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G = C5×D4⋊D4order 320 = 26·5

Direct product of C5 and D4⋊D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×D4⋊D4, D43(C5×D4), Q83(C5×D4), (C2×D8)⋊2C10, (C5×D4)⋊21D4, (C5×Q8)⋊21D4, (C10×D8)⋊16C2, C4⋊D42C10, C22⋊C85C10, C4.23(D4×C10), D4⋊C48C10, Q8⋊C44C10, (C2×SD16)⋊8C10, C20.384(C2×D4), (C2×C20).459D4, C23.13(C5×D4), C10.96C22≀C2, (C10×SD16)⋊25C2, C22.79(D4×C10), (C22×C10).31D4, C10.119(C4○D8), (C2×C20).914C23, (C2×C40).298C22, C10.132(C8⋊C22), (D4×C10).294C22, (Q8×C10).259C22, (C22×C20).421C22, C2.6(C5×C4○D8), (C2×C4○D4)⋊1C10, C4⋊C4.2(C2×C10), (C2×C8).1(C2×C10), C2.7(C5×C8⋊C22), (C10×C4○D4)⋊17C2, (C5×C4⋊D4)⋊29C2, (C5×C22⋊C8)⋊22C2, (C2×C4).105(C5×D4), (C5×D4⋊C4)⋊32C2, (C5×Q8⋊C4)⋊27C2, C2.10(C5×C22≀C2), (C2×D4).52(C2×C10), (C2×C10).635(C2×D4), (C2×Q8).44(C2×C10), (C5×C4⋊C4).224C22, (C22×C4).39(C2×C10), (C2×C4).89(C22×C10), SmallGroup(320,950)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×D4⋊D4
C1C2C22C2×C4C2×C20D4×C10C10×D8 — C5×D4⋊D4
C1C2C2×C4 — C5×D4⋊D4
C1C2×C10C22×C20 — C5×D4⋊D4

Generators and relations for C5×D4⋊D4
 G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 322 in 162 conjugacy classes, 58 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, D4⋊D4, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×D8, C5×SD16, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊C8, C5×D4⋊C4, C5×Q8⋊C4, C5×C4⋊D4, C10×D8, C10×SD16, C10×C4○D4, C5×D4⋊D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C22≀C2, C4○D8, C8⋊C22, C5×D4, C22×C10, D4⋊D4, D4×C10, C5×C22≀C2, C5×C4○D8, C5×C8⋊C22, C5×D4⋊D4

Smallest permutation representation of C5×D4⋊D4
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 54 12 45)(2 55 13 41)(3 51 14 42)(4 52 15 43)(5 53 11 44)(6 140 144 19)(7 136 145 20)(8 137 141 16)(9 138 142 17)(10 139 143 18)(21 155 159 34)(22 151 160 35)(23 152 156 31)(24 153 157 32)(25 154 158 33)(26 69 50 40)(27 70 46 36)(28 66 47 37)(29 67 48 38)(30 68 49 39)(56 81 94 65)(57 82 95 61)(58 83 91 62)(59 84 92 63)(60 85 93 64)(71 109 90 80)(72 110 86 76)(73 106 87 77)(74 107 88 78)(75 108 89 79)(96 105 134 121)(97 101 135 122)(98 102 131 123)(99 103 132 124)(100 104 133 125)(111 120 130 149)(112 116 126 150)(113 117 127 146)(114 118 128 147)(115 119 129 148)
(1 127)(2 128)(3 129)(4 130)(5 126)(6 72)(7 73)(8 74)(9 75)(10 71)(11 112)(12 113)(13 114)(14 115)(15 111)(16 107)(17 108)(18 109)(19 110)(20 106)(21 58)(22 59)(23 60)(24 56)(25 57)(26 124)(27 125)(28 121)(29 122)(30 123)(31 85)(32 81)(33 82)(34 83)(35 84)(36 100)(37 96)(38 97)(39 98)(40 99)(41 147)(42 148)(43 149)(44 150)(45 146)(46 104)(47 105)(48 101)(49 102)(50 103)(51 119)(52 120)(53 116)(54 117)(55 118)(61 154)(62 155)(63 151)(64 152)(65 153)(66 134)(67 135)(68 131)(69 132)(70 133)(76 140)(77 136)(78 137)(79 138)(80 139)(86 144)(87 145)(88 141)(89 142)(90 143)(91 159)(92 160)(93 156)(94 157)(95 158)
(1 56 37 87)(2 57 38 88)(3 58 39 89)(4 59 40 90)(5 60 36 86)(6 116 156 125)(7 117 157 121)(8 118 158 122)(9 119 159 123)(10 120 160 124)(11 93 70 72)(12 94 66 73)(13 95 67 74)(14 91 68 75)(15 92 69 71)(16 128 154 97)(17 129 155 98)(18 130 151 99)(19 126 152 100)(20 127 153 96)(21 102 142 148)(22 103 143 149)(23 104 144 150)(24 105 145 146)(25 101 141 147)(26 109 52 63)(27 110 53 64)(28 106 54 65)(29 107 55 61)(30 108 51 62)(31 133 140 112)(32 134 136 113)(33 135 137 114)(34 131 138 115)(35 132 139 111)(41 82 48 78)(42 83 49 79)(43 84 50 80)(44 85 46 76)(45 81 47 77)
(6 152)(7 153)(8 154)(9 155)(10 151)(16 158)(17 159)(18 160)(19 156)(20 157)(21 138)(22 139)(23 140)(24 136)(25 137)(26 50)(27 46)(28 47)(29 48)(30 49)(31 144)(32 145)(33 141)(34 142)(35 143)(41 55)(42 51)(43 52)(44 53)(45 54)(56 87)(57 88)(58 89)(59 90)(60 86)(61 78)(62 79)(63 80)(64 76)(65 77)(71 92)(72 93)(73 94)(74 95)(75 91)(81 106)(82 107)(83 108)(84 109)(85 110)(96 121)(97 122)(98 123)(99 124)(100 125)(101 135)(102 131)(103 132)(104 133)(105 134)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B5C5D8A8B8C8D10A···10L10M···10X10Y10Z10AA10AB20A···20P20Q···20X20Y20Z20AA20AB40A···40P
order1222222244444445555888810···1010···101010101020···2020···202020202040···40
size111144482222448111144441···14···488882···24···488884···4

95 irreducible representations

dim1111111111111111222222222244
type+++++++++++++
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10D4D4D4D4C4○D8C5×D4C5×D4C5×D4C5×D4C5×C4○D8C8⋊C22C5×C8⋊C22
kernelC5×D4⋊D4C5×C22⋊C8C5×D4⋊C4C5×Q8⋊C4C5×C4⋊D4C10×D8C10×SD16C10×C4○D4D4⋊D4C22⋊C8D4⋊C4Q8⋊C4C4⋊D4C2×D8C2×SD16C2×C4○D4C2×C20C5×D4C5×Q8C22×C10C10C2×C4D4Q8C23C2C10C2
# reps11111111444444441221448841614

Matrix representation of C5×D4⋊D4 in GL6(𝔽41)

1000000
0100000
001000
000100
000010
000001
,
4000000
0400000
000100
0040000
0000400
0000040
,
3440000
2970000
00122900
00292900
0000402
000001
,
100000
24400000
000900
009000
0000139
0000140
,
100000
24400000
001000
0004000
000010
0000140

G:=sub<GL(6,GF(41))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,29,0,0,0,0,4,7,0,0,0,0,0,0,12,29,0,0,0,0,29,29,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[1,24,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[1,24,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;

C5×D4⋊D4 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes D_4
% in TeX

G:=Group("C5xD4:D4");
// GroupNames label

G:=SmallGroup(320,950);
// by ID

G=gap.SmallGroup(320,950);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,856,7004,3511,172]);
// Polycyclic

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

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