metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.Q8⋊9D5, C4⋊C4.40D10, C4⋊D20.7C2, (C2×C8).139D10, D20⋊5C4⋊32C2, D10⋊1C8⋊31C2, D20⋊6C4⋊17C2, C4.74(C4○D20), C20.31(C4○D4), C10.56(C4○D8), C10.D8⋊16C2, (C22×D5).33D4, C22.218(D4×D5), C2.23(D40⋊C2), C10.71(C8⋊C22), (C2×C20).282C23, (C2×C40).286C22, C4.26(Q8⋊2D5), (C2×Dic5).219D4, (C2×D20).80C22, C5⋊4(C23.19D4), C4⋊Dic5.112C22, C2.23(SD16⋊3D5), C2.13(D10.13D4), C10.43(C22.D4), C4⋊C4⋊7D5⋊6C2, (C5×C4.Q8)⋊17C2, (C2×C4×D5).38C22, (C2×C10).287(C2×D4), (C5×C4⋊C4).75C22, (C2×C5⋊2C8).59C22, (C2×C4).385(C22×D5), SmallGroup(320,493)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Q8⋊D5
G = < a,b,c,d,e | a4=d5=e2=1, b4=a2, c2=a-1b2, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b3, bd=db, ebe=a-1b3, cd=dc, ece=a2c, ede=d-1 >
Subgroups: 526 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C23.19D4, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, C10.D8, D20⋊6C4, D10⋊1C8, D20⋊5C4, C5×C4.Q8, C4⋊C4⋊7D5, C4⋊D20, C4.Q8⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8⋊C22, C22×D5, C23.19D4, C4○D20, D4×D5, Q8⋊2D5, D10.13D4, D40⋊C2, SD16⋊3D5, C4.Q8⋊D5
►On 160 points
Generators in S160
(1 73 5 77)(2 74 6 78)(3 75 7 79)(4 76 8 80)(9 150 13 146)(10 151 14 147)(11 152 15 148)(12 145 16 149)(17 92 21 96)(18 93 22 89)(19 94 23 90)(20 95 24 91)(25 34 29 38)(26 35 30 39)(27 36 31 40)(28 37 32 33)(41 136 45 132)(42 129 46 133)(43 130 47 134)(44 131 48 135)(49 103 53 99)(50 104 54 100)(51 97 55 101)(52 98 56 102)(57 87 61 83)(58 88 62 84)(59 81 63 85)(60 82 64 86)(65 122 69 126)(66 123 70 127)(67 124 71 128)(68 125 72 121)(105 138 109 142)(106 139 110 143)(107 140 111 144)(108 141 112 137)(113 154 117 158)(114 155 118 159)(115 156 119 160)(116 157 120 153)
(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)
(1 118 79 157)(2 113 80 160)(3 116 73 155)(4 119 74 158)(5 114 75 153)(6 117 76 156)(7 120 77 159)(8 115 78 154)(9 124 148 69)(10 127 149 72)(11 122 150 67)(12 125 151 70)(13 128 152 65)(14 123 145 68)(15 126 146 71)(16 121 147 66)(17 46 90 131)(18 41 91 134)(19 44 92 129)(20 47 93 132)(21 42 94 135)(22 45 95 130)(23 48 96 133)(24 43 89 136)(25 110 40 141)(26 105 33 144)(27 108 34 139)(28 111 35 142)(29 106 36 137)(30 109 37 140)(31 112 38 143)(32 107 39 138)(49 81 101 61)(50 84 102 64)(51 87 103 59)(52 82 104 62)(53 85 97 57)(54 88 98 60)(55 83 99 63)(56 86 100 58)
(1 57 67 37 21)(2 58 68 38 22)(3 59 69 39 23)(4 60 70 40 24)(5 61 71 33 17)(6 62 72 34 18)(7 63 65 35 19)(8 64 66 36 20)(9 138 48 116 51)(10 139 41 117 52)(11 140 42 118 53)(12 141 43 119 54)(13 142 44 120 55)(14 143 45 113 56)(15 144 46 114 49)(16 137 47 115 50)(25 89 74 88 125)(26 90 75 81 126)(27 91 76 82 127)(28 92 77 83 128)(29 93 78 84 121)(30 94 79 85 122)(31 95 80 86 123)(32 96 73 87 124)(97 150 109 135 157)(98 151 110 136 158)(99 152 111 129 159)(100 145 112 130 160)(101 146 105 131 153)(102 147 106 132 154)(103 148 107 133 155)(104 149 108 134 156)
(1 132)(2 48)(3 134)(4 42)(5 136)(6 44)(7 130)(8 46)(9 68)(10 124)(11 70)(12 126)(13 72)(14 128)(15 66)(16 122)(17 158)(18 120)(19 160)(20 114)(21 154)(22 116)(23 156)(24 118)(25 97)(26 54)(27 99)(28 56)(29 101)(30 50)(31 103)(32 52)(33 98)(34 55)(35 100)(36 49)(37 102)(38 51)(39 104)(40 53)(41 73)(43 75)(45 77)(47 79)(57 106)(58 138)(59 108)(60 140)(61 110)(62 142)(63 112)(64 144)(65 145)(67 147)(69 149)(71 151)(74 135)(76 129)(78 131)(80 133)(81 141)(82 111)(83 143)(84 105)(85 137)(86 107)(87 139)(88 109)(89 157)(90 119)(91 159)(92 113)(93 153)(94 115)(95 155)(96 117)(121 146)(123 148)(125 150)(127 152)
G:=sub<Sym(160)| (1,73,5,77)(2,74,6,78)(3,75,7,79)(4,76,8,80)(9,150,13,146)(10,151,14,147)(11,152,15,148)(12,145,16,149)(17,92,21,96)(18,93,22,89)(19,94,23,90)(20,95,24,91)(25,34,29,38)(26,35,30,39)(27,36,31,40)(28,37,32,33)(41,136,45,132)(42,129,46,133)(43,130,47,134)(44,131,48,135)(49,103,53,99)(50,104,54,100)(51,97,55,101)(52,98,56,102)(57,87,61,83)(58,88,62,84)(59,81,63,85)(60,82,64,86)(65,122,69,126)(66,123,70,127)(67,124,71,128)(68,125,72,121)(105,138,109,142)(106,139,110,143)(107,140,111,144)(108,141,112,137)(113,154,117,158)(114,155,118,159)(115,156,119,160)(116,157,120,153), (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), (1,118,79,157)(2,113,80,160)(3,116,73,155)(4,119,74,158)(5,114,75,153)(6,117,76,156)(7,120,77,159)(8,115,78,154)(9,124,148,69)(10,127,149,72)(11,122,150,67)(12,125,151,70)(13,128,152,65)(14,123,145,68)(15,126,146,71)(16,121,147,66)(17,46,90,131)(18,41,91,134)(19,44,92,129)(20,47,93,132)(21,42,94,135)(22,45,95,130)(23,48,96,133)(24,43,89,136)(25,110,40,141)(26,105,33,144)(27,108,34,139)(28,111,35,142)(29,106,36,137)(30,109,37,140)(31,112,38,143)(32,107,39,138)(49,81,101,61)(50,84,102,64)(51,87,103,59)(52,82,104,62)(53,85,97,57)(54,88,98,60)(55,83,99,63)(56,86,100,58), (1,57,67,37,21)(2,58,68,38,22)(3,59,69,39,23)(4,60,70,40,24)(5,61,71,33,17)(6,62,72,34,18)(7,63,65,35,19)(8,64,66,36,20)(9,138,48,116,51)(10,139,41,117,52)(11,140,42,118,53)(12,141,43,119,54)(13,142,44,120,55)(14,143,45,113,56)(15,144,46,114,49)(16,137,47,115,50)(25,89,74,88,125)(26,90,75,81,126)(27,91,76,82,127)(28,92,77,83,128)(29,93,78,84,121)(30,94,79,85,122)(31,95,80,86,123)(32,96,73,87,124)(97,150,109,135,157)(98,151,110,136,158)(99,152,111,129,159)(100,145,112,130,160)(101,146,105,131,153)(102,147,106,132,154)(103,148,107,133,155)(104,149,108,134,156), (1,132)(2,48)(3,134)(4,42)(5,136)(6,44)(7,130)(8,46)(9,68)(10,124)(11,70)(12,126)(13,72)(14,128)(15,66)(16,122)(17,158)(18,120)(19,160)(20,114)(21,154)(22,116)(23,156)(24,118)(25,97)(26,54)(27,99)(28,56)(29,101)(30,50)(31,103)(32,52)(33,98)(34,55)(35,100)(36,49)(37,102)(38,51)(39,104)(40,53)(41,73)(43,75)(45,77)(47,79)(57,106)(58,138)(59,108)(60,140)(61,110)(62,142)(63,112)(64,144)(65,145)(67,147)(69,149)(71,151)(74,135)(76,129)(78,131)(80,133)(81,141)(82,111)(83,143)(84,105)(85,137)(86,107)(87,139)(88,109)(89,157)(90,119)(91,159)(92,113)(93,153)(94,115)(95,155)(96,117)(121,146)(123,148)(125,150)(127,152)>;
G:=Group( (1,73,5,77)(2,74,6,78)(3,75,7,79)(4,76,8,80)(9,150,13,146)(10,151,14,147)(11,152,15,148)(12,145,16,149)(17,92,21,96)(18,93,22,89)(19,94,23,90)(20,95,24,91)(25,34,29,38)(26,35,30,39)(27,36,31,40)(28,37,32,33)(41,136,45,132)(42,129,46,133)(43,130,47,134)(44,131,48,135)(49,103,53,99)(50,104,54,100)(51,97,55,101)(52,98,56,102)(57,87,61,83)(58,88,62,84)(59,81,63,85)(60,82,64,86)(65,122,69,126)(66,123,70,127)(67,124,71,128)(68,125,72,121)(105,138,109,142)(106,139,110,143)(107,140,111,144)(108,141,112,137)(113,154,117,158)(114,155,118,159)(115,156,119,160)(116,157,120,153), (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), (1,118,79,157)(2,113,80,160)(3,116,73,155)(4,119,74,158)(5,114,75,153)(6,117,76,156)(7,120,77,159)(8,115,78,154)(9,124,148,69)(10,127,149,72)(11,122,150,67)(12,125,151,70)(13,128,152,65)(14,123,145,68)(15,126,146,71)(16,121,147,66)(17,46,90,131)(18,41,91,134)(19,44,92,129)(20,47,93,132)(21,42,94,135)(22,45,95,130)(23,48,96,133)(24,43,89,136)(25,110,40,141)(26,105,33,144)(27,108,34,139)(28,111,35,142)(29,106,36,137)(30,109,37,140)(31,112,38,143)(32,107,39,138)(49,81,101,61)(50,84,102,64)(51,87,103,59)(52,82,104,62)(53,85,97,57)(54,88,98,60)(55,83,99,63)(56,86,100,58), (1,57,67,37,21)(2,58,68,38,22)(3,59,69,39,23)(4,60,70,40,24)(5,61,71,33,17)(6,62,72,34,18)(7,63,65,35,19)(8,64,66,36,20)(9,138,48,116,51)(10,139,41,117,52)(11,140,42,118,53)(12,141,43,119,54)(13,142,44,120,55)(14,143,45,113,56)(15,144,46,114,49)(16,137,47,115,50)(25,89,74,88,125)(26,90,75,81,126)(27,91,76,82,127)(28,92,77,83,128)(29,93,78,84,121)(30,94,79,85,122)(31,95,80,86,123)(32,96,73,87,124)(97,150,109,135,157)(98,151,110,136,158)(99,152,111,129,159)(100,145,112,130,160)(101,146,105,131,153)(102,147,106,132,154)(103,148,107,133,155)(104,149,108,134,156), (1,132)(2,48)(3,134)(4,42)(5,136)(6,44)(7,130)(8,46)(9,68)(10,124)(11,70)(12,126)(13,72)(14,128)(15,66)(16,122)(17,158)(18,120)(19,160)(20,114)(21,154)(22,116)(23,156)(24,118)(25,97)(26,54)(27,99)(28,56)(29,101)(30,50)(31,103)(32,52)(33,98)(34,55)(35,100)(36,49)(37,102)(38,51)(39,104)(40,53)(41,73)(43,75)(45,77)(47,79)(57,106)(58,138)(59,108)(60,140)(61,110)(62,142)(63,112)(64,144)(65,145)(67,147)(69,149)(71,151)(74,135)(76,129)(78,131)(80,133)(81,141)(82,111)(83,143)(84,105)(85,137)(86,107)(87,139)(88,109)(89,157)(90,119)(91,159)(92,113)(93,153)(94,115)(95,155)(96,117)(121,146)(123,148)(125,150)(127,152) );
G=PermutationGroup([[(1,73,5,77),(2,74,6,78),(3,75,7,79),(4,76,8,80),(9,150,13,146),(10,151,14,147),(11,152,15,148),(12,145,16,149),(17,92,21,96),(18,93,22,89),(19,94,23,90),(20,95,24,91),(25,34,29,38),(26,35,30,39),(27,36,31,40),(28,37,32,33),(41,136,45,132),(42,129,46,133),(43,130,47,134),(44,131,48,135),(49,103,53,99),(50,104,54,100),(51,97,55,101),(52,98,56,102),(57,87,61,83),(58,88,62,84),(59,81,63,85),(60,82,64,86),(65,122,69,126),(66,123,70,127),(67,124,71,128),(68,125,72,121),(105,138,109,142),(106,139,110,143),(107,140,111,144),(108,141,112,137),(113,154,117,158),(114,155,118,159),(115,156,119,160),(116,157,120,153)], [(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)], [(1,118,79,157),(2,113,80,160),(3,116,73,155),(4,119,74,158),(5,114,75,153),(6,117,76,156),(7,120,77,159),(8,115,78,154),(9,124,148,69),(10,127,149,72),(11,122,150,67),(12,125,151,70),(13,128,152,65),(14,123,145,68),(15,126,146,71),(16,121,147,66),(17,46,90,131),(18,41,91,134),(19,44,92,129),(20,47,93,132),(21,42,94,135),(22,45,95,130),(23,48,96,133),(24,43,89,136),(25,110,40,141),(26,105,33,144),(27,108,34,139),(28,111,35,142),(29,106,36,137),(30,109,37,140),(31,112,38,143),(32,107,39,138),(49,81,101,61),(50,84,102,64),(51,87,103,59),(52,82,104,62),(53,85,97,57),(54,88,98,60),(55,83,99,63),(56,86,100,58)], [(1,57,67,37,21),(2,58,68,38,22),(3,59,69,39,23),(4,60,70,40,24),(5,61,71,33,17),(6,62,72,34,18),(7,63,65,35,19),(8,64,66,36,20),(9,138,48,116,51),(10,139,41,117,52),(11,140,42,118,53),(12,141,43,119,54),(13,142,44,120,55),(14,143,45,113,56),(15,144,46,114,49),(16,137,47,115,50),(25,89,74,88,125),(26,90,75,81,126),(27,91,76,82,127),(28,92,77,83,128),(29,93,78,84,121),(30,94,79,85,122),(31,95,80,86,123),(32,96,73,87,124),(97,150,109,135,157),(98,151,110,136,158),(99,152,111,129,159),(100,145,112,130,160),(101,146,105,131,153),(102,147,106,132,154),(103,148,107,133,155),(104,149,108,134,156)], [(1,132),(2,48),(3,134),(4,42),(5,136),(6,44),(7,130),(8,46),(9,68),(10,124),(11,70),(12,126),(13,72),(14,128),(15,66),(16,122),(17,158),(18,120),(19,160),(20,114),(21,154),(22,116),(23,156),(24,118),(25,97),(26,54),(27,99),(28,56),(29,101),(30,50),(31,103),(32,52),(33,98),(34,55),(35,100),(36,49),(37,102),(38,51),(39,104),(40,53),(41,73),(43,75),(45,77),(47,79),(57,106),(58,138),(59,108),(60,140),(61,110),(62,142),(63,112),(64,144),(65,145),(67,147),(69,149),(71,151),(74,135),(76,129),(78,131),(80,133),(81,141),(82,111),(83,143),(84,105),(85,137),(86,107),(87,139),(88,109),(89,157),(90,119),(91,159),(92,113),(93,153),(94,115),(95,155),(96,117),(121,146),(123,148),(125,150),(127,152)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 40 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | C4○D20 | C8⋊C22 | Q8⋊2D5 | D4×D5 | D40⋊C2 | SD16⋊3D5 |
| kernel | C4.Q8⋊D5 | C10.D8 | D20⋊6C4 | D10⋊1C8 | D20⋊5C4 | C5×C4.Q8 | C4⋊C4⋊7D5 | C4⋊D20 | C2×Dic5 | C22×D5 | C4.Q8 | C20 | C4⋊C4 | C2×C8 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C4.Q8⋊D5 ►in GL4(𝔽41) generated by
| 40 | 39 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 30 | 30 | 0 | 0 |
| 26 | 0 | 0 | 0 |
| 0 | 0 | 23 | 6 |
| 0 | 0 | 35 | 18 |
| 0 | 24 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 6 |
| 32 | 23 | 0 | 0 |
| 9 | 9 | 0 | 0 |
| 0 | 0 | 28 | 39 |
| 0 | 0 | 2 | 13 |
G:=sub<GL(4,GF(41))| [40,1,0,0,39,1,0,0,0,0,40,0,0,0,0,40],[30,26,0,0,30,0,0,0,0,0,23,35,0,0,6,18],[0,12,0,0,24,0,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,6],[32,9,0,0,23,9,0,0,0,0,28,2,0,0,39,13] >;
C4.Q8⋊D5 in GAP, Magma, Sage, TeX
C_4.Q_8\rtimes D_5
% in TeX
G:=Group("C4.Q8:D5"); // GroupNames label
G:=SmallGroup(320,493);
// by ID
G=gap.SmallGroup(320,493);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,254,219,100,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=d^5=e^2=1,b^4=a^2,c^2=a^-1*b^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^3,b*d=d*b,e*b*e=a^-1*b^3,c*d=d*c,e*c*e=a^2*c,e*d*e=d^-1>;
// generators/relations