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G = C10.352+ 1+4order 320 = 26·5

35th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.352+ 1+4, C10.702- 1+4, C4⋊C4.91D10, C4⋊D4.8D5, (D4×Dic5)⋊18C2, C22⋊C4.6D10, (C2×D4).154D10, C20.48D443C2, (C2×C10).146C24, (C2×C20).625C23, (C22×C4).221D10, C4⋊Dic5.45C22, C2.37(D46D10), C23.12(C22×D5), Dic5.Q812C2, (D4×C10).120C22, C23.D1016C2, C22.6(D42D5), (C22×C10).17C23, (C2×Dic5).67C23, C22.167(C23×D5), Dic5.14D417C2, C23.D5.23C22, C23.18D1021C2, (C22×C20).311C22, C53(C22.33C24), (C2×Dic10).35C22, (C4×Dic5).101C22, C2.28(D4.10D10), C10.D4.159C22, (C22×Dic5).107C22, C10.82(C2×C4○D4), (C5×C4⋊D4).8C2, C2.34(C2×D42D5), (C2×C10).22(C4○D4), (C2×C10.D4)⋊29C2, (C5×C4⋊C4).142C22, (C2×C4).174(C22×D5), (C5×C22⋊C4).11C22, SmallGroup(320,1274)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.352+ 1+4
C1C5C10C2×C10C2×Dic5C22×Dic5D4×Dic5 — C10.352+ 1+4
C5C2×C10 — C10.352+ 1+4
C1C22C4⋊D4

Generators and relations for C10.352+ 1+4
 G = < a,b,c,d,e | a10=b4=c2=1, d2=a5b2, e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=b-1, bd=db, ebe-1=a5b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 670 in 218 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×14], D4 [×5], Q8, C23, C23 [×2], C10 [×3], C10 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×13], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2 [×2], C422C2 [×2], Dic10, C2×Dic5 [×8], C2×Dic5 [×5], C2×C20 [×2], C2×C20 [×2], C2×C20, C5×D4 [×5], C22×C10, C22×C10 [×2], C22.33C24, C4×Dic5 [×2], C10.D4 [×10], C4⋊Dic5, C4⋊Dic5 [×2], C23.D5 [×8], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, D4×C10, D4×C10 [×2], Dic5.14D4 [×2], C23.D10 [×2], Dic5.Q8 [×2], C2×C10.D4, C20.48D4, D4×Dic5 [×2], C23.18D10 [×4], C5×C4⋊D4, C10.352+ 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.33C24, D42D5 [×2], C23×D5, C2×D42D5, D46D10, D4.10D10, C10.352+ 1+4

Smallest permutation representation of C10.352+ 1+4
On 160 points
Generators in S160
(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 86 26 73)(2 87 27 74)(3 88 28 75)(4 89 29 76)(5 90 30 77)(6 81 21 78)(7 82 22 79)(8 83 23 80)(9 84 24 71)(10 85 25 72)(11 106 153 93)(12 107 154 94)(13 108 155 95)(14 109 156 96)(15 110 157 97)(16 101 158 98)(17 102 159 99)(18 103 160 100)(19 104 151 91)(20 105 152 92)(31 51 43 63)(32 52 44 64)(33 53 45 65)(34 54 46 66)(35 55 47 67)(36 56 48 68)(37 57 49 69)(38 58 50 70)(39 59 41 61)(40 60 42 62)(111 144 124 131)(112 145 125 132)(113 146 126 133)(114 147 127 134)(115 148 128 135)(116 149 129 136)(117 150 130 137)(118 141 121 138)(119 142 122 139)(120 143 123 140)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 61)(8 62)(9 63)(10 64)(11 126)(12 127)(13 128)(14 129)(15 130)(16 121)(17 122)(18 123)(19 124)(20 125)(21 58)(22 59)(23 60)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(50 81)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(1 132 21 150)(2 131 22 149)(3 140 23 148)(4 139 24 147)(5 138 25 146)(6 137 26 145)(7 136 27 144)(8 135 28 143)(9 134 29 142)(10 133 30 141)(11 49 158 32)(12 48 159 31)(13 47 160 40)(14 46 151 39)(15 45 152 38)(16 44 153 37)(17 43 154 36)(18 42 155 35)(19 41 156 34)(20 50 157 33)(51 107 68 99)(52 106 69 98)(53 105 70 97)(54 104 61 96)(55 103 62 95)(56 102 63 94)(57 101 64 93)(58 110 65 92)(59 109 66 91)(60 108 67 100)(71 127 89 119)(72 126 90 118)(73 125 81 117)(74 124 82 116)(75 123 83 115)(76 122 84 114)(77 121 85 113)(78 130 86 112)(79 129 87 111)(80 128 88 120)
(1 130 26 117)(2 129 27 116)(3 128 28 115)(4 127 29 114)(5 126 30 113)(6 125 21 112)(7 124 22 111)(8 123 23 120)(9 122 24 119)(10 121 25 118)(11 57 153 69)(12 56 154 68)(13 55 155 67)(14 54 156 66)(15 53 157 65)(16 52 158 64)(17 51 159 63)(18 60 160 62)(19 59 151 61)(20 58 152 70)(31 107 43 94)(32 106 44 93)(33 105 45 92)(34 104 46 91)(35 103 47 100)(36 102 48 99)(37 101 49 98)(38 110 50 97)(39 109 41 96)(40 108 42 95)(71 147 84 134)(72 146 85 133)(73 145 86 132)(74 144 87 131)(75 143 88 140)(76 142 89 139)(77 141 90 138)(78 150 81 137)(79 149 82 136)(80 148 83 135)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I···4N5A5B10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222222444444444···45510···10101010101010101020···2020202020
size1111224444441010101020···20222···2444488884···48888

50 irreducible representations

dim11111111122222244444
type+++++++++++++++---
imageC1C2C2C2C2C2C2C2C2D5C4○D4D10D10D10D102+ 1+42- 1+4D42D5D46D10D4.10D10
kernelC10.352+ 1+4Dic5.14D4C23.D10Dic5.Q8C2×C10.D4C20.48D4D4×Dic5C23.18D10C5×C4⋊D4C4⋊D4C2×C10C22⋊C4C4⋊C4C22×C4C2×D4C10C10C22C2C2
# reps12221124124422611444

Matrix representation of C10.352+ 1+4 in GL6(𝔽41)

4000000
0400000
0003400
0063500
003403434
001771
,
010000
100000
00712525
00318039
0035401740
001701740
,
100000
010000
0000401
0034403934
000010
001010
,
900000
090000
001422237
0038232815
0060396
002939396
,
090000
3200000
00272800
00121400
001802818
0027132713

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,6,34,1,0,0,34,35,0,7,0,0,0,0,34,7,0,0,0,0,34,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,3,35,17,0,0,1,18,40,0,0,0,25,0,17,17,0,0,25,39,40,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,34,0,1,0,0,0,40,0,0,0,0,40,39,1,1,0,0,1,34,0,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,14,38,6,29,0,0,2,23,0,39,0,0,22,28,39,39,0,0,37,15,6,6],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,27,12,18,27,0,0,28,14,0,13,0,0,0,0,28,27,0,0,0,0,18,13] >;

C10.352+ 1+4 in GAP, Magma, Sage, TeX

C_{10}._{35}2_+^{1+4}
% in TeX

G:=Group("C10.35ES+(2,2)");
// GroupNames label

G:=SmallGroup(320,1274);
// by ID

G=gap.SmallGroup(320,1274);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,219,675,297,136,12550]);
// Polycyclic

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

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