metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊8D4, C3⋊7(C8⋊D4), C8⋊5(C3⋊D4), (C2×C8).90D6, D6⋊3Q8⋊6C2, (C6×SD16)⋊3C2, (C2×SD16)⋊1S3, C24⋊1C4⋊27C2, (C2×D4).75D6, (C2×Q8).80D6, D6⋊3D4.9C2, C12.178(C2×D4), Q8⋊2Dic3⋊31C2, D4⋊Dic3⋊36C2, C2.30(Q8⋊3D6), C6.80(C8⋊C22), (C2×Dic3).73D4, (C22×S3).39D4, C22.271(S3×D4), (C6×Q8).80C22, C12.102(C4○D4), C4.33(D4⋊2S3), C2.20(D6⋊3D4), C6.117(C4⋊D4), (C2×C12).451C23, (C2×C24).115C22, C2.31(D4.D6), (C6×D4).100C22, C6.51(C8.C22), C4⋊Dic3.178C22, (C2×C8⋊S3)⋊3C2, C4.83(C2×C3⋊D4), (C2×C6).363(C2×D4), (S3×C2×C4).51C22, (C2×C3⋊C8).160C22, (C2×C4).540(C22×S3), SmallGroup(192,733)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊8D4
G = < a,b,c | a24=b4=c2=1, bab-1=a-1, cac=a5, cbc=b-1 >
Subgroups: 360 in 120 conjugacy classes, 41 normal (37 characteristic)
C1, 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, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C8⋊S3, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C2×C24, C3×SD16, S3×C2×C4, C2×C3⋊D4, C6×D4, C6×Q8, C8⋊D4, C24⋊1C4, D4⋊Dic3, Q8⋊2Dic3, C2×C8⋊S3, D6⋊3D4, D6⋊3Q8, C6×SD16, C24⋊8D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C8⋊C22, C8.C22, S3×D4, D4⋊2S3, C2×C3⋊D4, C8⋊D4, Q8⋊3D6, D4.D6, D6⋊3D4, C24⋊8D4
Character table of C24⋊8D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 8 | 12 | 2 | 2 | 2 | 8 | 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 | 0 | -1 | 2 | 2 | -2 | 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 | 0 | -1 | 2 | 2 | -2 | 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 | -2 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | -2 | -2 | 0 | -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 | -2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | -2 | 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 | 0 | -1 | 2 | 2 | 2 | 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 | 0 | 0 | 2 | 2 | -2 | 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 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 2 | -2 | 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 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 2 | -2 | 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 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 2 | -2 | 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 | 0 | -1 | 2 | -2 | 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 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 2 | -2 | 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 | 0 | 0 | -2i | 2i | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 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 |
| ρ24 | 4 | 4 | 4 | 4 | 0 | 0 | -2 | -4 | -4 | 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 | 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 | orthogonal lifted from Q8⋊3D6 |
| ρ26 | 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 | orthogonal lifted from Q8⋊3D6 |
| ρ27 | 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 |
| 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 | symplectic lifted from D4.D6, Schur index 2 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| 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 | 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 66 76 28)(2 65 77 27)(3 64 78 26)(4 63 79 25)(5 62 80 48)(6 61 81 47)(7 60 82 46)(8 59 83 45)(9 58 84 44)(10 57 85 43)(11 56 86 42)(12 55 87 41)(13 54 88 40)(14 53 89 39)(15 52 90 38)(16 51 91 37)(17 50 92 36)(18 49 93 35)(19 72 94 34)(20 71 95 33)(21 70 96 32)(22 69 73 31)(23 68 74 30)(24 67 75 29)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 51)(26 56)(27 61)(28 66)(29 71)(30 52)(31 57)(32 62)(33 67)(34 72)(35 53)(36 58)(37 63)(38 68)(39 49)(40 54)(41 59)(42 64)(43 69)(44 50)(45 55)(46 60)(47 65)(48 70)(73 85)(74 90)(75 95)(77 81)(78 86)(79 91)(80 96)(83 87)(84 92)(89 93)
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), (1,66,76,28)(2,65,77,27)(3,64,78,26)(4,63,79,25)(5,62,80,48)(6,61,81,47)(7,60,82,46)(8,59,83,45)(9,58,84,44)(10,57,85,43)(11,56,86,42)(12,55,87,41)(13,54,88,40)(14,53,89,39)(15,52,90,38)(16,51,91,37)(17,50,92,36)(18,49,93,35)(19,72,94,34)(20,71,95,33)(21,70,96,32)(22,69,73,31)(23,68,74,30)(24,67,75,29), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,51)(26,56)(27,61)(28,66)(29,71)(30,52)(31,57)(32,62)(33,67)(34,72)(35,53)(36,58)(37,63)(38,68)(39,49)(40,54)(41,59)(42,64)(43,69)(44,50)(45,55)(46,60)(47,65)(48,70)(73,85)(74,90)(75,95)(77,81)(78,86)(79,91)(80,96)(83,87)(84,92)(89,93)>;
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), (1,66,76,28)(2,65,77,27)(3,64,78,26)(4,63,79,25)(5,62,80,48)(6,61,81,47)(7,60,82,46)(8,59,83,45)(9,58,84,44)(10,57,85,43)(11,56,86,42)(12,55,87,41)(13,54,88,40)(14,53,89,39)(15,52,90,38)(16,51,91,37)(17,50,92,36)(18,49,93,35)(19,72,94,34)(20,71,95,33)(21,70,96,32)(22,69,73,31)(23,68,74,30)(24,67,75,29), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,51)(26,56)(27,61)(28,66)(29,71)(30,52)(31,57)(32,62)(33,67)(34,72)(35,53)(36,58)(37,63)(38,68)(39,49)(40,54)(41,59)(42,64)(43,69)(44,50)(45,55)(46,60)(47,65)(48,70)(73,85)(74,90)(75,95)(77,81)(78,86)(79,91)(80,96)(83,87)(84,92)(89,93) );
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)], [(1,66,76,28),(2,65,77,27),(3,64,78,26),(4,63,79,25),(5,62,80,48),(6,61,81,47),(7,60,82,46),(8,59,83,45),(9,58,84,44),(10,57,85,43),(11,56,86,42),(12,55,87,41),(13,54,88,40),(14,53,89,39),(15,52,90,38),(16,51,91,37),(17,50,92,36),(18,49,93,35),(19,72,94,34),(20,71,95,33),(21,70,96,32),(22,69,73,31),(23,68,74,30),(24,67,75,29)], [(2,6),(3,11),(4,16),(5,21),(8,12),(9,17),(10,22),(14,18),(15,23),(20,24),(25,51),(26,56),(27,61),(28,66),(29,71),(30,52),(31,57),(32,62),(33,67),(34,72),(35,53),(36,58),(37,63),(38,68),(39,49),(40,54),(41,59),(42,64),(43,69),(44,50),(45,55),(46,60),(47,65),(48,70),(73,85),(74,90),(75,95),(77,81),(78,86),(79,91),(80,96),(83,87),(84,92),(89,93)]])
Matrix representation of C24⋊8D4 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 39 | 39 | 34 |
| 0 | 0 | 34 | 68 | 39 | 5 |
| 0 | 0 | 34 | 39 | 34 | 39 |
| 0 | 0 | 34 | 68 | 34 | 68 |
| 46 | 71 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 30 | 50 | 27 |
| 0 | 0 | 15 | 58 | 50 | 23 |
| 0 | 0 | 50 | 27 | 58 | 43 |
| 0 | 0 | 50 | 23 | 58 | 15 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 46 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,34,34,34,34,0,0,39,68,39,68,0,0,39,39,34,34,0,0,34,5,39,68],[46,0,0,0,0,0,71,27,0,0,0,0,0,0,15,15,50,50,0,0,30,58,27,23,0,0,50,50,58,58,0,0,27,23,43,15],[1,46,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;
C24⋊8D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_8D_4 % in TeX
G:=Group("C24:8D4"); // GroupNames label
G:=SmallGroup(192,733);
// by ID
G=gap.SmallGroup(192,733);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,254,555,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
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