metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊9D4, C3⋊C8⋊10D4, C8⋊6(C3⋊D4), C3⋊5(C8⋊3D4), C24⋊C4⋊6C2, C4.26(S3×D4), (C2×C8).91D6, (C2×D24)⋊25C2, C12⋊3D4⋊6C2, (C6×SD16)⋊4C2, (C2×SD16)⋊2S3, (C2×D4).77D6, (C2×Q8).82D6, C12.179(C2×D4), C12.23D4⋊5C2, C6.32(C4⋊1D4), C2.31(Q8⋊3D6), C6.81(C8⋊C22), (C2×Dic3).74D4, C22.273(S3×D4), (C6×Q8).82C22, C2.23(C12⋊3D4), (C2×C12).453C23, (C2×C24).116C22, (C6×D4).102C22, (C2×D12).123C22, (C4×Dic3).52C22, (C2×D4⋊S3)⋊21C2, C4.10(C2×C3⋊D4), (C2×C6).365(C2×D4), (C2×Q8⋊2S3)⋊19C2, (C2×C3⋊C8).161C22, (C2×C4).542(C22×S3), SmallGroup(192,735)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊9D4
G = < a,b,c | a24=b4=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >
Subgroups: 536 in 144 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×SD16, C2×SD16, D24, C2×C3⋊C8, C4×Dic3, D6⋊C4, D4⋊S3, Q8⋊2S3, C2×C24, C3×SD16, C2×D12, C2×C3⋊D4, C6×D4, C6×Q8, C8⋊3D4, C24⋊C4, C2×D24, C2×D4⋊S3, C12⋊3D4, C2×Q8⋊2S3, C12.23D4, C6×SD16, C24⋊9D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C8⋊C22, S3×D4, C2×C3⋊D4, C8⋊3D4, Q8⋊3D6, C12⋊3D4, C24⋊9D4
Character table of C24⋊9D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 8 | 24 | 24 | 2 | 2 | 2 | 8 | 12 | 12 | 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 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | 2 | 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 | 0 | 0 | 2 | 2 | -2 | 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 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -2 | 2 | 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 | 0 | 0 | -1 | -2 | 2 | 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 | 0 | 0 | -1 | -2 | 2 | 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 | 0 | 0 | -1 | -2 | 2 | 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 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ24 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | -4 | -4 | 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 | 0 | 0 | -2 | 4 | -4 | 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 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | √6 | -√6 | -√6 | orthogonal lifted from Q8⋊3D6 |
| ρ28 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | -√6 | √6 | orthogonal lifted from Q8⋊3D6 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | √6 | -√6 | orthogonal lifted from Q8⋊3D6 |
| ρ30 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | -√6 | √6 | √6 | orthogonal lifted from Q8⋊3D6 |
►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 68 47 86)(2 49 48 91)(3 54 25 96)(4 59 26 77)(5 64 27 82)(6 69 28 87)(7 50 29 92)(8 55 30 73)(9 60 31 78)(10 65 32 83)(11 70 33 88)(12 51 34 93)(13 56 35 74)(14 61 36 79)(15 66 37 84)(16 71 38 89)(17 52 39 94)(18 57 40 75)(19 62 41 80)(20 67 42 85)(21 72 43 90)(22 53 44 95)(23 58 45 76)(24 63 46 81)
(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(46 48)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 96)(59 95)(60 94)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(71 83)(72 82)
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), (1,68,47,86)(2,49,48,91)(3,54,25,96)(4,59,26,77)(5,64,27,82)(6,69,28,87)(7,50,29,92)(8,55,30,73)(9,60,31,78)(10,65,32,83)(11,70,33,88)(12,51,34,93)(13,56,35,74)(14,61,36,79)(15,66,37,84)(16,71,38,89)(17,52,39,94)(18,57,40,75)(19,62,41,80)(20,67,42,85)(21,72,43,90)(22,53,44,95)(23,58,45,76)(24,63,46,81), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(46,48)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,96)(59,95)(60,94)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)>;
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), (1,68,47,86)(2,49,48,91)(3,54,25,96)(4,59,26,77)(5,64,27,82)(6,69,28,87)(7,50,29,92)(8,55,30,73)(9,60,31,78)(10,65,32,83)(11,70,33,88)(12,51,34,93)(13,56,35,74)(14,61,36,79)(15,66,37,84)(16,71,38,89)(17,52,39,94)(18,57,40,75)(19,62,41,80)(20,67,42,85)(21,72,43,90)(22,53,44,95)(23,58,45,76)(24,63,46,81), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(46,48)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,96)(59,95)(60,94)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82) );
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)], [(1,68,47,86),(2,49,48,91),(3,54,25,96),(4,59,26,77),(5,64,27,82),(6,69,28,87),(7,50,29,92),(8,55,30,73),(9,60,31,78),(10,65,32,83),(11,70,33,88),(12,51,34,93),(13,56,35,74),(14,61,36,79),(15,66,37,84),(16,71,38,89),(17,52,39,94),(18,57,40,75),(19,62,41,80),(20,67,42,85),(21,72,43,90),(22,53,44,95),(23,58,45,76),(24,63,46,81)], [(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(46,48),(49,81),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,96),(59,95),(60,94),(61,93),(62,92),(63,91),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,84),(71,83),(72,82)]])
Matrix representation of C24⋊9D4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 39 | 34 | 39 |
| 0 | 0 | 34 | 68 | 34 | 68 |
| 0 | 0 | 39 | 34 | 34 | 39 |
| 0 | 0 | 39 | 5 | 34 | 68 |
| 17 | 70 | 0 | 0 | 0 | 0 |
| 48 | 56 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 65 | 7 | 0 |
| 0 | 0 | 69 | 4 | 66 | 66 |
| 0 | 0 | 66 | 0 | 69 | 65 |
| 0 | 0 | 7 | 7 | 69 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 60 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,34,39,39,0,0,39,68,34,5,0,0,34,34,34,34,0,0,39,68,39,68],[17,48,0,0,0,0,70,56,0,0,0,0,0,0,69,69,66,7,0,0,65,4,0,7,0,0,7,66,69,69,0,0,0,66,65,4],[1,60,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1] >;
C24⋊9D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_9D_4 % in TeX
G:=Group("C24:9D4"); // GroupNames label
G:=SmallGroup(192,735);
// by ID
G=gap.SmallGroup(192,735);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,219,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export