direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D10⋊D6, D30⋊20D4, D30⋊11C23, C30.52C24, C6⋊3(D4×D5), C30⋊9(C2×D4), C10⋊3(S3×D4), D15⋊3(C2×D4), C5⋊D4⋊11D6, C23⋊6(S3×D5), C3⋊D4⋊11D10, (C2×C30)⋊5C23, (C6×D5)⋊5C23, D6⋊5(C22×D5), (C22×C10)⋊9D6, (C22×C6)⋊6D10, C15⋊10(C22×D4), (S3×C10)⋊5C23, (C2×Dic5)⋊17D6, (C22×D5)⋊13D6, D10⋊5(C22×S3), C6.52(C23×D5), (C2×Dic3)⋊17D10, (C22×S3)⋊12D10, (C23×D15)⋊10C2, C3⋊D20⋊19C22, C5⋊D12⋊20C22, C10.52(S3×C23), (C22×C30)⋊9C22, (C3×Dic5)⋊3C23, (C5×Dic3)⋊3C23, Dic5⋊3(C22×S3), Dic3⋊3(C22×D5), D30.C2⋊17C22, (C6×Dic5)⋊16C22, (C10×Dic3)⋊16C22, (C22×D15)⋊22C22, C3⋊4(C2×D4×D5), C5⋊4(C2×S3×D4), C22⋊5(C2×S3×D5), (C2×C5⋊D4)⋊12S3, (C2×C3⋊D4)⋊12D5, (C6×C5⋊D4)⋊14C2, (C2×S3×D5)⋊15C22, (C22×S3×D5)⋊11C2, (D5×C2×C6)⋊11C22, (C2×C6)⋊3(C22×D5), (C2×C5⋊D12)⋊23C2, (C10×C3⋊D4)⋊14C2, (C2×C3⋊D20)⋊23C2, C2.52(C22×S3×D5), (S3×C2×C10)⋊11C22, (C2×C10)⋊6(C22×S3), (C2×D30.C2)⋊24C2, (C5×C3⋊D4)⋊15C22, (C3×C5⋊D4)⋊15C22, SmallGroup(480,1124)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D10⋊D6
G = < a,b,c,d,e | a2=b10=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b5c, ece=b3c, ede=d-1 >
Subgroups: 2940 in 472 conjugacy classes, 124 normal (36 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, D15, D15, C30, C30, C30, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×C23, C5×Dic3, C3×Dic5, S3×D5, C6×D5, C6×D5, S3×C10, S3×C10, D30, D30, C2×C30, C2×C30, C2×C30, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, C23×D5, C2×S3×D4, D30.C2, C3⋊D20, C5⋊D12, C6×Dic5, C3×C5⋊D4, C10×Dic3, C5×C3⋊D4, C2×S3×D5, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C22×D15, C22×D15, C22×C30, C2×D4×D5, C2×D30.C2, C2×C3⋊D20, C2×C5⋊D12, D10⋊D6, C6×C5⋊D4, C10×C3⋊D4, C22×S3×D5, C23×D15, C2×D10⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C24, D10, C22×S3, C22×D4, C22×D5, S3×D4, S3×C23, S3×D5, D4×D5, C23×D5, C2×S3×D4, C2×S3×D5, C2×D4×D5, D10⋊D6, C22×S3×D5, C2×D10⋊D6
►On 120 points
Generators in S120
(1 87)(2 88)(3 89)(4 90)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(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 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 15)(12 14)(16 20)(17 19)(21 28)(22 27)(23 26)(24 25)(29 30)(31 38)(32 37)(33 36)(34 35)(39 40)(41 43)(44 50)(45 49)(46 48)(51 57)(52 56)(53 55)(58 60)(61 63)(64 70)(65 69)(66 68)(71 78)(72 77)(73 76)(74 75)(79 80)(81 82)(83 90)(84 89)(85 88)(86 87)(91 98)(92 97)(93 96)(94 95)(99 100)(101 103)(104 110)(105 109)(106 108)(111 117)(112 116)(113 115)(118 120)
(1 70 25 57 40 45)(2 61 26 58 31 46)(3 62 27 59 32 47)(4 63 28 60 33 48)(5 64 29 51 34 49)(6 65 30 52 35 50)(7 66 21 53 36 41)(8 67 22 54 37 42)(9 68 23 55 38 43)(10 69 24 56 39 44)(11 80 112 95 110 82)(12 71 113 96 101 83)(13 72 114 97 102 84)(14 73 115 98 103 85)(15 74 116 99 104 86)(16 75 117 100 105 87)(17 76 118 91 106 88)(18 77 119 92 107 89)(19 78 120 93 108 90)(20 79 111 94 109 81)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 116)(22 115)(23 114)(24 113)(25 112)(26 111)(27 120)(28 119)(29 118)(30 117)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 90)(48 89)(49 88)(50 87)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 80)(58 79)(59 78)(60 77)(61 94)(62 93)(63 92)(64 91)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)
G:=sub<Sym(120)| (1,87)(2,88)(3,89)(4,90)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (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)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120), (1,10)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,28)(22,27)(23,26)(24,25)(29,30)(31,38)(32,37)(33,36)(34,35)(39,40)(41,43)(44,50)(45,49)(46,48)(51,57)(52,56)(53,55)(58,60)(61,63)(64,70)(65,69)(66,68)(71,78)(72,77)(73,76)(74,75)(79,80)(81,82)(83,90)(84,89)(85,88)(86,87)(91,98)(92,97)(93,96)(94,95)(99,100)(101,103)(104,110)(105,109)(106,108)(111,117)(112,116)(113,115)(118,120), (1,70,25,57,40,45)(2,61,26,58,31,46)(3,62,27,59,32,47)(4,63,28,60,33,48)(5,64,29,51,34,49)(6,65,30,52,35,50)(7,66,21,53,36,41)(8,67,22,54,37,42)(9,68,23,55,38,43)(10,69,24,56,39,44)(11,80,112,95,110,82)(12,71,113,96,101,83)(13,72,114,97,102,84)(14,73,115,98,103,85)(15,74,116,99,104,86)(16,75,117,100,105,87)(17,76,118,91,106,88)(18,77,119,92,107,89)(19,78,120,93,108,90)(20,79,111,94,109,81), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,116)(22,115)(23,114)(24,113)(25,112)(26,111)(27,120)(28,119)(29,118)(30,117)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,90)(48,89)(49,88)(50,87)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,80)(58,79)(59,78)(60,77)(61,94)(62,93)(63,92)(64,91)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)>;
G:=Group( (1,87)(2,88)(3,89)(4,90)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (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)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120), (1,10)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,28)(22,27)(23,26)(24,25)(29,30)(31,38)(32,37)(33,36)(34,35)(39,40)(41,43)(44,50)(45,49)(46,48)(51,57)(52,56)(53,55)(58,60)(61,63)(64,70)(65,69)(66,68)(71,78)(72,77)(73,76)(74,75)(79,80)(81,82)(83,90)(84,89)(85,88)(86,87)(91,98)(92,97)(93,96)(94,95)(99,100)(101,103)(104,110)(105,109)(106,108)(111,117)(112,116)(113,115)(118,120), (1,70,25,57,40,45)(2,61,26,58,31,46)(3,62,27,59,32,47)(4,63,28,60,33,48)(5,64,29,51,34,49)(6,65,30,52,35,50)(7,66,21,53,36,41)(8,67,22,54,37,42)(9,68,23,55,38,43)(10,69,24,56,39,44)(11,80,112,95,110,82)(12,71,113,96,101,83)(13,72,114,97,102,84)(14,73,115,98,103,85)(15,74,116,99,104,86)(16,75,117,100,105,87)(17,76,118,91,106,88)(18,77,119,92,107,89)(19,78,120,93,108,90)(20,79,111,94,109,81), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,116)(22,115)(23,114)(24,113)(25,112)(26,111)(27,120)(28,119)(29,118)(30,117)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,90)(48,89)(49,88)(50,87)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,80)(58,79)(59,78)(60,77)(61,94)(62,93)(63,92)(64,91)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95) );
G=PermutationGroup([[(1,87),(2,88),(3,89),(4,90),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,61),(18,62),(19,63),(20,64),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120)], [(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),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,15),(12,14),(16,20),(17,19),(21,28),(22,27),(23,26),(24,25),(29,30),(31,38),(32,37),(33,36),(34,35),(39,40),(41,43),(44,50),(45,49),(46,48),(51,57),(52,56),(53,55),(58,60),(61,63),(64,70),(65,69),(66,68),(71,78),(72,77),(73,76),(74,75),(79,80),(81,82),(83,90),(84,89),(85,88),(86,87),(91,98),(92,97),(93,96),(94,95),(99,100),(101,103),(104,110),(105,109),(106,108),(111,117),(112,116),(113,115),(118,120)], [(1,70,25,57,40,45),(2,61,26,58,31,46),(3,62,27,59,32,47),(4,63,28,60,33,48),(5,64,29,51,34,49),(6,65,30,52,35,50),(7,66,21,53,36,41),(8,67,22,54,37,42),(9,68,23,55,38,43),(10,69,24,56,39,44),(11,80,112,95,110,82),(12,71,113,96,101,83),(13,72,114,97,102,84),(14,73,115,98,103,85),(15,74,116,99,104,86),(16,75,117,100,105,87),(17,76,118,91,106,88),(18,77,119,92,107,89),(19,78,120,93,108,90),(20,79,111,94,109,81)], [(1,110),(2,109),(3,108),(4,107),(5,106),(6,105),(7,104),(8,103),(9,102),(10,101),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,116),(22,115),(23,114),(24,113),(25,112),(26,111),(27,120),(28,119),(29,118),(30,117),(41,86),(42,85),(43,84),(44,83),(45,82),(46,81),(47,90),(48,89),(49,88),(50,87),(51,76),(52,75),(53,74),(54,73),(55,72),(56,71),(57,80),(58,79),(59,78),(60,77),(61,94),(62,93),(63,92),(64,91),(65,100),(66,99),(67,98),(68,97),(69,96),(70,95)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 10 | 10 | 15 | 15 | 15 | 15 | 30 | 30 | 2 | 6 | 6 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | S3×D4 | S3×D5 | D4×D5 | C2×S3×D5 | D10⋊D6 |
| kernel | C2×D10⋊D6 | C2×D30.C2 | C2×C3⋊D20 | C2×C5⋊D12 | D10⋊D6 | C6×C5⋊D4 | C10×C3⋊D4 | C22×S3×D5 | C23×D15 | C2×C5⋊D4 | D30 | C2×C3⋊D4 | C2×Dic5 | C5⋊D4 | C22×D5 | C22×C10 | C2×Dic3 | C3⋊D4 | C22×S3 | C22×C6 | C10 | C23 | C6 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 4 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 6 | 8 |
Matrix representation of C2×D10⋊D6 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 43 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 42 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 53 | 1 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 46 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 44 | 0 | 0 |
| 0 | 0 | 19 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,43,0,0,0,0,17,43,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,42,0,0,0,0,0,60,0,0,0,0,0,0,60,53,0,0,0,0,0,1],[0,1,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,46,1],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,43,19,0,0,0,0,44,18,0,0,0,0,0,0,1,0,0,0,0,0,15,60] >;
C2×D10⋊D6 in GAP, Magma, Sage, TeX
C_2\times D_{10}\rtimes D_6 % in TeX
G:=Group("C2xD10:D6"); // GroupNames label
G:=SmallGroup(480,1124);
// by ID
G=gap.SmallGroup(480,1124);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,675,346,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^5*c,e*c*e=b^3*c,e*d*e=d^-1>;
// generators/relations