metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D48, C3⋊1D16, C48⋊1C2, C16⋊1S3, C6.1D8, D24⋊1C2, C8.13D6, C4.1D12, C2.3D24, C12.24D4, C24.14C22, sometimes denoted D96 or Dih48 or Dih96, SmallGroup(96,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D48
G = < a,b | a48=b2=1, bab=a-1 >
Character table of D48
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 8A | 8B | 12A | 12B | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 48A | 48B | 48C | 48D | 48E | 48F | 48G | 48H | |
| size | 1 | 1 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | √2 | -√2 | -√2 | √2 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | -√2 | √2 | √2 | -√2 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | √3 | -√3 | √3 | -√3 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -√3 | √3 | -√3 | √3 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | √2 | -√2 | 0 | 0 | -ζ165+ζ163 | ζ1615-ζ169 | -ζ1615+ζ169 | ζ165-ζ163 | -√2 | -√2 | √2 | √2 | ζ1615-ζ169 | ζ165-ζ163 | ζ165-ζ163 | ζ1615-ζ169 | -ζ165+ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | -ζ1615+ζ169 | orthogonal lifted from D16 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | -√2 | √2 | 0 | 0 | ζ1615-ζ169 | ζ165-ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | √2 | √2 | -√2 | -√2 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ1615+ζ169 | ζ165-ζ163 | ζ1615-ζ169 | ζ1615-ζ169 | -ζ165+ζ163 | -ζ165+ζ163 | orthogonal lifted from D16 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | √2 | -√2 | 0 | 0 | ζ165-ζ163 | -ζ1615+ζ169 | ζ1615-ζ169 | -ζ165+ζ163 | -√2 | -√2 | √2 | √2 | -ζ1615+ζ169 | -ζ165+ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | ζ165-ζ163 | ζ165-ζ163 | ζ1615-ζ169 | ζ1615-ζ169 | orthogonal lifted from D16 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | -√2 | √2 | 0 | 0 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ165-ζ163 | ζ1615-ζ169 | √2 | √2 | -√2 | -√2 | -ζ165+ζ163 | ζ1615-ζ169 | ζ1615-ζ169 | -ζ165+ζ163 | -ζ1615+ζ169 | -ζ1615+ζ169 | ζ165-ζ163 | ζ165-ζ163 | orthogonal lifted from D16 |
| ρ16 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | 0 | 0 | 1 | 1 | -√2 | √2 | √2 | -√2 | √3 | -√3 | -√3 | √3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | orthogonal lifted from D24 |
| ρ17 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | 0 | 0 | 1 | 1 | -√2 | √2 | √2 | -√2 | -√3 | √3 | √3 | -√3 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | orthogonal lifted from D24 |
| ρ18 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | 0 | 0 | 1 | 1 | √2 | -√2 | -√2 | √2 | √3 | -√3 | -√3 | √3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | orthogonal lifted from D24 |
| ρ19 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | 0 | 0 | 1 | 1 | √2 | -√2 | -√2 | √2 | -√3 | √3 | √3 | -√3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | orthogonal lifted from D24 |
| ρ20 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | √2 | -√2 | -√3 | √3 | -ζ165+ζ163 | ζ1615-ζ169 | ζ167-ζ16 | ζ165-ζ163 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ167ζ3+ζ167+ζ16ζ3 | ζ165ζ32+ζ163ζ32+ζ163 | ζ1613ζ32+ζ1613+ζ1611ζ32 | ζ167ζ32+ζ167+ζ16ζ32 | ζ165ζ3+ζ165+ζ163ζ3 | ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ167ζ3+ζ16ζ3+ζ16 | ζ167ζ32+ζ16ζ32+ζ16 | orthogonal faithful |
| ρ21 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | -√2 | √2 | √3 | -√3 | ζ1615-ζ169 | ζ165-ζ163 | -ζ165+ζ163 | ζ167-ζ16 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ165ζ32+ζ163ζ32+ζ163 | ζ167ζ32+ζ16ζ32+ζ16 | ζ167ζ3+ζ16ζ3+ζ16 | ζ1613ζ32+ζ1613+ζ1611ζ32 | ζ167ζ3+ζ167+ζ16ζ3 | ζ167ζ32+ζ167+ζ16ζ32 | ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ165ζ3+ζ165+ζ163ζ3 | orthogonal faithful |
| ρ22 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | -√2 | √2 | -√3 | √3 | ζ1615-ζ169 | ζ165-ζ163 | -ζ165+ζ163 | ζ167-ζ16 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ1613ζ32+ζ1613+ζ1611ζ32 | ζ167ζ3+ζ16ζ3+ζ16 | ζ167ζ32+ζ16ζ32+ζ16 | ζ165ζ32+ζ163ζ32+ζ163 | ζ167ζ32+ζ167+ζ16ζ32 | ζ167ζ3+ζ167+ζ16ζ3 | ζ165ζ3+ζ165+ζ163ζ3 | ζ1613ζ3+ζ1611ζ3+ζ1611 | orthogonal faithful |
| ρ23 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | -√2 | √2 | √3 | -√3 | ζ167-ζ16 | -ζ165+ζ163 | ζ165-ζ163 | ζ1615-ζ169 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ165ζ3+ζ165+ζ163ζ3 | ζ167ζ3+ζ167+ζ16ζ3 | ζ167ζ32+ζ167+ζ16ζ32 | ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ167ζ32+ζ16ζ32+ζ16 | ζ167ζ3+ζ16ζ3+ζ16 | ζ1613ζ32+ζ1613+ζ1611ζ32 | ζ165ζ32+ζ163ζ32+ζ163 | orthogonal faithful |
| ρ24 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | -√2 | √2 | -√3 | √3 | ζ167-ζ16 | -ζ165+ζ163 | ζ165-ζ163 | ζ1615-ζ169 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ167ζ32+ζ167+ζ16ζ32 | ζ167ζ3+ζ167+ζ16ζ3 | ζ165ζ3+ζ165+ζ163ζ3 | ζ167ζ3+ζ16ζ3+ζ16 | ζ167ζ32+ζ16ζ32+ζ16 | ζ165ζ32+ζ163ζ32+ζ163 | ζ1613ζ32+ζ1613+ζ1611ζ32 | orthogonal faithful |
| ρ25 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | √2 | -√2 | -√3 | √3 | ζ165-ζ163 | ζ167-ζ16 | ζ1615-ζ169 | -ζ165+ζ163 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ166ζ3+ζ162ζ3+ζ162 | ζ166ζ32+ζ166+ζ162ζ32 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ167ζ32+ζ16ζ32+ζ16 | ζ165ζ3+ζ165+ζ163ζ3 | ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ167ζ3+ζ16ζ3+ζ16 | ζ165ζ32+ζ163ζ32+ζ163 | ζ1613ζ32+ζ1613+ζ1611ζ32 | ζ167ζ32+ζ167+ζ16ζ32 | ζ167ζ3+ζ167+ζ16ζ3 | orthogonal faithful |
| ρ26 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | √2 | -√2 | √3 | -√3 | ζ165-ζ163 | ζ167-ζ16 | ζ1615-ζ169 | -ζ165+ζ163 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ167ζ3+ζ16ζ3+ζ16 | ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ165ζ3+ζ165+ζ163ζ3 | ζ167ζ32+ζ16ζ32+ζ16 | ζ1613ζ32+ζ1613+ζ1611ζ32 | ζ165ζ32+ζ163ζ32+ζ163 | ζ167ζ3+ζ167+ζ16ζ3 | ζ167ζ32+ζ167+ζ16ζ32 | orthogonal faithful |
| ρ27 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | √2 | -√2 | √3 | -√3 | -ζ165+ζ163 | ζ1615-ζ169 | ζ167-ζ16 | ζ165-ζ163 | ζ166ζ3+ζ162ζ3+ζ162 | ζ1614ζ3+ζ1614+ζ1610ζ3 | ζ1614ζ32+ζ1610ζ32+ζ1610 | ζ166ζ32+ζ166+ζ162ζ32 | ζ167ζ32+ζ167+ζ16ζ32 | ζ1613ζ32+ζ1613+ζ1611ζ32 | ζ165ζ32+ζ163ζ32+ζ163 | ζ167ζ3+ζ167+ζ16ζ3 | ζ1613ζ3+ζ1611ζ3+ζ1611 | ζ165ζ3+ζ165+ζ163ζ3 | ζ167ζ32+ζ16ζ32+ζ16 | ζ167ζ3+ζ16ζ3+ζ16 | orthogonal faithful |
►On 48 points
Generators in S48
(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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25)]])
D48 is a maximal subgroup of
D96 C32⋊S3 C3⋊D32 C3⋊SD64 D48⋊7C2 C16⋊D6 S3×D16 D48⋊C2 D48⋊5C2 D144 C3⋊D48 C32⋊5D16 C5⋊D48 D240
D48 is a maximal quotient of
D96 C32⋊S3 Dic48 C48⋊5C4 C2.D48 D144 C3⋊D48 C32⋊5D16 C5⋊D48 D240
Matrix representation of D48 ►in GL2(𝔽47) generated by
| 37 | 16 |
| 31 | 2 |
| 2 | 38 |
| 16 | 45 |
G:=sub<GL(2,GF(47))| [37,31,16,2],[2,16,38,45] >;
D48 in GAP, Magma, Sage, TeX
D_{48} % in TeX
G:=Group("D48"); // GroupNames label
G:=SmallGroup(96,6);
// by ID
G=gap.SmallGroup(96,6);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,79,218,122,579,69,2309]);
// Polycyclic
G:=Group<a,b|a^48=b^2=1,b*a*b=a^-1>;
// generators/relations
Export
Subgroup lattice of D48 in TeX
Character table of D48 in TeX