metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.64D6, C3⋊C8⋊7D4, C3⋊2(C8⋊3D4), C4.11(S3×D4), C4⋊D12⋊9C2, (C2×D4).48D6, C4.4D4⋊2S3, (C2×C12).82D4, C12.25(C2×D4), (C2×Q8).62D6, C2.8(C12⋊3D4), C6.17(C4⋊1D4), C42.S3⋊9C2, (C6×D4).64C22, (C6×Q8).56C22, C2.19(D4⋊D6), C6.120(C8⋊C22), (C4×C12).107C22, (C2×C12).376C23, (C2×D12).101C22, (C2×D4⋊S3)⋊12C2, (C2×C6).507(C2×D4), (C3×C4.4D4)⋊2C2, (C2×Q8⋊2S3)⋊13C2, (C2×C4).62(C3⋊D4), (C2×C3⋊C8).122C22, (C2×C4).476(C22×S3), C22.182(C2×C3⋊D4), SmallGroup(192,617)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.64D6
G = < a,b,c,d | a4=b4=c6=1, d2=cbc-1=b-1, ab=ba, cac-1=a-1b2, dad-1=ab2, bd=db, dcd-1=b-1c-1 >
Subgroups: 528 in 144 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×C8, D8, SD16, C2×D4, C2×D4, C2×Q8, C3⋊C8, D12, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×SD16, C2×C3⋊C8, D4⋊S3, Q8⋊2S3, C4×C12, C3×C22⋊C4, C2×D12, C2×D12, C6×D4, C6×Q8, C8⋊3D4, C42.S3, C4⋊D12, C2×D4⋊S3, C2×Q8⋊2S3, C3×C4.4D4, C42.64D6
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, C12⋊3D4, D4⋊D6, C42.64D6
Character table of C42.64D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 1 | 1 | 8 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 8 | 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 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | -2 | -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 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | -2 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | -1 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | -1 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | -1 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | -1 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | 1 | √-3 | -√-3 | 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 | 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 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | orthogonal lifted from S3×D4 |
| 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 | 0 | 2 | 0 | -2 | 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 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊D6 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2√3 | 0 | 0 | 0 | -2√3 | 0 | 0 | 0 | orthogonal lifted from D4⋊D6 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -2 | 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 |
| ρ30 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2√3 | 0 | 0 | 0 | 2√3 | 0 | 0 | 0 | orthogonal lifted from D4⋊D6 |
►On 96 points
Generators in S96
(1 28 81 90)(2 25 82 95)(3 30 83 92)(4 27 84 89)(5 32 85 94)(6 29 86 91)(7 26 87 96)(8 31 88 93)(9 71 36 77)(10 68 37 74)(11 65 38 79)(12 70 39 76)(13 67 40 73)(14 72 33 78)(15 69 34 75)(16 66 35 80)(17 43 63 49)(18 48 64 54)(19 45 57 51)(20 42 58 56)(21 47 59 53)(22 44 60 50)(23 41 61 55)(24 46 62 52)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)(65 71 69 67)(66 72 70 68)(73 79 77 75)(74 80 78 76)(81 87 85 83)(82 88 86 84)(89 95 93 91)(90 96 94 92)
(1 20 72 65 19 2)(3 18 66 71 21 8)(4 7 22 70 67 17)(5 24 68 69 23 6)(9 43 93 26 54 39)(10 38 55 25 94 42)(11 41 95 32 56 37)(12 36 49 31 96 48)(13 47 89 30 50 35)(14 34 51 29 90 46)(15 45 91 28 52 33)(16 40 53 27 92 44)(57 82 81 58 78 79)(59 88 83 64 80 77)(60 76 73 63 84 87)(61 86 85 62 74 75)
(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)
G:=sub<Sym(96)| (1,28,81,90)(2,25,82,95)(3,30,83,92)(4,27,84,89)(5,32,85,94)(6,29,86,91)(7,26,87,96)(8,31,88,93)(9,71,36,77)(10,68,37,74)(11,65,38,79)(12,70,39,76)(13,67,40,73)(14,72,33,78)(15,69,34,75)(16,66,35,80)(17,43,63,49)(18,48,64,54)(19,45,57,51)(20,42,58,56)(21,47,59,53)(22,44,60,50)(23,41,61,55)(24,46,62,52), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,79,77,75)(74,80,78,76)(81,87,85,83)(82,88,86,84)(89,95,93,91)(90,96,94,92), (1,20,72,65,19,2)(3,18,66,71,21,8)(4,7,22,70,67,17)(5,24,68,69,23,6)(9,43,93,26,54,39)(10,38,55,25,94,42)(11,41,95,32,56,37)(12,36,49,31,96,48)(13,47,89,30,50,35)(14,34,51,29,90,46)(15,45,91,28,52,33)(16,40,53,27,92,44)(57,82,81,58,78,79)(59,88,83,64,80,77)(60,76,73,63,84,87)(61,86,85,62,74,75), (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)>;
G:=Group( (1,28,81,90)(2,25,82,95)(3,30,83,92)(4,27,84,89)(5,32,85,94)(6,29,86,91)(7,26,87,96)(8,31,88,93)(9,71,36,77)(10,68,37,74)(11,65,38,79)(12,70,39,76)(13,67,40,73)(14,72,33,78)(15,69,34,75)(16,66,35,80)(17,43,63,49)(18,48,64,54)(19,45,57,51)(20,42,58,56)(21,47,59,53)(22,44,60,50)(23,41,61,55)(24,46,62,52), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,79,77,75)(74,80,78,76)(81,87,85,83)(82,88,86,84)(89,95,93,91)(90,96,94,92), (1,20,72,65,19,2)(3,18,66,71,21,8)(4,7,22,70,67,17)(5,24,68,69,23,6)(9,43,93,26,54,39)(10,38,55,25,94,42)(11,41,95,32,56,37)(12,36,49,31,96,48)(13,47,89,30,50,35)(14,34,51,29,90,46)(15,45,91,28,52,33)(16,40,53,27,92,44)(57,82,81,58,78,79)(59,88,83,64,80,77)(60,76,73,63,84,87)(61,86,85,62,74,75), (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) );
G=PermutationGroup([[(1,28,81,90),(2,25,82,95),(3,30,83,92),(4,27,84,89),(5,32,85,94),(6,29,86,91),(7,26,87,96),(8,31,88,93),(9,71,36,77),(10,68,37,74),(11,65,38,79),(12,70,39,76),(13,67,40,73),(14,72,33,78),(15,69,34,75),(16,66,35,80),(17,43,63,49),(18,48,64,54),(19,45,57,51),(20,42,58,56),(21,47,59,53),(22,44,60,50),(23,41,61,55),(24,46,62,52)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60),(65,71,69,67),(66,72,70,68),(73,79,77,75),(74,80,78,76),(81,87,85,83),(82,88,86,84),(89,95,93,91),(90,96,94,92)], [(1,20,72,65,19,2),(3,18,66,71,21,8),(4,7,22,70,67,17),(5,24,68,69,23,6),(9,43,93,26,54,39),(10,38,55,25,94,42),(11,41,95,32,56,37),(12,36,49,31,96,48),(13,47,89,30,50,35),(14,34,51,29,90,46),(15,45,91,28,52,33),(16,40,53,27,92,44),(57,82,81,58,78,79),(59,88,83,64,80,77),(60,76,73,63,84,87),(61,86,85,62,74,75)], [(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)]])
Matrix representation of C42.64D6 ►in GL6(𝔽73)
| 32 | 3 | 0 | 0 | 0 | 0 |
| 72 | 41 | 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 |
| 3 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 33 | 33 | 21 |
| 0 | 0 | 40 | 61 | 52 | 12 |
| 0 | 0 | 33 | 21 | 52 | 40 |
| 0 | 0 | 52 | 12 | 33 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 21 | 52 | 40 |
| 0 | 0 | 61 | 40 | 61 | 21 |
| 0 | 0 | 21 | 33 | 33 | 21 |
| 0 | 0 | 12 | 52 | 61 | 40 |
G:=sub<GL(6,GF(73))| [32,72,0,0,0,0,3,41,0,0,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,66,14,0,0,0,0,59,7,0,0],[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],[1,3,0,0,0,0,0,72,0,0,0,0,0,0,21,40,33,52,0,0,33,61,21,12,0,0,33,52,52,33,0,0,21,12,40,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,61,21,12,0,0,21,40,33,52,0,0,52,61,33,61,0,0,40,21,21,40] >;
C42.64D6 in GAP, Magma, Sage, TeX
C_4^2._{64}D_6 % in TeX
G:=Group("C4^2.64D6"); // GroupNames label
G:=SmallGroup(192,617);
// by ID
G=gap.SmallGroup(192,617);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,555,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations
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