metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊1C4, C6.4D8, C8⋊1Dic3, C2.1D24, C6.2Q16, C12.5Q8, C4.5Dic6, C2.2Dic12, C22.9D12, (C2×C8).3S3, C3⋊2(C2.D8), C6.6(C4⋊C4), (C2×C24).5C2, (C2×C6).14D4, (C2×C4).69D6, C12.34(C2×C4), C4⋊Dic3.3C2, C4.7(C2×Dic3), C2.4(C4⋊Dic3), (C2×C12).82C22, SmallGroup(96,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊1C4
G = < a,b | a24=b4=1, bab-1=a-1 >
Character table of C24⋊1C4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 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 | i | i | -i | -i | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | -i | i | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | -i | i | i | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √2 | √2 | -√2 | -√2 | -√3 | -√3 | √3 | √3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | orthogonal lifted from D24 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 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 | 2 | 0 | 0 | 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 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 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 | -1 | 2 | 2 | 0 | 0 | 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 |
| ρ14 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 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 |
| ρ15 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 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 |
| ρ16 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √2 | √2 | -√2 | -√2 | √3 | √3 | -√3 | -√3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | orthogonal lifted from D24 |
| ρ17 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 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 |
| ρ18 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√2 | -√2 | √2 | √2 | √3 | √3 | -√3 | -√3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | orthogonal lifted from D24 |
| ρ19 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√2 | -√2 | √2 | √2 | -√3 | -√3 | √3 | √3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | orthogonal lifted from D24 |
| ρ20 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 2 | -2 | -2 | 2 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ21 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | 2 | 2 | -2 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ22 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ23 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ24 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -√3 | -√3 | √3 | √3 | √3 | -√3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ25 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ26 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | √3 | √3 | -√3 | -√3 | -√3 | √3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ27 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√2 | √2 | -√2 | √2 | -√3 | √3 | √3 | -√3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | symplectic lifted from Dic12, Schur index 2 |
| ρ28 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √2 | -√2 | √2 | -√2 | -√3 | √3 | √3 | -√3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | symplectic lifted from Dic12, Schur index 2 |
| ρ29 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √2 | -√2 | √2 | -√2 | √3 | -√3 | -√3 | √3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | symplectic lifted from Dic12, Schur index 2 |
| ρ30 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√2 | √2 | -√2 | √2 | √3 | -√3 | -√3 | √3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | symplectic lifted from Dic12, Schur index 2 |
►Regular action on 96 points
Generators in S96
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 29 85 60)(2 28 86 59)(3 27 87 58)(4 26 88 57)(5 25 89 56)(6 48 90 55)(7 47 91 54)(8 46 92 53)(9 45 93 52)(10 44 94 51)(11 43 95 50)(12 42 96 49)(13 41 73 72)(14 40 74 71)(15 39 75 70)(16 38 76 69)(17 37 77 68)(18 36 78 67)(19 35 79 66)(20 34 80 65)(21 33 81 64)(22 32 82 63)(23 31 83 62)(24 30 84 61)
G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,29,85,60)(2,28,86,59)(3,27,87,58)(4,26,88,57)(5,25,89,56)(6,48,90,55)(7,47,91,54)(8,46,92,53)(9,45,93,52)(10,44,94,51)(11,43,95,50)(12,42,96,49)(13,41,73,72)(14,40,74,71)(15,39,75,70)(16,38,76,69)(17,37,77,68)(18,36,78,67)(19,35,79,66)(20,34,80,65)(21,33,81,64)(22,32,82,63)(23,31,83,62)(24,30,84,61)>;
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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,29,85,60)(2,28,86,59)(3,27,87,58)(4,26,88,57)(5,25,89,56)(6,48,90,55)(7,47,91,54)(8,46,92,53)(9,45,93,52)(10,44,94,51)(11,43,95,50)(12,42,96,49)(13,41,73,72)(14,40,74,71)(15,39,75,70)(16,38,76,69)(17,37,77,68)(18,36,78,67)(19,35,79,66)(20,34,80,65)(21,33,81,64)(22,32,82,63)(23,31,83,62)(24,30,84,61) );
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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,29,85,60),(2,28,86,59),(3,27,87,58),(4,26,88,57),(5,25,89,56),(6,48,90,55),(7,47,91,54),(8,46,92,53),(9,45,93,52),(10,44,94,51),(11,43,95,50),(12,42,96,49),(13,41,73,72),(14,40,74,71),(15,39,75,70),(16,38,76,69),(17,37,77,68),(18,36,78,67),(19,35,79,66),(20,34,80,65),(21,33,81,64),(22,32,82,63),(23,31,83,62),(24,30,84,61)]])
C24⋊1C4 is a maximal subgroup of
C6.6D16 C6.SD32 C2.Dic24 C48⋊5C4 C48⋊6C4 C2.D48 D8⋊1Dic3 C6.5Q32 C24⋊8Q8 C24.13Q8 C4×D24 C4×Dic12 C8⋊Dic6 C42.16D6 C23.40D12 C23.15D12 C22.D24 C23.18D12 Dic3.D8 D4.2Dic6 D6.D8 C24⋊1C4⋊C2 Q8⋊3Dic6 Q8.3Dic6 D6.Q16 D6⋊C8.C2 Dic6.3Q8 D12⋊4Q8 D12.3Q8 Dic6⋊3Q8 C24⋊3Q8 C8⋊(C4×S3) C24⋊2Q8 C8.6Dic6 S3×C2.D8 C8.27(C4×S3) C23.27D12 C24⋊29D4 C24.82D4 C23.52D12 C24⋊2D4 Dic3×D8 D6⋊3D8 SD16⋊Dic3 C24⋊8D4 Dic3×Q16 D6⋊3Q16 C72⋊1C4 C6.18D24 C24⋊1Dic3 C60.8Q8 C120⋊9C4 D5.D24
C24⋊1C4 is a maximal quotient of
C24⋊1C8 C48⋊5C4 C48⋊6C4 C48.C4 C12.9C42 C72⋊1C4 C6.18D24 C24⋊1Dic3 C60.8Q8 C120⋊9C4 D5.D24
Matrix representation of C24⋊1C4 ►in GL4(𝔽73) generated by
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 50 | 55 |
| 0 | 0 | 18 | 68 |
| 18 | 71 | 0 | 0 |
| 53 | 55 | 0 | 0 |
| 0 | 0 | 53 | 55 |
| 0 | 0 | 2 | 20 |
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,50,18,0,0,55,68],[18,53,0,0,71,55,0,0,0,0,53,2,0,0,55,20] >;
C24⋊1C4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_1C_4 % in TeX
G:=Group("C24:1C4"); // GroupNames label
G:=SmallGroup(96,25);
// by ID
G=gap.SmallGroup(96,25);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,151,579,69,2309]);
// Polycyclic
G:=Group<a,b|a^24=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of C24⋊1C4 in TeX
Character table of C24⋊1C4 in TeX