metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊11D4, C3⋊C8⋊9D4, (C2×D8)⋊8S3, (C6×D8)⋊9C2, C8⋊3(C3⋊D4), C3⋊4(C8⋊3D4), C24⋊C4⋊9C2, C4.21(S3×D4), (C2×C8).85D6, C12⋊3D4⋊5C2, (C2×D4).62D6, C12.164(C2×D4), C23.12D6⋊4C2, C6.27(C4⋊1D4), C2.28(D8⋊S3), C6.49(C8⋊C22), (C2×Dic3).64D4, (C6×D4).81C22, C22.254(S3×D4), C2.18(C12⋊3D4), (C2×C12).431C23, (C2×C24).147C22, (C2×D12).115C22, (C4×Dic3).45C22, (C2×Dic6).120C22, C4.5(C2×C3⋊D4), (C2×D4⋊S3)⋊18C2, (C2×C24⋊C2)⋊23C2, (C2×D4.S3)⋊17C2, (C2×C6).344(C2×D4), (C2×C3⋊C8).148C22, (C2×C4).521(C22×S3), SmallGroup(192,713)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊11D4
G = < a,b,c | a24=b4=c2=1, bab-1=a5, cac=a11, cbc=b-1 >
Subgroups: 504 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, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×D8, C2×SD16, C24⋊C2, C2×C3⋊C8, C4×Dic3, D4⋊S3, D4.S3, C6.D4, C2×C24, C3×D8, C2×Dic6, C2×D12, C2×C3⋊D4, C6×D4, C8⋊3D4, C24⋊C4, C2×C24⋊C2, C2×D4⋊S3, C2×D4.S3, C23.12D6, C12⋊3D4, C6×D8, C24⋊11D4
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, D8⋊S3, C12⋊3D4, C24⋊11D4
Character table of C24⋊11D4
| 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 | 24 | 2 | 2 | 2 | 12 | 12 | 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 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 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 | 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 |
| ρ13 | 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 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 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 D6 |
| ρ17 | 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 |
| ρ18 | 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 |
| 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 | -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 |
| ρ22 | 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 |
| 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 | orthogonal lifted from S3×D4 |
| 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 70 88 37)(2 51 89 42)(3 56 90 47)(4 61 91 28)(5 66 92 33)(6 71 93 38)(7 52 94 43)(8 57 95 48)(9 62 96 29)(10 67 73 34)(11 72 74 39)(12 53 75 44)(13 58 76 25)(14 63 77 30)(15 68 78 35)(16 49 79 40)(17 54 80 45)(18 59 81 26)(19 64 82 31)(20 69 83 36)(21 50 84 41)(22 55 85 46)(23 60 86 27)(24 65 87 32)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 58)(26 69)(27 56)(28 67)(29 54)(30 65)(31 52)(32 63)(33 50)(34 61)(35 72)(36 59)(37 70)(38 57)(39 68)(40 55)(41 66)(42 53)(43 64)(44 51)(45 62)(46 49)(47 60)(48 71)(73 91)(74 78)(75 89)(77 87)(79 85)(80 96)(81 83)(82 94)(84 92)(86 90)(93 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), (1,70,88,37)(2,51,89,42)(3,56,90,47)(4,61,91,28)(5,66,92,33)(6,71,93,38)(7,52,94,43)(8,57,95,48)(9,62,96,29)(10,67,73,34)(11,72,74,39)(12,53,75,44)(13,58,76,25)(14,63,77,30)(15,68,78,35)(16,49,79,40)(17,54,80,45)(18,59,81,26)(19,64,82,31)(20,69,83,36)(21,50,84,41)(22,55,85,46)(23,60,86,27)(24,65,87,32), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,58)(26,69)(27,56)(28,67)(29,54)(30,65)(31,52)(32,63)(33,50)(34,61)(35,72)(36,59)(37,70)(38,57)(39,68)(40,55)(41,66)(42,53)(43,64)(44,51)(45,62)(46,49)(47,60)(48,71)(73,91)(74,78)(75,89)(77,87)(79,85)(80,96)(81,83)(82,94)(84,92)(86,90)(93,95)>;
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), (1,70,88,37)(2,51,89,42)(3,56,90,47)(4,61,91,28)(5,66,92,33)(6,71,93,38)(7,52,94,43)(8,57,95,48)(9,62,96,29)(10,67,73,34)(11,72,74,39)(12,53,75,44)(13,58,76,25)(14,63,77,30)(15,68,78,35)(16,49,79,40)(17,54,80,45)(18,59,81,26)(19,64,82,31)(20,69,83,36)(21,50,84,41)(22,55,85,46)(23,60,86,27)(24,65,87,32), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,58)(26,69)(27,56)(28,67)(29,54)(30,65)(31,52)(32,63)(33,50)(34,61)(35,72)(36,59)(37,70)(38,57)(39,68)(40,55)(41,66)(42,53)(43,64)(44,51)(45,62)(46,49)(47,60)(48,71)(73,91)(74,78)(75,89)(77,87)(79,85)(80,96)(81,83)(82,94)(84,92)(86,90)(93,95) );
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)], [(1,70,88,37),(2,51,89,42),(3,56,90,47),(4,61,91,28),(5,66,92,33),(6,71,93,38),(7,52,94,43),(8,57,95,48),(9,62,96,29),(10,67,73,34),(11,72,74,39),(12,53,75,44),(13,58,76,25),(14,63,77,30),(15,68,78,35),(16,49,79,40),(17,54,80,45),(18,59,81,26),(19,64,82,31),(20,69,83,36),(21,50,84,41),(22,55,85,46),(23,60,86,27),(24,65,87,32)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,58),(26,69),(27,56),(28,67),(29,54),(30,65),(31,52),(32,63),(33,50),(34,61),(35,72),(36,59),(37,70),(38,57),(39,68),(40,55),(41,66),(42,53),(43,64),(44,51),(45,62),(46,49),(47,60),(48,71),(73,91),(74,78),(75,89),(77,87),(79,85),(80,96),(81,83),(82,94),(84,92),(86,90),(93,95)]])
Matrix representation of C24⋊11D4 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 31 | 31 | 42 |
| 0 | 0 | 42 | 11 | 31 | 62 |
| 0 | 0 | 42 | 31 | 42 | 31 |
| 0 | 0 | 42 | 11 | 42 | 11 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 34 | 24 | 0 |
| 0 | 0 | 17 | 56 | 49 | 49 |
| 0 | 0 | 49 | 0 | 17 | 34 |
| 0 | 0 | 24 | 24 | 17 | 56 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,42,42,42,42,0,0,31,11,31,11,0,0,31,31,42,42,0,0,42,62,31,11],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,17,17,49,24,0,0,34,56,0,24,0,0,24,49,17,17,0,0,0,49,34,56],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1] >;
C24⋊11D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_{11}D_4 % in TeX
G:=Group("C24:11D4"); // GroupNames label
G:=SmallGroup(192,713);
// by ID
G=gap.SmallGroup(192,713);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,1094,135,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^11,c*b*c=b^-1>;
// generators/relations
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