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G = C32⋊D15order 270 = 2·33·5

2nd semidirect product of C32 and D15 acting via D15/C5=S3

Aliases: He32D5, C322D15, (C3×C15)⋊2S3, C5⋊(He3⋊C2), (C5×He3)⋊2C2, C15.2(C3⋊S3), C3.2(C3⋊D15), SmallGroup(270,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C5×He3 — C32⋊D15
 Chief series C1 — C3 — C15 — C3×C15 — C5×He3 — C32⋊D15
 Lower central C5×He3 — C32⋊D15
 Upper central C1 — C3

Generators and relations for C32⋊D15
G = < a,b,c,d | a3=b3=c15=d2=1, cac-1=ab=ba, dad=a-1b-1, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of C32⋊D15
On 45 points
Generators in S45
```(2 17 33)(3 34 18)(5 20 36)(6 37 21)(8 23 39)(9 40 24)(11 26 42)(12 43 27)(14 29 45)(15 31 30)
(1 16 32)(2 17 33)(3 18 34)(4 19 35)(5 20 36)(6 21 37)(7 22 38)(8 23 39)(9 24 40)(10 25 41)(11 26 42)(12 27 43)(13 28 44)(14 29 45)(15 30 31)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 32)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)```

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

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

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

32 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 5A 5B 6A 6B 15A 15B 15C 15D 15E ··· 15T order 1 2 3 3 3 3 3 3 5 5 6 6 15 15 15 15 15 ··· 15 size 1 45 1 1 6 6 6 6 2 2 45 45 2 2 2 2 6 ··· 6

32 irreducible representations

 dim 1 1 2 2 2 3 6 type + + + + + image C1 C2 S3 D5 D15 He3⋊C2 C32⋊D15 kernel C32⋊D15 C5×He3 C3×C15 He3 C32 C5 C1 # reps 1 1 4 2 16 4 4

Matrix representation of C32⋊D15 in GL5(𝔽31)

 0 30 0 0 0 1 30 0 0 0 0 0 1 0 0 0 0 26 5 0 0 0 1 0 25
,
 1 0 0 0 0 0 1 0 0 0 0 0 25 0 0 0 0 0 25 0 0 0 0 0 25
,
 3 6 0 0 0 25 9 0 0 0 0 0 1 24 0 0 0 0 30 1 0 0 0 30 0
,
 25 9 0 0 0 3 6 0 0 0 0 0 1 24 0 0 0 0 30 0 0 0 0 30 1

`G:=sub<GL(5,GF(31))| [0,1,0,0,0,30,30,0,0,0,0,0,1,26,1,0,0,0,5,0,0,0,0,0,25],[1,0,0,0,0,0,1,0,0,0,0,0,25,0,0,0,0,0,25,0,0,0,0,0,25],[3,25,0,0,0,6,9,0,0,0,0,0,1,0,0,0,0,24,30,30,0,0,0,1,0],[25,3,0,0,0,9,6,0,0,0,0,0,1,0,0,0,0,24,30,30,0,0,0,0,1] >;`

C32⋊D15 in GAP, Magma, Sage, TeX

`C_3^2\rtimes D_{15}`
`% in TeX`

`G:=Group("C3^2:D15");`
`// GroupNames label`

`G:=SmallGroup(270,19);`
`// by ID`

`G=gap.SmallGroup(270,19);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-5,-3,41,182,727,1443]);`
`// Polycyclic`

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

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