metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).16D6, C3⋊C8.33D4, C8.C22.S3, C4○D4.43D6, (C3×D4).16D4, C4.181(S3×D4), (C2×Q8).94D6, (C3×Q8).16D4, D4.Dic3.C2, C12.200(C2×D4), C3⋊5(D4.5D4), D4.7(C3⋊D4), C12.53D4⋊8C2, C12.47D4⋊7C2, (C2×C12).19C23, Q8.14D6.2C2, Q8.14(C3⋊D4), (C6×Q8).97C22, C12.10D4⋊10C2, C6.127(C4⋊D4), C4.Dic3.28C22, C2.33(C23.14D6), C22.16(D4⋊2S3), (C2×Dic6).136C22, (C3×M4(2)).13C22, C4.56(C2×C3⋊D4), (C2×C3⋊Q16)⋊22C2, (C2×C6).39(C4○D4), (C2×C3⋊C8).173C22, (C2×C4).19(C22×S3), (C3×C8.C22).1C2, (C3×C4○D4).17C22, SmallGroup(192,763)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).16D6
G = < a,b,c,d | a8=b2=c6=1, d2=a6, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a2c-1 >
Subgroups: 240 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C3⋊C8, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C8.C22, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, D4.S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C2×Dic6, C6×Q8, C3×C4○D4, D4.5D4, C12.53D4, C12.47D4, C12.10D4, C2×C3⋊Q16, D4.Dic3, Q8.14D6, C3×C8.C22, M4(2).16D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D4⋊2S3, C2×C3⋊D4, D4.5D4, C23.14D6, M4(2).16D6
Character table of M4(2).16D6
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 12C | 12D | 12E | 24A | 24B | |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 8 | 24 | 2 | 4 | 8 | 6 | 6 | 8 | 12 | 12 | 12 | 24 | 4 | 4 | 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 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | 2 | 0 | -1 | -1 | 1 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | -2 | 0 | -1 | -1 | 1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 2 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -√-3 | √-3 | -1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | 2 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | √-3 | -√-3 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | -2 | 2 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -√-3 | √-3 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 2 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | √-3 | -√-3 | -1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | 4 | -4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 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 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 2√2 | -2√2 | 0 | 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 | -2√2 | 2√2 | 0 | 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 22)(2 19)(3 24)(4 21)(5 18)(6 23)(7 20)(8 17)(9 92)(10 89)(11 94)(12 91)(13 96)(14 93)(15 90)(16 95)(25 50)(26 55)(27 52)(28 49)(29 54)(30 51)(31 56)(32 53)(33 65)(34 70)(35 67)(36 72)(37 69)(38 66)(39 71)(40 68)(41 62)(42 59)(43 64)(44 61)(45 58)(46 63)(47 60)(48 57)(73 86)(74 83)(75 88)(76 85)(77 82)(78 87)(79 84)(80 81)
(1 29 86)(2 32 87 4 30 81)(3 27 88 7 31 84)(5 25 82)(6 28 83 8 26 85)(9 66 60 15 68 58)(10 69 61)(11 72 62 13 70 64)(12 67 63 16 71 59)(14 65 57)(17 51 76 19 49 78)(18 54 77 22 50 73)(20 52 79)(21 55 80 23 53 74)(24 56 75)(33 44 93 37 48 89)(34 47 94 40 41 92)(35 42 95)(36 45 96 38 43 90)(39 46 91)
(1 13 7 11 5 9 3 15)(2 95 8 93 6 91 4 89)(10 23 16 21 14 19 12 17)(18 96 24 94 22 92 20 90)(25 60 31 58 29 64 27 62)(26 46 32 44 30 42 28 48)(33 83 39 81 37 87 35 85)(34 73 40 79 38 77 36 75)(41 54 47 52 45 50 43 56)(49 61 55 59 53 57 51 63)(65 78 71 76 69 74 67 80)(66 86 72 84 70 82 68 88)
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), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,92)(10,89)(11,94)(12,91)(13,96)(14,93)(15,90)(16,95)(25,50)(26,55)(27,52)(28,49)(29,54)(30,51)(31,56)(32,53)(33,65)(34,70)(35,67)(36,72)(37,69)(38,66)(39,71)(40,68)(41,62)(42,59)(43,64)(44,61)(45,58)(46,63)(47,60)(48,57)(73,86)(74,83)(75,88)(76,85)(77,82)(78,87)(79,84)(80,81), (1,29,86)(2,32,87,4,30,81)(3,27,88,7,31,84)(5,25,82)(6,28,83,8,26,85)(9,66,60,15,68,58)(10,69,61)(11,72,62,13,70,64)(12,67,63,16,71,59)(14,65,57)(17,51,76,19,49,78)(18,54,77,22,50,73)(20,52,79)(21,55,80,23,53,74)(24,56,75)(33,44,93,37,48,89)(34,47,94,40,41,92)(35,42,95)(36,45,96,38,43,90)(39,46,91), (1,13,7,11,5,9,3,15)(2,95,8,93,6,91,4,89)(10,23,16,21,14,19,12,17)(18,96,24,94,22,92,20,90)(25,60,31,58,29,64,27,62)(26,46,32,44,30,42,28,48)(33,83,39,81,37,87,35,85)(34,73,40,79,38,77,36,75)(41,54,47,52,45,50,43,56)(49,61,55,59,53,57,51,63)(65,78,71,76,69,74,67,80)(66,86,72,84,70,82,68,88)>;
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), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,92)(10,89)(11,94)(12,91)(13,96)(14,93)(15,90)(16,95)(25,50)(26,55)(27,52)(28,49)(29,54)(30,51)(31,56)(32,53)(33,65)(34,70)(35,67)(36,72)(37,69)(38,66)(39,71)(40,68)(41,62)(42,59)(43,64)(44,61)(45,58)(46,63)(47,60)(48,57)(73,86)(74,83)(75,88)(76,85)(77,82)(78,87)(79,84)(80,81), (1,29,86)(2,32,87,4,30,81)(3,27,88,7,31,84)(5,25,82)(6,28,83,8,26,85)(9,66,60,15,68,58)(10,69,61)(11,72,62,13,70,64)(12,67,63,16,71,59)(14,65,57)(17,51,76,19,49,78)(18,54,77,22,50,73)(20,52,79)(21,55,80,23,53,74)(24,56,75)(33,44,93,37,48,89)(34,47,94,40,41,92)(35,42,95)(36,45,96,38,43,90)(39,46,91), (1,13,7,11,5,9,3,15)(2,95,8,93,6,91,4,89)(10,23,16,21,14,19,12,17)(18,96,24,94,22,92,20,90)(25,60,31,58,29,64,27,62)(26,46,32,44,30,42,28,48)(33,83,39,81,37,87,35,85)(34,73,40,79,38,77,36,75)(41,54,47,52,45,50,43,56)(49,61,55,59,53,57,51,63)(65,78,71,76,69,74,67,80)(66,86,72,84,70,82,68,88) );
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)], [(1,22),(2,19),(3,24),(4,21),(5,18),(6,23),(7,20),(8,17),(9,92),(10,89),(11,94),(12,91),(13,96),(14,93),(15,90),(16,95),(25,50),(26,55),(27,52),(28,49),(29,54),(30,51),(31,56),(32,53),(33,65),(34,70),(35,67),(36,72),(37,69),(38,66),(39,71),(40,68),(41,62),(42,59),(43,64),(44,61),(45,58),(46,63),(47,60),(48,57),(73,86),(74,83),(75,88),(76,85),(77,82),(78,87),(79,84),(80,81)], [(1,29,86),(2,32,87,4,30,81),(3,27,88,7,31,84),(5,25,82),(6,28,83,8,26,85),(9,66,60,15,68,58),(10,69,61),(11,72,62,13,70,64),(12,67,63,16,71,59),(14,65,57),(17,51,76,19,49,78),(18,54,77,22,50,73),(20,52,79),(21,55,80,23,53,74),(24,56,75),(33,44,93,37,48,89),(34,47,94,40,41,92),(35,42,95),(36,45,96,38,43,90),(39,46,91)], [(1,13,7,11,5,9,3,15),(2,95,8,93,6,91,4,89),(10,23,16,21,14,19,12,17),(18,96,24,94,22,92,20,90),(25,60,31,58,29,64,27,62),(26,46,32,44,30,42,28,48),(33,83,39,81,37,87,35,85),(34,73,40,79,38,77,36,75),(41,54,47,52,45,50,43,56),(49,61,55,59,53,57,51,63),(65,78,71,76,69,74,67,80),(66,86,72,84,70,82,68,88)]])
Matrix representation of M4(2).16D6 ►in GL8(𝔽73)
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 2 | 1 | 0 | 0 | 0 | 0 |
| 48 | 48 | 72 | 0 | 0 | 0 | 0 | 0 |
| 49 | 48 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 46 |
| 0 | 0 | 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 | 0 | 0 |
| 1 | 0 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 49 | 50 | 1 | 1 | 0 | 0 | 0 | 0 |
| 48 | 49 | 72 | 0 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 46 | 46 | 19 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 22 |
| 0 | 0 | 0 | 0 | 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 0 | 63 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 | 0 | 0 |
G:=sub<GL(8,GF(73))| [0,1,48,49,0,0,0,0,0,1,48,48,0,0,0,0,1,2,72,72,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,0,0,0,46,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0],[0,1,49,48,0,0,0,0,72,72,50,49,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[46,0,0,0,0,0,0,0,46,0,0,27,0,0,0,0,19,46,27,27,0,0,0,0,46,27,0,0,0,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,10,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0] >;
M4(2).16D6 in GAP, Magma, Sage, TeX
M_4(2)._{16}D_6 % in TeX
G:=Group("M4(2).16D6"); // GroupNames label
G:=SmallGroup(192,763);
// by ID
G=gap.SmallGroup(192,763);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,219,184,1123,297,136,1684,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^3,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^-1>;
// generators/relations
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