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G = C7×C3⋊C16order 336 = 24·3·7

Direct product of C7 and C3⋊C16

Aliases: C7×C3⋊C16, C3⋊C112, C6.C56, C213C16, C84.8C4, C42.3C8, C56.4S3, C168.7C2, C12.2C28, C24.3C14, C28.5Dic3, C8.2(S3×C7), C14.2(C3⋊C8), C4.2(C7×Dic3), C2.(C7×C3⋊C8), SmallGroup(336,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C7×C3⋊C16
 Chief series C1 — C3 — C6 — C12 — C24 — C168 — C7×C3⋊C16
 Lower central C3 — C7×C3⋊C16
 Upper central C1 — C56

Generators and relations for C7×C3⋊C16
G = < a,b,c | a7=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C7×C3⋊C16
Regular action on 336 points
Generators in S336
(1 70 111 230 260 218 30)(2 71 112 231 261 219 31)(3 72 97 232 262 220 32)(4 73 98 233 263 221 17)(5 74 99 234 264 222 18)(6 75 100 235 265 223 19)(7 76 101 236 266 224 20)(8 77 102 237 267 209 21)(9 78 103 238 268 210 22)(10 79 104 239 269 211 23)(11 80 105 240 270 212 24)(12 65 106 225 271 213 25)(13 66 107 226 272 214 26)(14 67 108 227 257 215 27)(15 68 109 228 258 216 28)(16 69 110 229 259 217 29)(33 150 83 133 297 256 165)(34 151 84 134 298 241 166)(35 152 85 135 299 242 167)(36 153 86 136 300 243 168)(37 154 87 137 301 244 169)(38 155 88 138 302 245 170)(39 156 89 139 303 246 171)(40 157 90 140 304 247 172)(41 158 91 141 289 248 173)(42 159 92 142 290 249 174)(43 160 93 143 291 250 175)(44 145 94 144 292 251 176)(45 146 95 129 293 252 161)(46 147 96 130 294 253 162)(47 148 81 131 295 254 163)(48 149 82 132 296 255 164)(49 317 335 189 276 200 121)(50 318 336 190 277 201 122)(51 319 321 191 278 202 123)(52 320 322 192 279 203 124)(53 305 323 177 280 204 125)(54 306 324 178 281 205 126)(55 307 325 179 282 206 127)(56 308 326 180 283 207 128)(57 309 327 181 284 208 113)(58 310 328 182 285 193 114)(59 311 329 183 286 194 115)(60 312 330 184 287 195 116)(61 313 331 185 288 196 117)(62 314 332 186 273 197 118)(63 315 333 187 274 198 119)(64 316 334 188 275 199 120)
(1 147 288)(2 273 148)(3 149 274)(4 275 150)(5 151 276)(6 277 152)(7 153 278)(8 279 154)(9 155 280)(10 281 156)(11 157 282)(12 283 158)(13 159 284)(14 285 160)(15 145 286)(16 287 146)(17 188 33)(18 34 189)(19 190 35)(20 36 191)(21 192 37)(22 38 177)(23 178 39)(24 40 179)(25 180 41)(26 42 181)(27 182 43)(28 44 183)(29 184 45)(30 46 185)(31 186 47)(32 48 187)(49 234 298)(50 299 235)(51 236 300)(52 301 237)(53 238 302)(54 303 239)(55 240 304)(56 289 225)(57 226 290)(58 291 227)(59 228 292)(60 293 229)(61 230 294)(62 295 231)(63 232 296)(64 297 233)(65 207 91)(66 92 208)(67 193 93)(68 94 194)(69 195 95)(70 96 196)(71 197 81)(72 82 198)(73 199 83)(74 84 200)(75 201 85)(76 86 202)(77 203 87)(78 88 204)(79 205 89)(80 90 206)(97 132 119)(98 120 133)(99 134 121)(100 122 135)(101 136 123)(102 124 137)(103 138 125)(104 126 139)(105 140 127)(106 128 141)(107 142 113)(108 114 143)(109 144 115)(110 116 129)(111 130 117)(112 118 131)(161 217 330)(162 331 218)(163 219 332)(164 333 220)(165 221 334)(166 335 222)(167 223 336)(168 321 224)(169 209 322)(170 323 210)(171 211 324)(172 325 212)(173 213 326)(174 327 214)(175 215 328)(176 329 216)(241 317 264)(242 265 318)(243 319 266)(244 267 320)(245 305 268)(246 269 306)(247 307 270)(248 271 308)(249 309 272)(250 257 310)(251 311 258)(252 259 312)(253 313 260)(254 261 314)(255 315 262)(256 263 316)
(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)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)(177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)(209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)(225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256)(257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272)(273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288)(289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304)(305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320)(321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336)

G:=sub<Sym(336)| (1,70,111,230,260,218,30)(2,71,112,231,261,219,31)(3,72,97,232,262,220,32)(4,73,98,233,263,221,17)(5,74,99,234,264,222,18)(6,75,100,235,265,223,19)(7,76,101,236,266,224,20)(8,77,102,237,267,209,21)(9,78,103,238,268,210,22)(10,79,104,239,269,211,23)(11,80,105,240,270,212,24)(12,65,106,225,271,213,25)(13,66,107,226,272,214,26)(14,67,108,227,257,215,27)(15,68,109,228,258,216,28)(16,69,110,229,259,217,29)(33,150,83,133,297,256,165)(34,151,84,134,298,241,166)(35,152,85,135,299,242,167)(36,153,86,136,300,243,168)(37,154,87,137,301,244,169)(38,155,88,138,302,245,170)(39,156,89,139,303,246,171)(40,157,90,140,304,247,172)(41,158,91,141,289,248,173)(42,159,92,142,290,249,174)(43,160,93,143,291,250,175)(44,145,94,144,292,251,176)(45,146,95,129,293,252,161)(46,147,96,130,294,253,162)(47,148,81,131,295,254,163)(48,149,82,132,296,255,164)(49,317,335,189,276,200,121)(50,318,336,190,277,201,122)(51,319,321,191,278,202,123)(52,320,322,192,279,203,124)(53,305,323,177,280,204,125)(54,306,324,178,281,205,126)(55,307,325,179,282,206,127)(56,308,326,180,283,207,128)(57,309,327,181,284,208,113)(58,310,328,182,285,193,114)(59,311,329,183,286,194,115)(60,312,330,184,287,195,116)(61,313,331,185,288,196,117)(62,314,332,186,273,197,118)(63,315,333,187,274,198,119)(64,316,334,188,275,199,120), (1,147,288)(2,273,148)(3,149,274)(4,275,150)(5,151,276)(6,277,152)(7,153,278)(8,279,154)(9,155,280)(10,281,156)(11,157,282)(12,283,158)(13,159,284)(14,285,160)(15,145,286)(16,287,146)(17,188,33)(18,34,189)(19,190,35)(20,36,191)(21,192,37)(22,38,177)(23,178,39)(24,40,179)(25,180,41)(26,42,181)(27,182,43)(28,44,183)(29,184,45)(30,46,185)(31,186,47)(32,48,187)(49,234,298)(50,299,235)(51,236,300)(52,301,237)(53,238,302)(54,303,239)(55,240,304)(56,289,225)(57,226,290)(58,291,227)(59,228,292)(60,293,229)(61,230,294)(62,295,231)(63,232,296)(64,297,233)(65,207,91)(66,92,208)(67,193,93)(68,94,194)(69,195,95)(70,96,196)(71,197,81)(72,82,198)(73,199,83)(74,84,200)(75,201,85)(76,86,202)(77,203,87)(78,88,204)(79,205,89)(80,90,206)(97,132,119)(98,120,133)(99,134,121)(100,122,135)(101,136,123)(102,124,137)(103,138,125)(104,126,139)(105,140,127)(106,128,141)(107,142,113)(108,114,143)(109,144,115)(110,116,129)(111,130,117)(112,118,131)(161,217,330)(162,331,218)(163,219,332)(164,333,220)(165,221,334)(166,335,222)(167,223,336)(168,321,224)(169,209,322)(170,323,210)(171,211,324)(172,325,212)(173,213,326)(174,327,214)(175,215,328)(176,329,216)(241,317,264)(242,265,318)(243,319,266)(244,267,320)(245,305,268)(246,269,306)(247,307,270)(248,271,308)(249,309,272)(250,257,310)(251,311,258)(252,259,312)(253,313,260)(254,261,314)(255,315,262)(256,263,316), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336)>;

G:=Group( (1,70,111,230,260,218,30)(2,71,112,231,261,219,31)(3,72,97,232,262,220,32)(4,73,98,233,263,221,17)(5,74,99,234,264,222,18)(6,75,100,235,265,223,19)(7,76,101,236,266,224,20)(8,77,102,237,267,209,21)(9,78,103,238,268,210,22)(10,79,104,239,269,211,23)(11,80,105,240,270,212,24)(12,65,106,225,271,213,25)(13,66,107,226,272,214,26)(14,67,108,227,257,215,27)(15,68,109,228,258,216,28)(16,69,110,229,259,217,29)(33,150,83,133,297,256,165)(34,151,84,134,298,241,166)(35,152,85,135,299,242,167)(36,153,86,136,300,243,168)(37,154,87,137,301,244,169)(38,155,88,138,302,245,170)(39,156,89,139,303,246,171)(40,157,90,140,304,247,172)(41,158,91,141,289,248,173)(42,159,92,142,290,249,174)(43,160,93,143,291,250,175)(44,145,94,144,292,251,176)(45,146,95,129,293,252,161)(46,147,96,130,294,253,162)(47,148,81,131,295,254,163)(48,149,82,132,296,255,164)(49,317,335,189,276,200,121)(50,318,336,190,277,201,122)(51,319,321,191,278,202,123)(52,320,322,192,279,203,124)(53,305,323,177,280,204,125)(54,306,324,178,281,205,126)(55,307,325,179,282,206,127)(56,308,326,180,283,207,128)(57,309,327,181,284,208,113)(58,310,328,182,285,193,114)(59,311,329,183,286,194,115)(60,312,330,184,287,195,116)(61,313,331,185,288,196,117)(62,314,332,186,273,197,118)(63,315,333,187,274,198,119)(64,316,334,188,275,199,120), (1,147,288)(2,273,148)(3,149,274)(4,275,150)(5,151,276)(6,277,152)(7,153,278)(8,279,154)(9,155,280)(10,281,156)(11,157,282)(12,283,158)(13,159,284)(14,285,160)(15,145,286)(16,287,146)(17,188,33)(18,34,189)(19,190,35)(20,36,191)(21,192,37)(22,38,177)(23,178,39)(24,40,179)(25,180,41)(26,42,181)(27,182,43)(28,44,183)(29,184,45)(30,46,185)(31,186,47)(32,48,187)(49,234,298)(50,299,235)(51,236,300)(52,301,237)(53,238,302)(54,303,239)(55,240,304)(56,289,225)(57,226,290)(58,291,227)(59,228,292)(60,293,229)(61,230,294)(62,295,231)(63,232,296)(64,297,233)(65,207,91)(66,92,208)(67,193,93)(68,94,194)(69,195,95)(70,96,196)(71,197,81)(72,82,198)(73,199,83)(74,84,200)(75,201,85)(76,86,202)(77,203,87)(78,88,204)(79,205,89)(80,90,206)(97,132,119)(98,120,133)(99,134,121)(100,122,135)(101,136,123)(102,124,137)(103,138,125)(104,126,139)(105,140,127)(106,128,141)(107,142,113)(108,114,143)(109,144,115)(110,116,129)(111,130,117)(112,118,131)(161,217,330)(162,331,218)(163,219,332)(164,333,220)(165,221,334)(166,335,222)(167,223,336)(168,321,224)(169,209,322)(170,323,210)(171,211,324)(172,325,212)(173,213,326)(174,327,214)(175,215,328)(176,329,216)(241,317,264)(242,265,318)(243,319,266)(244,267,320)(245,305,268)(246,269,306)(247,307,270)(248,271,308)(249,309,272)(250,257,310)(251,311,258)(252,259,312)(253,313,260)(254,261,314)(255,315,262)(256,263,316), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336) );

G=PermutationGroup([(1,70,111,230,260,218,30),(2,71,112,231,261,219,31),(3,72,97,232,262,220,32),(4,73,98,233,263,221,17),(5,74,99,234,264,222,18),(6,75,100,235,265,223,19),(7,76,101,236,266,224,20),(8,77,102,237,267,209,21),(9,78,103,238,268,210,22),(10,79,104,239,269,211,23),(11,80,105,240,270,212,24),(12,65,106,225,271,213,25),(13,66,107,226,272,214,26),(14,67,108,227,257,215,27),(15,68,109,228,258,216,28),(16,69,110,229,259,217,29),(33,150,83,133,297,256,165),(34,151,84,134,298,241,166),(35,152,85,135,299,242,167),(36,153,86,136,300,243,168),(37,154,87,137,301,244,169),(38,155,88,138,302,245,170),(39,156,89,139,303,246,171),(40,157,90,140,304,247,172),(41,158,91,141,289,248,173),(42,159,92,142,290,249,174),(43,160,93,143,291,250,175),(44,145,94,144,292,251,176),(45,146,95,129,293,252,161),(46,147,96,130,294,253,162),(47,148,81,131,295,254,163),(48,149,82,132,296,255,164),(49,317,335,189,276,200,121),(50,318,336,190,277,201,122),(51,319,321,191,278,202,123),(52,320,322,192,279,203,124),(53,305,323,177,280,204,125),(54,306,324,178,281,205,126),(55,307,325,179,282,206,127),(56,308,326,180,283,207,128),(57,309,327,181,284,208,113),(58,310,328,182,285,193,114),(59,311,329,183,286,194,115),(60,312,330,184,287,195,116),(61,313,331,185,288,196,117),(62,314,332,186,273,197,118),(63,315,333,187,274,198,119),(64,316,334,188,275,199,120)], [(1,147,288),(2,273,148),(3,149,274),(4,275,150),(5,151,276),(6,277,152),(7,153,278),(8,279,154),(9,155,280),(10,281,156),(11,157,282),(12,283,158),(13,159,284),(14,285,160),(15,145,286),(16,287,146),(17,188,33),(18,34,189),(19,190,35),(20,36,191),(21,192,37),(22,38,177),(23,178,39),(24,40,179),(25,180,41),(26,42,181),(27,182,43),(28,44,183),(29,184,45),(30,46,185),(31,186,47),(32,48,187),(49,234,298),(50,299,235),(51,236,300),(52,301,237),(53,238,302),(54,303,239),(55,240,304),(56,289,225),(57,226,290),(58,291,227),(59,228,292),(60,293,229),(61,230,294),(62,295,231),(63,232,296),(64,297,233),(65,207,91),(66,92,208),(67,193,93),(68,94,194),(69,195,95),(70,96,196),(71,197,81),(72,82,198),(73,199,83),(74,84,200),(75,201,85),(76,86,202),(77,203,87),(78,88,204),(79,205,89),(80,90,206),(97,132,119),(98,120,133),(99,134,121),(100,122,135),(101,136,123),(102,124,137),(103,138,125),(104,126,139),(105,140,127),(106,128,141),(107,142,113),(108,114,143),(109,144,115),(110,116,129),(111,130,117),(112,118,131),(161,217,330),(162,331,218),(163,219,332),(164,333,220),(165,221,334),(166,335,222),(167,223,336),(168,321,224),(169,209,322),(170,323,210),(171,211,324),(172,325,212),(173,213,326),(174,327,214),(175,215,328),(176,329,216),(241,317,264),(242,265,318),(243,319,266),(244,267,320),(245,305,268),(246,269,306),(247,307,270),(248,271,308),(249,309,272),(250,257,310),(251,311,258),(252,259,312),(253,313,260),(254,261,314),(255,315,262),(256,263,316)], [(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),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176),(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208),(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224),(225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256),(257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272),(273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288),(289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304),(305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320),(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336)])

168 conjugacy classes

 class 1 2 3 4A 4B 6 7A ··· 7F 8A 8B 8C 8D 12A 12B 14A ··· 14F 16A ··· 16H 21A ··· 21F 24A 24B 24C 24D 28A ··· 28L 42A ··· 42F 56A ··· 56X 84A ··· 84L 112A ··· 112AV 168A ··· 168X order 1 2 3 4 4 6 7 ··· 7 8 8 8 8 12 12 14 ··· 14 16 ··· 16 21 ··· 21 24 24 24 24 28 ··· 28 42 ··· 42 56 ··· 56 84 ··· 84 112 ··· 112 168 ··· 168 size 1 1 2 1 1 2 1 ··· 1 1 1 1 1 2 2 1 ··· 1 3 ··· 3 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

168 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C4 C7 C8 C14 C16 C28 C56 C112 S3 Dic3 C3⋊C8 S3×C7 C3⋊C16 C7×Dic3 C7×C3⋊C8 C7×C3⋊C16 kernel C7×C3⋊C16 C168 C84 C3⋊C16 C42 C24 C21 C12 C6 C3 C56 C28 C14 C8 C7 C4 C2 C1 # reps 1 1 2 6 4 6 8 12 24 48 1 1 2 6 4 6 12 24

Matrix representation of C7×C3⋊C16 in GL2(𝔽337) generated by

 52 0 0 52
,
 0 336 1 336
,
 59 30 89 278
G:=sub<GL(2,GF(337))| [52,0,0,52],[0,1,336,336],[59,89,30,278] >;

C7×C3⋊C16 in GAP, Magma, Sage, TeX

C_7\times C_3\rtimes C_{16}
% in TeX

G:=Group("C7xC3:C16");
// GroupNames label

G:=SmallGroup(336,3);
// by ID

G=gap.SmallGroup(336,3);
# by ID

G:=PCGroup([6,-2,-7,-2,-2,-2,-3,84,50,69,8069]);
// Polycyclic

G:=Group<a,b,c|a^7=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

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