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G = D20.35C23order 320 = 26·5

16th non-split extension by D20 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.40C24, D20.35C23, 2- 1+44D5, Dic10.35C23, C55(Q8○D8), (C5×D4).39D4, (C5×Q8).39D4, C4○D4.17D10, C20.272(C2×D4), D4⋊D5.2C22, (C2×Q8).93D10, C4.40(C23×D5), Q8⋊D5.3C22, D4.21(C5⋊D4), D4.Dic513C2, C52C8.19C23, Q8.21(C5⋊D4), (C5×D4).28C23, D4.28(C22×D5), D4.D5.3C22, D4.9D1012C2, D4.8D1011C2, (C5×Q8).28C23, Q8.28(C22×D5), C20.C2312C2, C5⋊Q16.4C22, (C2×C20).121C23, C4○D20.34C22, C10.174(C22×D4), (C5×2- 1+4)⋊3C2, D4.10D1010C2, (Q8×C10).154C22, C4.Dic5.32C22, (C2×Dic10).211C22, C4.78(C2×C5⋊D4), (C2×C5⋊Q16)⋊32C2, (C2×C10).88(C2×D4), C22.9(C2×C5⋊D4), C2.47(C22×C5⋊D4), (C5×C4○D4).30C22, (C2×C4).105(C22×D5), (C2×C52C8).185C22, SmallGroup(320,1510)

Series: Derived Chief Lower central Upper central

C1C20 — D20.35C23
C1C5C10C20D20C4○D20D4.10D10 — D20.35C23
C5C10C20 — D20.35C23
C1C2C4○D42- 1+4

Generators and relations for D20.35C23
 G = < a,b,c,d,e | a20=b2=e2=1, c2=d2=a10, bab=a-1, ac=ca, ad=da, eae=a11, bc=cb, bd=db, ebe=a15b, dcd-1=a10c, ce=ec, de=ed >

Subgroups: 726 in 248 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×3], C2×C4 [×12], D4, D4 [×3], D4 [×7], Q8, Q8 [×3], Q8 [×9], D5, C10, C10 [×4], C2×C8 [×3], M4(2) [×3], D8, SD16 [×6], Q16 [×9], C2×Q8 [×3], C2×Q8 [×5], C4○D4, C4○D4 [×3], C4○D4 [×9], Dic5 [×3], C20, C20 [×3], C20 [×3], D10, C2×C10 [×3], C2×C10, C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22 [×6], 2- 1+4, 2- 1+4, C52C8, C52C8 [×3], Dic10 [×3], Dic10 [×3], C4×D5 [×3], D20, C2×Dic5 [×3], C5⋊D4 [×3], C2×C20 [×3], C2×C20 [×6], C5×D4, C5×D4 [×3], C5×D4 [×3], C5×Q8, C5×Q8 [×3], C5×Q8 [×3], Q8○D8, C2×C52C8 [×3], C4.Dic5 [×3], D4⋊D5, D4.D5 [×3], Q8⋊D5 [×3], C5⋊Q16 [×9], C2×Dic10 [×3], C4○D20 [×3], D42D5 [×3], Q8×D5, Q8×C10 [×3], Q8×C10, C5×C4○D4, C5×C4○D4 [×3], C5×C4○D4 [×3], C20.C23 [×3], C2×C5⋊Q16 [×3], D4.Dic5, D4.8D10 [×3], D4.9D10 [×3], D4.10D10, C5×2- 1+4, D20.35C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C5⋊D4 [×4], C22×D5 [×7], Q8○D8, C2×C5⋊D4 [×6], C23×D5, C22×C5⋊D4, D20.35C23

Smallest permutation representation of D20.35C23
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 27)(22 26)(23 25)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 69)(62 68)(63 67)(64 66)(70 80)(71 79)(72 78)(73 77)(74 76)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(95 100)(96 99)(97 98)(101 105)(102 104)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 126)(122 125)(123 124)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 151)(142 150)(143 149)(144 148)(145 147)(152 160)(153 159)(154 158)(155 157)
(1 98 11 88)(2 99 12 89)(3 100 13 90)(4 81 14 91)(5 82 15 92)(6 83 16 93)(7 84 17 94)(8 85 18 95)(9 86 19 96)(10 87 20 97)(21 120 31 110)(22 101 32 111)(23 102 33 112)(24 103 34 113)(25 104 35 114)(26 105 36 115)(27 106 37 116)(28 107 38 117)(29 108 39 118)(30 109 40 119)(41 124 51 134)(42 125 52 135)(43 126 53 136)(44 127 54 137)(45 128 55 138)(46 129 56 139)(47 130 57 140)(48 131 58 121)(49 132 59 122)(50 133 60 123)(61 152 71 142)(62 153 72 143)(63 154 73 144)(64 155 74 145)(65 156 75 146)(66 157 76 147)(67 158 77 148)(68 159 78 149)(69 160 79 150)(70 141 80 151)
(1 41 11 51)(2 42 12 52)(3 43 13 53)(4 44 14 54)(5 45 15 55)(6 46 16 56)(7 47 17 57)(8 48 18 58)(9 49 19 59)(10 50 20 60)(21 72 31 62)(22 73 32 63)(23 74 33 64)(24 75 34 65)(25 76 35 66)(26 77 36 67)(27 78 37 68)(28 79 38 69)(29 80 39 70)(30 61 40 71)(81 137 91 127)(82 138 92 128)(83 139 93 129)(84 140 94 130)(85 121 95 131)(86 122 96 132)(87 123 97 133)(88 124 98 134)(89 125 99 135)(90 126 100 136)(101 154 111 144)(102 155 112 145)(103 156 113 146)(104 157 114 147)(105 158 115 148)(106 159 116 149)(107 160 117 150)(108 141 118 151)(109 142 119 152)(110 143 120 153)
(1 32)(2 23)(3 34)(4 25)(5 36)(6 27)(7 38)(8 29)(9 40)(10 31)(11 22)(12 33)(13 24)(14 35)(15 26)(16 37)(17 28)(18 39)(19 30)(20 21)(41 63)(42 74)(43 65)(44 76)(45 67)(46 78)(47 69)(48 80)(49 71)(50 62)(51 73)(52 64)(53 75)(54 66)(55 77)(56 68)(57 79)(58 70)(59 61)(60 72)(81 104)(82 115)(83 106)(84 117)(85 108)(86 119)(87 110)(88 101)(89 112)(90 103)(91 114)(92 105)(93 116)(94 107)(95 118)(96 109)(97 120)(98 111)(99 102)(100 113)(121 141)(122 152)(123 143)(124 154)(125 145)(126 156)(127 147)(128 158)(129 149)(130 160)(131 151)(132 142)(133 153)(134 144)(135 155)(136 146)(137 157)(138 148)(139 159)(140 150)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E10A10B10C···10L20A···20T
order122222244444444445588888101010···1020···20
size112224202222444202020221010202020224···44···4

56 irreducible representations

dim11111111222222248
type+++++++++++++--
imageC1C2C2C2C2C2C2C2D4D4D5D10D10C5⋊D4C5⋊D4Q8○D8D20.35C23
kernelD20.35C23C20.C23C2×C5⋊Q16D4.Dic5D4.8D10D4.9D10D4.10D10C5×2- 1+4C5×D4C5×Q82- 1+4C2×Q8C4○D4D4Q8C5C1
# reps133133113126812422

Matrix representation of D20.35C23 in GL6(𝔽41)

3410000
4000000
0012100
00374000
001033118
00292940
,
1340000
0400000
0012100
0004000
00033118
0000040
,
100000
010000
001412130
0004017
0029372719
00040037
,
4000000
0400000
0030290
0001037
0032251110
00021040
,
100000
010000
0024600
00341700
003852411
0020242617

G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,1,37,10,29,0,0,21,40,33,2,0,0,0,0,1,9,0,0,0,0,18,40],[1,0,0,0,0,0,34,40,0,0,0,0,0,0,1,0,0,0,0,0,21,40,33,0,0,0,0,0,1,0,0,0,0,0,18,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,0,29,0,0,0,12,4,37,40,0,0,13,0,27,0,0,0,0,17,19,37],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,0,32,0,0,0,2,1,25,21,0,0,9,0,11,0,0,0,0,37,10,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,34,38,20,0,0,6,17,5,24,0,0,0,0,24,26,0,0,0,0,11,17] >;

D20.35C23 in GAP, Magma, Sage, TeX

D_{20}._{35}C_2^3
% in TeX

G:=Group("D20.35C2^3");
// GroupNames label

G:=SmallGroup(320,1510);
// by ID

G=gap.SmallGroup(320,1510);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,1684,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^20=b^2=e^2=1,c^2=d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^11,b*c=c*b,b*d=d*b,e*b*e=a^15*b,d*c*d^-1=a^10*c,c*e=e*c,d*e=e*d>;
// generators/relations

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