metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D15, C5⋊S3, C3⋊D5, C15⋊1C2, sometimes denoted D30 or Dih15 or Dih30, SmallGroup(30,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — D15 |
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 |
►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 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 15T2 | x15-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
Export
Subgroup lattice of D15 in TeX
Character table of D15 in TeX