direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D5×Q16, C40.33C23, C20.10C24, Dic20⋊15C22, Dic10.6C23, C10⋊2(C2×Q16), C4.46(D4×D5), C5⋊2(C22×Q16), (C10×Q16)⋊7C2, (C4×D5).69D4, C20.85(C2×D4), (C2×C8).246D10, (C5×Q16)⋊8C22, C5⋊Q16⋊8C22, C8.39(C22×D5), C4.10(C23×D5), (C2×Dic20)⋊20C2, D10.113(C2×D4), (Q8×D5).8C22, (C5×Q8).4C23, Q8.4(C22×D5), (C2×C40).98C22, C5⋊2C8.22C23, (C2×Q8).152D10, Dic5.25(C2×D4), (C4×D5).63C23, (C8×D5).44C22, C22.142(D4×D5), (C2×C20).527C23, (C2×Dic5).168D4, (C22×D5).160D4, C10.111(C22×D4), (Q8×C10).149C22, (C2×Dic10).204C22, (D5×C2×C8).6C2, C2.84(C2×D4×D5), (C2×Q8×D5).8C2, (C2×C5⋊Q16)⋊27C2, (C2×C10).400(C2×D4), (C2×C4×D5).329C22, (C2×C4).615(C22×D5), (C2×C5⋊2C8).293C22, SmallGroup(320,1435)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D5×Q16
G = < a,b,c,d,e | a2=b5=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 862 in 258 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, Q8, Q8, C23, D5, C10, C10, C2×C8, C2×C8, Q16, Q16, C22×C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C22×C8, C2×Q16, C2×Q16, C22×Q8, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×Q16, C8×D5, Dic20, C2×C5⋊2C8, C5⋊Q16, C2×C40, C5×Q16, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5, Q8×D5, Q8×C10, D5×C2×C8, C2×Dic20, D5×Q16, C2×C5⋊Q16, C10×Q16, C2×Q8×D5, C2×D5×Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C24, D10, C2×Q16, C22×D4, C22×D5, C22×Q16, D4×D5, C23×D5, D5×Q16, C2×D4×D5, C2×D5×Q16
►On 160 points
Generators in S160
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 33)(9 108)(10 109)(11 110)(12 111)(13 112)(14 105)(15 106)(16 107)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 97)(24 98)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 73)(32 74)(41 147)(42 148)(43 149)(44 150)(45 151)(46 152)(47 145)(48 146)(49 143)(50 144)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 153)(58 154)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 118)(66 119)(67 120)(68 113)(69 114)(70 115)(71 116)(72 117)(81 126)(82 127)(83 128)(84 121)(85 122)(86 123)(87 124)(88 125)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)
(1 138 150 71 12)(2 139 151 72 13)(3 140 152 65 14)(4 141 145 66 15)(5 142 146 67 16)(6 143 147 68 9)(7 144 148 69 10)(8 137 149 70 11)(17 80 159 96 87)(18 73 160 89 88)(19 74 153 90 81)(20 75 154 91 82)(21 76 155 92 83)(22 77 156 93 84)(23 78 157 94 85)(24 79 158 95 86)(25 58 131 127 102)(26 59 132 128 103)(27 60 133 121 104)(28 61 134 122 97)(29 62 135 123 98)(30 63 136 124 99)(31 64 129 125 100)(32 57 130 126 101)(33 51 43 115 110)(34 52 44 116 111)(35 53 45 117 112)(36 54 46 118 105)(37 55 47 119 106)(38 56 48 120 107)(39 49 41 113 108)(40 50 42 114 109)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 96)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)(33 110)(34 111)(35 112)(36 105)(37 106)(38 107)(39 108)(40 109)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(65 140)(66 141)(67 142)(68 143)(69 144)(70 137)(71 138)(72 139)(73 160)(74 153)(75 154)(76 155)(77 156)(78 157)(79 158)(80 159)(97 134)(98 135)(99 136)(100 129)(101 130)(102 131)(103 132)(104 133)
(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 58 5 62)(2 57 6 61)(3 64 7 60)(4 63 8 59)(9 28 13 32)(10 27 14 31)(11 26 15 30)(12 25 16 29)(17 115 21 119)(18 114 22 118)(19 113 23 117)(20 120 24 116)(33 155 37 159)(34 154 38 158)(35 153 39 157)(36 160 40 156)(41 85 45 81)(42 84 46 88)(43 83 47 87)(44 82 48 86)(49 94 53 90)(50 93 54 89)(51 92 55 96)(52 91 56 95)(65 100 69 104)(66 99 70 103)(67 98 71 102)(68 97 72 101)(73 109 77 105)(74 108 78 112)(75 107 79 111)(76 106 80 110)(121 152 125 148)(122 151 126 147)(123 150 127 146)(124 149 128 145)(129 144 133 140)(130 143 134 139)(131 142 135 138)(132 141 136 137)
G:=sub<Sym(160)| (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,108)(10,109)(11,110)(12,111)(13,112)(14,105)(15,106)(16,107)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,97)(24,98)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,145)(48,146)(49,143)(50,144)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(81,126)(82,127)(83,128)(84,121)(85,122)(86,123)(87,124)(88,125)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136), (1,138,150,71,12)(2,139,151,72,13)(3,140,152,65,14)(4,141,145,66,15)(5,142,146,67,16)(6,143,147,68,9)(7,144,148,69,10)(8,137,149,70,11)(17,80,159,96,87)(18,73,160,89,88)(19,74,153,90,81)(20,75,154,91,82)(21,76,155,92,83)(22,77,156,93,84)(23,78,157,94,85)(24,79,158,95,86)(25,58,131,127,102)(26,59,132,128,103)(27,60,133,121,104)(28,61,134,122,97)(29,62,135,123,98)(30,63,136,124,99)(31,64,129,125,100)(32,57,130,126,101)(33,51,43,115,110)(34,52,44,116,111)(35,53,45,117,112)(36,54,46,118,105)(37,55,47,119,106)(38,56,48,120,107)(39,49,41,113,108)(40,50,42,114,109), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,96)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(33,110)(34,111)(35,112)(36,105)(37,106)(38,107)(39,108)(40,109)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(65,140)(66,141)(67,142)(68,143)(69,144)(70,137)(71,138)(72,139)(73,160)(74,153)(75,154)(76,155)(77,156)(78,157)(79,158)(80,159)(97,134)(98,135)(99,136)(100,129)(101,130)(102,131)(103,132)(104,133), (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), (1,58,5,62)(2,57,6,61)(3,64,7,60)(4,63,8,59)(9,28,13,32)(10,27,14,31)(11,26,15,30)(12,25,16,29)(17,115,21,119)(18,114,22,118)(19,113,23,117)(20,120,24,116)(33,155,37,159)(34,154,38,158)(35,153,39,157)(36,160,40,156)(41,85,45,81)(42,84,46,88)(43,83,47,87)(44,82,48,86)(49,94,53,90)(50,93,54,89)(51,92,55,96)(52,91,56,95)(65,100,69,104)(66,99,70,103)(67,98,71,102)(68,97,72,101)(73,109,77,105)(74,108,78,112)(75,107,79,111)(76,106,80,110)(121,152,125,148)(122,151,126,147)(123,150,127,146)(124,149,128,145)(129,144,133,140)(130,143,134,139)(131,142,135,138)(132,141,136,137)>;
G:=Group( (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,108)(10,109)(11,110)(12,111)(13,112)(14,105)(15,106)(16,107)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,97)(24,98)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,73)(32,74)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,145)(48,146)(49,143)(50,144)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(81,126)(82,127)(83,128)(84,121)(85,122)(86,123)(87,124)(88,125)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136), (1,138,150,71,12)(2,139,151,72,13)(3,140,152,65,14)(4,141,145,66,15)(5,142,146,67,16)(6,143,147,68,9)(7,144,148,69,10)(8,137,149,70,11)(17,80,159,96,87)(18,73,160,89,88)(19,74,153,90,81)(20,75,154,91,82)(21,76,155,92,83)(22,77,156,93,84)(23,78,157,94,85)(24,79,158,95,86)(25,58,131,127,102)(26,59,132,128,103)(27,60,133,121,104)(28,61,134,122,97)(29,62,135,123,98)(30,63,136,124,99)(31,64,129,125,100)(32,57,130,126,101)(33,51,43,115,110)(34,52,44,116,111)(35,53,45,117,112)(36,54,46,118,105)(37,55,47,119,106)(38,56,48,120,107)(39,49,41,113,108)(40,50,42,114,109), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,96)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(33,110)(34,111)(35,112)(36,105)(37,106)(38,107)(39,108)(40,109)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(65,140)(66,141)(67,142)(68,143)(69,144)(70,137)(71,138)(72,139)(73,160)(74,153)(75,154)(76,155)(77,156)(78,157)(79,158)(80,159)(97,134)(98,135)(99,136)(100,129)(101,130)(102,131)(103,132)(104,133), (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), (1,58,5,62)(2,57,6,61)(3,64,7,60)(4,63,8,59)(9,28,13,32)(10,27,14,31)(11,26,15,30)(12,25,16,29)(17,115,21,119)(18,114,22,118)(19,113,23,117)(20,120,24,116)(33,155,37,159)(34,154,38,158)(35,153,39,157)(36,160,40,156)(41,85,45,81)(42,84,46,88)(43,83,47,87)(44,82,48,86)(49,94,53,90)(50,93,54,89)(51,92,55,96)(52,91,56,95)(65,100,69,104)(66,99,70,103)(67,98,71,102)(68,97,72,101)(73,109,77,105)(74,108,78,112)(75,107,79,111)(76,106,80,110)(121,152,125,148)(122,151,126,147)(123,150,127,146)(124,149,128,145)(129,144,133,140)(130,143,134,139)(131,142,135,138)(132,141,136,137) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,33),(9,108),(10,109),(11,110),(12,111),(13,112),(14,105),(15,106),(16,107),(17,99),(18,100),(19,101),(20,102),(21,103),(22,104),(23,97),(24,98),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,73),(32,74),(41,147),(42,148),(43,149),(44,150),(45,151),(46,152),(47,145),(48,146),(49,143),(50,144),(51,137),(52,138),(53,139),(54,140),(55,141),(56,142),(57,153),(58,154),(59,155),(60,156),(61,157),(62,158),(63,159),(64,160),(65,118),(66,119),(67,120),(68,113),(69,114),(70,115),(71,116),(72,117),(81,126),(82,127),(83,128),(84,121),(85,122),(86,123),(87,124),(88,125),(89,129),(90,130),(91,131),(92,132),(93,133),(94,134),(95,135),(96,136)], [(1,138,150,71,12),(2,139,151,72,13),(3,140,152,65,14),(4,141,145,66,15),(5,142,146,67,16),(6,143,147,68,9),(7,144,148,69,10),(8,137,149,70,11),(17,80,159,96,87),(18,73,160,89,88),(19,74,153,90,81),(20,75,154,91,82),(21,76,155,92,83),(22,77,156,93,84),(23,78,157,94,85),(24,79,158,95,86),(25,58,131,127,102),(26,59,132,128,103),(27,60,133,121,104),(28,61,134,122,97),(29,62,135,123,98),(30,63,136,124,99),(31,64,129,125,100),(32,57,130,126,101),(33,51,43,115,110),(34,52,44,116,111),(35,53,45,117,112),(36,54,46,118,105),(37,55,47,119,106),(38,56,48,120,107),(39,49,41,113,108),(40,50,42,114,109)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,96),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,57),(33,110),(34,111),(35,112),(36,105),(37,106),(38,107),(39,108),(40,109),(49,113),(50,114),(51,115),(52,116),(53,117),(54,118),(55,119),(56,120),(65,140),(66,141),(67,142),(68,143),(69,144),(70,137),(71,138),(72,139),(73,160),(74,153),(75,154),(76,155),(77,156),(78,157),(79,158),(80,159),(97,134),(98,135),(99,136),(100,129),(101,130),(102,131),(103,132),(104,133)], [(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)], [(1,58,5,62),(2,57,6,61),(3,64,7,60),(4,63,8,59),(9,28,13,32),(10,27,14,31),(11,26,15,30),(12,25,16,29),(17,115,21,119),(18,114,22,118),(19,113,23,117),(20,120,24,116),(33,155,37,159),(34,154,38,158),(35,153,39,157),(36,160,40,156),(41,85,45,81),(42,84,46,88),(43,83,47,87),(44,82,48,86),(49,94,53,90),(50,93,54,89),(51,92,55,96),(52,91,56,95),(65,100,69,104),(66,99,70,103),(67,98,71,102),(68,97,72,101),(73,109,77,105),(74,108,78,112),(75,107,79,111),(76,106,80,110),(121,152,125,148),(122,151,126,147),(123,150,127,146),(124,149,128,145),(129,144,133,140),(130,143,134,139),(131,142,135,138),(132,141,136,137)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | Q16 | D10 | D10 | D10 | D4×D5 | D4×D5 | D5×Q16 |
| kernel | C2×D5×Q16 | D5×C2×C8 | C2×Dic20 | D5×Q16 | C2×C5⋊Q16 | C10×Q16 | C2×Q8×D5 | C4×D5 | C2×Dic5 | C22×D5 | C2×Q16 | D10 | C2×C8 | Q16 | C2×Q8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 8 | 2 | 8 | 4 | 2 | 2 | 8 |
Matrix representation of C2×D5×Q16 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 1 |
| 0 | 0 | 33 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 33 | 1 |
| 24 | 35 | 0 | 0 |
| 7 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 37 | 40 | 0 | 0 |
| 17 | 4 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,33,0,0,1,7],[1,0,0,0,0,1,0,0,0,0,40,33,0,0,0,1],[24,7,0,0,35,0,0,0,0,0,40,0,0,0,0,40],[37,17,0,0,40,4,0,0,0,0,40,0,0,0,0,40] >;
C2×D5×Q16 in GAP, Magma, Sage, TeX
C_2\times D_5\times Q_{16} % in TeX
G:=Group("C2xD5xQ16"); // GroupNames label
G:=SmallGroup(320,1435);
// by ID
G=gap.SmallGroup(320,1435);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations