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## G = C2×D4⋊Dic5order 320 = 26·5

### Direct product of C2 and D4⋊Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D4⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4⋊Dic5 — C2×C4⋊Dic5 — C2×D4⋊Dic5
 Lower central C5 — C10 — C20 — C2×D4⋊Dic5
 Upper central C1 — C23 — C22×C4 — C22×D4

Generators and relations for C2×D4⋊Dic5
G = < a,b,c,d,e | a2=b4=c2=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 574 in 202 conjugacy classes, 87 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], D4 [×4], D4 [×6], C23, C23 [×10], C10 [×3], C10 [×4], C10 [×4], C4⋊C4 [×3], C2×C8 [×4], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×3], C24, Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], C2×C10 [×16], D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C52C8 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C5×D4 [×6], C22×C10, C22×C10 [×10], C2×D4⋊C4, C2×C52C8 [×2], C2×C52C8 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C22×Dic5, C22×C20, D4×C10 [×6], D4×C10 [×3], C23×C10, D4⋊Dic5 [×4], C22×C52C8, C2×C4⋊Dic5, D4×C2×C10, C2×D4⋊Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C2×D4⋊C4, D4⋊D5 [×2], D4.D5 [×2], C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D4⋊Dic5 [×4], C2×D4⋊D5, C2×D4.D5, C2×C23.D5, C2×D4⋊Dic5

Smallest permutation representation of C2×D4⋊Dic5
On 160 points
Generators in S160
(1 66)(2 67)(3 68)(4 69)(5 70)(6 61)(7 62)(8 63)(9 64)(10 65)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 121)(30 122)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)(41 89)(42 90)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(91 136)(92 137)(93 138)(94 139)(95 140)(96 131)(97 132)(98 133)(99 134)(100 135)(101 146)(102 147)(103 148)(104 149)(105 150)(106 141)(107 142)(108 143)(109 144)(110 145)(111 156)(112 157)(113 158)(114 159)(115 160)(116 151)(117 152)(118 153)(119 154)(120 155)
(1 47 31 20)(2 48 32 11)(3 49 33 12)(4 50 34 13)(5 41 35 14)(6 42 36 15)(7 43 37 16)(8 44 38 17)(9 45 39 18)(10 46 40 19)(21 138 150 159)(22 139 141 160)(23 140 142 151)(24 131 143 152)(25 132 144 153)(26 133 145 154)(27 134 146 155)(28 135 147 156)(29 136 148 157)(30 137 149 158)(51 62 81 72)(52 63 82 73)(53 64 83 74)(54 65 84 75)(55 66 85 76)(56 67 86 77)(57 68 87 78)(58 69 88 79)(59 70 89 80)(60 61 90 71)(91 103 112 121)(92 104 113 122)(93 105 114 123)(94 106 115 124)(95 107 116 125)(96 108 117 126)(97 109 118 127)(98 110 119 128)(99 101 120 129)(100 102 111 130)
(1 20)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(31 47)(32 48)(33 49)(34 50)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 61)(71 90)(72 81)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(100 111)(131 152)(132 153)(133 154)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 151)
(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)
(1 133 6 138)(2 132 7 137)(3 131 8 136)(4 140 9 135)(5 139 10 134)(11 144 16 149)(12 143 17 148)(13 142 18 147)(14 141 19 146)(15 150 20 145)(21 47 26 42)(22 46 27 41)(23 45 28 50)(24 44 29 49)(25 43 30 48)(31 154 36 159)(32 153 37 158)(33 152 38 157)(34 151 39 156)(35 160 40 155)(51 104 56 109)(52 103 57 108)(53 102 58 107)(54 101 59 106)(55 110 60 105)(61 93 66 98)(62 92 67 97)(63 91 68 96)(64 100 69 95)(65 99 70 94)(71 114 76 119)(72 113 77 118)(73 112 78 117)(74 111 79 116)(75 120 80 115)(81 122 86 127)(82 121 87 126)(83 130 88 125)(84 129 89 124)(85 128 90 123)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A ··· 8H 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 4 4 4 4 2 2 2 2 20 20 20 20 2 2 10 ··· 10 2 ··· 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 D4 D4 D5 D8 SD16 D10 Dic5 D10 C5⋊D4 C5⋊D4 D4⋊D5 D4.D5 kernel C2×D4⋊Dic5 D4⋊Dic5 C22×C5⋊2C8 C2×C4⋊Dic5 D4×C2×C10 D4×C10 C2×C20 C22×C10 C22×D4 C2×C10 C2×C10 C22×C4 C2×D4 C2×D4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 3 1 2 4 4 2 8 4 12 4 4 4

Matrix representation of C2×D4⋊Dic5 in GL7(𝔽41)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 40 0
,
 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 14 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 0 34 40 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40
,
 9 0 0 0 0 0 0 0 6 39 0 0 0 0 0 38 35 0 0 0 0 0 0 0 15 8 0 0 0 0 0 23 26 0 0 0 0 0 0 0 26 15 0 0 0 0 0 15 15

G:=sub<GL(7,GF(41))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0],[40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,14,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[40,0,0,0,0,0,0,0,34,1,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,0,6,38,0,0,0,0,0,39,35,0,0,0,0,0,0,0,15,23,0,0,0,0,0,8,26,0,0,0,0,0,0,0,26,15,0,0,0,0,0,15,15] >;

C2×D4⋊Dic5 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("C2xD4:Dic5");
// GroupNames label

G:=SmallGroup(320,841);
// by ID

G=gap.SmallGroup(320,841);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^10=1,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

׿
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