metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D24, C3⋊1D8, C8⋊1S3, C24⋊1C2, C4.9D6, C6.2D4, D12⋊1C2, C2.4D12, C12.9C22, sometimes denoted D48 or Dih24 or Dih48, SmallGroup(48,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D24
G = < a,b | a24=b2=1, bab=a-1 >
Character table of D24
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 8A | 8B | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 12 | 12 | 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 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 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 | -1 | -2 | -2 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | -√2 | √2 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ9 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | √2 | -√2 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | 0 | 0 | 1 | 1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | 0 | 0 | 1 | 1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ12 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | √2 | -√2 | -√3 | √3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | orthogonal faithful |
| ρ13 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | -√2 | √2 | -√3 | √3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | orthogonal faithful |
| ρ14 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | -√2 | √2 | √3 | -√3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | orthogonal faithful |
| ρ15 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | √2 | -√2 | √3 | -√3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | orthogonal faithful |
►On 24 points - transitive group 24T34
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)
G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,24),(17,23),(18,22),(19,21)]])
G:=TransitiveGroup(24,34);
D24 is a maximal subgroup of
D48 C48⋊C2 C3⋊D16 C8.6D6 C4○D24 C8⋊D6 S3×D8 Q8⋊3D6 D24⋊C2 D72 C3⋊D24 C32⋊5D8 A4⋊D8 C8.3S4 C5⋊D24 D120 C7⋊D24 D168 He3⋊D8 C32⋊2D24
D24 is a maximal quotient of
D48 C48⋊C2 Dic24 C24⋊1C4 C2.D24 D72 C3⋊D24 C32⋊5D8 A4⋊D8 C5⋊D24 D120 C7⋊D24 D168 C32⋊2D24
Matrix representation of D24 ►in GL2(𝔽23) generated by
| 13 | 18 |
| 5 | 7 |
| 7 | 5 |
| 18 | 16 |
G:=sub<GL(2,GF(23))| [13,5,18,7],[7,18,5,16] >;
D24 in GAP, Magma, Sage, TeX
D_{24} % in TeX
G:=Group("D24"); // GroupNames label
G:=SmallGroup(48,7);
// by ID
G=gap.SmallGroup(48,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,61,66,182,42,804]);
// Polycyclic
G:=Group<a,b|a^24=b^2=1,b*a*b=a^-1>;
// generators/relations
Export
Subgroup lattice of D24 in TeX
Character table of D24 in TeX