Copied to
clipboard

## G = D15⋊S3order 180 = 22·32·5

### The semidirect product of D15 and S3 acting via S3/C3=C2

Aliases: D15⋊S3, C153D6, C322D10, C52S32, C3⋊S3⋊D5, C33(S3×D5), (C3×D15)⋊3C2, (C3×C15)⋊4C22, (C5×C3⋊S3)⋊2C2, SmallGroup(180,30)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — D15⋊S3
 Chief series C1 — C5 — C15 — C3×C15 — C3×D15 — D15⋊S3
 Lower central C3×C15 — D15⋊S3
 Upper central C1

Generators and relations for D15⋊S3
G = < a,b,c,d | a15=b2=c3=d2=1, bab=a-1, ac=ca, dad=a11, bc=cb, dbd=a10b, dcd=c-1 >

Character table of D15⋊S3

 class 1 2A 2B 2C 3A 3B 3C 5A 5B 6A 6B 10A 10B 15A 15B 15C 15D 15E 15F 15G 15H size 1 9 15 15 2 2 4 2 2 30 30 18 18 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 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 -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 linear of order 2 ρ5 2 0 0 -2 2 -1 -1 2 2 0 1 0 0 -1 -1 -1 -1 2 2 -1 -1 orthogonal lifted from D6 ρ6 2 0 2 0 -1 2 -1 2 2 -1 0 0 0 -1 2 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ7 2 0 0 2 2 -1 -1 2 2 0 -1 0 0 -1 -1 -1 -1 2 2 -1 -1 orthogonal lifted from S3 ρ8 2 0 -2 0 -1 2 -1 2 2 1 0 0 0 -1 2 -1 -1 -1 -1 2 -1 orthogonal lifted from D6 ρ9 2 2 0 0 2 2 2 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ10 2 -2 0 0 2 2 2 -1+√5/2 -1-√5/2 0 0 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ11 2 -2 0 0 2 2 2 -1-√5/2 -1+√5/2 0 0 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ12 2 2 0 0 2 2 2 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ13 4 0 0 0 -2 -2 1 4 4 0 0 0 0 1 -2 1 1 -2 -2 -2 1 orthogonal lifted from S32 ρ14 4 0 0 0 -2 4 -2 -1-√5 -1+√5 0 0 0 0 1+√5/2 -1+√5 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1-√5 1-√5/2 orthogonal lifted from S3×D5 ρ15 4 0 0 0 4 -2 -2 -1+√5 -1-√5 0 0 0 0 1-√5/2 1+√5/2 1-√5/2 1+√5/2 -1-√5 -1+√5 1-√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ16 4 0 0 0 4 -2 -2 -1-√5 -1+√5 0 0 0 0 1+√5/2 1-√5/2 1+√5/2 1-√5/2 -1+√5 -1-√5 1+√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ17 4 0 0 0 -2 4 -2 -1+√5 -1-√5 0 0 0 0 1-√5/2 -1-√5 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1+√5 1+√5/2 orthogonal lifted from S3×D5 ρ18 4 0 0 0 -2 -2 1 -1+√5 -1-√5 0 0 0 0 2ζ54-ζ5 1+√5/2 -ζ54+2ζ5 2ζ53-ζ52 1+√5/2 1-√5/2 1-√5/2 -ζ53+2ζ52 complex faithful ρ19 4 0 0 0 -2 -2 1 -1-√5 -1+√5 0 0 0 0 2ζ53-ζ52 1-√5/2 -ζ53+2ζ52 -ζ54+2ζ5 1-√5/2 1+√5/2 1+√5/2 2ζ54-ζ5 complex faithful ρ20 4 0 0 0 -2 -2 1 -1+√5 -1-√5 0 0 0 0 -ζ54+2ζ5 1+√5/2 2ζ54-ζ5 -ζ53+2ζ52 1+√5/2 1-√5/2 1-√5/2 2ζ53-ζ52 complex faithful ρ21 4 0 0 0 -2 -2 1 -1-√5 -1+√5 0 0 0 0 -ζ53+2ζ52 1-√5/2 2ζ53-ζ52 2ζ54-ζ5 1-√5/2 1+√5/2 1+√5/2 -ζ54+2ζ5 complex faithful

Permutation representations of D15⋊S3
On 30 points - transitive group 30T43
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(2 12)(3 8)(5 15)(6 11)(9 14)(17 27)(18 23)(20 30)(21 26)(24 29)```

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

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

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

`G:=TransitiveGroup(30,43);`

D15⋊S3 is a maximal subgroup of   S32⋊D5  C32⋊D20  S32×D5
D15⋊S3 is a maximal quotient of   D30.S3  Dic15⋊S3  D30⋊S3  C323D20  C323Dic10

Matrix representation of D15⋊S3 in GL6(𝔽31)

 12 12 0 0 0 0 24 6 0 0 0 0 0 0 0 30 0 0 0 0 1 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 19 0 0 0 0 21 25 0 0 0 0 0 0 30 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 30
,
 30 0 0 0 0 0 0 30 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 30 30

`G:=sub<GL(6,GF(31))| [12,24,0,0,0,0,12,6,0,0,0,0,0,0,0,1,0,0,0,0,30,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,21,0,0,0,0,19,25,0,0,0,0,0,0,30,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,30,0,0,0,0,1,30],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,30,0,0,0,0,0,30] >;`

D15⋊S3 in GAP, Magma, Sage, TeX

`D_{15}\rtimes S_3`
`% in TeX`

`G:=Group("D15:S3");`
`// GroupNames label`

`G:=SmallGroup(180,30);`
`// by ID`

`G=gap.SmallGroup(180,30);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-3,-5,122,67,248,3604]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^15=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;`
`// generators/relations`

Export

׿
×
𝔽