metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.62D6, (C2xD4).45D6, (C2xC12).81D4, (C2xQ8).59D6, C12:2Q8:17C2, C4.4D4.5S3, C12.66(C4oD4), Q8:2Dic3:20C2, C42.S3:8C2, (C6xD4).61C22, (C6xQ8).53C22, C4.20(D4:2S3), C2.17(D4:D6), C6.118(C8:C22), (C2xC12).373C23, (C4xC12).104C22, D4:Dic3.12C2, C6.41(C4.4D4), C2.18(Q8.14D6), C2.8(C23.12D6), C6.119(C8.C22), C4:Dic3.150C22, C3:4(C42.28C22), (C2xC6).504(C2xD4), (C2xC4).60(C3:D4), (C2xC3:C8).120C22, (C3xC4.4D4).3C2, (C2xC4).473(C22xS3), C22.179(C2xC3:D4), SmallGroup(192,614)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.62D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >
Subgroups: 272 in 100 conjugacy classes, 39 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, C2xD4, C2xQ8, C2xQ8, C3:C8, Dic6, C2xDic3, C2xC12, C2xC12, C3xD4, C3xQ8, C22xC6, C8:C4, D4:C4, Q8:C4, C4.4D4, C4:Q8, C2xC3:C8, C4:Dic3, C4:Dic3, C4xC12, C3xC22:C4, C2xDic6, C6xD4, C6xQ8, C42.28C22, C42.S3, D4:Dic3, Q8:2Dic3, C12:2Q8, C3xC4.4D4, C42.62D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C4.4D4, C8:C22, C8.C22, D4:2S3, C2xC3:D4, C42.28C22, C23.12D6, D4:D6, Q8.14D6, C42.62D6
Character table of C42.62D6
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 4 | 4 | 8 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | -1 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | -1 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | 1 | √-3 | -√-3 | complex lifted from C3:D4 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | √-3 | -√-3 | complex lifted from C3:D4 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | -√-3 | √-3 | complex lifted from C3:D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | 1 | -√-3 | √-3 | complex lifted from C3:D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | complex lifted from C4oD4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | complex lifted from C4oD4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | complex lifted from C4oD4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | complex lifted from C4oD4 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4:D6 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 4 | 0 | 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 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4:D6 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | symplectic lifted from D4:2S3, Schur index 2 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 4 | 0 | 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 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2√3 | 0 | 0 | -2√3 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.14D6, Schur index 2 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | symplectic lifted from D4:2S3, Schur index 2 |
| ρ30 | 4 | -4 | -4 | 4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2√3 | 0 | 0 | 2√3 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.14D6, Schur index 2 |
►On 96 points
Generators in S96
(1 42 8 35)(2 40 9 33)(3 38 7 31)(4 32 11 39)(5 36 12 37)(6 34 10 41)(13 49 79 55)(14 69 80 63)(15 51 81 57)(16 71 82 65)(17 53 83 59)(18 67 84 61)(19 27 96 30)(20 47 91 44)(21 29 92 26)(22 43 93 46)(23 25 94 28)(24 45 95 48)(50 88 56 74)(52 90 58 76)(54 86 60 78)(62 73 68 87)(64 75 70 89)(66 77 72 85)
(1 93 4 96)(2 91 5 94)(3 95 6 92)(7 24 10 21)(8 22 11 19)(9 20 12 23)(13 16 87 90)(14 85 88 17)(15 18 89 86)(25 33 47 37)(26 38 48 34)(27 35 43 39)(28 40 44 36)(29 31 45 41)(30 42 46 32)(49 71 62 58)(50 59 63 72)(51 67 64 60)(52 55 65 68)(53 69 66 56)(54 57 61 70)(73 76 79 82)(74 83 80 77)(75 78 81 84)
(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 61 8 67)(2 63 9 69)(3 65 7 71)(4 54 11 60)(5 50 12 56)(6 52 10 58)(13 48 79 45)(14 40 80 33)(15 46 81 43)(16 38 82 31)(17 44 83 47)(18 42 84 35)(19 64 96 70)(20 53 91 59)(21 62 92 68)(22 51 93 57)(23 66 94 72)(24 49 95 55)(25 85 28 77)(26 73 29 87)(27 89 30 75)(32 78 39 86)(34 76 41 90)(36 74 37 88)
G:=sub<Sym(96)| (1,42,8,35)(2,40,9,33)(3,38,7,31)(4,32,11,39)(5,36,12,37)(6,34,10,41)(13,49,79,55)(14,69,80,63)(15,51,81,57)(16,71,82,65)(17,53,83,59)(18,67,84,61)(19,27,96,30)(20,47,91,44)(21,29,92,26)(22,43,93,46)(23,25,94,28)(24,45,95,48)(50,88,56,74)(52,90,58,76)(54,86,60,78)(62,73,68,87)(64,75,70,89)(66,77,72,85), (1,93,4,96)(2,91,5,94)(3,95,6,92)(7,24,10,21)(8,22,11,19)(9,20,12,23)(13,16,87,90)(14,85,88,17)(15,18,89,86)(25,33,47,37)(26,38,48,34)(27,35,43,39)(28,40,44,36)(29,31,45,41)(30,42,46,32)(49,71,62,58)(50,59,63,72)(51,67,64,60)(52,55,65,68)(53,69,66,56)(54,57,61,70)(73,76,79,82)(74,83,80,77)(75,78,81,84), (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), (1,61,8,67)(2,63,9,69)(3,65,7,71)(4,54,11,60)(5,50,12,56)(6,52,10,58)(13,48,79,45)(14,40,80,33)(15,46,81,43)(16,38,82,31)(17,44,83,47)(18,42,84,35)(19,64,96,70)(20,53,91,59)(21,62,92,68)(22,51,93,57)(23,66,94,72)(24,49,95,55)(25,85,28,77)(26,73,29,87)(27,89,30,75)(32,78,39,86)(34,76,41,90)(36,74,37,88)>;
G:=Group( (1,42,8,35)(2,40,9,33)(3,38,7,31)(4,32,11,39)(5,36,12,37)(6,34,10,41)(13,49,79,55)(14,69,80,63)(15,51,81,57)(16,71,82,65)(17,53,83,59)(18,67,84,61)(19,27,96,30)(20,47,91,44)(21,29,92,26)(22,43,93,46)(23,25,94,28)(24,45,95,48)(50,88,56,74)(52,90,58,76)(54,86,60,78)(62,73,68,87)(64,75,70,89)(66,77,72,85), (1,93,4,96)(2,91,5,94)(3,95,6,92)(7,24,10,21)(8,22,11,19)(9,20,12,23)(13,16,87,90)(14,85,88,17)(15,18,89,86)(25,33,47,37)(26,38,48,34)(27,35,43,39)(28,40,44,36)(29,31,45,41)(30,42,46,32)(49,71,62,58)(50,59,63,72)(51,67,64,60)(52,55,65,68)(53,69,66,56)(54,57,61,70)(73,76,79,82)(74,83,80,77)(75,78,81,84), (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), (1,61,8,67)(2,63,9,69)(3,65,7,71)(4,54,11,60)(5,50,12,56)(6,52,10,58)(13,48,79,45)(14,40,80,33)(15,46,81,43)(16,38,82,31)(17,44,83,47)(18,42,84,35)(19,64,96,70)(20,53,91,59)(21,62,92,68)(22,51,93,57)(23,66,94,72)(24,49,95,55)(25,85,28,77)(26,73,29,87)(27,89,30,75)(32,78,39,86)(34,76,41,90)(36,74,37,88) );
G=PermutationGroup([[(1,42,8,35),(2,40,9,33),(3,38,7,31),(4,32,11,39),(5,36,12,37),(6,34,10,41),(13,49,79,55),(14,69,80,63),(15,51,81,57),(16,71,82,65),(17,53,83,59),(18,67,84,61),(19,27,96,30),(20,47,91,44),(21,29,92,26),(22,43,93,46),(23,25,94,28),(24,45,95,48),(50,88,56,74),(52,90,58,76),(54,86,60,78),(62,73,68,87),(64,75,70,89),(66,77,72,85)], [(1,93,4,96),(2,91,5,94),(3,95,6,92),(7,24,10,21),(8,22,11,19),(9,20,12,23),(13,16,87,90),(14,85,88,17),(15,18,89,86),(25,33,47,37),(26,38,48,34),(27,35,43,39),(28,40,44,36),(29,31,45,41),(30,42,46,32),(49,71,62,58),(50,59,63,72),(51,67,64,60),(52,55,65,68),(53,69,66,56),(54,57,61,70),(73,76,79,82),(74,83,80,77),(75,78,81,84)], [(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)], [(1,61,8,67),(2,63,9,69),(3,65,7,71),(4,54,11,60),(5,50,12,56),(6,52,10,58),(13,48,79,45),(14,40,80,33),(15,46,81,43),(16,38,82,31),(17,44,83,47),(18,42,84,35),(19,64,96,70),(20,53,91,59),(21,62,92,68),(22,51,93,57),(23,66,94,72),(24,49,95,55),(25,85,28,77),(26,73,29,87),(27,89,30,75),(32,78,39,86),(34,76,41,90),(36,74,37,88)]])
Matrix representation of C42.62D6 ►in GL6(F73)
| 21 | 3 | 0 | 0 | 0 | 0 |
| 23 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 59 |
| 0 | 0 | 0 | 0 | 14 | 7 |
| 0 | 0 | 7 | 14 | 0 | 0 |
| 0 | 0 | 59 | 66 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 59 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 17 | 65 | 0 | 0 | 0 | 0 |
| 18 | 56 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 3 | 69 | 3 |
| 0 | 0 | 7 | 4 | 7 | 4 |
| 0 | 0 | 69 | 3 | 4 | 70 |
| 0 | 0 | 7 | 4 | 66 | 69 |
G:=sub<GL(6,GF(73))| [21,23,0,0,0,0,3,52,0,0,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,66,14,0,0,0,0,59,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[1,59,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[17,18,0,0,0,0,65,56,0,0,0,0,0,0,69,7,69,7,0,0,3,4,3,4,0,0,69,7,4,66,0,0,3,4,70,69] >;
C42.62D6 in GAP, Magma, Sage, TeX
C_4^2._{62}D_6 % in TeX
G:=Group("C4^2.62D6"); // GroupNames label
G:=SmallGroup(192,614);
// by ID
G=gap.SmallGroup(192,614);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,64,590,471,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b*c^-1>;
// generators/relations
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