direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C4.Dic5, C20.74C24, C24.5Dic5, (C23×C4).9D5, C5⋊2C8⋊13C23, C10⋊6(C2×M4(2)), C4.73(C23×D5), C5⋊6(C22×M4(2)), (C2×C10)⋊15M4(2), C10.62(C23×C4), (C23×C10).17C4, (C22×C20).56C4, (C23×C20).16C2, C2.3(C23×Dic5), (C2×C20).799C23, C20.239(C22×C4), (C22×C4).472D10, C23.35(C2×Dic5), C4.38(C22×Dic5), (C22×C4).18Dic5, (C22×C20).546C22, C22.28(C22×Dic5), (C2×C20).473(C2×C4), (C2×C5⋊2C8)⋊48C22, (C22×C5⋊2C8)⋊23C2, (C2×C4).86(C2×Dic5), (C2×C4).827(C22×D5), (C2×C10).306(C22×C4), (C22×C10).208(C2×C4), SmallGroup(320,1453)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 542 in 298 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×7], C22 [×11], C22 [×12], C5, C8 [×8], C2×C4 [×28], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C2×C8 [×12], M4(2) [×16], C22×C4 [×2], C22×C4 [×12], C24, C20, C20 [×7], C2×C10 [×11], C2×C10 [×12], C22×C8 [×2], C2×M4(2) [×12], C23×C4, C5⋊2C8 [×8], C2×C20 [×28], C22×C10, C22×C10 [×6], C22×C10 [×4], C22×M4(2), C2×C5⋊2C8 [×12], C4.Dic5 [×16], C22×C20 [×2], C22×C20 [×12], C23×C10, C22×C5⋊2C8 [×2], C2×C4.Dic5 [×12], C23×C20, C22×C4.Dic5
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, M4(2) [×4], C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C2×M4(2) [×6], C23×C4, C2×Dic5 [×28], C22×D5 [×7], C22×M4(2), C4.Dic5 [×4], C22×Dic5 [×14], C23×D5, C2×C4.Dic5 [×6], C23×Dic5, C22×C4.Dic5
Generators and relations
G = < a,b,c,d,e | a2=b2=c4=1, d10=c2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d9 >
►On 160 points
Generators in S160
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 76)(22 77)(23 78)(24 79)(25 80)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 133)(92 134)(93 135)(94 136)(95 137)(96 138)(97 139)(98 140)(99 121)(100 122)(101 157)(102 158)(103 159)(104 160)(105 141)(106 142)(107 143)(108 144)(109 145)(110 146)(111 147)(112 148)(113 149)(114 150)(115 151)(116 152)(117 153)(118 154)(119 155)(120 156)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 21)(17 22)(18 23)(19 24)(20 25)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 101)(99 102)(100 103)(121 158)(122 159)(123 160)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(131 148)(132 149)(133 150)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 157)
(1 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 66 51 76)(42 67 52 77)(43 68 53 78)(44 69 54 79)(45 70 55 80)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)(81 119 91 109)(82 120 92 110)(83 101 93 111)(84 102 94 112)(85 103 95 113)(86 104 96 114)(87 105 97 115)(88 106 98 116)(89 107 99 117)(90 108 100 118)(121 153 131 143)(122 154 132 144)(123 155 133 145)(124 156 134 146)(125 157 135 147)(126 158 136 148)(127 159 137 149)(128 160 138 150)(129 141 139 151)(130 142 140 152)
(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 94 6 99 11 84 16 89)(2 83 7 88 12 93 17 98)(3 92 8 97 13 82 18 87)(4 81 9 86 14 91 19 96)(5 90 10 95 15 100 20 85)(21 112 26 117 31 102 36 107)(22 101 27 106 32 111 37 116)(23 110 28 115 33 120 38 105)(24 119 29 104 34 109 39 114)(25 108 30 113 35 118 40 103)(41 136 46 121 51 126 56 131)(42 125 47 130 52 135 57 140)(43 134 48 139 53 124 58 129)(44 123 49 128 54 133 59 138)(45 132 50 137 55 122 60 127)(61 153 66 158 71 143 76 148)(62 142 67 147 72 152 77 157)(63 151 68 156 73 141 78 146)(64 160 69 145 74 150 79 155)(65 149 70 154 75 159 80 144)
G:=sub<Sym(160)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,76)(22,77)(23,78)(24,79)(25,80)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,133)(92,134)(93,135)(94,136)(95,137)(96,138)(97,139)(98,140)(99,121)(100,122)(101,157)(102,158)(103,159)(104,160)(105,141)(106,142)(107,143)(108,144)(109,145)(110,146)(111,147)(112,148)(113,149)(114,150)(115,151)(116,152)(117,153)(118,154)(119,155)(120,156), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,21)(17,22)(18,23)(19,24)(20,25)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,101)(99,102)(100,103)(121,158)(122,159)(123,160)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157), (1,31,11,21)(2,32,12,22)(3,33,13,23)(4,34,14,24)(5,35,15,25)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,66,51,76)(42,67,52,77)(43,68,53,78)(44,69,54,79)(45,70,55,80)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65)(81,119,91,109)(82,120,92,110)(83,101,93,111)(84,102,94,112)(85,103,95,113)(86,104,96,114)(87,105,97,115)(88,106,98,116)(89,107,99,117)(90,108,100,118)(121,153,131,143)(122,154,132,144)(123,155,133,145)(124,156,134,146)(125,157,135,147)(126,158,136,148)(127,159,137,149)(128,160,138,150)(129,141,139,151)(130,142,140,152), (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), (1,94,6,99,11,84,16,89)(2,83,7,88,12,93,17,98)(3,92,8,97,13,82,18,87)(4,81,9,86,14,91,19,96)(5,90,10,95,15,100,20,85)(21,112,26,117,31,102,36,107)(22,101,27,106,32,111,37,116)(23,110,28,115,33,120,38,105)(24,119,29,104,34,109,39,114)(25,108,30,113,35,118,40,103)(41,136,46,121,51,126,56,131)(42,125,47,130,52,135,57,140)(43,134,48,139,53,124,58,129)(44,123,49,128,54,133,59,138)(45,132,50,137,55,122,60,127)(61,153,66,158,71,143,76,148)(62,142,67,147,72,152,77,157)(63,151,68,156,73,141,78,146)(64,160,69,145,74,150,79,155)(65,149,70,154,75,159,80,144)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,76)(22,77)(23,78)(24,79)(25,80)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,133)(92,134)(93,135)(94,136)(95,137)(96,138)(97,139)(98,140)(99,121)(100,122)(101,157)(102,158)(103,159)(104,160)(105,141)(106,142)(107,143)(108,144)(109,145)(110,146)(111,147)(112,148)(113,149)(114,150)(115,151)(116,152)(117,153)(118,154)(119,155)(120,156), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,21)(17,22)(18,23)(19,24)(20,25)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,101)(99,102)(100,103)(121,158)(122,159)(123,160)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157), (1,31,11,21)(2,32,12,22)(3,33,13,23)(4,34,14,24)(5,35,15,25)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,66,51,76)(42,67,52,77)(43,68,53,78)(44,69,54,79)(45,70,55,80)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65)(81,119,91,109)(82,120,92,110)(83,101,93,111)(84,102,94,112)(85,103,95,113)(86,104,96,114)(87,105,97,115)(88,106,98,116)(89,107,99,117)(90,108,100,118)(121,153,131,143)(122,154,132,144)(123,155,133,145)(124,156,134,146)(125,157,135,147)(126,158,136,148)(127,159,137,149)(128,160,138,150)(129,141,139,151)(130,142,140,152), (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), (1,94,6,99,11,84,16,89)(2,83,7,88,12,93,17,98)(3,92,8,97,13,82,18,87)(4,81,9,86,14,91,19,96)(5,90,10,95,15,100,20,85)(21,112,26,117,31,102,36,107)(22,101,27,106,32,111,37,116)(23,110,28,115,33,120,38,105)(24,119,29,104,34,109,39,114)(25,108,30,113,35,118,40,103)(41,136,46,121,51,126,56,131)(42,125,47,130,52,135,57,140)(43,134,48,139,53,124,58,129)(44,123,49,128,54,133,59,138)(45,132,50,137,55,122,60,127)(61,153,66,158,71,143,76,148)(62,142,67,147,72,152,77,157)(63,151,68,156,73,141,78,146)(64,160,69,145,74,150,79,155)(65,149,70,154,75,159,80,144) );
G=PermutationGroup([(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,76),(22,77),(23,78),(24,79),(25,80),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(81,123),(82,124),(83,125),(84,126),(85,127),(86,128),(87,129),(88,130),(89,131),(90,132),(91,133),(92,134),(93,135),(94,136),(95,137),(96,138),(97,139),(98,140),(99,121),(100,122),(101,157),(102,158),(103,159),(104,160),(105,141),(106,142),(107,143),(108,144),(109,145),(110,146),(111,147),(112,148),(113,149),(114,150),(115,151),(116,152),(117,153),(118,154),(119,155),(120,156)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,21),(17,22),(18,23),(19,24),(20,25),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(97,120),(98,101),(99,102),(100,103),(121,158),(122,159),(123,160),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146),(130,147),(131,148),(132,149),(133,150),(134,151),(135,152),(136,153),(137,154),(138,155),(139,156),(140,157)], [(1,31,11,21),(2,32,12,22),(3,33,13,23),(4,34,14,24),(5,35,15,25),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30),(41,66,51,76),(42,67,52,77),(43,68,53,78),(44,69,54,79),(45,70,55,80),(46,71,56,61),(47,72,57,62),(48,73,58,63),(49,74,59,64),(50,75,60,65),(81,119,91,109),(82,120,92,110),(83,101,93,111),(84,102,94,112),(85,103,95,113),(86,104,96,114),(87,105,97,115),(88,106,98,116),(89,107,99,117),(90,108,100,118),(121,153,131,143),(122,154,132,144),(123,155,133,145),(124,156,134,146),(125,157,135,147),(126,158,136,148),(127,159,137,149),(128,160,138,150),(129,141,139,151),(130,142,140,152)], [(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)], [(1,94,6,99,11,84,16,89),(2,83,7,88,12,93,17,98),(3,92,8,97,13,82,18,87),(4,81,9,86,14,91,19,96),(5,90,10,95,15,100,20,85),(21,112,26,117,31,102,36,107),(22,101,27,106,32,111,37,116),(23,110,28,115,33,120,38,105),(24,119,29,104,34,109,39,114),(25,108,30,113,35,118,40,103),(41,136,46,121,51,126,56,131),(42,125,47,130,52,135,57,140),(43,134,48,139,53,124,58,129),(44,123,49,128,54,133,59,138),(45,132,50,137,55,122,60,127),(61,153,66,158,71,143,76,148),(62,142,67,147,72,152,77,157),(63,151,68,156,73,141,78,146),(64,160,69,145,74,150,79,155),(65,149,70,154,75,159,80,144)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 19 | 1 | 0 | 0 |
| 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 39 | 0 |
| 0 | 0 | 0 | 0 | 21 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 37 | 15 | 0 | 0 |
| 0 | 18 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,32],[1,0,0,0,0,0,19,16,0,0,0,1,16,0,0,0,0,0,39,0,0,0,0,0,21],[40,0,0,0,0,0,37,18,0,0,0,15,4,0,0,0,0,0,0,9,0,0,0,1,0] >;
104 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8P | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D5 | M4(2) | Dic5 | D10 | Dic5 | C4.Dic5 |
| kernel | C22×C4.Dic5 | C22×C5⋊2C8 | C2×C4.Dic5 | C23×C20 | C22×C20 | C23×C10 | C23×C4 | C2×C10 | C22×C4 | C22×C4 | C24 | C22 |
| # reps | 1 | 2 | 12 | 1 | 14 | 2 | 2 | 8 | 14 | 14 | 2 | 32 |
In GAP, Magma, Sage, TeX
C_2^2\times C_4.Dic_5
% in TeX
G:=Group("C2^2xC4.Dic5"); // GroupNames label
G:=SmallGroup(320,1453);
// by ID
G=gap.SmallGroup(320,1453);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1123,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^10=c^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^9>;
// generators/relations
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