metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D12, C24⋊10D4, C3⋊3(C8⋊D4), C4⋊C4.51D6, (C2×C8).66D6, C2.D8⋊11S3, C4.D12⋊7C2, C4.54(C2×D12), C12⋊D4.9C2, C6.D8⋊23C2, C12.134(C2×D4), C12.41(C4○D4), C6.SD16⋊20C2, C2.24(D8⋊S3), C6.47(C4⋊D4), C6.43(C8⋊C22), (C2×Dic3).50D4, (C22×S3).29D4, C22.232(S3×D4), C2.20(C12⋊D4), (C2×C12).302C23, (C2×C24).144C22, C4.10(Q8⋊3S3), C2.23(Q16⋊S3), (C2×D12).84C22, C6.71(C8.C22), (C2×Dic6).90C22, (C3×C2.D8)⋊8C2, (C2×C8⋊S3)⋊6C2, (C2×C24⋊C2)⋊22C2, (C2×C6).307(C2×D4), (C2×C3⋊C8).72C22, (S3×C2×C4).41C22, (C3×C4⋊C4).95C22, (C2×C4).405(C22×S3), SmallGroup(192,445)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊3D12
G = < a,b,c | a8=b12=c2=1, bab-1=a-1, cac=a3, cbc=b-1 >
Subgroups: 416 in 120 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C8⋊S3, C24⋊C2, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C2×D12, C8⋊D4, C6.D8, C6.SD16, C3×C2.D8, C12⋊D4, C4.D12, C2×C8⋊S3, C2×C24⋊C2, C8⋊3D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C8⋊C22, C8.C22, C2×D12, S3×D4, Q8⋊3S3, C8⋊D4, C12⋊D4, D8⋊S3, Q16⋊S3, C8⋊3D12
Character table of C8⋊3D12
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 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 | 0 | 2 | -2 | 2 | 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 | 0 | -1 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 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 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | 2 | 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 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | -√3 | √3 | -√3 | √3 | 1 | -1 | -1 | 1 | orthogonal lifted from D12 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | √3 | -√3 | √3 | -√3 | 1 | -1 | -1 | 1 | orthogonal lifted from D12 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | √3 | -√3 | -√3 | √3 | -1 | 1 | 1 | -1 | orthogonal lifted from D12 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | -√3 | √3 | √3 | -√3 | -1 | 1 | 1 | -1 | orthogonal lifted from D12 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | 4 | 4 | 4 | 0 | 0 | -2 | -4 | -4 | 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 | 0 | 4 | 0 | 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 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 4 | 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 | 0 | -2 | 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 | 0 | -2 | 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 D8⋊S3 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 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 | 0 | -2 | 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 D8⋊S3 |
►On 96 points
Generators in S96
(1 56 39 86 69 83 14 27)(2 28 15 84 70 87 40 57)(3 58 41 88 71 73 16 29)(4 30 17 74 72 89 42 59)(5 60 43 90 61 75 18 31)(6 32 19 76 62 91 44 49)(7 50 45 92 63 77 20 33)(8 34 21 78 64 93 46 51)(9 52 47 94 65 79 22 35)(10 36 23 80 66 95 48 53)(11 54 37 96 67 81 24 25)(12 26 13 82 68 85 38 55)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 75)(26 74)(27 73)(28 84)(29 83)(30 82)(31 81)(32 80)(33 79)(34 78)(35 77)(36 76)(49 95)(50 94)(51 93)(52 92)(53 91)(54 90)(55 89)(56 88)(57 87)(58 86)(59 85)(60 96)(61 67)(62 66)(63 65)(68 72)(69 71)
G:=sub<Sym(96)| (1,56,39,86,69,83,14,27)(2,28,15,84,70,87,40,57)(3,58,41,88,71,73,16,29)(4,30,17,74,72,89,42,59)(5,60,43,90,61,75,18,31)(6,32,19,76,62,91,44,49)(7,50,45,92,63,77,20,33)(8,34,21,78,64,93,46,51)(9,52,47,94,65,79,22,35)(10,36,23,80,66,95,48,53)(11,54,37,96,67,81,24,25)(12,26,13,82,68,85,38,55), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,75)(26,74)(27,73)(28,84)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,88)(57,87)(58,86)(59,85)(60,96)(61,67)(62,66)(63,65)(68,72)(69,71)>;
G:=Group( (1,56,39,86,69,83,14,27)(2,28,15,84,70,87,40,57)(3,58,41,88,71,73,16,29)(4,30,17,74,72,89,42,59)(5,60,43,90,61,75,18,31)(6,32,19,76,62,91,44,49)(7,50,45,92,63,77,20,33)(8,34,21,78,64,93,46,51)(9,52,47,94,65,79,22,35)(10,36,23,80,66,95,48,53)(11,54,37,96,67,81,24,25)(12,26,13,82,68,85,38,55), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,75)(26,74)(27,73)(28,84)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,88)(57,87)(58,86)(59,85)(60,96)(61,67)(62,66)(63,65)(68,72)(69,71) );
G=PermutationGroup([[(1,56,39,86,69,83,14,27),(2,28,15,84,70,87,40,57),(3,58,41,88,71,73,16,29),(4,30,17,74,72,89,42,59),(5,60,43,90,61,75,18,31),(6,32,19,76,62,91,44,49),(7,50,45,92,63,77,20,33),(8,34,21,78,64,93,46,51),(9,52,47,94,65,79,22,35),(10,36,23,80,66,95,48,53),(11,54,37,96,67,81,24,25),(12,26,13,82,68,85,38,55)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,42),(14,41),(15,40),(16,39),(17,38),(18,37),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,75),(26,74),(27,73),(28,84),(29,83),(30,82),(31,81),(32,80),(33,79),(34,78),(35,77),(36,76),(49,95),(50,94),(51,93),(52,92),(53,91),(54,90),(55,89),(56,88),(57,87),(58,86),(59,85),(60,96),(61,67),(62,66),(63,65),(68,72),(69,71)]])
Matrix representation of C8⋊3D12 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 22 | 20 | 20 |
| 0 | 0 | 31 | 11 | 0 | 20 |
| 0 | 0 | 62 | 11 | 0 | 0 |
| 0 | 0 | 42 | 11 | 20 | 20 |
| 52 | 3 | 0 | 0 | 0 | 0 |
| 23 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 63 | 66 | 0 | 30 |
| 0 | 0 | 7 | 13 | 43 | 0 |
| 0 | 0 | 64 | 4 | 60 | 7 |
| 0 | 0 | 8 | 64 | 66 | 10 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 59 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 24 | 23 | 1 | 1 |
| 0 | 0 | 25 | 24 | 0 | 72 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,42,31,62,42,0,0,22,11,11,11,0,0,20,0,0,20,0,0,20,20,0,20],[52,23,0,0,0,0,3,21,0,0,0,0,0,0,63,7,64,8,0,0,66,13,4,64,0,0,0,43,60,66,0,0,30,0,7,10],[72,59,0,0,0,0,0,1,0,0,0,0,0,0,1,0,24,25,0,0,72,72,23,24,0,0,0,0,1,0,0,0,0,0,1,72] >;
C8⋊3D12 in GAP, Magma, Sage, TeX
C_8\rtimes_3D_{12} % in TeX
G:=Group("C8:3D12"); // GroupNames label
G:=SmallGroup(192,445);
// by ID
G=gap.SmallGroup(192,445);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,254,219,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=b^12=c^2=1,b*a*b^-1=a^-1,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations
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