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## G = Dic10⋊5C8order 320 = 26·5

### 3rd semidirect product of Dic10 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Dic10⋊5C8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C5⋊2C8 — C4×C5⋊2C8 — Dic10⋊5C8
 Lower central C5 — C10 — Dic10⋊5C8
 Upper central C1 — C2×C4 — C4⋊C8

Generators and relations for Dic105C8
G = < a,b,c | a20=c8=1, b2=a10, bab-1=a-1, cac-1=a11, bc=cb >

Subgroups: 254 in 102 conjugacy classes, 61 normal (31 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, C20, C2×C10, C4×C8, C4⋊C8, C4⋊C8, C4×Q8, C52C8, C52C8, C40, Dic10, C2×Dic5, C2×C20, C8×Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C4×C52C8, C8×Dic5, C20.8Q8, C5×C4⋊C8, C4×Dic10, Dic105C8
Quotients: C1, C2, C4, C22, C8, C2×C4, Q8, C23, D5, C2×C8, C22×C4, C2×Q8, C4○D4, D10, C4×Q8, C22×C8, C8○D4, C4×D5, C22×D5, C8×Q8, C8×D5, C2×C4×D5, D42D5, Q8×D5, Dic53Q8, D5×C2×C8, D20.2C4, Dic105C8

Smallest permutation representation of Dic105C8
Regular action on 320 points
Generators in S320
(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)
(1 254 11 244)(2 253 12 243)(3 252 13 242)(4 251 14 241)(5 250 15 260)(6 249 16 259)(7 248 17 258)(8 247 18 257)(9 246 19 256)(10 245 20 255)(21 192 31 182)(22 191 32 181)(23 190 33 200)(24 189 34 199)(25 188 35 198)(26 187 36 197)(27 186 37 196)(28 185 38 195)(29 184 39 194)(30 183 40 193)(41 82 51 92)(42 81 52 91)(43 100 53 90)(44 99 54 89)(45 98 55 88)(46 97 56 87)(47 96 57 86)(48 95 58 85)(49 94 59 84)(50 93 60 83)(61 114 71 104)(62 113 72 103)(63 112 73 102)(64 111 74 101)(65 110 75 120)(66 109 76 119)(67 108 77 118)(68 107 78 117)(69 106 79 116)(70 105 80 115)(121 300 131 290)(122 299 132 289)(123 298 133 288)(124 297 134 287)(125 296 135 286)(126 295 136 285)(127 294 137 284)(128 293 138 283)(129 292 139 282)(130 291 140 281)(141 180 151 170)(142 179 152 169)(143 178 153 168)(144 177 154 167)(145 176 155 166)(146 175 156 165)(147 174 157 164)(148 173 158 163)(149 172 159 162)(150 171 160 161)(201 301 211 311)(202 320 212 310)(203 319 213 309)(204 318 214 308)(205 317 215 307)(206 316 216 306)(207 315 217 305)(208 314 218 304)(209 313 219 303)(210 312 220 302)(221 274 231 264)(222 273 232 263)(223 272 233 262)(224 271 234 261)(225 270 235 280)(226 269 236 279)(227 268 237 278)(228 267 238 277)(229 266 239 276)(230 265 240 275)
(1 84 162 315 113 138 228 183)(2 95 163 306 114 129 229 194)(3 86 164 317 115 140 230 185)(4 97 165 308 116 131 231 196)(5 88 166 319 117 122 232 187)(6 99 167 310 118 133 233 198)(7 90 168 301 119 124 234 189)(8 81 169 312 120 135 235 200)(9 92 170 303 101 126 236 191)(10 83 171 314 102 137 237 182)(11 94 172 305 103 128 238 193)(12 85 173 316 104 139 239 184)(13 96 174 307 105 130 240 195)(14 87 175 318 106 121 221 186)(15 98 176 309 107 132 222 197)(16 89 177 320 108 123 223 188)(17 100 178 311 109 134 224 199)(18 91 179 302 110 125 225 190)(19 82 180 313 111 136 226 181)(20 93 161 304 112 127 227 192)(21 245 50 160 218 63 284 278)(22 256 51 151 219 74 285 269)(23 247 52 142 220 65 286 280)(24 258 53 153 201 76 287 271)(25 249 54 144 202 67 288 262)(26 260 55 155 203 78 289 273)(27 251 56 146 204 69 290 264)(28 242 57 157 205 80 291 275)(29 253 58 148 206 71 292 266)(30 244 59 159 207 62 293 277)(31 255 60 150 208 73 294 268)(32 246 41 141 209 64 295 279)(33 257 42 152 210 75 296 270)(34 248 43 143 211 66 297 261)(35 259 44 154 212 77 298 272)(36 250 45 145 213 68 299 263)(37 241 46 156 214 79 300 274)(38 252 47 147 215 70 281 265)(39 243 48 158 216 61 282 276)(40 254 49 149 217 72 283 267)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 5A 5B 8A ··· 8H 8I ··· 8P 8Q 8R 8S 8T 10A ··· 10F 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 5 5 8 ··· 8 8 ··· 8 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 1 1 1 1 2 2 2 2 10 ··· 10 2 2 2 ··· 2 5 ··· 5 10 10 10 10 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 Q8 D5 C4○D4 D10 D10 C8○D4 C4×D5 C8×D5 D4⋊2D5 Q8×D5 D20.2C4 kernel Dic10⋊5C8 C4×C5⋊2C8 C8×Dic5 C20.8Q8 C5×C4⋊C8 C4×Dic10 C10.D4 C4⋊Dic5 C2×Dic10 Dic10 C5⋊2C8 C4⋊C8 C20 C42 C2×C8 C10 C2×C4 C4 C4 C4 C2 # reps 1 1 2 2 1 1 4 2 2 16 2 2 2 2 4 4 8 16 2 2 4

Matrix representation of Dic105C8 in GL4(𝔽41) generated by

 0 40 0 0 1 35 0 0 0 0 0 33 0 0 36 0
,
 6 40 0 0 35 35 0 0 0 0 40 35 0 0 14 1
,
 14 0 0 0 0 14 0 0 0 0 27 39 0 0 32 14
G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,0,36,0,0,33,0],[6,35,0,0,40,35,0,0,0,0,40,14,0,0,35,1],[14,0,0,0,0,14,0,0,0,0,27,32,0,0,39,14] >;

Dic105C8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_5C_8
% in TeX

G:=Group("Dic10:5C8");
// GroupNames label

G:=SmallGroup(320,457);
// by ID

G=gap.SmallGroup(320,457);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,64,135,142,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=c^8=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,b*c=c*b>;
// generators/relations

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