metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊2C4, C8⋊2Dic3, C12.4Q8, C4.4Dic6, C6.2SD16, C22.8D12, (C2×C8).6S3, C3⋊2(C4.Q8), C6.5(C4⋊C4), (C2×C24).8C2, (C2×C6).13D4, (C2×C4).68D6, C12.33(C2×C4), C4⋊Dic3.2C2, C4.6(C2×Dic3), C2.2(C24⋊C2), C2.3(C4⋊Dic3), (C2×C12).81C22, SmallGroup(96,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊Dic3
G = < a,b,c | a8=b6=1, c2=b3, ab=ba, cac-1=a3, cbc-1=b-1 >
Character table of C8⋊Dic3
| 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 | 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 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| 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 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 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 |
| ρ19 | 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 | complex lifted from SD16 |
| ρ20 | 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 | complex lifted from SD16 |
| ρ21 | 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
| ρ23 | 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 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ87ζ32+ζ85ζ32+ζ85 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ83ζ3+ζ8ζ3+ζ8 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ83ζ3+ζ8ζ3+ζ8 | complex lifted from C24⋊C2 |
| ρ24 | 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 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ87ζ32+ζ85ζ32+ζ85 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ83ζ3+ζ8ζ3+ζ8 | complex lifted from C24⋊C2 |
| ρ25 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-2 | -√-2 | √-2 | √-2 | -√3 | -√3 | √3 | √3 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ87ζ32+ζ85ζ32+ζ85 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ87ζ32+ζ85ζ32+ζ85 | -ζ83ζ3+ζ8ζ3+ζ8 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ83ζ3+ζ8ζ3+ζ8 | -ζ83ζ32+ζ8ζ32+ζ8 | complex lifted from C24⋊C2 |
| ρ26 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√-2 | √-2 | -√-2 | √-2 | √3 | -√3 | -√3 | √3 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ83ζ3+ζ8ζ3+ζ8 | -ζ87ζ32+ζ85ζ32+ζ85 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ87ζ32+ζ85ζ32+ζ85 | -ζ83ζ3+ζ8ζ3+ζ8 | -ζ83ζ32+ζ8ζ32+ζ8 | complex lifted from C24⋊C2 |
| ρ27 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-2 | √-2 | -√-2 | -√-2 | √3 | √3 | -√3 | -√3 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ83ζ3+ζ8ζ3+ζ8 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ83ζ3+ζ8ζ3+ζ8 | -ζ87ζ32+ζ85ζ32+ζ85 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ87ζ32+ζ85ζ32+ζ85 | -ζ87ζ3+ζ85ζ3+ζ85 | complex lifted from C24⋊C2 |
| 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 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ83ζ3+ζ8ζ3+ζ8 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ87ζ32+ζ85ζ32+ζ85 | complex lifted from C24⋊C2 |
| ρ29 | 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 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ83ζ3+ζ8ζ3+ζ8 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ87ζ32+ζ85ζ32+ζ85 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ87ζ32+ζ85ζ32+ζ85 | complex lifted from C24⋊C2 |
| ρ30 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √-2 | -√-2 | √-2 | -√-2 | -√3 | √3 | √3 | -√3 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ87ζ3+ζ85ζ3+ζ85 | -ζ87ζ32+ζ85ζ32+ζ85 | -ζ83ζ3+ζ8ζ3+ζ8 | -ζ83ζ32+ζ8ζ32+ζ8 | -ζ83ζ3+ζ8ζ3+ζ8 | -ζ87ζ32+ζ85ζ32+ζ85 | -ζ87ζ3+ζ85ζ3+ζ85 | complex lifted from C24⋊C2 |
►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 30 79 24 59 68)(2 31 80 17 60 69)(3 32 73 18 61 70)(4 25 74 19 62 71)(5 26 75 20 63 72)(6 27 76 21 64 65)(7 28 77 22 57 66)(8 29 78 23 58 67)(9 33 43 50 85 90)(10 34 44 51 86 91)(11 35 45 52 87 92)(12 36 46 53 88 93)(13 37 47 54 81 94)(14 38 48 55 82 95)(15 39 41 56 83 96)(16 40 42 49 84 89)
(1 50 24 9)(2 53 17 12)(3 56 18 15)(4 51 19 10)(5 54 20 13)(6 49 21 16)(7 52 22 11)(8 55 23 14)(25 44 62 91)(26 47 63 94)(27 42 64 89)(28 45 57 92)(29 48 58 95)(30 43 59 90)(31 46 60 93)(32 41 61 96)(33 68 85 79)(34 71 86 74)(35 66 87 77)(36 69 88 80)(37 72 81 75)(38 67 82 78)(39 70 83 73)(40 65 84 76)
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), (1,30,79,24,59,68)(2,31,80,17,60,69)(3,32,73,18,61,70)(4,25,74,19,62,71)(5,26,75,20,63,72)(6,27,76,21,64,65)(7,28,77,22,57,66)(8,29,78,23,58,67)(9,33,43,50,85,90)(10,34,44,51,86,91)(11,35,45,52,87,92)(12,36,46,53,88,93)(13,37,47,54,81,94)(14,38,48,55,82,95)(15,39,41,56,83,96)(16,40,42,49,84,89), (1,50,24,9)(2,53,17,12)(3,56,18,15)(4,51,19,10)(5,54,20,13)(6,49,21,16)(7,52,22,11)(8,55,23,14)(25,44,62,91)(26,47,63,94)(27,42,64,89)(28,45,57,92)(29,48,58,95)(30,43,59,90)(31,46,60,93)(32,41,61,96)(33,68,85,79)(34,71,86,74)(35,66,87,77)(36,69,88,80)(37,72,81,75)(38,67,82,78)(39,70,83,73)(40,65,84,76)>;
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), (1,30,79,24,59,68)(2,31,80,17,60,69)(3,32,73,18,61,70)(4,25,74,19,62,71)(5,26,75,20,63,72)(6,27,76,21,64,65)(7,28,77,22,57,66)(8,29,78,23,58,67)(9,33,43,50,85,90)(10,34,44,51,86,91)(11,35,45,52,87,92)(12,36,46,53,88,93)(13,37,47,54,81,94)(14,38,48,55,82,95)(15,39,41,56,83,96)(16,40,42,49,84,89), (1,50,24,9)(2,53,17,12)(3,56,18,15)(4,51,19,10)(5,54,20,13)(6,49,21,16)(7,52,22,11)(8,55,23,14)(25,44,62,91)(26,47,63,94)(27,42,64,89)(28,45,57,92)(29,48,58,95)(30,43,59,90)(31,46,60,93)(32,41,61,96)(33,68,85,79)(34,71,86,74)(35,66,87,77)(36,69,88,80)(37,72,81,75)(38,67,82,78)(39,70,83,73)(40,65,84,76) );
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)], [(1,30,79,24,59,68),(2,31,80,17,60,69),(3,32,73,18,61,70),(4,25,74,19,62,71),(5,26,75,20,63,72),(6,27,76,21,64,65),(7,28,77,22,57,66),(8,29,78,23,58,67),(9,33,43,50,85,90),(10,34,44,51,86,91),(11,35,45,52,87,92),(12,36,46,53,88,93),(13,37,47,54,81,94),(14,38,48,55,82,95),(15,39,41,56,83,96),(16,40,42,49,84,89)], [(1,50,24,9),(2,53,17,12),(3,56,18,15),(4,51,19,10),(5,54,20,13),(6,49,21,16),(7,52,22,11),(8,55,23,14),(25,44,62,91),(26,47,63,94),(27,42,64,89),(28,45,57,92),(29,48,58,95),(30,43,59,90),(31,46,60,93),(32,41,61,96),(33,68,85,79),(34,71,86,74),(35,66,87,77),(36,69,88,80),(37,72,81,75),(38,67,82,78),(39,70,83,73),(40,65,84,76)]])
C8⋊Dic3 is a maximal subgroup of
C24.6Q8 C24.Q8 D24⋊2C4 D8⋊2Dic3 C24⋊9Q8 C24.13Q8 C4×C24⋊C2 C8⋊Dic6 D24⋊C4 Dic12⋊C4 C23.39D12 C23.15D12 C23.43D12 C23.18D12 D4⋊Dic6 D4.Dic6 D6.SD16 D6⋊C8⋊11C2 Q8⋊2Dic6 Q8.4Dic6 D6.1SD16 C8⋊Dic3⋊C2 Dic6.3Q8 D12⋊3Q8 D12.3Q8 Dic6⋊4Q8 C24⋊5Q8 C8.8Dic6 S3×C4.Q8 (S3×C8)⋊C4 C24⋊4Q8 C8⋊S3⋊C4 C23.27D12 C24⋊30D4 C23.52D12 C24⋊3D4 C24.4D4 D8⋊Dic3 C24⋊12D4 Dic3×SD16 C24⋊14D4 Q16⋊Dic3 C24.36D4 C8⋊Dic9 C12.Dic6 C24⋊2Dic3 C60.7Q8 C120⋊10C4 C120⋊C4
C8⋊Dic3 is a maximal quotient of
C24⋊2C8 C24.Q8 C12.9C42 C8⋊Dic9 C12.Dic6 C24⋊2Dic3 C60.7Q8 C120⋊10C4 C120⋊C4
Matrix representation of C8⋊Dic3 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 64 | 63 |
| 0 | 72 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 39 | 47 | 0 | 0 |
| 8 | 34 | 0 | 0 |
| 0 | 0 | 67 | 3 |
| 0 | 0 | 37 | 6 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,22,64,0,0,0,63],[0,1,0,0,72,1,0,0,0,0,1,0,0,0,0,1],[39,8,0,0,47,34,0,0,0,0,67,37,0,0,3,6] >;
C8⋊Dic3 in GAP, Magma, Sage, TeX
C_8\rtimes {\rm Dic}_3 % in TeX
G:=Group("C8:Dic3"); // GroupNames label
G:=SmallGroup(96,24);
// by ID
G=gap.SmallGroup(96,24);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,55,579,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^8=b^6=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C8⋊Dic3 in TeX
Character table of C8⋊Dic3 in TeX