metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.5D8, D12:2C4, C2.2D24, C12.45D4, C6.3SD16, C22.10D12, (C2xC8):2S3, (C2xC24):2C2, C4.8(C4xS3), C4:Dic3:1C2, (C2xC4).71D6, (C2xC6).15D4, C3:2(D4:C4), C12.18(C2xC4), C2.8(D6:C4), (C2xD12).1C2, C2.3(C24:C2), C4.20(C3:D4), C6.7(C22:C4), (C2xC12).83C22, SmallGroup(96,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D24
G = < a,b,c | a2=b24=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D6, C22:C4, D8, SD16, C4xS3, D12, C3:D4, D4:C4, C24:C2, D24, D6:C4, C2.D24
Character table of C2.D24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 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 | 2 | 2 | 2 | 12 | 12 | 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 | i | -i | -1 | 1 | -1 | -i | -i | i | i | 1 | -1 | -1 | 1 | -i | i | i | -i | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | i | -i | -1 | 1 | -1 | i | i | -i | -i | 1 | -1 | -1 | 1 | i | -i | -i | i | -i | -i | i | i | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -i | i | -1 | 1 | -1 | i | i | -i | -i | 1 | -1 | -1 | 1 | i | -i | -i | i | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -i | i | -1 | 1 | -1 | -i | -i | i | i | 1 | -1 | -1 | 1 | -i | i | i | -i | i | i | -i | -i | linear of order 4 |
| ρ9 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 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 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -√3 | -√3 | -√3 | √3 | √3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 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 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √2 | -√2 | √2 | -√2 | √3 | √3 | -√3 | -√3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | orthogonal lifted from D24 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 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 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √2 | -√2 | √2 | -√2 | -√3 | -√3 | √3 | √3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | orthogonal lifted from D24 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | √3 | √3 | √3 | -√3 | -√3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√2 | √2 | -√2 | √2 | √3 | √3 | -√3 | -√3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | orthogonal lifted from D24 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√2 | √2 | -√2 | √2 | -√3 | -√3 | √3 | √3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | orthogonal lifted from D24 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | 2i | 2i | -2i | -2i | -1 | 1 | 1 | -1 | -i | i | i | -i | i | i | -i | -i | complex lifted from C4xS3 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | -2i | -2i | 2i | 2i | -1 | 1 | 1 | -1 | i | -i | -i | i | -i | -i | i | i | complex lifted from C4xS3 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | √-2 | -√-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √-2 | -√-2 | -√-2 | √-2 | √3 | -√3 | √3 | -√3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | complex lifted from C24:C2 |
| ρ25 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | -√-3 | √-3 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | √-3 | complex lifted from C3:D4 |
| ρ26 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | √-3 | -√-3 | -√-3 | -√-3 | √-3 | √-3 | √-3 | -√-3 | complex lifted from C3:D4 |
| ρ27 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√-2 | √-2 | √-2 | -√-2 | -√3 | √3 | -√3 | √3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | complex lifted from C24:C2 |
| ρ28 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√-2 | √-2 | √-2 | -√-2 | √3 | -√3 | √3 | -√3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | complex lifted from C24:C2 |
| ρ29 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -√-2 | √-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | √-2 | -√-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ30 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √-2 | -√-2 | -√-2 | √-2 | -√3 | √3 | -√3 | √3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | complex lifted from C24:C2 |
►On 48 points
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)
(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 42 43 24)(2 23 44 41)(3 40 45 22)(4 21 46 39)(5 38 47 20)(6 19 48 37)(7 36 25 18)(8 17 26 35)(9 34 27 16)(10 15 28 33)(11 32 29 14)(12 13 30 31)
G:=sub<Sym(48)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42), (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), (1,42,43,24)(2,23,44,41)(3,40,45,22)(4,21,46,39)(5,38,47,20)(6,19,48,37)(7,36,25,18)(8,17,26,35)(9,34,27,16)(10,15,28,33)(11,32,29,14)(12,13,30,31)>;
G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42), (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), (1,42,43,24)(2,23,44,41)(3,40,45,22)(4,21,46,39)(5,38,47,20)(6,19,48,37)(7,36,25,18)(8,17,26,35)(9,34,27,16)(10,15,28,33)(11,32,29,14)(12,13,30,31) );
G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42)], [(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)], [(1,42,43,24),(2,23,44,41),(3,40,45,22),(4,21,46,39),(5,38,47,20),(6,19,48,37),(7,36,25,18),(8,17,26,35),(9,34,27,16),(10,15,28,33),(11,32,29,14),(12,13,30,31)]])
C2.D24 is a maximal subgroup of
C4xC24:C2 C4xD24 C4.5D24 C42.264D6 C42.16D6 D24:C4 C42.19D6 C42.20D6 D12.31D4 D12:13D4 D12.32D4 D12:14D4 C23.43D12 C22.D24 C23.18D12 Dic3:4D8 Dic3.SD16 C4:C4.D6 S3xD4:C4 C4:C4:19D6 D6:D8 C3:C8:D4 D4:S3:C4 Dic3:7SD16 (C2xC8).D6 Q8:C4:S3 Q8:7(C4xS3) C4:C4.150D6 D6:2SD16 C3:(C8:D4) Q8:3(C4xS3) D12:3Q8 C4:D24 D12.19D4 C42.36D6 D12:4Q8 D12.3Q8 Dic6:8D4 D6.4SD16 C4.Q8:S3 D12:Q8 D12.Q8 D6.5D8 C2.D8:S3 D12:2Q8 D12.2Q8 C23.28D12 C24:30D4 C24:29D4 C23.53D12 C23.54D12 C24:2D4 C24:3D4 Dic3:D8 D12:D4 Dic3:5SD16 (C3xD4).D4 D6:6SD16 D12:7D4 (C2xQ16):S3 D12.17D4 C2.D72 C6.16D24 C6.17D24 C62.84D4 C10.D24 D60:15C4 D60:8C4 D12:F5
C2.D24 is a maximal quotient of
C4.17D24 C22.2D24 C4.D24 C12.2D8 C2.Dic24 C2.D48 D24.1C4 M5(2):S3 C12.4D8 D24:2C4 C12.9C42 C2.D72 C6.16D24 C6.17D24 C62.84D4 C10.D24 D60:15C4 D60:8C4 D12:F5
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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 |
| 0 | 0 | 0 | 35 | 32 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 |
| 0 | 0 | 0 | 38 | 0 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[46,0,0,0,0,0,0,1,0,0,0,72,1,0,0,0,0,0,0,35,0,0,0,25,32],[27,0,0,0,0,0,0,72,0,0,0,72,0,0,0,0,0,0,0,38,0,0,0,25,0] >;
C2.D24 in GAP, Magma, Sage, TeX
C_2.D_{24} % in TeX
G:=Group("C2.D24"); // GroupNames label
G:=SmallGroup(96,28);
// by ID
G=gap.SmallGroup(96,28);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,79,362,86,2309]);
// Polycyclic
G:=Group<a,b,c|a^2=b^24=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations
Export
Subgroup lattice of C2.D24 in TeX
Character table of C2.D24 in TeX