direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D4×Dic5, Q8⋊6(C2×Dic5), D4⋊7(C2×Dic5), (Q8×Dic5)⋊33C2, (D4×Dic5)⋊45C2, (C2×D4).252D10, C10.71(C23×C4), (C2×Q8).208D10, C20.158(C22×C4), (C2×C10).312C24, (C2×C20).888C23, (C22×C4).388D10, C2.12(C23×Dic5), C4.21(C22×Dic5), C22.48(C23×D5), (D4×C10).274C22, C4⋊Dic5.391C22, (Q8×C10).241C22, C23.209(C22×D5), C23.21D10⋊36C2, C22.3(C22×Dic5), (C22×C10).238C23, (C22×C20).294C22, (C4×Dic5).361C22, (C2×Dic5).301C23, C23.D5.134C22, (C22×Dic5).258C22, C5⋊6(C4×C4○D4), C2.7(D5×C4○D4), (C2×C20)⋊29(C2×C4), (C5×C4○D4)⋊11C4, (C5×D4)⋊31(C2×C4), (C5×Q8)⋊28(C2×C4), (C2×C4×Dic5)⋊14C2, (C2×C4)⋊8(C2×Dic5), (C2×C4○D4).14D5, (C10×C4○D4).13C2, C10.214(C2×C4○D4), (C2×C4).637(C22×D5), (C2×C10).131(C22×C4), SmallGroup(320,1498)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×Dic5
G = < a,b,c,d,e | a4=c2=d10=1, b2=a2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 734 in 310 conjugacy classes, 195 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C4×C4○D4, C4×Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C22×Dic5, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C2×C4×Dic5, C23.21D10, D4×Dic5, Q8×Dic5, C10×C4○D4, C4○D4×Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, C24, Dic5, D10, C23×C4, C2×C4○D4, C2×Dic5, C22×D5, C4×C4○D4, C22×Dic5, C23×D5, D5×C4○D4, C23×Dic5, C4○D4×Dic5
►On 160 points
Generators in S160
(1 68 27 59)(2 69 28 60)(3 70 29 51)(4 61 30 52)(5 62 21 53)(6 63 22 54)(7 64 23 55)(8 65 24 56)(9 66 25 57)(10 67 26 58)(11 111 156 127)(12 112 157 128)(13 113 158 129)(14 114 159 130)(15 115 160 121)(16 116 151 122)(17 117 152 123)(18 118 153 124)(19 119 154 125)(20 120 155 126)(31 76 50 82)(32 77 41 83)(33 78 42 84)(34 79 43 85)(35 80 44 86)(36 71 45 87)(37 72 46 88)(38 73 47 89)(39 74 48 90)(40 75 49 81)(91 136 107 142)(92 137 108 143)(93 138 109 144)(94 139 110 145)(95 140 101 146)(96 131 102 147)(97 132 103 148)(98 133 104 149)(99 134 105 150)(100 135 106 141)
(1 54 27 63)(2 55 28 64)(3 56 29 65)(4 57 30 66)(5 58 21 67)(6 59 22 68)(7 60 23 69)(8 51 24 70)(9 52 25 61)(10 53 26 62)(11 116 156 122)(12 117 157 123)(13 118 158 124)(14 119 159 125)(15 120 160 126)(16 111 151 127)(17 112 152 128)(18 113 153 129)(19 114 154 130)(20 115 155 121)(31 71 50 87)(32 72 41 88)(33 73 42 89)(34 74 43 90)(35 75 44 81)(36 76 45 82)(37 77 46 83)(38 78 47 84)(39 79 48 85)(40 80 49 86)(91 147 107 131)(92 148 108 132)(93 149 109 133)(94 150 110 134)(95 141 101 135)(96 142 102 136)(97 143 103 137)(98 144 104 138)(99 145 105 139)(100 146 106 140)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 41)(10 42)(11 142)(12 143)(13 144)(14 145)(15 146)(16 147)(17 148)(18 149)(19 150)(20 141)(21 38)(22 39)(23 40)(24 31)(25 32)(26 33)(27 34)(28 35)(29 36)(30 37)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 88)(62 89)(63 90)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(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)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 131 6 136)(2 140 7 135)(3 139 8 134)(4 138 9 133)(5 137 10 132)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)(21 143 26 148)(22 142 27 147)(23 141 28 146)(24 150 29 145)(25 149 30 144)(41 153 46 158)(42 152 47 157)(43 151 48 156)(44 160 49 155)(45 159 50 154)(51 94 56 99)(52 93 57 98)(53 92 58 97)(54 91 59 96)(55 100 60 95)(61 109 66 104)(62 108 67 103)(63 107 68 102)(64 106 69 101)(65 105 70 110)(71 114 76 119)(72 113 77 118)(73 112 78 117)(74 111 79 116)(75 120 80 115)(81 126 86 121)(82 125 87 130)(83 124 88 129)(84 123 89 128)(85 122 90 127)
G:=sub<Sym(160)| (1,68,27,59)(2,69,28,60)(3,70,29,51)(4,61,30,52)(5,62,21,53)(6,63,22,54)(7,64,23,55)(8,65,24,56)(9,66,25,57)(10,67,26,58)(11,111,156,127)(12,112,157,128)(13,113,158,129)(14,114,159,130)(15,115,160,121)(16,116,151,122)(17,117,152,123)(18,118,153,124)(19,119,154,125)(20,120,155,126)(31,76,50,82)(32,77,41,83)(33,78,42,84)(34,79,43,85)(35,80,44,86)(36,71,45,87)(37,72,46,88)(38,73,47,89)(39,74,48,90)(40,75,49,81)(91,136,107,142)(92,137,108,143)(93,138,109,144)(94,139,110,145)(95,140,101,146)(96,131,102,147)(97,132,103,148)(98,133,104,149)(99,134,105,150)(100,135,106,141), (1,54,27,63)(2,55,28,64)(3,56,29,65)(4,57,30,66)(5,58,21,67)(6,59,22,68)(7,60,23,69)(8,51,24,70)(9,52,25,61)(10,53,26,62)(11,116,156,122)(12,117,157,123)(13,118,158,124)(14,119,159,125)(15,120,160,126)(16,111,151,127)(17,112,152,128)(18,113,153,129)(19,114,154,130)(20,115,155,121)(31,71,50,87)(32,72,41,88)(33,73,42,89)(34,74,43,90)(35,75,44,81)(36,76,45,82)(37,77,46,83)(38,78,47,84)(39,79,48,85)(40,80,49,86)(91,147,107,131)(92,148,108,132)(93,149,109,133)(94,150,110,134)(95,141,101,135)(96,142,102,136)(97,143,103,137)(98,144,104,138)(99,145,105,139)(100,146,106,140), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,41)(10,42)(11,142)(12,143)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,141)(21,38)(22,39)(23,40)(24,31)(25,32)(26,33)(27,34)(28,35)(29,36)(30,37)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,88)(62,89)(63,90)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (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)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,131,6,136)(2,140,7,135)(3,139,8,134)(4,138,9,133)(5,137,10,132)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35)(21,143,26,148)(22,142,27,147)(23,141,28,146)(24,150,29,145)(25,149,30,144)(41,153,46,158)(42,152,47,157)(43,151,48,156)(44,160,49,155)(45,159,50,154)(51,94,56,99)(52,93,57,98)(53,92,58,97)(54,91,59,96)(55,100,60,95)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,114,76,119)(72,113,77,118)(73,112,78,117)(74,111,79,116)(75,120,80,115)(81,126,86,121)(82,125,87,130)(83,124,88,129)(84,123,89,128)(85,122,90,127)>;
G:=Group( (1,68,27,59)(2,69,28,60)(3,70,29,51)(4,61,30,52)(5,62,21,53)(6,63,22,54)(7,64,23,55)(8,65,24,56)(9,66,25,57)(10,67,26,58)(11,111,156,127)(12,112,157,128)(13,113,158,129)(14,114,159,130)(15,115,160,121)(16,116,151,122)(17,117,152,123)(18,118,153,124)(19,119,154,125)(20,120,155,126)(31,76,50,82)(32,77,41,83)(33,78,42,84)(34,79,43,85)(35,80,44,86)(36,71,45,87)(37,72,46,88)(38,73,47,89)(39,74,48,90)(40,75,49,81)(91,136,107,142)(92,137,108,143)(93,138,109,144)(94,139,110,145)(95,140,101,146)(96,131,102,147)(97,132,103,148)(98,133,104,149)(99,134,105,150)(100,135,106,141), (1,54,27,63)(2,55,28,64)(3,56,29,65)(4,57,30,66)(5,58,21,67)(6,59,22,68)(7,60,23,69)(8,51,24,70)(9,52,25,61)(10,53,26,62)(11,116,156,122)(12,117,157,123)(13,118,158,124)(14,119,159,125)(15,120,160,126)(16,111,151,127)(17,112,152,128)(18,113,153,129)(19,114,154,130)(20,115,155,121)(31,71,50,87)(32,72,41,88)(33,73,42,89)(34,74,43,90)(35,75,44,81)(36,76,45,82)(37,77,46,83)(38,78,47,84)(39,79,48,85)(40,80,49,86)(91,147,107,131)(92,148,108,132)(93,149,109,133)(94,150,110,134)(95,141,101,135)(96,142,102,136)(97,143,103,137)(98,144,104,138)(99,145,105,139)(100,146,106,140), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,41)(10,42)(11,142)(12,143)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,141)(21,38)(22,39)(23,40)(24,31)(25,32)(26,33)(27,34)(28,35)(29,36)(30,37)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,88)(62,89)(63,90)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (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)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,131,6,136)(2,140,7,135)(3,139,8,134)(4,138,9,133)(5,137,10,132)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35)(21,143,26,148)(22,142,27,147)(23,141,28,146)(24,150,29,145)(25,149,30,144)(41,153,46,158)(42,152,47,157)(43,151,48,156)(44,160,49,155)(45,159,50,154)(51,94,56,99)(52,93,57,98)(53,92,58,97)(54,91,59,96)(55,100,60,95)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,114,76,119)(72,113,77,118)(73,112,78,117)(74,111,79,116)(75,120,80,115)(81,126,86,121)(82,125,87,130)(83,124,88,129)(84,123,89,128)(85,122,90,127) );
G=PermutationGroup([[(1,68,27,59),(2,69,28,60),(3,70,29,51),(4,61,30,52),(5,62,21,53),(6,63,22,54),(7,64,23,55),(8,65,24,56),(9,66,25,57),(10,67,26,58),(11,111,156,127),(12,112,157,128),(13,113,158,129),(14,114,159,130),(15,115,160,121),(16,116,151,122),(17,117,152,123),(18,118,153,124),(19,119,154,125),(20,120,155,126),(31,76,50,82),(32,77,41,83),(33,78,42,84),(34,79,43,85),(35,80,44,86),(36,71,45,87),(37,72,46,88),(38,73,47,89),(39,74,48,90),(40,75,49,81),(91,136,107,142),(92,137,108,143),(93,138,109,144),(94,139,110,145),(95,140,101,146),(96,131,102,147),(97,132,103,148),(98,133,104,149),(99,134,105,150),(100,135,106,141)], [(1,54,27,63),(2,55,28,64),(3,56,29,65),(4,57,30,66),(5,58,21,67),(6,59,22,68),(7,60,23,69),(8,51,24,70),(9,52,25,61),(10,53,26,62),(11,116,156,122),(12,117,157,123),(13,118,158,124),(14,119,159,125),(15,120,160,126),(16,111,151,127),(17,112,152,128),(18,113,153,129),(19,114,154,130),(20,115,155,121),(31,71,50,87),(32,72,41,88),(33,73,42,89),(34,74,43,90),(35,75,44,81),(36,76,45,82),(37,77,46,83),(38,78,47,84),(39,79,48,85),(40,80,49,86),(91,147,107,131),(92,148,108,132),(93,149,109,133),(94,150,110,134),(95,141,101,135),(96,142,102,136),(97,143,103,137),(98,144,104,138),(99,145,105,139),(100,146,106,140)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,41),(10,42),(11,142),(12,143),(13,144),(14,145),(15,146),(16,147),(17,148),(18,149),(19,150),(20,141),(21,38),(22,39),(23,40),(24,31),(25,32),(26,33),(27,34),(28,35),(29,36),(30,37),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,88),(62,89),(63,90),(64,81),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,127),(108,128),(109,129),(110,130),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)], [(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),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,131,6,136),(2,140,7,135),(3,139,8,134),(4,138,9,133),(5,137,10,132),(11,34,16,39),(12,33,17,38),(13,32,18,37),(14,31,19,36),(15,40,20,35),(21,143,26,148),(22,142,27,147),(23,141,28,146),(24,150,29,145),(25,149,30,144),(41,153,46,158),(42,152,47,157),(43,151,48,156),(44,160,49,155),(45,159,50,154),(51,94,56,99),(52,93,57,98),(53,92,58,97),(54,91,59,96),(55,100,60,95),(61,109,66,104),(62,108,67,103),(63,107,68,102),(64,106,69,101),(65,105,70,110),(71,114,76,119),(72,113,77,118),(73,112,78,117),(74,111,79,116),(75,120,80,115),(81,126,86,121),(82,125,87,130),(83,124,88,129),(84,123,89,128),(85,122,90,127)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4R | 4S | ··· | 4AD | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | C4○D4 | D10 | D10 | D10 | Dic5 | D5×C4○D4 |
| kernel | C4○D4×Dic5 | C2×C4×Dic5 | C23.21D10 | D4×Dic5 | Q8×Dic5 | C10×C4○D4 | C5×C4○D4 | C2×C4○D4 | Dic5 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 16 | 2 | 8 | 6 | 6 | 2 | 16 | 8 |
Matrix representation of C4○D4×Dic5 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 40 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 23 | 13 | 0 | 0 |
| 16 | 18 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,40,7,0,0,0,0,40,0,0,0,0,40],[23,16,0,0,13,18,0,0,0,0,9,0,0,0,0,9] >;
C4○D4×Dic5 in GAP, Magma, Sage, TeX
C_4\circ D_4\times {\rm Dic}_5 % in TeX
G:=Group("C4oD4xDic5"); // GroupNames label
G:=SmallGroup(320,1498);
// by ID
G=gap.SmallGroup(320,1498);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,570,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^2=d^10=1,b^2=a^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^2*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations