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## G = C10.342+ 1+4order 320 = 26·5

### 34th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.342+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C10.D4 — C10.342+ 1+4
 Lower central C5 — C2×C10 — C10.342+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C10.342+ 1+4
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, dbd-1=ebe=a5b, cd=dc, ce=ec, ede=a5b2d >

Subgroups: 790 in 238 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C422C2, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.47C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23.11D10, C23.D10, Dic54D4, D10.12D4, Dic5.Q8, C4⋊C4⋊D5, C2×C10.D4, C4×C5⋊D4, D4×Dic5, C23.18D10, Dic5⋊D4, C5×C4⋊D4, C10.342+ 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.47C24, D42D5, C23×D5, C2×D42D5, D46D10, D5×C4○D4, C10.342+ 1+4

Smallest permutation representation of C10.342+ 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)
(1 110 25 95)(2 109 26 94)(3 108 27 93)(4 107 28 92)(5 106 29 91)(6 105 30 100)(7 104 21 99)(8 103 22 98)(9 102 23 97)(10 101 24 96)(11 85 160 80)(12 84 151 79)(13 83 152 78)(14 82 153 77)(15 81 154 76)(16 90 155 75)(17 89 156 74)(18 88 157 73)(19 87 158 72)(20 86 159 71)(31 124 46 119)(32 123 47 118)(33 122 48 117)(34 121 49 116)(35 130 50 115)(36 129 41 114)(37 128 42 113)(38 127 43 112)(39 126 44 111)(40 125 45 120)(51 149 66 134)(52 148 67 133)(53 147 68 132)(54 146 69 131)(55 145 70 140)(56 144 61 139)(57 143 62 138)(58 142 63 137)(59 141 64 136)(60 150 65 135)
(1 65 6 70)(2 64 7 69)(3 63 8 68)(4 62 9 67)(5 61 10 66)(11 125 16 130)(12 124 17 129)(13 123 18 128)(14 122 19 127)(15 121 20 126)(21 54 26 59)(22 53 27 58)(23 52 28 57)(24 51 29 56)(25 60 30 55)(31 89 36 84)(32 88 37 83)(33 87 38 82)(34 86 39 81)(35 85 40 90)(41 79 46 74)(42 78 47 73)(43 77 48 72)(44 76 49 71)(45 75 50 80)(91 139 96 134)(92 138 97 133)(93 137 98 132)(94 136 99 131)(95 135 100 140)(101 149 106 144)(102 148 107 143)(103 147 108 142)(104 146 109 141)(105 145 110 150)(111 154 116 159)(112 153 117 158)(113 152 118 157)(114 151 119 156)(115 160 120 155)
(1 120 25 125)(2 111 26 126)(3 112 27 127)(4 113 28 128)(5 114 29 129)(6 115 30 130)(7 116 21 121)(8 117 22 122)(9 118 23 123)(10 119 24 124)(11 70 160 55)(12 61 151 56)(13 62 152 57)(14 63 153 58)(15 64 154 59)(16 65 155 60)(17 66 156 51)(18 67 157 52)(19 68 158 53)(20 69 159 54)(31 91 46 106)(32 92 47 107)(33 93 48 108)(34 94 49 109)(35 95 50 110)(36 96 41 101)(37 97 42 102)(38 98 43 103)(39 99 44 104)(40 100 45 105)(71 141 86 136)(72 142 87 137)(73 143 88 138)(74 144 89 139)(75 145 90 140)(76 146 81 131)(77 147 82 132)(78 148 83 133)(79 149 84 134)(80 150 85 135)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 156)(22 157)(23 158)(24 159)(25 160)(26 151)(27 152)(28 153)(29 154)(30 155)(31 146)(32 147)(33 148)(34 149)(35 150)(36 141)(37 142)(38 143)(39 144)(40 145)(41 136)(42 137)(43 138)(44 139)(45 140)(46 131)(47 132)(48 133)(49 134)(50 135)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)```

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 2+ 1+4 D4⋊2D5 D4⋊6D10 D5×C4○D4 kernel C10.342+ 1+4 C23.11D10 C23.D10 Dic5⋊4D4 D10.12D4 Dic5.Q8 C4⋊C4⋊D5 C2×C10.D4 C4×C5⋊D4 D4×Dic5 C23.18D10 Dic5⋊D4 C5×C4⋊D4 C4⋊D4 Dic5 C2×C10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 3 1 2 4 4 4 2 2 6 1 4 4 4

Matrix representation of C10.342+ 1+4 in GL6(𝔽41)

 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 35 0 0 0 0 6 35
,
 0 9 0 0 0 0 9 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 35 6 0 0 0 0 1 6
,
 9 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 35 6 0 0 0 0 1 6
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 9 0 0 0 0 32 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,6,0,0,0,0,35,35],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,35,1,0,0,0,0,6,6],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,35,1,0,0,0,0,6,6],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C10.342+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{34}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.34ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1273);`
`// by ID`

`G=gap.SmallGroup(320,1273);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,100,794,297,12550]);`
`// Polycyclic`

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

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