direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D24, C8⋊7D6, C6⋊1D8, C4.7D12, C24⋊8C22, C12.30D4, D12⋊3C22, C12.29C23, C22.13D12, C3⋊1(C2×D8), (C2×C8)⋊3S3, (C2×C24)⋊5C2, (C2×D12)⋊5C2, (C2×C4).80D6, (C2×C6).17D4, C6.10(C2×D4), C2.12(C2×D12), C4.27(C22×S3), (C2×C12).89C22, SmallGroup(96,110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D24
G = < a,b,c | a2=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 258 in 76 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, C23, C12, D6, C2×C6, C2×C8, D8, C2×D4, C24, D12, D12, C2×C12, C22×S3, C2×D8, D24, C2×C24, C2×D12, C2×D24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C22×S3, C2×D8, D24, C2×D12, C2×D24
Character table of C2×D24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 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 | 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 | 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 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 1 | 1 | -1 | 2 | -2 | -2 | 2 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 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 |
| ρ12 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 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 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 1 | 1 | -1 | -2 | 2 | 2 | -2 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | -√2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 2 | -2 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | 0 | √2 | √2 | √2 | √2 | -√2 | -√2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 2 | -2 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | 0 | -√2 | -√2 | -√2 | -√2 | √2 | √2 | √2 | √2 | orthogonal lifted from D8 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | √2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | √3 | -√3 | √3 | -√3 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -√3 | √3 | √3 | -√3 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -√3 | √3 | -√3 | √3 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | √3 | -√3 | -√3 | √3 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -1 | 1 | √2 | -√2 | √2 | -√2 | -√3 | √3 | -√3 | √3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ32+ζ8ζ32+ζ8 | orthogonal lifted from D24 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -1 | 1 | -√2 | √2 | -√2 | √2 | √3 | -√3 | √3 | -√3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ3+ζ85ζ3+ζ85 | orthogonal lifted from D24 |
| ρ25 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | 1 | -√2 | -√2 | √2 | √2 | √3 | -√3 | -√3 | √3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ32+ζ8ζ32+ζ8 | orthogonal lifted from D24 |
| ρ26 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -1 | 1 | -√2 | √2 | -√2 | √2 | -√3 | √3 | -√3 | √3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ3+ζ83+ζ8ζ3 | orthogonal lifted from D24 |
| ρ27 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | 1 | √2 | √2 | -√2 | -√2 | √3 | -√3 | -√3 | √3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ3+ζ83+ζ8ζ3 | orthogonal lifted from D24 |
| ρ28 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | -1 | 1 | √2 | -√2 | √2 | -√2 | √3 | -√3 | √3 | -√3 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ32+ζ87+ζ85ζ32 | orthogonal lifted from D24 |
| ρ29 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | 1 | √2 | √2 | -√2 | -√2 | -√3 | √3 | √3 | -√3 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ3+ζ85ζ3+ζ85 | orthogonal lifted from D24 |
| ρ30 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | 1 | -√2 | -√2 | √2 | √2 | -√3 | √3 | √3 | -√3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ32+ζ87+ζ85ζ32 | orthogonal lifted from D24 |
►On 48 points
Generators in S48
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 25)
(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 9)(2 8)(3 7)(4 6)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(25 35)(26 34)(27 33)(28 32)(29 31)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)
G:=sub<Sym(48)| (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,25), (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), (1,9)(2,8)(3,7)(4,6)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(25,35)(26,34)(27,33)(28,32)(29,31)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)>;
G:=Group( (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,25), (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), (1,9)(2,8)(3,7)(4,6)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(25,35)(26,34)(27,33)(28,32)(29,31)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43) );
G=PermutationGroup([[(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,25)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(25,35),(26,34),(27,33),(28,32),(29,31),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43)]])
C2×D24 is a maximal subgroup of
C6.D16 D24.C4 C2.D48 M5(2)⋊S3 C12⋊4D8 C8.8D12 D24⋊C4 C8⋊D12 D12⋊13D4 D12⋊14D4 D4⋊D12 D12⋊3D4 Q8⋊4D12 D12.12D4 C4⋊D24 D12.19D4 C24⋊7D4 D24⋊9C4 Dic3⋊5D8 D6⋊2D8 C24.19D4 C16⋊D6 C24⋊29D4 C24⋊3D4 Q8.9D12 C24⋊5D4 C24⋊9D4 C24.28D4 Q16⋊D6 D4.12D12 C2×S3×D8 D8⋊15D6
C2×D24 is a maximal quotient of
C24⋊8Q8 C4.5D24 C12⋊4D8 D12⋊13D4 C22.D24 C4⋊D24 D12⋊4Q8 D48⋊7C2 C16⋊D6 C16.D6 C24⋊29D4
Matrix representation of C2×D24 ►in GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 48 |
| 0 | 0 | 0 | 38 | 41 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 48 | 72 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,0,38,0,0,0,48,41],[72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,48,0,0,0,0,72] >;
C2×D24 in GAP, Magma, Sage, TeX
C_2\times D_{24} % in TeX
G:=Group("C2xD24"); // GroupNames label
G:=SmallGroup(96,110);
// by ID
G=gap.SmallGroup(96,110);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,122,579,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^2=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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