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G = C32×C39order 351 = 33·13

Abelian group of type [3,3,39]

direct product, abelian, monomial, 3-elementary

Aliases: C32×C39, SmallGroup(351,14)

Series: Derived Chief Lower central Upper central

C1 — C32×C39
C1C13C39C3×C39 — C32×C39
C1 — C32×C39
C1 — C32×C39

Generators and relations for C32×C39
 G = < a,b,c | a3=b3=c39=1, ab=ba, ac=ca, bc=cb >

Subgroups: 56, all normal (4 characteristic)
C1, C3 [×13], C32 [×13], C13, C33, C39 [×13], C3×C39 [×13], C32×C39
Quotients: C1, C3 [×13], C32 [×13], C13, C33, C39 [×13], C3×C39 [×13], C32×C39

Smallest permutation representation of C32×C39
Regular action on 351 points
Generators in S351
(1 266 150)(2 267 151)(3 268 152)(4 269 153)(5 270 154)(6 271 155)(7 272 156)(8 273 118)(9 235 119)(10 236 120)(11 237 121)(12 238 122)(13 239 123)(14 240 124)(15 241 125)(16 242 126)(17 243 127)(18 244 128)(19 245 129)(20 246 130)(21 247 131)(22 248 132)(23 249 133)(24 250 134)(25 251 135)(26 252 136)(27 253 137)(28 254 138)(29 255 139)(30 256 140)(31 257 141)(32 258 142)(33 259 143)(34 260 144)(35 261 145)(36 262 146)(37 263 147)(38 264 148)(39 265 149)(40 302 170)(41 303 171)(42 304 172)(43 305 173)(44 306 174)(45 307 175)(46 308 176)(47 309 177)(48 310 178)(49 311 179)(50 312 180)(51 274 181)(52 275 182)(53 276 183)(54 277 184)(55 278 185)(56 279 186)(57 280 187)(58 281 188)(59 282 189)(60 283 190)(61 284 191)(62 285 192)(63 286 193)(64 287 194)(65 288 195)(66 289 157)(67 290 158)(68 291 159)(69 292 160)(70 293 161)(71 294 162)(72 295 163)(73 296 164)(74 297 165)(75 298 166)(76 299 167)(77 300 168)(78 301 169)(79 345 219)(80 346 220)(81 347 221)(82 348 222)(83 349 223)(84 350 224)(85 351 225)(86 313 226)(87 314 227)(88 315 228)(89 316 229)(90 317 230)(91 318 231)(92 319 232)(93 320 233)(94 321 234)(95 322 196)(96 323 197)(97 324 198)(98 325 199)(99 326 200)(100 327 201)(101 328 202)(102 329 203)(103 330 204)(104 331 205)(105 332 206)(106 333 207)(107 334 208)(108 335 209)(109 336 210)(110 337 211)(111 338 212)(112 339 213)(113 340 214)(114 341 215)(115 342 216)(116 343 217)(117 344 218)
(1 94 48)(2 95 49)(3 96 50)(4 97 51)(5 98 52)(6 99 53)(7 100 54)(8 101 55)(9 102 56)(10 103 57)(11 104 58)(12 105 59)(13 106 60)(14 107 61)(15 108 62)(16 109 63)(17 110 64)(18 111 65)(19 112 66)(20 113 67)(21 114 68)(22 115 69)(23 116 70)(24 117 71)(25 79 72)(26 80 73)(27 81 74)(28 82 75)(29 83 76)(30 84 77)(31 85 78)(32 86 40)(33 87 41)(34 88 42)(35 89 43)(36 90 44)(37 91 45)(38 92 46)(39 93 47)(118 202 185)(119 203 186)(120 204 187)(121 205 188)(122 206 189)(123 207 190)(124 208 191)(125 209 192)(126 210 193)(127 211 194)(128 212 195)(129 213 157)(130 214 158)(131 215 159)(132 216 160)(133 217 161)(134 218 162)(135 219 163)(136 220 164)(137 221 165)(138 222 166)(139 223 167)(140 224 168)(141 225 169)(142 226 170)(143 227 171)(144 228 172)(145 229 173)(146 230 174)(147 231 175)(148 232 176)(149 233 177)(150 234 178)(151 196 179)(152 197 180)(153 198 181)(154 199 182)(155 200 183)(156 201 184)(235 329 279)(236 330 280)(237 331 281)(238 332 282)(239 333 283)(240 334 284)(241 335 285)(242 336 286)(243 337 287)(244 338 288)(245 339 289)(246 340 290)(247 341 291)(248 342 292)(249 343 293)(250 344 294)(251 345 295)(252 346 296)(253 347 297)(254 348 298)(255 349 299)(256 350 300)(257 351 301)(258 313 302)(259 314 303)(260 315 304)(261 316 305)(262 317 306)(263 318 307)(264 319 308)(265 320 309)(266 321 310)(267 322 311)(268 323 312)(269 324 274)(270 325 275)(271 326 276)(272 327 277)(273 328 278)
(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)(313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351)

G:=sub<Sym(351)| (1,266,150)(2,267,151)(3,268,152)(4,269,153)(5,270,154)(6,271,155)(7,272,156)(8,273,118)(9,235,119)(10,236,120)(11,237,121)(12,238,122)(13,239,123)(14,240,124)(15,241,125)(16,242,126)(17,243,127)(18,244,128)(19,245,129)(20,246,130)(21,247,131)(22,248,132)(23,249,133)(24,250,134)(25,251,135)(26,252,136)(27,253,137)(28,254,138)(29,255,139)(30,256,140)(31,257,141)(32,258,142)(33,259,143)(34,260,144)(35,261,145)(36,262,146)(37,263,147)(38,264,148)(39,265,149)(40,302,170)(41,303,171)(42,304,172)(43,305,173)(44,306,174)(45,307,175)(46,308,176)(47,309,177)(48,310,178)(49,311,179)(50,312,180)(51,274,181)(52,275,182)(53,276,183)(54,277,184)(55,278,185)(56,279,186)(57,280,187)(58,281,188)(59,282,189)(60,283,190)(61,284,191)(62,285,192)(63,286,193)(64,287,194)(65,288,195)(66,289,157)(67,290,158)(68,291,159)(69,292,160)(70,293,161)(71,294,162)(72,295,163)(73,296,164)(74,297,165)(75,298,166)(76,299,167)(77,300,168)(78,301,169)(79,345,219)(80,346,220)(81,347,221)(82,348,222)(83,349,223)(84,350,224)(85,351,225)(86,313,226)(87,314,227)(88,315,228)(89,316,229)(90,317,230)(91,318,231)(92,319,232)(93,320,233)(94,321,234)(95,322,196)(96,323,197)(97,324,198)(98,325,199)(99,326,200)(100,327,201)(101,328,202)(102,329,203)(103,330,204)(104,331,205)(105,332,206)(106,333,207)(107,334,208)(108,335,209)(109,336,210)(110,337,211)(111,338,212)(112,339,213)(113,340,214)(114,341,215)(115,342,216)(116,343,217)(117,344,218), (1,94,48)(2,95,49)(3,96,50)(4,97,51)(5,98,52)(6,99,53)(7,100,54)(8,101,55)(9,102,56)(10,103,57)(11,104,58)(12,105,59)(13,106,60)(14,107,61)(15,108,62)(16,109,63)(17,110,64)(18,111,65)(19,112,66)(20,113,67)(21,114,68)(22,115,69)(23,116,70)(24,117,71)(25,79,72)(26,80,73)(27,81,74)(28,82,75)(29,83,76)(30,84,77)(31,85,78)(32,86,40)(33,87,41)(34,88,42)(35,89,43)(36,90,44)(37,91,45)(38,92,46)(39,93,47)(118,202,185)(119,203,186)(120,204,187)(121,205,188)(122,206,189)(123,207,190)(124,208,191)(125,209,192)(126,210,193)(127,211,194)(128,212,195)(129,213,157)(130,214,158)(131,215,159)(132,216,160)(133,217,161)(134,218,162)(135,219,163)(136,220,164)(137,221,165)(138,222,166)(139,223,167)(140,224,168)(141,225,169)(142,226,170)(143,227,171)(144,228,172)(145,229,173)(146,230,174)(147,231,175)(148,232,176)(149,233,177)(150,234,178)(151,196,179)(152,197,180)(153,198,181)(154,199,182)(155,200,183)(156,201,184)(235,329,279)(236,330,280)(237,331,281)(238,332,282)(239,333,283)(240,334,284)(241,335,285)(242,336,286)(243,337,287)(244,338,288)(245,339,289)(246,340,290)(247,341,291)(248,342,292)(249,343,293)(250,344,294)(251,345,295)(252,346,296)(253,347,297)(254,348,298)(255,349,299)(256,350,300)(257,351,301)(258,313,302)(259,314,303)(260,315,304)(261,316,305)(262,317,306)(263,318,307)(264,319,308)(265,320,309)(266,321,310)(267,322,311)(268,323,312)(269,324,274)(270,325,275)(271,326,276)(272,327,277)(273,328,278), (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)(313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351)>;

G:=Group( (1,266,150)(2,267,151)(3,268,152)(4,269,153)(5,270,154)(6,271,155)(7,272,156)(8,273,118)(9,235,119)(10,236,120)(11,237,121)(12,238,122)(13,239,123)(14,240,124)(15,241,125)(16,242,126)(17,243,127)(18,244,128)(19,245,129)(20,246,130)(21,247,131)(22,248,132)(23,249,133)(24,250,134)(25,251,135)(26,252,136)(27,253,137)(28,254,138)(29,255,139)(30,256,140)(31,257,141)(32,258,142)(33,259,143)(34,260,144)(35,261,145)(36,262,146)(37,263,147)(38,264,148)(39,265,149)(40,302,170)(41,303,171)(42,304,172)(43,305,173)(44,306,174)(45,307,175)(46,308,176)(47,309,177)(48,310,178)(49,311,179)(50,312,180)(51,274,181)(52,275,182)(53,276,183)(54,277,184)(55,278,185)(56,279,186)(57,280,187)(58,281,188)(59,282,189)(60,283,190)(61,284,191)(62,285,192)(63,286,193)(64,287,194)(65,288,195)(66,289,157)(67,290,158)(68,291,159)(69,292,160)(70,293,161)(71,294,162)(72,295,163)(73,296,164)(74,297,165)(75,298,166)(76,299,167)(77,300,168)(78,301,169)(79,345,219)(80,346,220)(81,347,221)(82,348,222)(83,349,223)(84,350,224)(85,351,225)(86,313,226)(87,314,227)(88,315,228)(89,316,229)(90,317,230)(91,318,231)(92,319,232)(93,320,233)(94,321,234)(95,322,196)(96,323,197)(97,324,198)(98,325,199)(99,326,200)(100,327,201)(101,328,202)(102,329,203)(103,330,204)(104,331,205)(105,332,206)(106,333,207)(107,334,208)(108,335,209)(109,336,210)(110,337,211)(111,338,212)(112,339,213)(113,340,214)(114,341,215)(115,342,216)(116,343,217)(117,344,218), (1,94,48)(2,95,49)(3,96,50)(4,97,51)(5,98,52)(6,99,53)(7,100,54)(8,101,55)(9,102,56)(10,103,57)(11,104,58)(12,105,59)(13,106,60)(14,107,61)(15,108,62)(16,109,63)(17,110,64)(18,111,65)(19,112,66)(20,113,67)(21,114,68)(22,115,69)(23,116,70)(24,117,71)(25,79,72)(26,80,73)(27,81,74)(28,82,75)(29,83,76)(30,84,77)(31,85,78)(32,86,40)(33,87,41)(34,88,42)(35,89,43)(36,90,44)(37,91,45)(38,92,46)(39,93,47)(118,202,185)(119,203,186)(120,204,187)(121,205,188)(122,206,189)(123,207,190)(124,208,191)(125,209,192)(126,210,193)(127,211,194)(128,212,195)(129,213,157)(130,214,158)(131,215,159)(132,216,160)(133,217,161)(134,218,162)(135,219,163)(136,220,164)(137,221,165)(138,222,166)(139,223,167)(140,224,168)(141,225,169)(142,226,170)(143,227,171)(144,228,172)(145,229,173)(146,230,174)(147,231,175)(148,232,176)(149,233,177)(150,234,178)(151,196,179)(152,197,180)(153,198,181)(154,199,182)(155,200,183)(156,201,184)(235,329,279)(236,330,280)(237,331,281)(238,332,282)(239,333,283)(240,334,284)(241,335,285)(242,336,286)(243,337,287)(244,338,288)(245,339,289)(246,340,290)(247,341,291)(248,342,292)(249,343,293)(250,344,294)(251,345,295)(252,346,296)(253,347,297)(254,348,298)(255,349,299)(256,350,300)(257,351,301)(258,313,302)(259,314,303)(260,315,304)(261,316,305)(262,317,306)(263,318,307)(264,319,308)(265,320,309)(266,321,310)(267,322,311)(268,323,312)(269,324,274)(270,325,275)(271,326,276)(272,327,277)(273,328,278), (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)(313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351) );

G=PermutationGroup([(1,266,150),(2,267,151),(3,268,152),(4,269,153),(5,270,154),(6,271,155),(7,272,156),(8,273,118),(9,235,119),(10,236,120),(11,237,121),(12,238,122),(13,239,123),(14,240,124),(15,241,125),(16,242,126),(17,243,127),(18,244,128),(19,245,129),(20,246,130),(21,247,131),(22,248,132),(23,249,133),(24,250,134),(25,251,135),(26,252,136),(27,253,137),(28,254,138),(29,255,139),(30,256,140),(31,257,141),(32,258,142),(33,259,143),(34,260,144),(35,261,145),(36,262,146),(37,263,147),(38,264,148),(39,265,149),(40,302,170),(41,303,171),(42,304,172),(43,305,173),(44,306,174),(45,307,175),(46,308,176),(47,309,177),(48,310,178),(49,311,179),(50,312,180),(51,274,181),(52,275,182),(53,276,183),(54,277,184),(55,278,185),(56,279,186),(57,280,187),(58,281,188),(59,282,189),(60,283,190),(61,284,191),(62,285,192),(63,286,193),(64,287,194),(65,288,195),(66,289,157),(67,290,158),(68,291,159),(69,292,160),(70,293,161),(71,294,162),(72,295,163),(73,296,164),(74,297,165),(75,298,166),(76,299,167),(77,300,168),(78,301,169),(79,345,219),(80,346,220),(81,347,221),(82,348,222),(83,349,223),(84,350,224),(85,351,225),(86,313,226),(87,314,227),(88,315,228),(89,316,229),(90,317,230),(91,318,231),(92,319,232),(93,320,233),(94,321,234),(95,322,196),(96,323,197),(97,324,198),(98,325,199),(99,326,200),(100,327,201),(101,328,202),(102,329,203),(103,330,204),(104,331,205),(105,332,206),(106,333,207),(107,334,208),(108,335,209),(109,336,210),(110,337,211),(111,338,212),(112,339,213),(113,340,214),(114,341,215),(115,342,216),(116,343,217),(117,344,218)], [(1,94,48),(2,95,49),(3,96,50),(4,97,51),(5,98,52),(6,99,53),(7,100,54),(8,101,55),(9,102,56),(10,103,57),(11,104,58),(12,105,59),(13,106,60),(14,107,61),(15,108,62),(16,109,63),(17,110,64),(18,111,65),(19,112,66),(20,113,67),(21,114,68),(22,115,69),(23,116,70),(24,117,71),(25,79,72),(26,80,73),(27,81,74),(28,82,75),(29,83,76),(30,84,77),(31,85,78),(32,86,40),(33,87,41),(34,88,42),(35,89,43),(36,90,44),(37,91,45),(38,92,46),(39,93,47),(118,202,185),(119,203,186),(120,204,187),(121,205,188),(122,206,189),(123,207,190),(124,208,191),(125,209,192),(126,210,193),(127,211,194),(128,212,195),(129,213,157),(130,214,158),(131,215,159),(132,216,160),(133,217,161),(134,218,162),(135,219,163),(136,220,164),(137,221,165),(138,222,166),(139,223,167),(140,224,168),(141,225,169),(142,226,170),(143,227,171),(144,228,172),(145,229,173),(146,230,174),(147,231,175),(148,232,176),(149,233,177),(150,234,178),(151,196,179),(152,197,180),(153,198,181),(154,199,182),(155,200,183),(156,201,184),(235,329,279),(236,330,280),(237,331,281),(238,332,282),(239,333,283),(240,334,284),(241,335,285),(242,336,286),(243,337,287),(244,338,288),(245,339,289),(246,340,290),(247,341,291),(248,342,292),(249,343,293),(250,344,294),(251,345,295),(252,346,296),(253,347,297),(254,348,298),(255,349,299),(256,350,300),(257,351,301),(258,313,302),(259,314,303),(260,315,304),(261,316,305),(262,317,306),(263,318,307),(264,319,308),(265,320,309),(266,321,310),(267,322,311),(268,323,312),(269,324,274),(270,325,275),(271,326,276),(272,327,277),(273,328,278)], [(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),(313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351)])

351 conjugacy classes

class 1 3A···3Z13A···13L39A···39KZ
order13···313···1339···39
size11···11···11···1

351 irreducible representations

dim1111
type+
imageC1C3C13C39
kernelC32×C39C3×C39C33C32
# reps12612312

Matrix representation of C32×C39 in GL3(𝔽79) generated by

5500
010
0023
,
100
010
0055
,
7600
0760
0020
G:=sub<GL(3,GF(79))| [55,0,0,0,1,0,0,0,23],[1,0,0,0,1,0,0,0,55],[76,0,0,0,76,0,0,0,20] >;

C32×C39 in GAP, Magma, Sage, TeX

C_3^2\times C_{39}
% in TeX

G:=Group("C3^2xC39");
// GroupNames label

G:=SmallGroup(351,14);
// by ID

G=gap.SmallGroup(351,14);
# by ID

G:=PCGroup([4,-3,-3,-3,-13]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^39=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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