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## G = D15order 30 = 2·3·5

### Dihedral group

Aliases: D15, C5⋊S3, C3⋊D5, C151C2, sometimes denoted D30 or Dih15 or Dih30, SmallGroup(30,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — D15
 Chief series C1 — C5 — C15 — D15
 Lower central C15 — D15
 Upper central C1

Generators and relations for D15
G = < a,b | a15=b2=1, bab=a-1 >

Character table of D15

 class 1 2 3 5A 5B 15A 15B 15C 15D size 1 15 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 linear of order 2 ρ3 2 0 -1 2 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ4 2 0 2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ5 2 0 2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ6 2 0 -1 -1-√5/2 -1+√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 ζ3ζ54-ζ3ζ5-ζ5 ζ32ζ54-ζ32ζ5-ζ5 orthogonal faithful ρ7 2 0 -1 -1+√5/2 -1-√5/2 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ54-ζ3ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 orthogonal faithful ρ8 2 0 -1 -1-√5/2 -1+√5/2 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ54-ζ3ζ5-ζ5 orthogonal faithful ρ9 2 0 -1 -1+√5/2 -1-√5/2 ζ3ζ54-ζ3ζ5-ζ5 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 orthogonal faithful

Permutation representations of D15
On 15 points - transitive group 15T2
Generators in S15
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)```

`G:=sub<Sym(15)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9)])`

`G:=TransitiveGroup(15,2);`

Regular action on 30 points - transitive group 30T3
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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 30)(14 29)(15 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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,30)(14,29)(15,28)>;`

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

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

`G:=TransitiveGroup(30,3);`

D15 is a maximal subgroup of
C3⋊D15  C5⋊S4  D75  C24⋊D15
C5⋊D3p: S3×D5  D45  C5⋊D15  D105  D165  D195 ...
D15 is a maximal quotient of
Dic15  C3⋊D15  C5⋊S4  C5⋊D15  C24⋊D15
D15p: D45  D75  D105  D165  D195 ...

Polynomial with Galois group D15 over ℚ
actionf(x)Disc(f)
15T2x15-x14+17x13+138x12-188x11+875x10+6364x9-23852x8-50961x7+192039x6+95751x5-707345x4+775903x3-478177x2+162804x-22825-54·192·2310·477·836·2292·22872·38772·840672·11085072·12519112·1300732692

Matrix representation of D15 in GL2(𝔽29) generated by

 15 4 25 28
,
 28 0 4 1
`G:=sub<GL(2,GF(29))| [15,25,4,28],[28,4,0,1] >;`

D15 in GAP, Magma, Sage, TeX

`D_{15}`
`% in TeX`

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

`G:=SmallGroup(30,3);`
`// by ID`

`G=gap.SmallGroup(30,3);`
`# by ID`

`G:=PCGroup([3,-2,-3,-5,25,218]);`
`// Polycyclic`

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

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