direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xD24, C8:7D6, C6:1D8, C4.7D12, C24:8C22, C12.30D4, D12:3C22, C12.29C23, C22.13D12, C3:1(C2xD8), (C2xC8):3S3, (C2xC24):5C2, (C2xD12):5C2, (C2xC4).80D6, (C2xC6).17D4, C6.10(C2xD4), C2.12(C2xD12), C4.27(C22xS3), (C2xC12).89C22, SmallGroup(96,110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xD24
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, C2xC4, D4, C23, C12, D6, C2xC6, C2xC8, D8, C2xD4, C24, D12, D12, C2xC12, C22xS3, C2xD8, D24, C2xC24, C2xD12, C2xD24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, D12, C22xS3, C2xD8, D24, C2xD12, C2xD24
Character table of C2xD24
| 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)]])
C2xD24 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 C2xS3xD8 D8:15D6
C2xD24 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 C2xD24 ►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] >;
C2xD24 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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