metabelian, supersoluble, monomial, A-group
Aliases: D15⋊D5, C52⋊4D6, C15⋊2D10, C3⋊2D52, C5⋊D5⋊2S3, C5⋊3(S3×D5), (C5×D15)⋊3C2, (C5×C15)⋊4C22, (C3×C5⋊D5)⋊2C2, SmallGroup(300,40)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C15 — D15⋊D5 |
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 |
►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
Export
Subgroup lattice of D15⋊D5 in TeX
Character table of D15⋊D5 in TeX