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G = C2×D4.10D10order 320 = 26·5

Direct product of C2 and D4.10D10

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D4.10D10, C10.13C25, C20.48C24, D10.7C24, D20.40C23, C1022- 1+4, Dic5.8C24, Dic10.37C23, C4○D418D10, (C2×C10).4C24, (Q8×D5)⋊14C22, C2.14(D5×C24), C4.63(C23×D5), C5⋊D4.1C23, (C2×D4).254D10, C52(C2×2- 1+4), C4○D2026C22, (C2×Q8).212D10, D4.29(C22×D5), (C4×D5).19C23, (C5×D4).29C23, (C5×Q8).30C23, Q8.30(C22×D5), D42D513C22, (C2×C20).567C23, (C22×C4).292D10, C22.57(C23×D5), (C2×Dic10)⋊75C22, (C22×Dic10)⋊25C2, (D4×C10).279C22, (C2×D20).298C22, (Q8×C10).247C22, C23.215(C22×D5), (C22×C10).249C23, (C22×C20).303C22, (C2×Dic5).167C23, (C22×D5).260C23, (C22×Dic5).169C22, (C2×Q8×D5)⋊21C2, (C2×C4○D4)⋊14D5, (C10×C4○D4)⋊15C2, (C2×C4○D20)⋊38C2, (C2×D42D5)⋊30C2, (C5×C4○D4)⋊21C22, (C2×C4×D5).182C22, (C2×C4).253(C22×D5), (C2×C5⋊D4).152C22, SmallGroup(320,1620)

Series: Derived Chief Lower central Upper central

C1C10 — C2×D4.10D10
C1C5C10D10C22×D5C2×C4×D5C2×Q8×D5 — C2×D4.10D10
C5C10 — C2×D4.10D10
C1C22C2×C4○D4

Generators and relations for C2×D4.10D10
 G = < a,b,c,d,e | a2=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=b2c, ce=ec, ede-1=d9 >

Subgroups: 2126 in 794 conjugacy classes, 447 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C4 [×12], C22, C22 [×6], C22 [×14], C5, C2×C4, C2×C4 [×15], C2×C4 [×54], D4 [×12], D4 [×28], Q8 [×4], Q8 [×36], C23 [×3], C23 [×2], D5 [×4], C10, C10 [×2], C10 [×6], C22×C4 [×3], C22×C4 [×12], C2×D4 [×3], C2×D4 [×7], C2×Q8, C2×Q8 [×49], C4○D4 [×8], C4○D4 [×72], Dic5 [×12], C20 [×8], D10 [×4], D10 [×4], C2×C10, C2×C10 [×6], C2×C10 [×6], C22×Q8 [×5], C2×C4○D4, C2×C4○D4 [×9], 2- 1+4 [×16], Dic10 [×36], C4×D5 [×24], D20 [×4], C2×Dic5 [×30], C5⋊D4 [×24], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×D5 [×2], C22×C10 [×3], C2×2- 1+4, C2×Dic10 [×33], C2×C4×D5 [×6], C2×D20, C4○D20 [×24], D42D5 [×48], Q8×D5 [×16], C22×Dic5 [×6], C2×C5⋊D4 [×6], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C22×Dic10 [×3], C2×C4○D20 [×3], C2×D42D5 [×6], C2×Q8×D5 [×2], D4.10D10 [×16], C10×C4○D4, C2×D4.10D10
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], 2- 1+4 [×2], C25, C22×D5 [×35], C2×2- 1+4, C23×D5 [×15], D4.10D10 [×2], D5×C24, C2×D4.10D10

Smallest permutation representation of C2×D4.10D10
On 160 points
Generators in S160
(1 139)(2 140)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 141)(58 142)(59 143)(60 144)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 101)(69 102)(70 103)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 111)(79 112)(80 113)
(1 155 11 145)(2 146 12 156)(3 157 13 147)(4 148 14 158)(5 159 15 149)(6 150 16 160)(7 141 17 151)(8 152 18 142)(9 143 19 153)(10 154 20 144)(21 109 31 119)(22 120 32 110)(23 111 33 101)(24 102 34 112)(25 113 35 103)(26 104 36 114)(27 115 37 105)(28 106 38 116)(29 117 39 107)(30 108 40 118)(41 139 51 129)(42 130 52 140)(43 121 53 131)(44 132 54 122)(45 123 55 133)(46 134 56 124)(47 125 57 135)(48 136 58 126)(49 127 59 137)(50 138 60 128)(61 94 71 84)(62 85 72 95)(63 96 73 86)(64 87 74 97)(65 98 75 88)(66 89 76 99)(67 100 77 90)(68 91 78 81)(69 82 79 92)(70 93 80 83)
(1 67)(2 78)(3 69)(4 80)(5 71)(6 62)(7 73)(8 64)(9 75)(10 66)(11 77)(12 68)(13 79)(14 70)(15 61)(16 72)(17 63)(18 74)(19 65)(20 76)(21 60)(22 51)(23 42)(24 53)(25 44)(26 55)(27 46)(28 57)(29 48)(30 59)(31 50)(32 41)(33 52)(34 43)(35 54)(36 45)(37 56)(38 47)(39 58)(40 49)(81 156)(82 147)(83 158)(84 149)(85 160)(86 151)(87 142)(88 153)(89 144)(90 155)(91 146)(92 157)(93 148)(94 159)(95 150)(96 141)(97 152)(98 143)(99 154)(100 145)(101 130)(102 121)(103 132)(104 123)(105 134)(106 125)(107 136)(108 127)(109 138)(110 129)(111 140)(112 131)(113 122)(114 133)(115 124)(116 135)(117 126)(118 137)(119 128)(120 139)
(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)
(1 128 11 138)(2 137 12 127)(3 126 13 136)(4 135 14 125)(5 124 15 134)(6 133 16 123)(7 122 17 132)(8 131 18 121)(9 140 19 130)(10 129 20 139)(21 100 31 90)(22 89 32 99)(23 98 33 88)(24 87 34 97)(25 96 35 86)(26 85 36 95)(27 94 37 84)(28 83 38 93)(29 92 39 82)(30 81 40 91)(41 154 51 144)(42 143 52 153)(43 152 53 142)(44 141 54 151)(45 150 55 160)(46 159 56 149)(47 148 57 158)(48 157 58 147)(49 146 59 156)(50 155 60 145)(61 105 71 115)(62 114 72 104)(63 103 73 113)(64 112 74 102)(65 101 75 111)(66 110 76 120)(67 119 77 109)(68 108 78 118)(69 117 79 107)(70 106 80 116)

G:=sub<Sym(160)| (1,139)(2,140)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,141)(58,142)(59,143)(60,144)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,113), (1,155,11,145)(2,146,12,156)(3,157,13,147)(4,148,14,158)(5,159,15,149)(6,150,16,160)(7,141,17,151)(8,152,18,142)(9,143,19,153)(10,154,20,144)(21,109,31,119)(22,120,32,110)(23,111,33,101)(24,102,34,112)(25,113,35,103)(26,104,36,114)(27,115,37,105)(28,106,38,116)(29,117,39,107)(30,108,40,118)(41,139,51,129)(42,130,52,140)(43,121,53,131)(44,132,54,122)(45,123,55,133)(46,134,56,124)(47,125,57,135)(48,136,58,126)(49,127,59,137)(50,138,60,128)(61,94,71,84)(62,85,72,95)(63,96,73,86)(64,87,74,97)(65,98,75,88)(66,89,76,99)(67,100,77,90)(68,91,78,81)(69,82,79,92)(70,93,80,83), (1,67)(2,78)(3,69)(4,80)(5,71)(6,62)(7,73)(8,64)(9,75)(10,66)(11,77)(12,68)(13,79)(14,70)(15,61)(16,72)(17,63)(18,74)(19,65)(20,76)(21,60)(22,51)(23,42)(24,53)(25,44)(26,55)(27,46)(28,57)(29,48)(30,59)(31,50)(32,41)(33,52)(34,43)(35,54)(36,45)(37,56)(38,47)(39,58)(40,49)(81,156)(82,147)(83,158)(84,149)(85,160)(86,151)(87,142)(88,153)(89,144)(90,155)(91,146)(92,157)(93,148)(94,159)(95,150)(96,141)(97,152)(98,143)(99,154)(100,145)(101,130)(102,121)(103,132)(104,123)(105,134)(106,125)(107,136)(108,127)(109,138)(110,129)(111,140)(112,131)(113,122)(114,133)(115,124)(116,135)(117,126)(118,137)(119,128)(120,139), (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), (1,128,11,138)(2,137,12,127)(3,126,13,136)(4,135,14,125)(5,124,15,134)(6,133,16,123)(7,122,17,132)(8,131,18,121)(9,140,19,130)(10,129,20,139)(21,100,31,90)(22,89,32,99)(23,98,33,88)(24,87,34,97)(25,96,35,86)(26,85,36,95)(27,94,37,84)(28,83,38,93)(29,92,39,82)(30,81,40,91)(41,154,51,144)(42,143,52,153)(43,152,53,142)(44,141,54,151)(45,150,55,160)(46,159,56,149)(47,148,57,158)(48,157,58,147)(49,146,59,156)(50,155,60,145)(61,105,71,115)(62,114,72,104)(63,103,73,113)(64,112,74,102)(65,101,75,111)(66,110,76,120)(67,119,77,109)(68,108,78,118)(69,117,79,107)(70,106,80,116)>;

G:=Group( (1,139)(2,140)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,100)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,141)(58,142)(59,143)(60,144)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,113), (1,155,11,145)(2,146,12,156)(3,157,13,147)(4,148,14,158)(5,159,15,149)(6,150,16,160)(7,141,17,151)(8,152,18,142)(9,143,19,153)(10,154,20,144)(21,109,31,119)(22,120,32,110)(23,111,33,101)(24,102,34,112)(25,113,35,103)(26,104,36,114)(27,115,37,105)(28,106,38,116)(29,117,39,107)(30,108,40,118)(41,139,51,129)(42,130,52,140)(43,121,53,131)(44,132,54,122)(45,123,55,133)(46,134,56,124)(47,125,57,135)(48,136,58,126)(49,127,59,137)(50,138,60,128)(61,94,71,84)(62,85,72,95)(63,96,73,86)(64,87,74,97)(65,98,75,88)(66,89,76,99)(67,100,77,90)(68,91,78,81)(69,82,79,92)(70,93,80,83), (1,67)(2,78)(3,69)(4,80)(5,71)(6,62)(7,73)(8,64)(9,75)(10,66)(11,77)(12,68)(13,79)(14,70)(15,61)(16,72)(17,63)(18,74)(19,65)(20,76)(21,60)(22,51)(23,42)(24,53)(25,44)(26,55)(27,46)(28,57)(29,48)(30,59)(31,50)(32,41)(33,52)(34,43)(35,54)(36,45)(37,56)(38,47)(39,58)(40,49)(81,156)(82,147)(83,158)(84,149)(85,160)(86,151)(87,142)(88,153)(89,144)(90,155)(91,146)(92,157)(93,148)(94,159)(95,150)(96,141)(97,152)(98,143)(99,154)(100,145)(101,130)(102,121)(103,132)(104,123)(105,134)(106,125)(107,136)(108,127)(109,138)(110,129)(111,140)(112,131)(113,122)(114,133)(115,124)(116,135)(117,126)(118,137)(119,128)(120,139), (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), (1,128,11,138)(2,137,12,127)(3,126,13,136)(4,135,14,125)(5,124,15,134)(6,133,16,123)(7,122,17,132)(8,131,18,121)(9,140,19,130)(10,129,20,139)(21,100,31,90)(22,89,32,99)(23,98,33,88)(24,87,34,97)(25,96,35,86)(26,85,36,95)(27,94,37,84)(28,83,38,93)(29,92,39,82)(30,81,40,91)(41,154,51,144)(42,143,52,153)(43,152,53,142)(44,141,54,151)(45,150,55,160)(46,159,56,149)(47,148,57,158)(48,157,58,147)(49,146,59,156)(50,155,60,145)(61,105,71,115)(62,114,72,104)(63,103,73,113)(64,112,74,102)(65,101,75,111)(66,110,76,120)(67,119,77,109)(68,108,78,118)(69,117,79,107)(70,106,80,116) );

G=PermutationGroup([(1,139),(2,140),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,133),(16,134),(17,135),(18,136),(19,137),(20,138),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,97),(30,98),(31,99),(32,100),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,154),(51,155),(52,156),(53,157),(54,158),(55,159),(56,160),(57,141),(58,142),(59,143),(60,144),(61,114),(62,115),(63,116),(64,117),(65,118),(66,119),(67,120),(68,101),(69,102),(70,103),(71,104),(72,105),(73,106),(74,107),(75,108),(76,109),(77,110),(78,111),(79,112),(80,113)], [(1,155,11,145),(2,146,12,156),(3,157,13,147),(4,148,14,158),(5,159,15,149),(6,150,16,160),(7,141,17,151),(8,152,18,142),(9,143,19,153),(10,154,20,144),(21,109,31,119),(22,120,32,110),(23,111,33,101),(24,102,34,112),(25,113,35,103),(26,104,36,114),(27,115,37,105),(28,106,38,116),(29,117,39,107),(30,108,40,118),(41,139,51,129),(42,130,52,140),(43,121,53,131),(44,132,54,122),(45,123,55,133),(46,134,56,124),(47,125,57,135),(48,136,58,126),(49,127,59,137),(50,138,60,128),(61,94,71,84),(62,85,72,95),(63,96,73,86),(64,87,74,97),(65,98,75,88),(66,89,76,99),(67,100,77,90),(68,91,78,81),(69,82,79,92),(70,93,80,83)], [(1,67),(2,78),(3,69),(4,80),(5,71),(6,62),(7,73),(8,64),(9,75),(10,66),(11,77),(12,68),(13,79),(14,70),(15,61),(16,72),(17,63),(18,74),(19,65),(20,76),(21,60),(22,51),(23,42),(24,53),(25,44),(26,55),(27,46),(28,57),(29,48),(30,59),(31,50),(32,41),(33,52),(34,43),(35,54),(36,45),(37,56),(38,47),(39,58),(40,49),(81,156),(82,147),(83,158),(84,149),(85,160),(86,151),(87,142),(88,153),(89,144),(90,155),(91,146),(92,157),(93,148),(94,159),(95,150),(96,141),(97,152),(98,143),(99,154),(100,145),(101,130),(102,121),(103,132),(104,123),(105,134),(106,125),(107,136),(108,127),(109,138),(110,129),(111,140),(112,131),(113,122),(114,133),(115,124),(116,135),(117,126),(118,137),(119,128),(120,139)], [(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)], [(1,128,11,138),(2,137,12,127),(3,126,13,136),(4,135,14,125),(5,124,15,134),(6,133,16,123),(7,122,17,132),(8,131,18,121),(9,140,19,130),(10,129,20,139),(21,100,31,90),(22,89,32,99),(23,98,33,88),(24,87,34,97),(25,96,35,86),(26,85,36,95),(27,94,37,84),(28,83,38,93),(29,92,39,82),(30,81,40,91),(41,154,51,144),(42,143,52,153),(43,152,53,142),(44,141,54,151),(45,150,55,160),(46,159,56,149),(47,148,57,158),(48,157,58,147),(49,146,59,156),(50,155,60,145),(61,105,71,115),(62,114,72,104),(63,103,73,113),(64,112,74,102),(65,101,75,111),(66,110,76,120),(67,119,77,109),(68,108,78,118),(69,117,79,107),(70,106,80,116)])

74 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A···4H4I···4T5A5B10A···10F10G···10R20A···20H20I···20T
order12222···222224···44···45510···1010···1020···2020···20
size11112···2101010102···210···10222···24···42···24···4

74 irreducible representations

dim11111112222244
type++++++++++++--
imageC1C2C2C2C2C2C2D5D10D10D10D102- 1+4D4.10D10
kernelC2×D4.10D10C22×Dic10C2×C4○D20C2×D42D5C2×Q8×D5D4.10D10C10×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C10C2
# reps1336216126621628

Matrix representation of C2×D4.10D10 in GL6(𝔽41)

4000000
0400000
0040000
0004000
0000400
0000040
,
4000000
0400000
0000400
0000040
001000
000100
,
4000000
0400000
0025372226
0032163819
002226164
003819925
,
3470000
3410000
00029036
0014144040
00036012
0040402727
,
010000
100000
0020350
00253976
00350390
0076162

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,25,32,22,38,0,0,37,16,26,19,0,0,22,38,16,9,0,0,26,19,4,25],[34,34,0,0,0,0,7,1,0,0,0,0,0,0,0,14,0,40,0,0,29,14,36,40,0,0,0,40,0,27,0,0,36,40,12,27],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,25,35,7,0,0,0,39,0,6,0,0,35,7,39,16,0,0,0,6,0,2] >;

C2×D4.10D10 in GAP, Magma, Sage, TeX

C_2\times D_4._{10}D_{10}
% in TeX

G:=Group("C2xD4.10D10");
// GroupNames label

G:=SmallGroup(320,1620);
// by ID

G=gap.SmallGroup(320,1620);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,297,136,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^10=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b^2*c,c*e=e*c,e*d*e^-1=d^9>;
// generators/relations

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