metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.31D4, C3⋊C8.9D4, C24⋊C4⋊5C2, C4.23(S3×D4), (C2×C8).89D6, (C2×D4).70D6, C8.3(C3⋊D4), C3⋊4(C8.2D4), (C2×Q8).76D6, C12.174(C2×D4), (C2×SD16).2S3, (C6×SD16).2C2, Dic3⋊Q8⋊4C2, (C2×Dic12)⋊25C2, C6.29(C4⋊1D4), (C2×Dic3).70D4, (C6×D4).93C22, C22.264(S3×D4), (C6×Q8).74C22, C2.20(C12⋊3D4), (C2×C12).444C23, (C2×C24).114C22, C23.12D6.6C2, C2.28(D4.D6), C6.48(C8.C22), (C4×Dic3).51C22, (C2×Dic6).125C22, C4.7(C2×C3⋊D4), (C2×C3⋊Q16)⋊17C2, (C2×C6).356(C2×D4), (C2×D4.S3).9C2, (C2×C3⋊C8).156C22, (C2×C4).533(C22×S3), SmallGroup(192,726)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.31D4
G = < a,b,c | a24=b4=1, c2=a12, bab-1=a5, cac-1=a-1, cbc-1=b-1 >
Subgroups: 344 in 124 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×SD16, C2×Q16, Dic12, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D4.S3, C3⋊Q16, C6.D4, C2×C24, C3×SD16, C2×Dic6, C6×D4, C6×Q8, C8.2D4, C24⋊C4, C2×Dic12, C2×D4.S3, C23.12D6, C2×C3⋊Q16, Dic3⋊Q8, C6×SD16, C24.31D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C8.C22, S3×D4, C2×C3⋊D4, C8.2D4, D4.D6, C12⋊3D4, C24.31D4
Character table of C24.31D4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 8 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | |
| 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 | -2 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ16 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | 2 | -2 | 0 | 0 | -1 | 1 | -√-3 | √-3 | 1 | 1 | -1 | -1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | -2 | 2 | 0 | 0 | -1 | 1 | √-3 | -√-3 | -1 | -1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | -2 | 2 | 0 | 0 | -1 | 1 | -√-3 | √-3 | -1 | -1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | 2 | -2 | 0 | 0 | -1 | 1 | √-3 | -√-3 | 1 | 1 | -1 | -1 | complex lifted from C3⋊D4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | 4 | 4 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | -√6 | √6 | symplectic lifted from D4.D6, Schur index 2 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | -√6 | √6 | symplectic lifted from D4.D6, Schur index 2 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | √6 | -√6 | symplectic lifted from D4.D6, Schur index 2 |
| ρ30 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | √6 | -√6 | symplectic lifted from D4.D6, Schur index 2 |
►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 55 38 80)(2 60 39 85)(3 65 40 90)(4 70 41 95)(5 51 42 76)(6 56 43 81)(7 61 44 86)(8 66 45 91)(9 71 46 96)(10 52 47 77)(11 57 48 82)(12 62 25 87)(13 67 26 92)(14 72 27 73)(15 53 28 78)(16 58 29 83)(17 63 30 88)(18 68 31 93)(19 49 32 74)(20 54 33 79)(21 59 34 84)(22 64 35 89)(23 69 36 94)(24 50 37 75)
(1 95 13 83)(2 94 14 82)(3 93 15 81)(4 92 16 80)(5 91 17 79)(6 90 18 78)(7 89 19 77)(8 88 20 76)(9 87 21 75)(10 86 22 74)(11 85 23 73)(12 84 24 96)(25 59 37 71)(26 58 38 70)(27 57 39 69)(28 56 40 68)(29 55 41 67)(30 54 42 66)(31 53 43 65)(32 52 44 64)(33 51 45 63)(34 50 46 62)(35 49 47 61)(36 72 48 60)
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), (1,55,38,80)(2,60,39,85)(3,65,40,90)(4,70,41,95)(5,51,42,76)(6,56,43,81)(7,61,44,86)(8,66,45,91)(9,71,46,96)(10,52,47,77)(11,57,48,82)(12,62,25,87)(13,67,26,92)(14,72,27,73)(15,53,28,78)(16,58,29,83)(17,63,30,88)(18,68,31,93)(19,49,32,74)(20,54,33,79)(21,59,34,84)(22,64,35,89)(23,69,36,94)(24,50,37,75), (1,95,13,83)(2,94,14,82)(3,93,15,81)(4,92,16,80)(5,91,17,79)(6,90,18,78)(7,89,19,77)(8,88,20,76)(9,87,21,75)(10,86,22,74)(11,85,23,73)(12,84,24,96)(25,59,37,71)(26,58,38,70)(27,57,39,69)(28,56,40,68)(29,55,41,67)(30,54,42,66)(31,53,43,65)(32,52,44,64)(33,51,45,63)(34,50,46,62)(35,49,47,61)(36,72,48,60)>;
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), (1,55,38,80)(2,60,39,85)(3,65,40,90)(4,70,41,95)(5,51,42,76)(6,56,43,81)(7,61,44,86)(8,66,45,91)(9,71,46,96)(10,52,47,77)(11,57,48,82)(12,62,25,87)(13,67,26,92)(14,72,27,73)(15,53,28,78)(16,58,29,83)(17,63,30,88)(18,68,31,93)(19,49,32,74)(20,54,33,79)(21,59,34,84)(22,64,35,89)(23,69,36,94)(24,50,37,75), (1,95,13,83)(2,94,14,82)(3,93,15,81)(4,92,16,80)(5,91,17,79)(6,90,18,78)(7,89,19,77)(8,88,20,76)(9,87,21,75)(10,86,22,74)(11,85,23,73)(12,84,24,96)(25,59,37,71)(26,58,38,70)(27,57,39,69)(28,56,40,68)(29,55,41,67)(30,54,42,66)(31,53,43,65)(32,52,44,64)(33,51,45,63)(34,50,46,62)(35,49,47,61)(36,72,48,60) );
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)], [(1,55,38,80),(2,60,39,85),(3,65,40,90),(4,70,41,95),(5,51,42,76),(6,56,43,81),(7,61,44,86),(8,66,45,91),(9,71,46,96),(10,52,47,77),(11,57,48,82),(12,62,25,87),(13,67,26,92),(14,72,27,73),(15,53,28,78),(16,58,29,83),(17,63,30,88),(18,68,31,93),(19,49,32,74),(20,54,33,79),(21,59,34,84),(22,64,35,89),(23,69,36,94),(24,50,37,75)], [(1,95,13,83),(2,94,14,82),(3,93,15,81),(4,92,16,80),(5,91,17,79),(6,90,18,78),(7,89,19,77),(8,88,20,76),(9,87,21,75),(10,86,22,74),(11,85,23,73),(12,84,24,96),(25,59,37,71),(26,58,38,70),(27,57,39,69),(28,56,40,68),(29,55,41,67),(30,54,42,66),(31,53,43,65),(32,52,44,64),(33,51,45,63),(34,50,46,62),(35,49,47,61),(36,72,48,60)]])
Matrix representation of C24.31D4 ►in GL8(𝔽73)
| 68 | 5 | 48 | 50 | 0 | 0 | 0 | 0 |
| 68 | 63 | 48 | 48 | 0 | 0 | 0 | 0 |
| 1 | 37 | 10 | 68 | 0 | 0 | 0 | 0 |
| 37 | 1 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 68 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 68 | 63 |
| 0 | 0 | 0 | 0 | 39 | 34 | 5 | 68 |
| 0 | 0 | 0 | 0 | 39 | 5 | 5 | 10 |
| 67 | 8 | 43 | 70 | 0 | 0 | 0 | 0 |
| 14 | 6 | 70 | 33 | 0 | 0 | 0 | 0 |
| 56 | 25 | 65 | 67 | 0 | 0 | 0 | 0 |
| 25 | 42 | 59 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 48 | 11 | 23 | 22 |
| 0 | 0 | 0 | 0 | 36 | 25 | 72 | 50 |
| 0 | 0 | 0 | 0 | 25 | 62 | 25 | 62 |
| 0 | 0 | 0 | 0 | 37 | 48 | 37 | 48 |
| 55 | 50 | 0 | 0 | 0 | 0 | 0 | 0 |
| 68 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 68 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 70 | 57 |
| 0 | 0 | 0 | 0 | 8 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 65 | 0 | 0 |
G:=sub<GL(8,GF(73))| [68,68,1,37,0,0,0,0,5,63,37,1,0,0,0,0,48,48,10,5,0,0,0,0,50,48,68,5,0,0,0,0,0,0,0,0,0,0,39,39,0,0,0,0,0,0,34,5,0,0,0,0,68,68,5,5,0,0,0,0,5,63,68,10],[67,14,56,25,0,0,0,0,8,6,25,42,0,0,0,0,43,70,65,59,0,0,0,0,70,33,67,8,0,0,0,0,0,0,0,0,48,36,25,37,0,0,0,0,11,25,62,48,0,0,0,0,23,72,25,37,0,0,0,0,22,50,62,48],[55,68,0,0,0,0,0,0,50,18,0,0,0,0,0,0,0,0,50,68,0,0,0,0,0,0,18,23,0,0,0,0,0,0,0,0,0,0,8,35,0,0,0,0,0,0,43,65,0,0,0,0,16,70,0,0,0,0,0,0,13,57,0,0] >;
C24.31D4 in GAP, Magma, Sage, TeX
C_{24}._{31}D_4 % in TeX
G:=Group("C24.31D4"); // GroupNames label
G:=SmallGroup(192,726);
// by ID
G=gap.SmallGroup(192,726);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,232,1094,135,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=a^12,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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