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## G = C11×Q32order 352 = 25·11

### Direct product of C11 and Q32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C11×Q32, C16.C22, Q16.C22, C176.3C2, C22.17D8, C44.38D4, C88.26C22, C8.4(C2×C22), C4.3(D4×C11), C2.5(C11×D8), (C11×Q16).2C2, SmallGroup(352,62)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C11×Q32
 Chief series C1 — C2 — C4 — C8 — C88 — C11×Q16 — C11×Q32
 Lower central C1 — C2 — C4 — C8 — C11×Q32
 Upper central C1 — C22 — C44 — C88 — C11×Q32

Generators and relations for C11×Q32
G = < a,b,c | a11=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C11×Q32
Regular action on 352 points
Generators in S352
(1 200 270 186 173 228 336 73 35 133 156)(2 201 271 187 174 229 321 74 36 134 157)(3 202 272 188 175 230 322 75 37 135 158)(4 203 257 189 176 231 323 76 38 136 159)(5 204 258 190 161 232 324 77 39 137 160)(6 205 259 191 162 233 325 78 40 138 145)(7 206 260 192 163 234 326 79 41 139 146)(8 207 261 177 164 235 327 80 42 140 147)(9 208 262 178 165 236 328 65 43 141 148)(10 193 263 179 166 237 329 66 44 142 149)(11 194 264 180 167 238 330 67 45 143 150)(12 195 265 181 168 239 331 68 46 144 151)(13 196 266 182 169 240 332 69 47 129 152)(14 197 267 183 170 225 333 70 48 130 153)(15 198 268 184 171 226 334 71 33 131 154)(16 199 269 185 172 227 335 72 34 132 155)(17 212 105 248 94 288 338 122 320 56 301)(18 213 106 249 95 273 339 123 305 57 302)(19 214 107 250 96 274 340 124 306 58 303)(20 215 108 251 81 275 341 125 307 59 304)(21 216 109 252 82 276 342 126 308 60 289)(22 217 110 253 83 277 343 127 309 61 290)(23 218 111 254 84 278 344 128 310 62 291)(24 219 112 255 85 279 345 113 311 63 292)(25 220 97 256 86 280 346 114 312 64 293)(26 221 98 241 87 281 347 115 313 49 294)(27 222 99 242 88 282 348 116 314 50 295)(28 223 100 243 89 283 349 117 315 51 296)(29 224 101 244 90 284 350 118 316 52 297)(30 209 102 245 91 285 351 119 317 53 298)(31 210 103 246 92 286 352 120 318 54 299)(32 211 104 247 93 287 337 121 319 55 300)
(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)(337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352)
(1 341 9 349)(2 340 10 348)(3 339 11 347)(4 338 12 346)(5 337 13 345)(6 352 14 344)(7 351 15 343)(8 350 16 342)(17 239 25 231)(18 238 26 230)(19 237 27 229)(20 236 28 228)(21 235 29 227)(22 234 30 226)(23 233 31 225)(24 232 32 240)(33 253 41 245)(34 252 42 244)(35 251 43 243)(36 250 44 242)(37 249 45 241)(38 248 46 256)(39 247 47 255)(40 246 48 254)(49 188 57 180)(50 187 58 179)(51 186 59 178)(52 185 60 177)(53 184 61 192)(54 183 62 191)(55 182 63 190)(56 181 64 189)(65 100 73 108)(66 99 74 107)(67 98 75 106)(68 97 76 105)(69 112 77 104)(70 111 78 103)(71 110 79 102)(72 109 80 101)(81 141 89 133)(82 140 90 132)(83 139 91 131)(84 138 92 130)(85 137 93 129)(86 136 94 144)(87 135 95 143)(88 134 96 142)(113 204 121 196)(114 203 122 195)(115 202 123 194)(116 201 124 193)(117 200 125 208)(118 199 126 207)(119 198 127 206)(120 197 128 205)(145 286 153 278)(146 285 154 277)(147 284 155 276)(148 283 156 275)(149 282 157 274)(150 281 158 273)(151 280 159 288)(152 279 160 287)(161 300 169 292)(162 299 170 291)(163 298 171 290)(164 297 172 289)(165 296 173 304)(166 295 174 303)(167 294 175 302)(168 293 176 301)(209 334 217 326)(210 333 218 325)(211 332 219 324)(212 331 220 323)(213 330 221 322)(214 329 222 321)(215 328 223 336)(216 327 224 335)(257 320 265 312)(258 319 266 311)(259 318 267 310)(260 317 268 309)(261 316 269 308)(262 315 270 307)(263 314 271 306)(264 313 272 305)

G:=sub<Sym(352)| (1,200,270,186,173,228,336,73,35,133,156)(2,201,271,187,174,229,321,74,36,134,157)(3,202,272,188,175,230,322,75,37,135,158)(4,203,257,189,176,231,323,76,38,136,159)(5,204,258,190,161,232,324,77,39,137,160)(6,205,259,191,162,233,325,78,40,138,145)(7,206,260,192,163,234,326,79,41,139,146)(8,207,261,177,164,235,327,80,42,140,147)(9,208,262,178,165,236,328,65,43,141,148)(10,193,263,179,166,237,329,66,44,142,149)(11,194,264,180,167,238,330,67,45,143,150)(12,195,265,181,168,239,331,68,46,144,151)(13,196,266,182,169,240,332,69,47,129,152)(14,197,267,183,170,225,333,70,48,130,153)(15,198,268,184,171,226,334,71,33,131,154)(16,199,269,185,172,227,335,72,34,132,155)(17,212,105,248,94,288,338,122,320,56,301)(18,213,106,249,95,273,339,123,305,57,302)(19,214,107,250,96,274,340,124,306,58,303)(20,215,108,251,81,275,341,125,307,59,304)(21,216,109,252,82,276,342,126,308,60,289)(22,217,110,253,83,277,343,127,309,61,290)(23,218,111,254,84,278,344,128,310,62,291)(24,219,112,255,85,279,345,113,311,63,292)(25,220,97,256,86,280,346,114,312,64,293)(26,221,98,241,87,281,347,115,313,49,294)(27,222,99,242,88,282,348,116,314,50,295)(28,223,100,243,89,283,349,117,315,51,296)(29,224,101,244,90,284,350,118,316,52,297)(30,209,102,245,91,285,351,119,317,53,298)(31,210,103,246,92,286,352,120,318,54,299)(32,211,104,247,93,287,337,121,319,55,300), (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)(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352), (1,341,9,349)(2,340,10,348)(3,339,11,347)(4,338,12,346)(5,337,13,345)(6,352,14,344)(7,351,15,343)(8,350,16,342)(17,239,25,231)(18,238,26,230)(19,237,27,229)(20,236,28,228)(21,235,29,227)(22,234,30,226)(23,233,31,225)(24,232,32,240)(33,253,41,245)(34,252,42,244)(35,251,43,243)(36,250,44,242)(37,249,45,241)(38,248,46,256)(39,247,47,255)(40,246,48,254)(49,188,57,180)(50,187,58,179)(51,186,59,178)(52,185,60,177)(53,184,61,192)(54,183,62,191)(55,182,63,190)(56,181,64,189)(65,100,73,108)(66,99,74,107)(67,98,75,106)(68,97,76,105)(69,112,77,104)(70,111,78,103)(71,110,79,102)(72,109,80,101)(81,141,89,133)(82,140,90,132)(83,139,91,131)(84,138,92,130)(85,137,93,129)(86,136,94,144)(87,135,95,143)(88,134,96,142)(113,204,121,196)(114,203,122,195)(115,202,123,194)(116,201,124,193)(117,200,125,208)(118,199,126,207)(119,198,127,206)(120,197,128,205)(145,286,153,278)(146,285,154,277)(147,284,155,276)(148,283,156,275)(149,282,157,274)(150,281,158,273)(151,280,159,288)(152,279,160,287)(161,300,169,292)(162,299,170,291)(163,298,171,290)(164,297,172,289)(165,296,173,304)(166,295,174,303)(167,294,175,302)(168,293,176,301)(209,334,217,326)(210,333,218,325)(211,332,219,324)(212,331,220,323)(213,330,221,322)(214,329,222,321)(215,328,223,336)(216,327,224,335)(257,320,265,312)(258,319,266,311)(259,318,267,310)(260,317,268,309)(261,316,269,308)(262,315,270,307)(263,314,271,306)(264,313,272,305)>;

G:=Group( (1,200,270,186,173,228,336,73,35,133,156)(2,201,271,187,174,229,321,74,36,134,157)(3,202,272,188,175,230,322,75,37,135,158)(4,203,257,189,176,231,323,76,38,136,159)(5,204,258,190,161,232,324,77,39,137,160)(6,205,259,191,162,233,325,78,40,138,145)(7,206,260,192,163,234,326,79,41,139,146)(8,207,261,177,164,235,327,80,42,140,147)(9,208,262,178,165,236,328,65,43,141,148)(10,193,263,179,166,237,329,66,44,142,149)(11,194,264,180,167,238,330,67,45,143,150)(12,195,265,181,168,239,331,68,46,144,151)(13,196,266,182,169,240,332,69,47,129,152)(14,197,267,183,170,225,333,70,48,130,153)(15,198,268,184,171,226,334,71,33,131,154)(16,199,269,185,172,227,335,72,34,132,155)(17,212,105,248,94,288,338,122,320,56,301)(18,213,106,249,95,273,339,123,305,57,302)(19,214,107,250,96,274,340,124,306,58,303)(20,215,108,251,81,275,341,125,307,59,304)(21,216,109,252,82,276,342,126,308,60,289)(22,217,110,253,83,277,343,127,309,61,290)(23,218,111,254,84,278,344,128,310,62,291)(24,219,112,255,85,279,345,113,311,63,292)(25,220,97,256,86,280,346,114,312,64,293)(26,221,98,241,87,281,347,115,313,49,294)(27,222,99,242,88,282,348,116,314,50,295)(28,223,100,243,89,283,349,117,315,51,296)(29,224,101,244,90,284,350,118,316,52,297)(30,209,102,245,91,285,351,119,317,53,298)(31,210,103,246,92,286,352,120,318,54,299)(32,211,104,247,93,287,337,121,319,55,300), (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)(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352), (1,341,9,349)(2,340,10,348)(3,339,11,347)(4,338,12,346)(5,337,13,345)(6,352,14,344)(7,351,15,343)(8,350,16,342)(17,239,25,231)(18,238,26,230)(19,237,27,229)(20,236,28,228)(21,235,29,227)(22,234,30,226)(23,233,31,225)(24,232,32,240)(33,253,41,245)(34,252,42,244)(35,251,43,243)(36,250,44,242)(37,249,45,241)(38,248,46,256)(39,247,47,255)(40,246,48,254)(49,188,57,180)(50,187,58,179)(51,186,59,178)(52,185,60,177)(53,184,61,192)(54,183,62,191)(55,182,63,190)(56,181,64,189)(65,100,73,108)(66,99,74,107)(67,98,75,106)(68,97,76,105)(69,112,77,104)(70,111,78,103)(71,110,79,102)(72,109,80,101)(81,141,89,133)(82,140,90,132)(83,139,91,131)(84,138,92,130)(85,137,93,129)(86,136,94,144)(87,135,95,143)(88,134,96,142)(113,204,121,196)(114,203,122,195)(115,202,123,194)(116,201,124,193)(117,200,125,208)(118,199,126,207)(119,198,127,206)(120,197,128,205)(145,286,153,278)(146,285,154,277)(147,284,155,276)(148,283,156,275)(149,282,157,274)(150,281,158,273)(151,280,159,288)(152,279,160,287)(161,300,169,292)(162,299,170,291)(163,298,171,290)(164,297,172,289)(165,296,173,304)(166,295,174,303)(167,294,175,302)(168,293,176,301)(209,334,217,326)(210,333,218,325)(211,332,219,324)(212,331,220,323)(213,330,221,322)(214,329,222,321)(215,328,223,336)(216,327,224,335)(257,320,265,312)(258,319,266,311)(259,318,267,310)(260,317,268,309)(261,316,269,308)(262,315,270,307)(263,314,271,306)(264,313,272,305) );

G=PermutationGroup([[(1,200,270,186,173,228,336,73,35,133,156),(2,201,271,187,174,229,321,74,36,134,157),(3,202,272,188,175,230,322,75,37,135,158),(4,203,257,189,176,231,323,76,38,136,159),(5,204,258,190,161,232,324,77,39,137,160),(6,205,259,191,162,233,325,78,40,138,145),(7,206,260,192,163,234,326,79,41,139,146),(8,207,261,177,164,235,327,80,42,140,147),(9,208,262,178,165,236,328,65,43,141,148),(10,193,263,179,166,237,329,66,44,142,149),(11,194,264,180,167,238,330,67,45,143,150),(12,195,265,181,168,239,331,68,46,144,151),(13,196,266,182,169,240,332,69,47,129,152),(14,197,267,183,170,225,333,70,48,130,153),(15,198,268,184,171,226,334,71,33,131,154),(16,199,269,185,172,227,335,72,34,132,155),(17,212,105,248,94,288,338,122,320,56,301),(18,213,106,249,95,273,339,123,305,57,302),(19,214,107,250,96,274,340,124,306,58,303),(20,215,108,251,81,275,341,125,307,59,304),(21,216,109,252,82,276,342,126,308,60,289),(22,217,110,253,83,277,343,127,309,61,290),(23,218,111,254,84,278,344,128,310,62,291),(24,219,112,255,85,279,345,113,311,63,292),(25,220,97,256,86,280,346,114,312,64,293),(26,221,98,241,87,281,347,115,313,49,294),(27,222,99,242,88,282,348,116,314,50,295),(28,223,100,243,89,283,349,117,315,51,296),(29,224,101,244,90,284,350,118,316,52,297),(30,209,102,245,91,285,351,119,317,53,298),(31,210,103,246,92,286,352,120,318,54,299),(32,211,104,247,93,287,337,121,319,55,300)], [(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),(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352)], [(1,341,9,349),(2,340,10,348),(3,339,11,347),(4,338,12,346),(5,337,13,345),(6,352,14,344),(7,351,15,343),(8,350,16,342),(17,239,25,231),(18,238,26,230),(19,237,27,229),(20,236,28,228),(21,235,29,227),(22,234,30,226),(23,233,31,225),(24,232,32,240),(33,253,41,245),(34,252,42,244),(35,251,43,243),(36,250,44,242),(37,249,45,241),(38,248,46,256),(39,247,47,255),(40,246,48,254),(49,188,57,180),(50,187,58,179),(51,186,59,178),(52,185,60,177),(53,184,61,192),(54,183,62,191),(55,182,63,190),(56,181,64,189),(65,100,73,108),(66,99,74,107),(67,98,75,106),(68,97,76,105),(69,112,77,104),(70,111,78,103),(71,110,79,102),(72,109,80,101),(81,141,89,133),(82,140,90,132),(83,139,91,131),(84,138,92,130),(85,137,93,129),(86,136,94,144),(87,135,95,143),(88,134,96,142),(113,204,121,196),(114,203,122,195),(115,202,123,194),(116,201,124,193),(117,200,125,208),(118,199,126,207),(119,198,127,206),(120,197,128,205),(145,286,153,278),(146,285,154,277),(147,284,155,276),(148,283,156,275),(149,282,157,274),(150,281,158,273),(151,280,159,288),(152,279,160,287),(161,300,169,292),(162,299,170,291),(163,298,171,290),(164,297,172,289),(165,296,173,304),(166,295,174,303),(167,294,175,302),(168,293,176,301),(209,334,217,326),(210,333,218,325),(211,332,219,324),(212,331,220,323),(213,330,221,322),(214,329,222,321),(215,328,223,336),(216,327,224,335),(257,320,265,312),(258,319,266,311),(259,318,267,310),(260,317,268,309),(261,316,269,308),(262,315,270,307),(263,314,271,306),(264,313,272,305)]])

121 conjugacy classes

 class 1 2 4A 4B 4C 8A 8B 11A ··· 11J 16A 16B 16C 16D 22A ··· 22J 44A ··· 44J 44K ··· 44AD 88A ··· 88T 176A ··· 176AN order 1 2 4 4 4 8 8 11 ··· 11 16 16 16 16 22 ··· 22 44 ··· 44 44 ··· 44 88 ··· 88 176 ··· 176 size 1 1 2 8 8 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2 2 ··· 2

121 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C11 C22 C22 D4 D8 Q32 D4×C11 C11×D8 C11×Q32 kernel C11×Q32 C176 C11×Q16 Q32 C16 Q16 C44 C22 C11 C4 C2 C1 # reps 1 1 2 10 10 20 1 2 4 10 20 40

Matrix representation of C11×Q32 in GL2(𝔽353) generated by

 22 0 0 22
,
 183 20 333 183
,
 302 218 218 51
G:=sub<GL(2,GF(353))| [22,0,0,22],[183,333,20,183],[302,218,218,51] >;

C11×Q32 in GAP, Magma, Sage, TeX

C_{11}\times Q_{32}
% in TeX

G:=Group("C11xQ32");
// GroupNames label

G:=SmallGroup(352,62);
// by ID

G=gap.SmallGroup(352,62);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,1056,553,1063,3171,1593,165,7924,3970,88]);
// Polycyclic

G:=Group<a,b,c|a^11=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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