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## G = D8.F5order 320 = 26·5

### 1st non-split extension by D8 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — D8.F5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C8×D5 — D10.Q8 — D8.F5
 Lower central C5 — C10 — C20 — C40 — D8.F5
 Upper central C1 — C2 — C4 — C8 — D8

Generators and relations for D8.F5
G = < a,b,c,d | a8=b2=c5=1, d4=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c3 >

Character table of D8.F5

 class 1 2A 2B 2C 4A 4B 4C 4D 5 8A 8B 8C 8D 8E 8F 10A 10B 10C 16A 16B 16C 16D 16E 16F 16G 16H 20 40A 40B size 1 1 8 10 2 5 5 40 4 2 2 10 10 40 40 4 16 16 10 10 10 10 10 10 10 10 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 -1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 1 1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ5 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 -1 i -i 1 1 1 i -i -i -i -i i i i 1 1 1 linear of order 4 ρ6 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 i -i 1 -1 -1 -i i i i i -i -i -i 1 1 1 linear of order 4 ρ7 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 -i i 1 -1 -1 i -i -i -i -i i i i 1 1 1 linear of order 4 ρ8 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 -1 -i i 1 1 1 -i i i i i -i -i -i 1 1 1 linear of order 4 ρ9 2 2 0 -2 2 -2 -2 0 2 -2 -2 2 2 0 0 2 0 0 0 0 0 0 0 0 0 0 2 -2 -2 orthogonal lifted from D4 ρ10 2 2 0 2 2 2 2 0 2 -2 -2 -2 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 2 -2 -2 orthogonal lifted from D4 ρ11 2 2 0 -2 -2 2 2 0 2 0 0 0 0 0 0 2 0 0 -√2 √2 √2 -√2 -√2 √2 √2 -√2 -2 0 0 orthogonal lifted from D8 ρ12 2 2 0 -2 -2 2 2 0 2 0 0 0 0 0 0 2 0 0 √2 -√2 -√2 √2 √2 -√2 -√2 √2 -2 0 0 orthogonal lifted from D8 ρ13 2 2 0 2 -2 -2 -2 0 2 0 0 0 0 0 0 2 0 0 -√-2 -√-2 -√-2 √-2 √-2 √-2 √-2 -√-2 -2 0 0 complex lifted from SD16 ρ14 2 2 0 2 -2 -2 -2 0 2 0 0 0 0 0 0 2 0 0 √-2 √-2 √-2 -√-2 -√-2 -√-2 -√-2 √-2 -2 0 0 complex lifted from SD16 ρ15 2 -2 0 0 0 -2i 2i 0 2 -√2 √2 √-2 -√-2 0 0 -2 0 0 ζ1613+ζ167 ζ167+ζ165 ζ1615+ζ1613 ζ169+ζ163 ζ1611+ζ16 ζ163+ζ16 ζ1611+ζ169 ζ1615+ζ165 0 √2 -√2 complex lifted from D8.C4 ρ16 2 -2 0 0 0 2i -2i 0 2 -√2 √2 -√-2 √-2 0 0 -2 0 0 ζ1611+ζ16 ζ163+ζ16 ζ1611+ζ169 ζ1615+ζ165 ζ1613+ζ167 ζ167+ζ165 ζ1615+ζ1613 ζ169+ζ163 0 √2 -√2 complex lifted from D8.C4 ρ17 2 -2 0 0 0 2i -2i 0 2 √2 -√2 √-2 -√-2 0 0 -2 0 0 ζ1615+ζ1613 ζ1613+ζ167 ζ1615+ζ165 ζ163+ζ16 ζ1611+ζ169 ζ1611+ζ16 ζ169+ζ163 ζ167+ζ165 0 -√2 √2 complex lifted from D8.C4 ρ18 2 -2 0 0 0 -2i 2i 0 2 -√2 √2 √-2 -√-2 0 0 -2 0 0 ζ1615+ζ165 ζ1615+ζ1613 ζ167+ζ165 ζ1611+ζ16 ζ169+ζ163 ζ1611+ζ169 ζ163+ζ16 ζ1613+ζ167 0 √2 -√2 complex lifted from D8.C4 ρ19 2 -2 0 0 0 -2i 2i 0 2 √2 -√2 -√-2 √-2 0 0 -2 0 0 ζ163+ζ16 ζ169+ζ163 ζ1611+ζ16 ζ1615+ζ1613 ζ167+ζ165 ζ1615+ζ165 ζ1613+ζ167 ζ1611+ζ169 0 -√2 √2 complex lifted from D8.C4 ρ20 2 -2 0 0 0 2i -2i 0 2 -√2 √2 -√-2 √-2 0 0 -2 0 0 ζ169+ζ163 ζ1611+ζ169 ζ163+ζ16 ζ1613+ζ167 ζ1615+ζ165 ζ1615+ζ1613 ζ167+ζ165 ζ1611+ζ16 0 √2 -√2 complex lifted from D8.C4 ρ21 2 -2 0 0 0 2i -2i 0 2 √2 -√2 √-2 -√-2 0 0 -2 0 0 ζ167+ζ165 ζ1615+ζ165 ζ1613+ζ167 ζ1611+ζ169 ζ163+ζ16 ζ169+ζ163 ζ1611+ζ16 ζ1615+ζ1613 0 -√2 √2 complex lifted from D8.C4 ρ22 2 -2 0 0 0 -2i 2i 0 2 √2 -√2 -√-2 √-2 0 0 -2 0 0 ζ1611+ζ169 ζ1611+ζ16 ζ169+ζ163 ζ167+ζ165 ζ1615+ζ1613 ζ1613+ζ167 ζ1615+ζ165 ζ163+ζ16 0 -√2 √2 complex lifted from D8.C4 ρ23 4 4 4 0 4 0 0 0 -1 4 4 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from F5 ρ24 4 4 -4 0 4 0 0 0 -1 4 4 0 0 0 0 -1 1 1 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from C2×F5 ρ25 4 4 0 0 4 0 0 0 -1 -4 -4 0 0 0 0 -1 √5 -√5 0 0 0 0 0 0 0 0 -1 1 1 orthogonal lifted from C22⋊F5 ρ26 4 4 0 0 4 0 0 0 -1 -4 -4 0 0 0 0 -1 -√5 √5 0 0 0 0 0 0 0 0 -1 1 1 orthogonal lifted from C22⋊F5 ρ27 8 8 0 0 -8 0 0 0 -2 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 0 0 orthogonal lifted from D20⋊C4, Schur index 2 ρ28 8 -8 0 0 0 0 0 0 -2 -4√2 4√2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 symplectic faithful, Schur index 2 ρ29 8 -8 0 0 0 0 0 0 -2 4√2 -4√2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 symplectic faithful, Schur index 2

Smallest permutation representation of D8.F5
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 97)(2 104)(3 103)(4 102)(5 101)(6 100)(7 99)(8 98)(9 145)(10 152)(11 151)(12 150)(13 149)(14 148)(15 147)(16 146)(17 129)(18 136)(19 135)(20 134)(21 133)(22 132)(23 131)(24 130)(25 144)(26 143)(27 142)(28 141)(29 140)(30 139)(31 138)(32 137)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 128)(40 127)(41 88)(42 87)(43 86)(44 85)(45 84)(46 83)(47 82)(48 81)(49 90)(50 89)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(57 75)(58 74)(59 73)(60 80)(61 79)(62 78)(63 77)(64 76)(65 106)(66 105)(67 112)(68 111)(69 110)(70 109)(71 108)(72 107)(113 157)(114 156)(115 155)(116 154)(117 153)(118 160)(119 159)(120 158)
(1 69 57 46 53)(2 70 58 47 54)(3 71 59 48 55)(4 72 60 41 56)(5 65 61 42 49)(6 66 62 43 50)(7 67 63 44 51)(8 68 64 45 52)(9 36 22 155 25)(10 37 23 156 26)(11 38 24 157 27)(12 39 17 158 28)(13 40 18 159 29)(14 33 19 160 30)(15 34 20 153 31)(16 35 21 154 32)(73 81 92 103 108)(74 82 93 104 109)(75 83 94 97 110)(76 84 95 98 111)(77 85 96 99 112)(78 86 89 100 105)(79 87 90 101 106)(80 88 91 102 107)(113 142 151 121 130)(114 143 152 122 131)(115 144 145 123 132)(116 137 146 124 133)(117 138 147 125 134)(118 139 148 126 135)(119 140 149 127 136)(120 141 150 128 129)
(1 131 3 129 5 135 7 133)(2 130 4 136 6 134 8 132)(9 112 27 94 13 108 31 90)(10 111 28 93 14 107 32 89)(11 110 29 92 15 106 25 96)(12 109 30 91 16 105 26 95)(17 104 19 102 21 100 23 98)(18 103 20 101 22 99 24 97)(33 88 154 78 37 84 158 74)(34 87 155 77 38 83 159 73)(35 86 156 76 39 82 160 80)(36 85 157 75 40 81 153 79)(41 119 62 125 45 115 58 121)(42 118 63 124 46 114 59 128)(43 117 64 123 47 113 60 127)(44 116 57 122 48 120 61 126)(49 148 67 137 53 152 71 141)(50 147 68 144 54 151 72 140)(51 146 69 143 55 150 65 139)(52 145 70 142 56 149 66 138)```

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

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

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

Matrix representation of D8.F5 in GL6(𝔽241)

 211 0 0 0 0 0 0 8 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 0 8 0 0 0 0 211 0 0 0 0 0 0 0 117 0 234 234 0 0 7 124 7 0 0 0 0 7 124 7 0 0 234 234 0 117
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 240 240 240 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 0 165 0 0 0 0 115 0 0 0 0 0 0 0 8 0 136 136 0 0 136 136 0 8 0 0 105 113 105 0 0 0 233 128 128 233

`G:=sub<GL(6,GF(241))| [211,0,0,0,0,0,0,8,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,211,0,0,0,0,8,0,0,0,0,0,0,0,117,7,0,234,0,0,0,124,7,234,0,0,234,7,124,0,0,0,234,0,7,117],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[0,115,0,0,0,0,165,0,0,0,0,0,0,0,8,136,105,233,0,0,0,136,113,128,0,0,136,0,105,128,0,0,136,8,0,233] >;`

D8.F5 in GAP, Magma, Sage, TeX

`D_8.F_5`
`% in TeX`

`G:=Group("D8.F5");`
`// GroupNames label`

`G:=SmallGroup(320,243);`
`// by ID`

`G=gap.SmallGroup(320,243);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,184,675,346,192,1684,851,102,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^5=1,d^4=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^3>;`
`// generators/relations`

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