metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊12D4, (C2×D8)⋊9S3, (C6×D8)⋊10C2, C8⋊4(C3⋊D4), C3⋊5(C8⋊2D4), (C2×C8).86D6, D6⋊3D4⋊6C2, C8⋊Dic3⋊21C2, (C2×D4).66D6, C12.168(C2×D4), C12.94(C4○D4), D4⋊Dic3⋊31C2, C2.31(D8⋊S3), C6.52(C8⋊C22), (C2×Dic3).66D4, (C6×D4).85C22, (C22×S3).36D4, C22.259(S3×D4), C4.29(D4⋊2S3), C2.17(D6⋊3D4), C6.110(C4⋊D4), (C2×C12).436C23, (C2×C24).148C22, C4⋊Dic3.167C22, (C2×C8⋊S3)⋊7C2, C4.80(C2×C3⋊D4), (C2×C6).349(C2×D4), (S3×C2×C4).46C22, (C2×C3⋊C8).150C22, (C2×C4).526(C22×S3), SmallGroup(192,718)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊12D4
G = < a,b,c | a24=b4=c2=1, bab-1=a11, cac=a5, cbc=b-1 >
Subgroups: 424 in 130 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C8⋊S3, C2×C3⋊C8, C4⋊Dic3, C6.D4, C2×C24, C3×D8, S3×C2×C4, C2×C3⋊D4, C6×D4, C8⋊2D4, C8⋊Dic3, D4⋊Dic3, C2×C8⋊S3, D6⋊3D4, C6×D8, C24⋊12D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C8⋊C22, S3×D4, D4⋊2S3, C2×C3⋊D4, C8⋊2D4, D8⋊S3, D6⋊3D4, C24⋊12D4
Character table of C24⋊12D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 12 | 2 | 2 | 2 | 12 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 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 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | 2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 2 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | √-3 | √-3 | -√-3 | -2 | 2 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | √-3 | -√-3 | √-3 | 2 | -2 | 0 | 0 | 1 | -1 | -1 | 1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | -√-3 | √-3 | -√-3 | 2 | -2 | 0 | 0 | 1 | -1 | -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 | -√-3 | √-3 | -2 | 2 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○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 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 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 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| 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 | complex lifted from D8⋊S3 |
| 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 | complex lifted from D8⋊S3 |
| 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 | complex lifted from D8⋊S3 |
| 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 | complex lifted from D8⋊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 84 68)(2 30 85 55)(3 41 86 66)(4 28 87 53)(5 39 88 64)(6 26 89 51)(7 37 90 62)(8 48 91 49)(9 35 92 60)(10 46 93 71)(11 33 94 58)(12 44 95 69)(13 31 96 56)(14 42 73 67)(15 29 74 54)(16 40 75 65)(17 27 76 52)(18 38 77 63)(19 25 78 50)(20 36 79 61)(21 47 80 72)(22 34 81 59)(23 45 82 70)(24 32 83 57)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 50)(26 55)(27 60)(28 65)(29 70)(30 51)(31 56)(32 61)(33 66)(34 71)(35 52)(36 57)(37 62)(38 67)(39 72)(40 53)(41 58)(42 63)(43 68)(44 49)(45 54)(46 59)(47 64)(48 69)(73 77)(74 82)(75 87)(76 92)(79 83)(80 88)(81 93)(85 89)(86 94)(91 95)
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), (1,43,84,68)(2,30,85,55)(3,41,86,66)(4,28,87,53)(5,39,88,64)(6,26,89,51)(7,37,90,62)(8,48,91,49)(9,35,92,60)(10,46,93,71)(11,33,94,58)(12,44,95,69)(13,31,96,56)(14,42,73,67)(15,29,74,54)(16,40,75,65)(17,27,76,52)(18,38,77,63)(19,25,78,50)(20,36,79,61)(21,47,80,72)(22,34,81,59)(23,45,82,70)(24,32,83,57), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,50)(26,55)(27,60)(28,65)(29,70)(30,51)(31,56)(32,61)(33,66)(34,71)(35,52)(36,57)(37,62)(38,67)(39,72)(40,53)(41,58)(42,63)(43,68)(44,49)(45,54)(46,59)(47,64)(48,69)(73,77)(74,82)(75,87)(76,92)(79,83)(80,88)(81,93)(85,89)(86,94)(91,95)>;
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), (1,43,84,68)(2,30,85,55)(3,41,86,66)(4,28,87,53)(5,39,88,64)(6,26,89,51)(7,37,90,62)(8,48,91,49)(9,35,92,60)(10,46,93,71)(11,33,94,58)(12,44,95,69)(13,31,96,56)(14,42,73,67)(15,29,74,54)(16,40,75,65)(17,27,76,52)(18,38,77,63)(19,25,78,50)(20,36,79,61)(21,47,80,72)(22,34,81,59)(23,45,82,70)(24,32,83,57), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,50)(26,55)(27,60)(28,65)(29,70)(30,51)(31,56)(32,61)(33,66)(34,71)(35,52)(36,57)(37,62)(38,67)(39,72)(40,53)(41,58)(42,63)(43,68)(44,49)(45,54)(46,59)(47,64)(48,69)(73,77)(74,82)(75,87)(76,92)(79,83)(80,88)(81,93)(85,89)(86,94)(91,95) );
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)], [(1,43,84,68),(2,30,85,55),(3,41,86,66),(4,28,87,53),(5,39,88,64),(6,26,89,51),(7,37,90,62),(8,48,91,49),(9,35,92,60),(10,46,93,71),(11,33,94,58),(12,44,95,69),(13,31,96,56),(14,42,73,67),(15,29,74,54),(16,40,75,65),(17,27,76,52),(18,38,77,63),(19,25,78,50),(20,36,79,61),(21,47,80,72),(22,34,81,59),(23,45,82,70),(24,32,83,57)], [(2,6),(3,11),(4,16),(5,21),(8,12),(9,17),(10,22),(14,18),(15,23),(20,24),(25,50),(26,55),(27,60),(28,65),(29,70),(30,51),(31,56),(32,61),(33,66),(34,71),(35,52),(36,57),(37,62),(38,67),(39,72),(40,53),(41,58),(42,63),(43,68),(44,49),(45,54),(46,59),(47,64),(48,69),(73,77),(74,82),(75,87),(76,92),(79,83),(80,88),(81,93),(85,89),(86,94),(91,95)]])
Matrix representation of C24⋊12D4 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 11 |
| 0 | 0 | 0 | 0 | 62 | 62 |
| 0 | 0 | 11 | 31 | 51 | 11 |
| 0 | 0 | 42 | 42 | 62 | 62 |
| 59 | 3 | 0 | 0 | 0 | 0 |
| 56 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 40 | 24 | 61 |
| 0 | 0 | 33 | 7 | 12 | 49 |
| 0 | 0 | 5 | 34 | 7 | 33 |
| 0 | 0 | 39 | 68 | 40 | 66 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 15 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,11,42,0,0,0,0,31,42,0,0,51,62,51,62,0,0,11,62,11,62],[59,56,0,0,0,0,3,14,0,0,0,0,0,0,66,33,5,39,0,0,40,7,34,68,0,0,24,12,7,40,0,0,61,49,33,66],[72,15,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C24⋊12D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_{12}D_4 % in TeX
G:=Group("C24:12D4"); // GroupNames label
G:=SmallGroup(192,718);
// by ID
G=gap.SmallGroup(192,718);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,1094,135,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^11,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
Export