful ρ8 3 1 0 2ζ54+ζ52 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 1+√5/2 1-√5/2 ζ53 ζ5 ζ54 ζ52 complex faithful ρ9 3 -1 0 2ζ54+ζ52 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 1+√5/2 1-√5/2 -ζ53 -ζ5 -ζ54 -ζ52 complex faithful ρ10 3 -1 0 ζ53+2ζ5 ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 1+√5/2 1-√5/2 -ζ52 -ζ54 -ζ5 -ζ53 complex faithful ρ11 3 -1 0 ζ54+2ζ53 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 1-√5/2 1+√5/2 -ζ5 -ζ52 -ζ53 -ζ54 complex faithful ρ12 6 0 0 1+√5 1-√5 1-√5 1+√5 -3-√5/2 -3+√5/2 0 0 0 0 orthogonal faithful ρ13 6 0 0 1-√5 1+√5 1+√5 1-√5 -3+√5/2 -3-√5/2 0 0 0 0 orthogonal faithful

Permutation representations of C52⋊S3
On 15 points - transitive group 15T13
Generators in S15
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(1 3 5 2 4)(11 15 14 13 12)
(1 9 14)(2 7 12)(3 10 15)(4 8 13)(5 6 11)
(6 11)(7 12)(8 13)(9 14)(10 15)```

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

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

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

`G:=TransitiveGroup(15,13);`

On 15 points - transitive group 15T14
Generators in S15
```(6 7 8 9 10)(11 12 13 14 15)
(1 5 4 3 2)(6 7 8 9 10)(11 14 12 15 13)
(1 6 12)(2 9 14)(3 7 11)(4 10 13)(5 8 15)
(2 5)(3 4)(6 12)(7 13)(8 14)(9 15)(10 11)```

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

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

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

`G:=TransitiveGroup(15,14);`

On 25 points: primitive - transitive group 25T16
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 17 15)(3 10 22)(4 24 7)(5 11 18)(6 14 12)(8 20 9)(13 25 21)(16 23 19)
(6 23)(7 24)(8 25)(9 21)(10 22)(11 18)(12 19)(13 20)(14 16)(15 17)```

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

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

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

`G:=TransitiveGroup(25,16);`

On 30 points - transitive group 30T37
Generators in S30
```(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 5 4 3 10)(2 8 6 9 7)(11 14 12 15 13)(16 17 18 19 20)(21 24 22 25 23)(26 27 28 29 30)
(1 18 11)(2 26 23)(3 19 15)(4 17 12)(5 20 14)(6 30 24)(7 29 25)(8 28 21)(9 27 22)(10 16 13)
(1 2)(3 6)(4 9)(5 7)(8 10)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)```

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

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

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

`G:=TransitiveGroup(30,37);`

On 30 points - transitive group 30T38
Generators in S30
```(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)
(6 10 9 8 7)(11 13 15 12 14)(16 20 19 18 17)(26 28 30 27 29)
(1 9 13)(2 10 15)(3 6 12)(4 7 14)(5 8 11)(16 27 21)(17 29 22)(18 26 23)(19 28 24)(20 30 25)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)```

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

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

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

`G:=TransitiveGroup(30,38);`

C52⋊S3 is a maximal subgroup of   C52⋊D6  C52⋊(C3⋊S3)
C52⋊S3 is a maximal quotient of   C522Dic3  C52⋊D9  C52⋊(C3⋊S3)

Polynomial with Galois group C52⋊S3 over ℚ
actionf(x)Disc(f)
15T13x15-60x13-90x12+1095x11+1224x10-14450x9-26775x8+59640x7+177525x6+12810x5-335025x4-386555x3-246465x2-192375x-92889-353·524·136·172·732
15T14x15+5x14+50x13+220x12+970x11+1142x10+7935x9-2815x8-156480x7+1127980x6-3389737x5+7033715x4-9360980x3+9369840x2-5090420x+1432484-229·518·618·20832·1390212

Matrix representation of C52⋊S3 in GL3(𝔽11) generated by

 4 5 10 0 7 6 0 1 6
,
 4 3 2 4 5 10 9 5 10
,
 10 9 5 6 0 1 0 0 1
,
 1 2 6 0 10 0 0 0 10
`G:=sub<GL(3,GF(11))| [4,0,0,5,7,1,10,6,6],[4,4,9,3,5,5,2,10,10],[10,6,0,9,0,0,5,1,1],[1,0,0,2,10,0,6,0,10] >;`

C52⋊S3 in GAP, Magma, Sage, TeX

`C_5^2\rtimes S_3`
`% in TeX`

`G:=Group("C5^2:S3");`
`// GroupNames label`

`G:=SmallGroup(150,5);`
`// by ID`

`G=gap.SmallGroup(150,5);`
`# by ID`

`G:=PCGroup([4,-2,-3,-5,5,33,582,2307,919]);`
`// Polycyclic`

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

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