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## G = C5⋊D15⋊C3order 450 = 2·32·52

### The semidirect product of C5⋊D15 and C3 acting faithfully

Aliases: C5⋊D15⋊C3, C3⋊(C52⋊C6), (C5×C15)⋊1C6, C52⋊C32S3, C522(C3×S3), (C3×C52⋊C3)⋊1C2, SmallGroup(450,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C15 — C5⋊D15⋊C3
 Chief series C1 — C52 — C5×C15 — C3×C52⋊C3 — C5⋊D15⋊C3
 Lower central C5×C15 — C5⋊D15⋊C3
 Upper central C1

Generators and relations for C5⋊D15⋊C3
G = < a,b,c,d | a5=b15=c2=d3=1, dbd-1=ab=ba, cac=a-1, dad-1=a3b12, cbc=b-1, dcd-1=ac >

75C2
25C3
50C3
3C5
3C5
25S3
75C6
25C32
45D5
45D5
3C15
3C15
25C3×S3
15D15
15D15

Character table of C5⋊D15⋊C3

 class 1 2 3A 3B 3C 3D 3E 5A 5B 5C 5D 6A 6B 15A 15B 15C 15D 15E 15F 15G 15H size 1 75 2 25 25 50 50 6 6 6 6 75 75 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 trivial ρ2 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 1 -1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 ζ65 ζ6 1 1 1 1 1 1 1 1 linear of order 6 ρ4 1 -1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 ζ6 ζ65 1 1 1 1 1 1 1 1 linear of order 6 ρ5 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ7 2 0 -1 2 2 -1 -1 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ9 2 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ10 6 0 6 0 0 0 0 1-√5 -3+√5/2 -3-√5/2 1+√5 0 0 -3+√5/2 1+√5 1-√5 1-√5 1+√5 -3-√5/2 -3+√5/2 -3-√5/2 orthogonal lifted from C52⋊C6 ρ11 6 0 6 0 0 0 0 -3-√5/2 1-√5 1+√5 -3+√5/2 0 0 1-√5 -3+√5/2 -3-√5/2 -3-√5/2 -3+√5/2 1+√5 1-√5 1+√5 orthogonal lifted from C52⋊C6 ρ12 6 0 6 0 0 0 0 -3+√5/2 1+√5 1-√5 -3-√5/2 0 0 1+√5 -3-√5/2 -3+√5/2 -3+√5/2 -3-√5/2 1-√5 1+√5 1-√5 orthogonal lifted from C52⋊C6 ρ13 6 0 6 0 0 0 0 1+√5 -3-√5/2 -3+√5/2 1-√5 0 0 -3-√5/2 1-√5 1+√5 1+√5 1-√5 -3+√5/2 -3-√5/2 -3+√5/2 orthogonal lifted from C52⋊C6 ρ14 6 0 -3 0 0 0 0 -3+√5/2 1+√5 1-√5 -3-√5/2 0 0 -1-√5/2 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal faithful ρ15 6 0 -3 0 0 0 0 -3-√5/2 1-√5 1+√5 -3+√5/2 0 0 -1+√5/2 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal faithful ρ16 6 0 -3 0 0 0 0 1+√5 -3-√5/2 -3+√5/2 1-√5 0 0 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 -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 orthogonal faithful ρ17 6 0 -3 0 0 0 0 1-√5 -3+√5/2 -3-√5/2 1+√5 0 0 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 -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 orthogonal faithful ρ18 6 0 -3 0 0 0 0 -3+√5/2 1+√5 1-√5 -3-√5/2 0 0 -1-√5/2 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal faithful ρ19 6 0 -3 0 0 0 0 1+√5 -3-√5/2 -3+√5/2 1-√5 0 0 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 -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 orthogonal faithful ρ20 6 0 -3 0 0 0 0 -3-√5/2 1-√5 1+√5 -3+√5/2 0 0 -1+√5/2 -3ζ3ζ54-2ζ3ζ52+ζ3ζ5-ζ3-2ζ54-ζ52 -2ζ3ζ54-3ζ3ζ53+ζ3ζ52-ζ3-ζ54-2ζ53 ζ3ζ53-3ζ3ζ52-2ζ3ζ5-ζ3-2ζ52-ζ5 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal faithful ρ21 6 0 -3 0 0 0 0 1-√5 -3+√5/2 -3-√5/2 1+√5 0 0 -3ζ32ζ54-2ζ32ζ52+ζ32ζ5-ζ32-2ζ54-ζ52 -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 orthogonal faithful

Smallest permutation representation of C5⋊D15⋊C3
On 45 points
Generators in S45
```(1 15 5 11 8)(2 13 6 12 9)(3 14 4 10 7)(16 25 19 28 22)(17 26 20 29 23)(18 27 21 30 24)(31 34 37 40 43)(32 35 38 41 44)(33 36 39 42 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 11)(2 10)(3 12)(4 13)(5 15)(6 14)(7 9)(16 20)(17 19)(21 30)(22 29)(23 28)(24 27)(25 26)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(43 45)
(1 38 30)(2 33 25)(3 43 20)(4 40 23)(5 35 18)(6 45 28)(7 37 26)(8 32 21)(9 42 16)(10 31 17)(11 41 27)(12 36 22)(13 39 19)(14 34 29)(15 44 24)```

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

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

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

Matrix representation of C5⋊D15⋊C3 in GL6(𝔽31)

 12 1 0 0 0 0 30 0 0 0 0 0 0 0 12 1 0 0 0 0 30 0 0 0 19 18 19 18 18 19 1 13 1 13 13 0
,
 16 26 0 0 0 0 5 14 0 0 0 0 0 0 26 17 0 0 0 0 14 8 0 0 0 5 5 19 27 16 15 17 17 4 24 20
,
 8 20 0 0 0 0 17 23 0 0 0 0 0 0 20 15 0 0 0 0 23 11 0 0 0 8 8 28 12 17 23 20 20 28 8 19
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 30 30 30 30 29 30 0 0 0 0 1 0 1 0 0 0 19 0

`G:=sub<GL(6,GF(31))| [12,30,0,0,19,1,1,0,0,0,18,13,0,0,12,30,19,1,0,0,1,0,18,13,0,0,0,0,18,13,0,0,0,0,19,0],[16,5,0,0,0,15,26,14,0,0,5,17,0,0,26,14,5,17,0,0,17,8,19,4,0,0,0,0,27,24,0,0,0,0,16,20],[8,17,0,0,0,23,20,23,0,0,8,20,0,0,20,23,8,20,0,0,15,11,28,28,0,0,0,0,12,8,0,0,0,0,17,19],[0,0,0,30,0,1,0,0,0,30,0,0,1,0,0,30,0,0,0,1,0,30,0,0,0,0,12,29,1,19,0,0,1,30,0,0] >;`

C5⋊D15⋊C3 in GAP, Magma, Sage, TeX

`C_5\rtimes D_{15}\rtimes C_3`
`% in TeX`

`G:=Group("C5:D15:C3");`
`// GroupNames label`

`G:=SmallGroup(450,24);`
`// by ID`

`G=gap.SmallGroup(450,24);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-5,5,182,1443,2348,9004,1359]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^15=c^2=d^3=1,d*b*d^-1=a*b=b*a,c*a*c=a^-1,d*a*d^-1=a^3*b^12,c*b*c=b^-1,d*c*d^-1=a*c>;`
`// generators/relations`

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