metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.82D6, C4⋊Q8⋊7S3, C4⋊C4.85D6, (C2×C12).296D4, C12.84(C4○D4), C6.SD16⋊42C2, C6.99(C8⋊C22), C42⋊7S3.9C2, C6.D8.16C2, (C2×C12).407C23, (C4×C12).136C22, C42.S3⋊15C2, C4.17(Q8⋊3S3), C6.59(C4.4D4), C6.98(C8.C22), C2.20(D12⋊6C22), (C2×D12).110C22, C2.19(Q8.11D6), C2.12(C12.23D4), C3⋊5(C42.28C22), (C2×Dic6).114C22, (C3×C4⋊Q8)⋊7C2, (C2×C6).538(C2×D4), (C2×C4).74(C3⋊D4), (C2×C3⋊C8).139C22, (C3×C4⋊C4).132C22, (C2×C4).504(C22×S3), C22.210(C2×C3⋊D4), SmallGroup(192,648)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.82D6
G = < a,b,c,d | a4=b4=1, c6=a2b2, d2=a2b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c5 >
Subgroups: 304 in 100 conjugacy classes, 39 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C3⋊C8, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C2×C3⋊C8, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×D12, C6×Q8, C42.28C22, C42.S3, C6.D8, C6.SD16, C42⋊7S3, C3×C4⋊Q8, C42.82D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, C8⋊C22, C8.C22, Q8⋊3S3, C2×C3⋊D4, C42.28C22, D12⋊6C22, Q8.11D6, C12.23D4, C42.82D6
Character table of C42.82D6
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 24 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | -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 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | -√-3 | -√-3 | √-3 | √-3 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | -√-3 | √-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | √-3 | -√-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | √-3 | √-3 | -√-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2i | 0 | -2i | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 2i | 0 | -2i | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | -2i | 0 | 2i | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2i | 0 | 2i | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| 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 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ26 | 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 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2√-3 | 0 | 0 | 0 | 0 | -2√-3 | 0 | 0 | 0 | 0 | complex lifted from D12⋊6C22 |
| ρ28 | 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 | complex lifted from Q8.11D6 |
| ρ29 | 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 | complex lifted from Q8.11D6 |
| ρ30 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2√-3 | 0 | 0 | 0 | 0 | 2√-3 | 0 | 0 | 0 | 0 | complex lifted from D12⋊6C22 |
►On 96 points
Generators in S96
(1 24 83 85)(2 86 84 13)(3 14 73 87)(4 88 74 15)(5 16 75 89)(6 90 76 17)(7 18 77 91)(8 92 78 19)(9 20 79 93)(10 94 80 21)(11 22 81 95)(12 96 82 23)(25 62 43 59)(26 60 44 63)(27 64 45 49)(28 50 46 65)(29 66 47 51)(30 52 48 67)(31 68 37 53)(32 54 38 69)(33 70 39 55)(34 56 40 71)(35 72 41 57)(36 58 42 61)
(1 51 77 72)(2 61 78 52)(3 53 79 62)(4 63 80 54)(5 55 81 64)(6 65 82 56)(7 57 83 66)(8 67 84 58)(9 59 73 68)(10 69 74 60)(11 49 75 70)(12 71 76 50)(13 42 92 30)(14 31 93 43)(15 44 94 32)(16 33 95 45)(17 46 96 34)(18 35 85 47)(19 48 86 36)(20 25 87 37)(21 38 88 26)(22 27 89 39)(23 40 90 28)(24 29 91 41)
(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 6 66 50 77 82 57 71)(2 70 58 81 78 49 67 5)(3 4 68 60 79 80 59 69)(7 12 72 56 83 76 51 65)(8 64 52 75 84 55 61 11)(9 10 62 54 73 74 53 63)(13 45 36 89 92 33 48 22)(14 21 37 32 93 88 25 44)(15 43 26 87 94 31 38 20)(16 19 39 30 95 86 27 42)(17 41 28 85 96 29 40 18)(23 47 34 91 90 35 46 24)
G:=sub<Sym(96)| (1,24,83,85)(2,86,84,13)(3,14,73,87)(4,88,74,15)(5,16,75,89)(6,90,76,17)(7,18,77,91)(8,92,78,19)(9,20,79,93)(10,94,80,21)(11,22,81,95)(12,96,82,23)(25,62,43,59)(26,60,44,63)(27,64,45,49)(28,50,46,65)(29,66,47,51)(30,52,48,67)(31,68,37,53)(32,54,38,69)(33,70,39,55)(34,56,40,71)(35,72,41,57)(36,58,42,61), (1,51,77,72)(2,61,78,52)(3,53,79,62)(4,63,80,54)(5,55,81,64)(6,65,82,56)(7,57,83,66)(8,67,84,58)(9,59,73,68)(10,69,74,60)(11,49,75,70)(12,71,76,50)(13,42,92,30)(14,31,93,43)(15,44,94,32)(16,33,95,45)(17,46,96,34)(18,35,85,47)(19,48,86,36)(20,25,87,37)(21,38,88,26)(22,27,89,39)(23,40,90,28)(24,29,91,41), (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), (1,6,66,50,77,82,57,71)(2,70,58,81,78,49,67,5)(3,4,68,60,79,80,59,69)(7,12,72,56,83,76,51,65)(8,64,52,75,84,55,61,11)(9,10,62,54,73,74,53,63)(13,45,36,89,92,33,48,22)(14,21,37,32,93,88,25,44)(15,43,26,87,94,31,38,20)(16,19,39,30,95,86,27,42)(17,41,28,85,96,29,40,18)(23,47,34,91,90,35,46,24)>;
G:=Group( (1,24,83,85)(2,86,84,13)(3,14,73,87)(4,88,74,15)(5,16,75,89)(6,90,76,17)(7,18,77,91)(8,92,78,19)(9,20,79,93)(10,94,80,21)(11,22,81,95)(12,96,82,23)(25,62,43,59)(26,60,44,63)(27,64,45,49)(28,50,46,65)(29,66,47,51)(30,52,48,67)(31,68,37,53)(32,54,38,69)(33,70,39,55)(34,56,40,71)(35,72,41,57)(36,58,42,61), (1,51,77,72)(2,61,78,52)(3,53,79,62)(4,63,80,54)(5,55,81,64)(6,65,82,56)(7,57,83,66)(8,67,84,58)(9,59,73,68)(10,69,74,60)(11,49,75,70)(12,71,76,50)(13,42,92,30)(14,31,93,43)(15,44,94,32)(16,33,95,45)(17,46,96,34)(18,35,85,47)(19,48,86,36)(20,25,87,37)(21,38,88,26)(22,27,89,39)(23,40,90,28)(24,29,91,41), (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), (1,6,66,50,77,82,57,71)(2,70,58,81,78,49,67,5)(3,4,68,60,79,80,59,69)(7,12,72,56,83,76,51,65)(8,64,52,75,84,55,61,11)(9,10,62,54,73,74,53,63)(13,45,36,89,92,33,48,22)(14,21,37,32,93,88,25,44)(15,43,26,87,94,31,38,20)(16,19,39,30,95,86,27,42)(17,41,28,85,96,29,40,18)(23,47,34,91,90,35,46,24) );
G=PermutationGroup([[(1,24,83,85),(2,86,84,13),(3,14,73,87),(4,88,74,15),(5,16,75,89),(6,90,76,17),(7,18,77,91),(8,92,78,19),(9,20,79,93),(10,94,80,21),(11,22,81,95),(12,96,82,23),(25,62,43,59),(26,60,44,63),(27,64,45,49),(28,50,46,65),(29,66,47,51),(30,52,48,67),(31,68,37,53),(32,54,38,69),(33,70,39,55),(34,56,40,71),(35,72,41,57),(36,58,42,61)], [(1,51,77,72),(2,61,78,52),(3,53,79,62),(4,63,80,54),(5,55,81,64),(6,65,82,56),(7,57,83,66),(8,67,84,58),(9,59,73,68),(10,69,74,60),(11,49,75,70),(12,71,76,50),(13,42,92,30),(14,31,93,43),(15,44,94,32),(16,33,95,45),(17,46,96,34),(18,35,85,47),(19,48,86,36),(20,25,87,37),(21,38,88,26),(22,27,89,39),(23,40,90,28),(24,29,91,41)], [(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)], [(1,6,66,50,77,82,57,71),(2,70,58,81,78,49,67,5),(3,4,68,60,79,80,59,69),(7,12,72,56,83,76,51,65),(8,64,52,75,84,55,61,11),(9,10,62,54,73,74,53,63),(13,45,36,89,92,33,48,22),(14,21,37,32,93,88,25,44),(15,43,26,87,94,31,38,20),(16,19,39,30,95,86,27,42),(17,41,28,85,96,29,40,18),(23,47,34,91,90,35,46,24)]])
Matrix representation of C42.82D6 ►in GL6(𝔽73)
| 54 | 6 | 0 | 0 | 0 | 0 |
| 37 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 60 |
| 0 | 0 | 0 | 0 | 13 | 43 |
| 0 | 0 | 43 | 13 | 0 | 0 |
| 0 | 0 | 60 | 30 | 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 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 25 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 31 | 31 | 42 |
| 0 | 0 | 42 | 11 | 31 | 62 |
| 0 | 0 | 31 | 42 | 31 | 42 |
| 0 | 0 | 31 | 62 | 31 | 62 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 42 | 31 | 42 |
| 0 | 0 | 11 | 42 | 11 | 42 |
| 0 | 0 | 42 | 31 | 31 | 42 |
| 0 | 0 | 62 | 31 | 11 | 42 |
G:=sub<GL(6,GF(73))| [54,37,0,0,0,0,6,19,0,0,0,0,0,0,0,0,43,60,0,0,0,0,13,30,0,0,30,13,0,0,0,0,60,43,0,0],[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],[27,25,0,0,0,0,0,46,0,0,0,0,0,0,42,42,31,31,0,0,31,11,42,62,0,0,31,31,31,31,0,0,42,62,42,62],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,31,11,42,62,0,0,42,42,31,31,0,0,31,11,31,11,0,0,42,42,42,42] >;
C42.82D6 in GAP, Magma, Sage, TeX
C_4^2._{82}D_6 % in TeX
G:=Group("C4^2.82D6"); // GroupNames label
G:=SmallGroup(192,648);
// by ID
G=gap.SmallGroup(192,648);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,555,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations
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