direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D30, C2×D15, C6⋊D5, C10⋊S3, C5⋊2D6, C3⋊2D10, C30⋊1C2, C15⋊2C22, sometimes denoted D60 or Dih30 or Dih60, SmallGroup(60,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — D30 |
Generators and relations for D30
G = < a,b | a30=b2=1, bab=a-1 >
Character table of D30
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 6 | 10A | 10B | 15A | 15B | 15C | 15D | 30A | 30B | 30C | 30D | |
| size | 1 | 1 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | 1 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | -2 | 0 | 0 | 2 | -1-√5/2 | -1+√5/2 | -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 | 1-√5/2 | orthogonal lifted from D10 |
| ρ8 | 2 | -2 | 0 | 0 | 2 | -1+√5/2 | -1-√5/2 | -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 | 1+√5/2 | orthogonal lifted from D10 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | -1-√5/2 | -1+√5/2 | 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 | -1+√5/2 | orthogonal lifted from D5 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | -1+√5/2 | -1-√5/2 | 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 | -1-√5/2 | orthogonal lifted from D5 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | -1-√5/2 | -1+√5/2 | -1 | -1+√5/2 | -1-√5/2 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | orthogonal lifted from D15 |
| ρ12 | 2 | 2 | 0 | 0 | -1 | -1-√5/2 | -1+√5/2 | -1 | -1+√5/2 | -1-√5/2 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | orthogonal lifted from D15 |
| ρ13 | 2 | 2 | 0 | 0 | -1 | -1+√5/2 | -1-√5/2 | -1 | -1-√5/2 | -1+√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | orthogonal lifted from D15 |
| ρ14 | 2 | -2 | 0 | 0 | -1 | -1+√5/2 | -1-√5/2 | 1 | 1+√5/2 | 1-√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52+ζ52 | ζ32ζ54-ζ32ζ5+ζ54 | ζ3ζ54-ζ3ζ5+ζ54 | ζ3ζ53-ζ3ζ52+ζ53 | orthogonal faithful |
| ρ15 | 2 | -2 | 0 | 0 | -1 | -1-√5/2 | -1+√5/2 | 1 | 1-√5/2 | 1+√5/2 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | ζ3ζ54-ζ3ζ5+ζ54 | -ζ3ζ53+ζ3ζ52+ζ52 | ζ3ζ53-ζ3ζ52+ζ53 | ζ32ζ54-ζ32ζ5+ζ54 | orthogonal faithful |
| ρ16 | 2 | -2 | 0 | 0 | -1 | -1-√5/2 | -1+√5/2 | 1 | 1-√5/2 | 1+√5/2 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ32ζ54-ζ32ζ5+ζ54 | ζ3ζ53-ζ3ζ52+ζ53 | -ζ3ζ53+ζ3ζ52+ζ52 | ζ3ζ54-ζ3ζ5+ζ54 | orthogonal faithful |
| ρ17 | 2 | -2 | 0 | 0 | -1 | -1+√5/2 | -1-√5/2 | 1 | 1+√5/2 | 1-√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52+ζ53 | ζ3ζ54-ζ3ζ5+ζ54 | ζ32ζ54-ζ32ζ5+ζ54 | -ζ3ζ53+ζ3ζ52+ζ52 | orthogonal faithful |
| ρ18 | 2 | 2 | 0 | 0 | -1 | -1+√5/2 | -1-√5/2 | -1 | -1-√5/2 | -1+√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | orthogonal lifted from D15 |
►On 30 points - transitive group 30T14
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24)]])
G:=TransitiveGroup(30,14);
D30 is a maximal subgroup of
D30.C2 C3⋊D20 C5⋊D12 D60 C15⋊7D4 C2×S3×D5 Q8⋊D15
D30 is a maximal quotient of Dic30 D60 C15⋊7D4
Matrix representation of D30 ►in GL2(𝔽29) generated by
| 0 | 28 |
| 1 | 9 |
| 9 | 22 |
| 28 | 20 |
G:=sub<GL(2,GF(29))| [0,1,28,9],[9,28,22,20] >;
D30 in GAP, Magma, Sage, TeX
D_{30} % in TeX
G:=Group("D30"); // GroupNames label
G:=SmallGroup(60,12);
// by ID
G=gap.SmallGroup(60,12);
# by ID
G:=PCGroup([4,-2,-2,-3,-5,98,771]);
// Polycyclic
G:=Group<a,b|a^30=b^2=1,b*a*b=a^-1>;
// generators/relations
Export
Subgroup lattice of D30 in TeX
Character table of D30 in TeX