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## G = C52⋊(C3⋊S3)  order 450 = 2·32·52

### The semidirect product of C52 and C3⋊S3 acting via C3⋊S3/C3=S3

Aliases: C52⋊(C3⋊S3), C3⋊(C52⋊S3), (C5×C15)⋊1S3, C52⋊C31S3, (C3×C52⋊C3)⋊2C2, SmallGroup(450,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C3×C52⋊C3 — C52⋊(C3⋊S3)
 Chief series C1 — C52 — C5×C15 — C3×C52⋊C3 — C52⋊(C3⋊S3)
 Lower central C3×C52⋊C3 — C52⋊(C3⋊S3)
 Upper central C1

Generators and relations for C52⋊(C3⋊S3)
G = < a,b,c,d,e | a5=b5=c3=d3=e2=1, ab=ba, ac=ca, dad-1=ab3, ae=ea, bc=cb, dbd-1=a-1b3, ebe=a-1b-1, cd=dc, ece=c-1, ede=d-1 >

45C2
25C3
25C3
25C3
3C5
3C5
15S3
75S3
75S3
75S3
25C32
9D5
45C10
3C15
3C15
25C3⋊S3
3D15
15C5×S3

Character table of C52⋊(C3⋊S3)

 class 1 2 3A 3B 3C 3D 5A 5B 5C 5D 5E 5F 10A 10B 10C 10D 15A 15B 15C 15D 15E 15F 15G 15H size 1 45 2 50 50 50 3 3 3 3 6 6 45 45 45 45 6 6 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 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 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 -1 -1 2 -1 2 2 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 0 2 -1 -1 -1 2 2 2 2 2 2 0 0 0 0 2 2 2 2 2 2 2 2 orthogonal lifted from S3 ρ5 2 0 -1 2 -1 -1 2 2 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 0 -1 -1 -1 2 2 2 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 3 -1 3 0 0 0 ζ54+2ζ53 2ζ52+ζ5 ζ53+2ζ5 2ζ54+ζ52 1-√5/2 1+√5/2 -ζ5 -ζ53 -ζ52 -ζ54 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 ζ53+2ζ5 1+√5/2 1-√5/2 1+√5/2 1-√5/2 complex lifted from C52⋊S3 ρ8 3 1 3 0 0 0 ζ54+2ζ53 2ζ52+ζ5 ζ53+2ζ5 2ζ54+ζ52 1-√5/2 1+√5/2 ζ5 ζ53 ζ52 ζ54 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 ζ53+2ζ5 1+√5/2 1-√5/2 1+√5/2 1-√5/2 complex lifted from C52⋊S3 ρ9 3 1 3 0 0 0 ζ53+2ζ5 2ζ54+ζ52 2ζ52+ζ5 ζ54+2ζ53 1+√5/2 1-√5/2 ζ52 ζ5 ζ54 ζ53 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 2ζ52+ζ5 1-√5/2 1+√5/2 1-√5/2 1+√5/2 complex lifted from C52⋊S3 ρ10 3 1 3 0 0 0 2ζ54+ζ52 ζ53+2ζ5 ζ54+2ζ53 2ζ52+ζ5 1+√5/2 1-√5/2 ζ53 ζ54 ζ5 ζ52 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 ζ54+2ζ53 1-√5/2 1+√5/2 1-√5/2 1+√5/2 complex lifted from C52⋊S3 ρ11 3 1 3 0 0 0 2ζ52+ζ5 ζ54+2ζ53 2ζ54+ζ52 ζ53+2ζ5 1-√5/2 1+√5/2 ζ54 ζ52 ζ53 ζ5 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 2ζ54+ζ52 1+√5/2 1-√5/2 1+√5/2 1-√5/2 complex lifted from C52⋊S3 ρ12 3 -1 3 0 0 0 ζ53+2ζ5 2ζ54+ζ52 2ζ52+ζ5 ζ54+2ζ53 1+√5/2 1-√5/2 -ζ52 -ζ5 -ζ54 -ζ53 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 2ζ52+ζ5 1-√5/2 1+√5/2 1-√5/2 1+√5/2 complex lifted from C52⋊S3 ρ13 3 -1 3 0 0 0 2ζ54+ζ52 ζ53+2ζ5 ζ54+2ζ53 2ζ52+ζ5 1+√5/2 1-√5/2 -ζ53 -ζ54 -ζ5 -ζ52 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 ζ54+2ζ53 1-√5/2 1+√5/2 1-√5/2 1+√5/2 complex lifted from C52⋊S3 ρ14 3 -1 3 0 0 0 2ζ52+ζ5 ζ54+2ζ53 2ζ54+ζ52 ζ53+2ζ5 1-√5/2 1+√5/2 -ζ54 -ζ52 -ζ53 -ζ5 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 2ζ54+ζ52 1+√5/2 1-√5/2 1+√5/2 1-√5/2 complex lifted from C52⋊S3 ρ15 6 0 6 0 0 0 1+√5 1+√5 1-√5 1-√5 -3-√5/2 -3+√5/2 0 0 0 0 1-√5 1+√5 1+√5 1-√5 -3+√5/2 -3-√5/2 -3+√5/2 -3-√5/2 orthogonal lifted from C52⋊S3 ρ16 6 0 -3 0 0 0 1-√5 1-√5 1+√5 1+√5 -3+√5/2 -3-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 orthogonal faithful ρ17 6 0 -3 0 0 0 1+√5 1+√5 1-√5 1-√5 -3-√5/2 -3+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 orthogonal faithful ρ18 6 0 6 0 0 0 1-√5 1-√5 1+√5 1+√5 -3+√5/2 -3-√5/2 0 0 0 0 1+√5 1-√5 1-√5 1+√5 -3-√5/2 -3+√5/2 -3-√5/2 -3+√5/2 orthogonal lifted from C52⋊S3 ρ19 6 0 -3 0 0 0 1+√5 1+√5 1-√5 1-√5 -3-√5/2 -3+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 orthogonal faithful ρ20 6 0 -3 0 0 0 1-√5 1-√5 1+√5 1+√5 -3+√5/2 -3-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 orthogonal faithful ρ21 6 0 -3 0 0 0 2ζ54+4ζ53 4ζ52+2ζ5 2ζ53+4ζ5 4ζ54+2ζ52 1-√5 1+√5 0 0 0 0 -2ζ54-ζ52 -ζ54-2ζ53 -2ζ52-ζ5 -ζ53-2ζ5 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 complex faithful ρ22 6 0 -3 0 0 0 4ζ54+2ζ52 2ζ53+4ζ5 2ζ54+4ζ53 4ζ52+2ζ5 1+√5 1-√5 0 0 0 0 -2ζ52-ζ5 -2ζ54-ζ52 -ζ53-2ζ5 -ζ54-2ζ53 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 complex faithful ρ23 6 0 -3 0 0 0 4ζ52+2ζ5 2ζ54+4ζ53 4ζ54+2ζ52 2ζ53+4ζ5 1-√5 1+√5 0 0 0 0 -ζ53-2ζ5 -2ζ52-ζ5 -ζ54-2ζ53 -2ζ54-ζ52 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 complex faithful ρ24 6 0 -3 0 0 0 2ζ53+4ζ5 4ζ54+2ζ52 4ζ52+2ζ5 2ζ54+4ζ53 1+√5 1-√5 0 0 0 0 -ζ54-2ζ53 -ζ53-2ζ5 -2ζ54-ζ52 -2ζ52-ζ5 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 complex faithful

Smallest permutation representation of C52⋊(C3⋊S3)
On 45 points
Generators in S45
```(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)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)
(1 3 5 2 4)(6 8 10 7 9)(11 13 15 12 14)(31 35 34 33 32)(36 40 39 38 37)(41 45 44 43 42)
(1 13 8)(2 14 9)(3 15 10)(4 11 6)(5 12 7)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)
(1 19 34)(2 17 32)(3 20 35)(4 18 33)(5 16 31)(6 23 38)(7 21 36)(8 24 39)(9 22 37)(10 25 40)(11 28 43)(12 26 41)(13 29 44)(14 27 42)(15 30 45)
(6 11)(7 12)(8 13)(9 14)(10 15)(16 31)(17 32)(18 33)(19 34)(20 35)(21 41)(22 42)(23 43)(24 44)(25 45)(26 36)(27 37)(28 38)(29 39)(30 40)```

`G:=sub<Sym(45)| (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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (1,3,5,2,4)(6,8,10,7,9)(11,13,15,12,14)(31,35,34,33,32)(36,40,39,38,37)(41,45,44,43,42), (1,13,8)(2,14,9)(3,15,10)(4,11,6)(5,12,7)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (1,19,34)(2,17,32)(3,20,35)(4,18,33)(5,16,31)(6,23,38)(7,21,36)(8,24,39)(9,22,37)(10,25,40)(11,28,43)(12,26,41)(13,29,44)(14,27,42)(15,30,45), (6,11)(7,12)(8,13)(9,14)(10,15)(16,31)(17,32)(18,33)(19,34)(20,35)(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40)>;`

`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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (1,3,5,2,4)(6,8,10,7,9)(11,13,15,12,14)(31,35,34,33,32)(36,40,39,38,37)(41,45,44,43,42), (1,13,8)(2,14,9)(3,15,10)(4,11,6)(5,12,7)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (1,19,34)(2,17,32)(3,20,35)(4,18,33)(5,16,31)(6,23,38)(7,21,36)(8,24,39)(9,22,37)(10,25,40)(11,28,43)(12,26,41)(13,29,44)(14,27,42)(15,30,45), (6,11)(7,12)(8,13)(9,14)(10,15)(16,31)(17,32)(18,33)(19,34)(20,35)(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40) );`

`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),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45)], [(1,3,5,2,4),(6,8,10,7,9),(11,13,15,12,14),(31,35,34,33,32),(36,40,39,38,37),(41,45,44,43,42)], [(1,13,8),(2,14,9),(3,15,10),(4,11,6),(5,12,7),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40)], [(1,19,34),(2,17,32),(3,20,35),(4,18,33),(5,16,31),(6,23,38),(7,21,36),(8,24,39),(9,22,37),(10,25,40),(11,28,43),(12,26,41),(13,29,44),(14,27,42),(15,30,45)], [(6,11),(7,12),(8,13),(9,14),(10,15),(16,31),(17,32),(18,33),(19,34),(20,35),(21,41),(22,42),(23,43),(24,44),(25,45),(26,36),(27,37),(28,38),(29,39),(30,40)]])`

Matrix representation of C52⋊(C3⋊S3) in GL5(𝔽31)

 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 0 4 0 0 0 0 0 4
,
 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 8
,
 29 30 0 0 0 3 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 30 30 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

`G:=sub<GL(5,GF(31))| [1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,8],[29,3,0,0,0,30,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[30,0,0,0,0,30,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;`

C52⋊(C3⋊S3) in GAP, Magma, Sage, TeX

`C_5^2\rtimes (C_3\rtimes S_3)`
`% in TeX`

`G:=Group("C5^2:(C3:S3)");`
`// GroupNames label`

`G:=SmallGroup(450,21);`
`// by ID`

`G=gap.SmallGroup(450,21);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-5,5,41,182,2888,10804,4284]);`
`// Polycyclic`

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

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