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## G = D15⋊D5order 300 = 22·3·52

### The semidirect product of D15 and D5 acting via D5/C5=C2

Aliases: D15⋊D5, C524D6, C152D10, C32D52, C5⋊D52S3, C53(S3×D5), (C5×D15)⋊3C2, (C5×C15)⋊4C22, (C3×C5⋊D5)⋊2C2, SmallGroup(300,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C15 — D15⋊D5
 Chief series C1 — C5 — C52 — C5×C15 — C5×D15 — D15⋊D5
 Lower central C5×C15 — D15⋊D5
 Upper central C1

Generators and relations for D15⋊D5
G = < a,b,c,d | a15=b2=c5=d2=1, bab=a-1, ac=ca, dad=a4, bc=cb, dbd=a3b, dcd=c-1 >

15C2
15C2
25C2
2C5
2C5
75C22
5S3
5S3
25C6
3D5
3D5
5D5
5D5
10D5
10D5
15C10
15C10
2C15
2C15
25D6
15D10
15D10
10C3×D5
10C3×D5
3D52

Character table of D15⋊D5

 class 1 2A 2B 2C 3 5A 5B 5C 5D 5E 5F 5G 5H 6 10A 10B 10C 10D 15A 15B 15C 15D 15E 15F 15G 15H 15I 15J 15K 15L size 1 15 15 25 2 2 2 2 2 4 4 4 4 50 30 30 30 30 4 4 4 4 4 4 4 4 4 4 4 4 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 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 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 -1 1 -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 -1 -1 -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 2 2 2 2 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 0 0 -2 -1 2 2 2 2 2 2 2 2 1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ7 2 0 -2 0 2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 0 0 1-√5/2 0 1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 orthogonal lifted from D10 ρ8 2 -2 0 0 2 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 0 1-√5/2 0 1+√5/2 0 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 2 2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ9 2 -2 0 0 2 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 0 1+√5/2 0 1-√5/2 0 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 2 2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ10 2 2 0 0 2 -1+√5/2 -1-√5/2 2 2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 0 -1-√5/2 0 -1+√5/2 0 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 2 2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ11 2 0 2 0 2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 0 0 -1+√5/2 0 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 orthogonal lifted from D5 ρ12 2 0 -2 0 2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 0 0 1+√5/2 0 1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 orthogonal lifted from D10 ρ13 2 0 2 0 2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 0 0 -1-√5/2 0 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 2 orthogonal lifted from D5 ρ14 2 2 0 0 2 -1-√5/2 -1+√5/2 2 2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 0 -1+√5/2 0 -1-√5/2 0 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 2 2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ15 4 0 0 0 -2 -1+√5 -1-√5 4 4 -1+√5 -1+√5 -1-√5 -1-√5 0 0 0 0 0 1+√5/2 1+√5/2 1+√5/2 1+√5/2 -2 -2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ16 4 0 0 0 4 -1+√5 -1-√5 -1+√5 -1-√5 -1 3-√5/2 -1 3+√5/2 0 0 0 0 0 -1 3+√5/2 3+√5/2 -1 -1+√5 -1-√5 -1+√5 3-√5/2 -1 -1 3-√5/2 -1-√5 orthogonal lifted from D52 ρ17 4 0 0 0 -2 4 4 -1+√5 -1-√5 -1-√5 -1+√5 -1+√5 -1-√5 0 0 0 0 0 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -2 orthogonal lifted from S3×D5 ρ18 4 0 0 0 4 -1+√5 -1-√5 -1-√5 -1+√5 3-√5/2 -1 3+√5/2 -1 0 0 0 0 0 3+√5/2 -1 -1 3+√5/2 -1-√5 -1+√5 -1+√5 -1 3-√5/2 3-√5/2 -1 -1-√5 orthogonal lifted from D52 ρ19 4 0 0 0 -2 4 4 -1-√5 -1+√5 -1+√5 -1-√5 -1-√5 -1+√5 0 0 0 0 0 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -2 orthogonal lifted from S3×D5 ρ20 4 0 0 0 4 -1-√5 -1+√5 -1+√5 -1-√5 3+√5/2 -1 3-√5/2 -1 0 0 0 0 0 3-√5/2 -1 -1 3-√5/2 -1+√5 -1-√5 -1-√5 -1 3+√5/2 3+√5/2 -1 -1+√5 orthogonal lifted from D52 ρ21 4 0 0 0 4 -1-√5 -1+√5 -1-√5 -1+√5 -1 3+√5/2 -1 3-√5/2 0 0 0 0 0 -1 3-√5/2 3-√5/2 -1 -1-√5 -1+√5 -1-√5 3+√5/2 -1 -1 3+√5/2 -1+√5 orthogonal lifted from D52 ρ22 4 0 0 0 -2 -1-√5 -1+√5 4 4 -1-√5 -1-√5 -1+√5 -1+√5 0 0 0 0 0 1-√5/2 1-√5/2 1-√5/2 1-√5/2 -2 -2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ23 4 0 0 0 -2 -1+√5 -1-√5 -1-√5 -1+√5 3-√5/2 -1 3+√5/2 -1 0 0 0 0 0 ζ3ζ54+ζ3ζ5-2ζ3-2 1+√-15/2 1-√-15/2 ζ32ζ54+ζ32ζ5-2ζ32-2 1+√5/2 1-√5/2 1-√5/2 1-√-15/2 ζ32ζ53+ζ32ζ52-2ζ32-2 ζ3ζ53+ζ3ζ52-2ζ3-2 1+√-15/2 1+√5/2 complex faithful ρ24 4 0 0 0 -2 -1+√5 -1-√5 -1+√5 -1-√5 -1 3-√5/2 -1 3+√5/2 0 0 0 0 0 1+√-15/2 ζ32ζ54+ζ32ζ5-2ζ32-2 ζ3ζ54+ζ3ζ5-2ζ3-2 1-√-15/2 1-√5/2 1+√5/2 1-√5/2 ζ32ζ53+ζ32ζ52-2ζ32-2 1+√-15/2 1-√-15/2 ζ3ζ53+ζ3ζ52-2ζ3-2 1+√5/2 complex faithful ρ25 4 0 0 0 -2 -1-√5 -1+√5 -1-√5 -1+√5 -1 3+√5/2 -1 3-√5/2 0 0 0 0 0 1-√-15/2 ζ32ζ53+ζ32ζ52-2ζ32-2 ζ3ζ53+ζ3ζ52-2ζ3-2 1+√-15/2 1+√5/2 1-√5/2 1+√5/2 ζ32ζ54+ζ32ζ5-2ζ32-2 1-√-15/2 1+√-15/2 ζ3ζ54+ζ3ζ5-2ζ3-2 1-√5/2 complex faithful ρ26 4 0 0 0 -2 -1-√5 -1+√5 -1+√5 -1-√5 3+√5/2 -1 3-√5/2 -1 0 0 0 0 0 ζ32ζ53+ζ32ζ52-2ζ32-2 1+√-15/2 1-√-15/2 ζ3ζ53+ζ3ζ52-2ζ3-2 1-√5/2 1+√5/2 1+√5/2 1-√-15/2 ζ3ζ54+ζ3ζ5-2ζ3-2 ζ32ζ54+ζ32ζ5-2ζ32-2 1+√-15/2 1-√5/2 complex faithful ρ27 4 0 0 0 -2 -1-√5 -1+√5 -1-√5 -1+√5 -1 3+√5/2 -1 3-√5/2 0 0 0 0 0 1+√-15/2 ζ3ζ53+ζ3ζ52-2ζ3-2 ζ32ζ53+ζ32ζ52-2ζ32-2 1-√-15/2 1+√5/2 1-√5/2 1+√5/2 ζ3ζ54+ζ3ζ5-2ζ3-2 1+√-15/2 1-√-15/2 ζ32ζ54+ζ32ζ5-2ζ32-2 1-√5/2 complex faithful ρ28 4 0 0 0 -2 -1-√5 -1+√5 -1+√5 -1-√5 3+√5/2 -1 3-√5/2 -1 0 0 0 0 0 ζ3ζ53+ζ3ζ52-2ζ3-2 1-√-15/2 1+√-15/2 ζ32ζ53+ζ32ζ52-2ζ32-2 1-√5/2 1+√5/2 1+√5/2 1+√-15/2 ζ32ζ54+ζ32ζ5-2ζ32-2 ζ3ζ54+ζ3ζ5-2ζ3-2 1-√-15/2 1-√5/2 complex faithful ρ29 4 0 0 0 -2 -1+√5 -1-√5 -1+√5 -1-√5 -1 3-√5/2 -1 3+√5/2 0 0 0 0 0 1-√-15/2 ζ3ζ54+ζ3ζ5-2ζ3-2 ζ32ζ54+ζ32ζ5-2ζ32-2 1+√-15/2 1-√5/2 1+√5/2 1-√5/2 ζ3ζ53+ζ3ζ52-2ζ3-2 1-√-15/2 1+√-15/2 ζ32ζ53+ζ32ζ52-2ζ32-2 1+√5/2 complex faithful ρ30 4 0 0 0 -2 -1+√5 -1-√5 -1-√5 -1+√5 3-√5/2 -1 3+√5/2 -1 0 0 0 0 0 ζ32ζ54+ζ32ζ5-2ζ32-2 1-√-15/2 1+√-15/2 ζ3ζ54+ζ3ζ5-2ζ3-2 1+√5/2 1-√5/2 1-√5/2 1+√-15/2 ζ3ζ53+ζ3ζ52-2ζ3-2 ζ32ζ53+ζ32ζ52-2ζ32-2 1-√-15/2 1+√5/2 complex faithful

Permutation representations of D15⋊D5
On 30 points - transitive group 30T79
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)
(1 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 30)(12 29)(13 28)(14 27)(15 26)
(1 7 13 4 10)(2 8 14 5 11)(3 9 15 6 12)(16 25 19 28 22)(17 26 20 29 23)(18 27 21 30 24)
(1 10)(2 14)(4 7)(5 11)(6 15)(9 12)(16 22)(17 26)(18 30)(20 23)(21 27)(25 28)```

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

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

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

`G:=TransitiveGroup(30,79);`

Matrix representation of D15⋊D5 in GL6(𝔽31)

 30 1 0 0 0 0 30 0 0 0 0 0 0 0 30 13 0 0 0 0 18 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 17 4 0 0 0 0 21 14 0 0 0 0 0 0 13 18 0 0 0 0 1 18 0 0 0 0 0 0 30 0 0 0 0 0 0 30
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 0 0 0 0 1 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 12 0 0 0 0 0 30

`G:=sub<GL(6,GF(31))| [30,30,0,0,0,0,1,0,0,0,0,0,0,0,30,18,0,0,0,0,13,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,21,0,0,0,0,4,14,0,0,0,0,0,0,13,1,0,0,0,0,18,18,0,0,0,0,0,0,30,0,0,0,0,0,0,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,30,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,12,30] >;`

D15⋊D5 in GAP, Magma, Sage, TeX

`D_{15}\rtimes D_5`
`% in TeX`

`G:=Group("D15:D5");`
`// GroupNames label`

`G:=SmallGroup(300,40);`
`// by ID`

`G=gap.SmallGroup(300,40);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-5,-5,122,963,488,3009]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^15=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^4,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

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