Aliases: He3⋊D8, C6.2S3≀C2, C3.(C32⋊D8), He3⋊2C8⋊1C2, He3⋊2D4⋊1C2, C2.4(He3⋊D4), (C2×He3).2D4, He3⋊3C4.2C22, SmallGroup(432,235)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, eae=ab=ba, cac-1=dcd-1=ab-1, dad-1=bc-1, bc=cb, bd=db, ebe=b-1, ece=c-1, ede=d-1 >
Subgroups: 679 in 65 conjugacy classes, 11 normal (9 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, Dic3, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C24, D12, C3⋊D4, He3, C3×Dic3, S3×C6, C2×C3⋊S3, D24, C32⋊C6, C2×He3, C3⋊D12, He3⋊3C4, C2×C32⋊C6, He3⋊2C8, He3⋊2D4, He3⋊D8
Quotients: C1, C2, C22, D4, D8, S3≀C2, C32⋊D8, He3⋊D4, He3⋊D8
Character table of He3⋊D8
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 36 | 36 | 2 | 12 | 12 | 18 | 2 | 12 | 12 | 36 | 36 | 36 | 36 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -√2 | √2 | 0 | 0 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ7 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | √2 | -√2 | 0 | 0 | -√2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ8 | 4 | 4 | 0 | 2 | 4 | -2 | 1 | 0 | 4 | -2 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ9 | 4 | 4 | 0 | -2 | 4 | -2 | 1 | 0 | 4 | -2 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ10 | 4 | 4 | 2 | 0 | 4 | 1 | -2 | 0 | 4 | 1 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ11 | 4 | 4 | -2 | 0 | 4 | 1 | -2 | 0 | 4 | 1 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ12 | 4 | -4 | 0 | 0 | 4 | 1 | -2 | 0 | -4 | -1 | 2 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊D8 |
| ρ13 | 4 | -4 | 0 | 0 | 4 | -2 | 1 | 0 | -4 | 2 | -1 | 0 | 0 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊D8 |
| ρ14 | 4 | -4 | 0 | 0 | 4 | -2 | 1 | 0 | -4 | 2 | -1 | 0 | 0 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊D8 |
| ρ15 | 4 | -4 | 0 | 0 | 4 | 1 | -2 | 0 | -4 | -1 | 2 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊D8 |
| ρ16 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | -2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from He3⋊D4 |
| ρ17 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | -2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from He3⋊D4 |
| ρ18 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from He3⋊D4 |
| ρ19 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from He3⋊D4 |
| ρ20 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√3 | √3 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | orthogonal faithful |
| ρ21 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√3 | √3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | orthogonal faithful |
| ρ22 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √3 | -√3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | orthogonal faithful |
| ρ23 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √3 | -√3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | orthogonal faithful |
►On 72 points
Generators in S72
(1 57 13)(2 54 17)(3 59 15)(4 69 37)(5 61 9)(6 50 21)(7 63 11)(8 65 33)(10 26 71)(12 44 52)(14 30 67)(16 48 56)(18 68 47)(19 32 60)(22 72 43)(23 28 64)(25 38 49)(29 34 53)(35 46 58)(39 42 62)
(1 57 13)(2 58 14)(3 59 15)(4 60 16)(5 61 9)(6 62 10)(7 63 11)(8 64 12)(17 46 67)(18 47 68)(19 48 69)(20 41 70)(21 42 71)(22 43 72)(23 44 65)(24 45 66)(25 49 38)(26 50 39)(27 51 40)(28 52 33)(29 53 34)(30 54 35)(31 55 36)(32 56 37)
(1 45 29)(3 31 47)(5 41 25)(7 27 43)(9 20 38)(11 40 22)(13 24 34)(15 36 18)(17 67 46)(19 48 69)(21 71 42)(23 44 65)(26 50 39)(28 33 52)(30 54 35)(32 37 56)(49 61 70)(51 72 63)(53 57 66)(55 68 59)
(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)
(2 8)(3 7)(4 6)(9 61)(10 60)(11 59)(12 58)(13 57)(14 64)(15 63)(16 62)(17 52)(18 51)(19 50)(20 49)(21 56)(22 55)(23 54)(24 53)(25 41)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 67)(34 66)(35 65)(36 72)(37 71)(38 70)(39 69)(40 68)
G:=sub<Sym(72)| (1,57,13)(2,54,17)(3,59,15)(4,69,37)(5,61,9)(6,50,21)(7,63,11)(8,65,33)(10,26,71)(12,44,52)(14,30,67)(16,48,56)(18,68,47)(19,32,60)(22,72,43)(23,28,64)(25,38,49)(29,34,53)(35,46,58)(39,42,62), (1,57,13)(2,58,14)(3,59,15)(4,60,16)(5,61,9)(6,62,10)(7,63,11)(8,64,12)(17,46,67)(18,47,68)(19,48,69)(20,41,70)(21,42,71)(22,43,72)(23,44,65)(24,45,66)(25,49,38)(26,50,39)(27,51,40)(28,52,33)(29,53,34)(30,54,35)(31,55,36)(32,56,37), (1,45,29)(3,31,47)(5,41,25)(7,27,43)(9,20,38)(11,40,22)(13,24,34)(15,36,18)(17,67,46)(19,48,69)(21,71,42)(23,44,65)(26,50,39)(28,33,52)(30,54,35)(32,37,56)(49,61,70)(51,72,63)(53,57,66)(55,68,59), (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), (2,8)(3,7)(4,6)(9,61)(10,60)(11,59)(12,58)(13,57)(14,64)(15,63)(16,62)(17,52)(18,51)(19,50)(20,49)(21,56)(22,55)(23,54)(24,53)(25,41)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,67)(34,66)(35,65)(36,72)(37,71)(38,70)(39,69)(40,68)>;
G:=Group( (1,57,13)(2,54,17)(3,59,15)(4,69,37)(5,61,9)(6,50,21)(7,63,11)(8,65,33)(10,26,71)(12,44,52)(14,30,67)(16,48,56)(18,68,47)(19,32,60)(22,72,43)(23,28,64)(25,38,49)(29,34,53)(35,46,58)(39,42,62), (1,57,13)(2,58,14)(3,59,15)(4,60,16)(5,61,9)(6,62,10)(7,63,11)(8,64,12)(17,46,67)(18,47,68)(19,48,69)(20,41,70)(21,42,71)(22,43,72)(23,44,65)(24,45,66)(25,49,38)(26,50,39)(27,51,40)(28,52,33)(29,53,34)(30,54,35)(31,55,36)(32,56,37), (1,45,29)(3,31,47)(5,41,25)(7,27,43)(9,20,38)(11,40,22)(13,24,34)(15,36,18)(17,67,46)(19,48,69)(21,71,42)(23,44,65)(26,50,39)(28,33,52)(30,54,35)(32,37,56)(49,61,70)(51,72,63)(53,57,66)(55,68,59), (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), (2,8)(3,7)(4,6)(9,61)(10,60)(11,59)(12,58)(13,57)(14,64)(15,63)(16,62)(17,52)(18,51)(19,50)(20,49)(21,56)(22,55)(23,54)(24,53)(25,41)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,67)(34,66)(35,65)(36,72)(37,71)(38,70)(39,69)(40,68) );
G=PermutationGroup([[(1,57,13),(2,54,17),(3,59,15),(4,69,37),(5,61,9),(6,50,21),(7,63,11),(8,65,33),(10,26,71),(12,44,52),(14,30,67),(16,48,56),(18,68,47),(19,32,60),(22,72,43),(23,28,64),(25,38,49),(29,34,53),(35,46,58),(39,42,62)], [(1,57,13),(2,58,14),(3,59,15),(4,60,16),(5,61,9),(6,62,10),(7,63,11),(8,64,12),(17,46,67),(18,47,68),(19,48,69),(20,41,70),(21,42,71),(22,43,72),(23,44,65),(24,45,66),(25,49,38),(26,50,39),(27,51,40),(28,52,33),(29,53,34),(30,54,35),(31,55,36),(32,56,37)], [(1,45,29),(3,31,47),(5,41,25),(7,27,43),(9,20,38),(11,40,22),(13,24,34),(15,36,18),(17,67,46),(19,48,69),(21,71,42),(23,44,65),(26,50,39),(28,33,52),(30,54,35),(32,37,56),(49,61,70),(51,72,63),(53,57,66),(55,68,59)], [(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)], [(2,8),(3,7),(4,6),(9,61),(10,60),(11,59),(12,58),(13,57),(14,64),(15,63),(16,62),(17,52),(18,51),(19,50),(20,49),(21,56),(22,55),(23,54),(24,53),(25,41),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,67),(34,66),(35,65),(36,72),(37,71),(38,70),(39,69),(40,68)]])
Matrix representation of He3⋊D8 ►in GL10(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 30 | 57 | 72 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 6 | 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 72 |
| 72 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 41 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 46 | 36 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 48 | 64 | 56 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 20 | 39 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 57 | 57 | 56 | 0 | 0 | 0 | 0 | 0 | 0 |
| 50 | 22 | 7 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 48 | 24 | 48 | 24 | 48 |
| 0 | 0 | 0 | 0 | 25 | 49 | 25 | 49 | 25 | 49 |
| 0 | 0 | 0 | 0 | 24 | 48 | 25 | 49 | 24 | 49 |
| 0 | 0 | 0 | 0 | 25 | 49 | 24 | 49 | 24 | 48 |
| 0 | 0 | 0 | 0 | 24 | 48 | 24 | 49 | 25 | 49 |
| 0 | 0 | 0 | 0 | 25 | 49 | 24 | 48 | 24 | 49 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 37 | 59 | 43 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 52 | 13 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 |
G:=sub<GL(10,GF(73))| [1,0,30,10,0,0,0,0,0,0,0,1,57,6,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[72,72,1,46,0,0,0,0,0,0,1,0,41,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[48,72,35,50,0,0,0,0,0,0,64,20,57,22,0,0,0,0,0,0,56,39,57,7,0,0,0,0,0,0,34,17,56,21,0,0,0,0,0,0,0,0,0,0,24,25,24,25,24,25,0,0,0,0,48,49,48,49,48,49,0,0,0,0,24,25,25,24,24,24,0,0,0,0,48,49,49,49,49,48,0,0,0,0,24,25,24,24,25,24,0,0,0,0,48,49,49,48,49,49],[0,1,37,9,0,0,0,0,0,0,1,0,59,52,0,0,0,0,0,0,0,0,43,13,0,0,0,0,0,0,0,0,60,30,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0] >;
He3⋊D8 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes D_8 % in TeX
G:=Group("He3:D8"); // GroupNames label
G:=SmallGroup(432,235);
// by ID
G=gap.SmallGroup(432,235);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,254,135,58,1124,851,298,348,1027,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,e*a*e=a*b=b*a,c*a*c^-1=d*c*d^-1=a*b^-1,d*a*d^-1=b*c^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations
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