metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊7D4, C8⋊2D12, C4.Q8⋊3S3, C3⋊2(C8⋊2D4), C4⋊C4.39D6, (C2×C8).61D6, (C2×D24)⋊24C2, C12⋊D4⋊6C2, C4.51(C2×D12), C12.131(C2×D4), C6.D8⋊16C2, C12.30(C4○D4), C2.22(Q8⋊3D6), C6.44(C4⋊D4), C6.70(C8⋊C22), C4.4(Q8⋊3S3), (C2×Dic3).42D4, (C22×S3).24D4, C22.217(S3×D4), C2.17(C12⋊D4), (C2×C12).281C23, (C2×C24).110C22, (C2×D12).75C22, (C3×C4.Q8)⋊3C2, (C2×C8⋊S3)⋊2C2, (C2×C6).286(C2×D4), (C2×C3⋊C8).58C22, (S3×C2×C4).33C22, (C3×C4⋊C4).74C22, (C2×C4).384(C22×S3), SmallGroup(192,424)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊7D4
G = < a,b,c | a24=b4=c2=1, bab-1=a19, cac=a-1, cbc=b-1 >
Subgroups: 512 in 130 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C8⋊S3, D24, C2×C3⋊C8, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C2×D12, C8⋊2D4, C6.D8, C3×C4.Q8, C12⋊D4, C2×C8⋊S3, C2×D24, C24⋊7D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C8⋊C22, C2×D12, S3×D4, Q8⋊3S3, C8⋊2D4, C12⋊D4, Q8⋊3D6, C24⋊7D4
Character table of C24⋊7D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 12 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 12 | 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 | 0 | -1 | 2 | 2 | -2 | 2 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -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 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 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 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | -2 | 2 | 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 | 0 | -1 | -2 | 2 | 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 | 0 | -1 | -2 | 2 | 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 | 0 | -1 | -2 | 2 | 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 | 0 | 2 | 2 | -2 | 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 | 0 | 2 | 2 | -2 | 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 | 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 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ25 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | -4 | -4 | 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 |
| 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 74 40 70)(2 93 41 65)(3 88 42 60)(4 83 43 55)(5 78 44 50)(6 73 45 69)(7 92 46 64)(8 87 47 59)(9 82 48 54)(10 77 25 49)(11 96 26 68)(12 91 27 63)(13 86 28 58)(14 81 29 53)(15 76 30 72)(16 95 31 67)(17 90 32 62)(18 85 33 57)(19 80 34 52)(20 75 35 71)(21 94 36 66)(22 89 37 61)(23 84 38 56)(24 79 39 51)
(1 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(49 67)(50 66)(51 65)(52 64)(53 63)(54 62)(55 61)(56 60)(57 59)(68 72)(69 71)(73 75)(76 96)(77 95)(78 94)(79 93)(80 92)(81 91)(82 90)(83 89)(84 88)(85 87)
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), (1,74,40,70)(2,93,41,65)(3,88,42,60)(4,83,43,55)(5,78,44,50)(6,73,45,69)(7,92,46,64)(8,87,47,59)(9,82,48,54)(10,77,25,49)(11,96,26,68)(12,91,27,63)(13,86,28,58)(14,81,29,53)(15,76,30,72)(16,95,31,67)(17,90,32,62)(18,85,33,57)(19,80,34,52)(20,75,35,71)(21,94,36,66)(22,89,37,61)(23,84,38,56)(24,79,39,51), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(68,72)(69,71)(73,75)(76,96)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)>;
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), (1,74,40,70)(2,93,41,65)(3,88,42,60)(4,83,43,55)(5,78,44,50)(6,73,45,69)(7,92,46,64)(8,87,47,59)(9,82,48,54)(10,77,25,49)(11,96,26,68)(12,91,27,63)(13,86,28,58)(14,81,29,53)(15,76,30,72)(16,95,31,67)(17,90,32,62)(18,85,33,57)(19,80,34,52)(20,75,35,71)(21,94,36,66)(22,89,37,61)(23,84,38,56)(24,79,39,51), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(68,72)(69,71)(73,75)(76,96)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87) );
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)], [(1,74,40,70),(2,93,41,65),(3,88,42,60),(4,83,43,55),(5,78,44,50),(6,73,45,69),(7,92,46,64),(8,87,47,59),(9,82,48,54),(10,77,25,49),(11,96,26,68),(12,91,27,63),(13,86,28,58),(14,81,29,53),(15,76,30,72),(16,95,31,67),(17,90,32,62),(18,85,33,57),(19,80,34,52),(20,75,35,71),(21,94,36,66),(22,89,37,61),(23,84,38,56),(24,79,39,51)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,48),(18,47),(19,46),(20,45),(21,44),(22,43),(23,42),(24,41),(49,67),(50,66),(51,65),(52,64),(53,63),(54,62),(55,61),(56,60),(57,59),(68,72),(69,71),(73,75),(76,96),(77,95),(78,94),(79,93),(80,92),(81,91),(82,90),(83,89),(84,88),(85,87)]])
Matrix representation of C24⋊7D4 ►in GL8(𝔽73)
| 55 | 18 | 32 | 32 | 0 | 0 | 0 | 0 |
| 55 | 37 | 41 | 0 | 0 | 0 | 0 | 0 |
| 60 | 60 | 18 | 55 | 0 | 0 | 0 | 0 |
| 13 | 0 | 18 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 11 | 20 |
| 0 | 0 | 0 | 0 | 60 | 1 | 0 | 20 |
| 0 | 0 | 0 | 0 | 32 | 32 | 59 | 21 |
| 0 | 0 | 0 | 0 | 56 | 59 | 40 | 12 |
| 7 | 14 | 71 | 0 | 0 | 0 | 0 | 0 |
| 59 | 66 | 0 | 71 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 21 | 21 | 72 | 3 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 48 | 48 | 0 | 52 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 59 | 0 | 25 | 72 |
G:=sub<GL(8,GF(73))| [55,55,60,13,0,0,0,0,18,37,60,0,0,0,0,0,32,41,18,18,0,0,0,0,32,0,55,36,0,0,0,0,0,0,0,0,1,60,32,56,0,0,0,0,1,1,32,59,0,0,0,0,11,0,59,40,0,0,0,0,20,20,21,12],[7,59,0,0,0,0,0,0,14,66,0,0,0,0,0,0,71,0,66,14,0,0,0,0,0,71,59,7,0,0,0,0,0,0,0,0,0,21,1,48,0,0,0,0,0,21,0,48,0,0,0,0,1,72,0,0,0,0,0,0,0,3,0,52],[72,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,59,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,25,0,0,0,0,0,0,0,72] >;
C24⋊7D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_7D_4 % in TeX
G:=Group("C24:7D4"); // GroupNames label
G:=SmallGroup(192,424);
// by ID
G=gap.SmallGroup(192,424);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,254,555,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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