metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.7D4, Dic6.7D4, M4(2).6D6, C3⋊C8.27D4, D12.C4.C2, C4.153(S3×D4), (C2×Q8).32D6, C12.100(C2×D4), C4.10D4⋊4S3, C3⋊1(D4.5D4), C8.D6.1C2, C12.53D4⋊4C2, C6.12(C4⋊D4), (C2×C12).12C23, C4○D12.8C22, (C6×Q8).10C22, C12.47D4⋊12C2, C2.15(Dic3⋊D4), Q8.11D6.1C2, C4.Dic3.7C22, C22.16(C4○D12), (C2×Dic6).48C22, (C3×M4(2)).23C22, (C2×C3⋊Q16)⋊1C2, (C2×C3⋊C8).4C22, (C2×C6).33(C4○D4), (C3×C4.10D4)⋊2C2, (C2×C4).12(C22×S3), SmallGroup(192,314)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.7D4
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=a6b, dbd-1=a3b, dcd-1=a9c3 >
Subgroups: 272 in 100 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6, C6, C8 [×5], C2×C4, C2×C4 [×3], D4 [×2], Q8 [×5], Dic3 [×2], C12 [×2], C12, D6, C2×C6, C2×C8 [×2], M4(2) [×2], M4(2) [×2], SD16 [×2], Q16 [×4], C2×Q8, C2×Q8, C4○D4, C3⋊C8 [×2], C3⋊C8, C24 [×2], Dic6, Dic6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8 [×2], C4.10D4, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22 [×2], S3×C8, C8⋊S3, C24⋊C2, Dic12, C2×C3⋊C8, C4.Dic3, Q8⋊2S3, C3⋊Q16 [×3], C3×M4(2) [×2], C2×Dic6, C4○D12, C6×Q8, D4.5D4, C12.53D4, C12.47D4, C3×C4.10D4, D12.C4, C8.D6, Q8.11D6, C2×C3⋊Q16, D12.7D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C22×S3, C4⋊D4, C4○D12, S3×D4 [×2], D4.5D4, Dic3⋊D4, D12.7D4
Character table of D12.7D4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 12 | 2 | 2 | 2 | 8 | 12 | 24 | 2 | 4 | 4 | 4 | 6 | 6 | 8 | 12 | 24 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 2 | 2 | -2 | 2 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ17 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -2i | 0 | 0 | 2i | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2i | 0 | 0 | -2i | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√-3 | √-3 | -i | -√3 | √3 | i | complex lifted from C4○D12 |
| ρ20 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√-3 | √-3 | i | √3 | -√3 | -i | complex lifted from C4○D12 |
| ρ21 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √-3 | -√-3 | -i | √3 | -√3 | i | complex lifted from C4○D12 |
| ρ22 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √-3 | -√-3 | i | -√3 | √3 | -i | complex lifted from C4○D12 |
| ρ23 | 4 | 4 | -4 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | -4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
| ρ27 | 8 | -8 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, 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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 17)(14 16)(18 24)(19 23)(20 22)(25 33)(26 32)(27 31)(28 30)(34 36)(37 39)(40 48)(41 47)(42 46)(43 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 67)(62 66)(63 65)(68 72)(69 71)(73 78)(74 77)(75 76)(79 84)(80 83)(81 82)(85 92)(86 91)(87 90)(88 89)(93 96)(94 95)
(1 49 89 76 7 55 95 82)(2 56 90 83 8 50 96 77)(3 51 91 78 9 57 85 84)(4 58 92 73 10 52 86 79)(5 53 93 80 11 59 87 74)(6 60 94 75 12 54 88 81)(13 36 62 39 19 30 68 45)(14 31 63 46 20 25 69 40)(15 26 64 41 21 32 70 47)(16 33 65 48 22 27 71 42)(17 28 66 43 23 34 72 37)(18 35 67 38 24 29 61 44)
(1 66 10 63 7 72 4 69)(2 67 11 64 8 61 5 70)(3 68 12 65 9 62 6 71)(13 94 22 91 19 88 16 85)(14 95 23 92 20 89 17 86)(15 96 24 93 21 90 18 87)(25 52 34 49 31 58 28 55)(26 53 35 50 32 59 29 56)(27 54 36 51 33 60 30 57)(37 82 46 79 43 76 40 73)(38 83 47 80 44 77 41 74)(39 84 48 81 45 78 42 75)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,17)(14,16)(18,24)(19,23)(20,22)(25,33)(26,32)(27,31)(28,30)(34,36)(37,39)(40,48)(41,47)(42,46)(43,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,67)(62,66)(63,65)(68,72)(69,71)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(85,92)(86,91)(87,90)(88,89)(93,96)(94,95), (1,49,89,76,7,55,95,82)(2,56,90,83,8,50,96,77)(3,51,91,78,9,57,85,84)(4,58,92,73,10,52,86,79)(5,53,93,80,11,59,87,74)(6,60,94,75,12,54,88,81)(13,36,62,39,19,30,68,45)(14,31,63,46,20,25,69,40)(15,26,64,41,21,32,70,47)(16,33,65,48,22,27,71,42)(17,28,66,43,23,34,72,37)(18,35,67,38,24,29,61,44), (1,66,10,63,7,72,4,69)(2,67,11,64,8,61,5,70)(3,68,12,65,9,62,6,71)(13,94,22,91,19,88,16,85)(14,95,23,92,20,89,17,86)(15,96,24,93,21,90,18,87)(25,52,34,49,31,58,28,55)(26,53,35,50,32,59,29,56)(27,54,36,51,33,60,30,57)(37,82,46,79,43,76,40,73)(38,83,47,80,44,77,41,74)(39,84,48,81,45,78,42,75)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,17)(14,16)(18,24)(19,23)(20,22)(25,33)(26,32)(27,31)(28,30)(34,36)(37,39)(40,48)(41,47)(42,46)(43,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,67)(62,66)(63,65)(68,72)(69,71)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(85,92)(86,91)(87,90)(88,89)(93,96)(94,95), (1,49,89,76,7,55,95,82)(2,56,90,83,8,50,96,77)(3,51,91,78,9,57,85,84)(4,58,92,73,10,52,86,79)(5,53,93,80,11,59,87,74)(6,60,94,75,12,54,88,81)(13,36,62,39,19,30,68,45)(14,31,63,46,20,25,69,40)(15,26,64,41,21,32,70,47)(16,33,65,48,22,27,71,42)(17,28,66,43,23,34,72,37)(18,35,67,38,24,29,61,44), (1,66,10,63,7,72,4,69)(2,67,11,64,8,61,5,70)(3,68,12,65,9,62,6,71)(13,94,22,91,19,88,16,85)(14,95,23,92,20,89,17,86)(15,96,24,93,21,90,18,87)(25,52,34,49,31,58,28,55)(26,53,35,50,32,59,29,56)(27,54,36,51,33,60,30,57)(37,82,46,79,43,76,40,73)(38,83,47,80,44,77,41,74)(39,84,48,81,45,78,42,75) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,17),(14,16),(18,24),(19,23),(20,22),(25,33),(26,32),(27,31),(28,30),(34,36),(37,39),(40,48),(41,47),(42,46),(43,45),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,67),(62,66),(63,65),(68,72),(69,71),(73,78),(74,77),(75,76),(79,84),(80,83),(81,82),(85,92),(86,91),(87,90),(88,89),(93,96),(94,95)], [(1,49,89,76,7,55,95,82),(2,56,90,83,8,50,96,77),(3,51,91,78,9,57,85,84),(4,58,92,73,10,52,86,79),(5,53,93,80,11,59,87,74),(6,60,94,75,12,54,88,81),(13,36,62,39,19,30,68,45),(14,31,63,46,20,25,69,40),(15,26,64,41,21,32,70,47),(16,33,65,48,22,27,71,42),(17,28,66,43,23,34,72,37),(18,35,67,38,24,29,61,44)], [(1,66,10,63,7,72,4,69),(2,67,11,64,8,61,5,70),(3,68,12,65,9,62,6,71),(13,94,22,91,19,88,16,85),(14,95,23,92,20,89,17,86),(15,96,24,93,21,90,18,87),(25,52,34,49,31,58,28,55),(26,53,35,50,32,59,29,56),(27,54,36,51,33,60,30,57),(37,82,46,79,43,76,40,73),(38,83,47,80,44,77,41,74),(39,84,48,81,45,78,42,75)])
Matrix representation of D12.7D4 ►in GL8(𝔽73)
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 72 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 72 |
| 67 | 0 | 16 | 32 | 0 | 0 | 0 | 0 |
| 0 | 67 | 41 | 57 | 0 | 0 | 0 | 0 |
| 57 | 41 | 6 | 0 | 0 | 0 | 0 | 0 |
| 32 | 16 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 16 | 0 | 41 |
| 0 | 0 | 0 | 0 | 57 | 16 | 32 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 57 | 16 |
| 0 | 0 | 0 | 0 | 0 | 57 | 57 | 16 |
| 57 | 41 | 6 | 0 | 0 | 0 | 0 | 0 |
| 32 | 16 | 0 | 6 | 0 | 0 | 0 | 0 |
| 67 | 0 | 16 | 32 | 0 | 0 | 0 | 0 |
| 0 | 67 | 41 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 20 | 20 | 6 |
| 0 | 0 | 0 | 0 | 53 | 0 | 67 | 53 |
| 0 | 0 | 0 | 0 | 10 | 3 | 6 | 0 |
| 0 | 0 | 0 | 0 | 70 | 63 | 0 | 67 |
G:=sub<GL(8,GF(73))| [1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,2,72,0,0,0,0,0,2,0,0,72],[1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72],[67,0,57,32,0,0,0,0,0,67,41,16,0,0,0,0,16,41,6,0,0,0,0,0,32,57,0,6,0,0,0,0,0,0,0,0,57,57,16,0,0,0,0,0,16,16,0,57,0,0,0,0,0,32,57,57,0,0,0,0,41,0,16,16],[57,32,67,0,0,0,0,0,41,16,0,67,0,0,0,0,6,0,16,41,0,0,0,0,0,6,32,57,0,0,0,0,0,0,0,0,0,53,10,70,0,0,0,0,20,0,3,63,0,0,0,0,20,67,6,0,0,0,0,0,6,53,0,67] >;
D12.7D4 in GAP, Magma, Sage, TeX
D_{12}._7D_4 % in TeX
G:=Group("D12.7D4"); // GroupNames label
G:=SmallGroup(192,314);
// by ID
G=gap.SmallGroup(192,314);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,555,184,297,136,1684,851,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^3>;
// generators/relations
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