metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.37D4, C3⋊C8.10D4, (C2×Q16)⋊9S3, C4.28(S3×D4), (C2×C8).96D6, (C6×Q16)⋊10C2, C24⋊C4⋊10C2, C8.6(C3⋊D4), C3⋊5(C8.2D4), (C2×Q8).90D6, C12.188(C2×D4), Dic3⋊Q8⋊6C2, C6.34(C4⋊1D4), (C2×Dic3).78D4, C22.281(S3×D4), (C6×Q8).93C22, C2.25(C12⋊3D4), (C2×C12).464C23, (C2×C24).151C22, C12.23D4.6C2, C2.31(Q16⋊S3), C6.81(C8.C22), (C2×D12).126C22, (C4×Dic3).56C22, (C2×Dic6).133C22, C4.15(C2×C3⋊D4), (C2×C24⋊C2).8C2, (C2×C3⋊Q16)⋊21C2, (C2×C6).375(C2×D4), (C2×C3⋊C8).168C22, (C2×Q8⋊2S3).9C2, (C2×C4).552(C22×S3), SmallGroup(192,749)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.37D4
G = < a,b,c | a24=b4=c2=1, bab-1=a5, cac=a11, cbc=a12b-1 >
Subgroups: 376 in 124 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C2×Q16, C24⋊C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, Q8⋊2S3, C3⋊Q16, C2×C24, C3×Q16, C2×Dic6, C2×D12, C6×Q8, C8.2D4, C24⋊C4, C2×C24⋊C2, C2×Q8⋊2S3, C2×C3⋊Q16, Dic3⋊Q8, C12.23D4, C6×Q16, C24.37D4
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, Q16⋊S3, C12⋊3D4, C24.37D4
Character table of C24.37D4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | 2 | 8 | 8 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 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 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -2 | 2 | 0 | 0 | -1 | 1 | √-3 | -√-3 | √-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 | 2 | -2 | 0 | 0 | -1 | 1 | -√-3 | √-3 | √-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 | -2 | 2 | 0 | 0 | -1 | 1 | -√-3 | √-3 | -√-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 | 2 | -2 | 0 | 0 | -1 | 1 | √-3 | -√-3 | -√-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 | 2 | 2 | 0 | 0 | 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 | 2 | -2 | 0 | 0 | 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 | complex lifted from Q16⋊S3 |
| 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 | complex lifted from Q16⋊S3 |
| 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 | complex lifted from Q16⋊S3 |
| 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 | complex lifted from Q16⋊S3 |
►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 43 63 74)(2 48 64 79)(3 29 65 84)(4 34 66 89)(5 39 67 94)(6 44 68 75)(7 25 69 80)(8 30 70 85)(9 35 71 90)(10 40 72 95)(11 45 49 76)(12 26 50 81)(13 31 51 86)(14 36 52 91)(15 41 53 96)(16 46 54 77)(17 27 55 82)(18 32 56 87)(19 37 57 92)(20 42 58 73)(21 47 59 78)(22 28 60 83)(23 33 61 88)(24 38 62 93)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 80)(26 91)(27 78)(28 89)(29 76)(30 87)(31 74)(32 85)(33 96)(34 83)(35 94)(36 81)(37 92)(38 79)(39 90)(40 77)(41 88)(42 75)(43 86)(44 73)(45 84)(46 95)(47 82)(48 93)(49 53)(50 64)(52 62)(54 60)(55 71)(56 58)(57 69)(59 67)(61 65)(66 72)(68 70)
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), (1,43,63,74)(2,48,64,79)(3,29,65,84)(4,34,66,89)(5,39,67,94)(6,44,68,75)(7,25,69,80)(8,30,70,85)(9,35,71,90)(10,40,72,95)(11,45,49,76)(12,26,50,81)(13,31,51,86)(14,36,52,91)(15,41,53,96)(16,46,54,77)(17,27,55,82)(18,32,56,87)(19,37,57,92)(20,42,58,73)(21,47,59,78)(22,28,60,83)(23,33,61,88)(24,38,62,93), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,80)(26,91)(27,78)(28,89)(29,76)(30,87)(31,74)(32,85)(33,96)(34,83)(35,94)(36,81)(37,92)(38,79)(39,90)(40,77)(41,88)(42,75)(43,86)(44,73)(45,84)(46,95)(47,82)(48,93)(49,53)(50,64)(52,62)(54,60)(55,71)(56,58)(57,69)(59,67)(61,65)(66,72)(68,70)>;
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), (1,43,63,74)(2,48,64,79)(3,29,65,84)(4,34,66,89)(5,39,67,94)(6,44,68,75)(7,25,69,80)(8,30,70,85)(9,35,71,90)(10,40,72,95)(11,45,49,76)(12,26,50,81)(13,31,51,86)(14,36,52,91)(15,41,53,96)(16,46,54,77)(17,27,55,82)(18,32,56,87)(19,37,57,92)(20,42,58,73)(21,47,59,78)(22,28,60,83)(23,33,61,88)(24,38,62,93), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,80)(26,91)(27,78)(28,89)(29,76)(30,87)(31,74)(32,85)(33,96)(34,83)(35,94)(36,81)(37,92)(38,79)(39,90)(40,77)(41,88)(42,75)(43,86)(44,73)(45,84)(46,95)(47,82)(48,93)(49,53)(50,64)(52,62)(54,60)(55,71)(56,58)(57,69)(59,67)(61,65)(66,72)(68,70) );
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)], [(1,43,63,74),(2,48,64,79),(3,29,65,84),(4,34,66,89),(5,39,67,94),(6,44,68,75),(7,25,69,80),(8,30,70,85),(9,35,71,90),(10,40,72,95),(11,45,49,76),(12,26,50,81),(13,31,51,86),(14,36,52,91),(15,41,53,96),(16,46,54,77),(17,27,55,82),(18,32,56,87),(19,37,57,92),(20,42,58,73),(21,47,59,78),(22,28,60,83),(23,33,61,88),(24,38,62,93)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,80),(26,91),(27,78),(28,89),(29,76),(30,87),(31,74),(32,85),(33,96),(34,83),(35,94),(36,81),(37,92),(38,79),(39,90),(40,77),(41,88),(42,75),(43,86),(44,73),(45,84),(46,95),(47,82),(48,93),(49,53),(50,64),(52,62),(54,60),(55,71),(56,58),(57,69),(59,67),(61,65),(66,72),(68,70)]])
Matrix representation of C24.37D4 ►in GL6(𝔽73)
| 72 | 3 | 0 | 0 | 0 | 0 |
| 48 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 62 | 62 | 11 |
| 0 | 0 | 11 | 22 | 62 | 51 |
| 0 | 0 | 42 | 31 | 0 | 0 |
| 0 | 0 | 42 | 11 | 0 | 0 |
| 1 | 70 | 0 | 0 | 0 | 0 |
| 25 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 55 | 18 | 36 |
| 0 | 0 | 51 | 69 | 18 | 55 |
| 0 | 0 | 64 | 55 | 22 | 18 |
| 0 | 0 | 64 | 9 | 69 | 51 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 48 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 72 | 0 |
| 0 | 0 | 72 | 72 | 1 | 1 |
G:=sub<GL(6,GF(73))| [72,48,0,0,0,0,3,1,0,0,0,0,0,0,11,11,42,42,0,0,62,22,31,11,0,0,62,62,0,0,0,0,11,51,0,0],[1,25,0,0,0,0,70,72,0,0,0,0,0,0,4,51,64,64,0,0,55,69,55,9,0,0,18,18,22,69,0,0,36,55,18,51],[72,48,0,0,0,0,0,1,0,0,0,0,0,0,1,72,1,72,0,0,0,72,0,72,0,0,0,0,72,1,0,0,0,0,0,1] >;
C24.37D4 in GAP, Magma, Sage, TeX
C_{24}._{37}D_4 % in TeX
G:=Group("C24.37D4"); // GroupNames label
G:=SmallGroup(192,749);
// by ID
G=gap.SmallGroup(192,749);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,219,184,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^11,c*b*c=a^12*b^-1>;
// generators/relations
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