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## G = C14×C3⋊C8order 336 = 24·3·7

### Direct product of C14 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C14×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C84 — C7×C3⋊C8 — C14×C3⋊C8
 Lower central C3 — C14×C3⋊C8
 Upper central C1 — C2×C28

Generators and relations for C14×C3⋊C8
G = < a,b,c | a14=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C14×C3⋊C8
Regular action on 336 points
Generators in S336
(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)
(1 109 224)(2 110 211)(3 111 212)(4 112 213)(5 99 214)(6 100 215)(7 101 216)(8 102 217)(9 103 218)(10 104 219)(11 105 220)(12 106 221)(13 107 222)(14 108 223)(15 314 274)(16 315 275)(17 316 276)(18 317 277)(19 318 278)(20 319 279)(21 320 280)(22 321 267)(23 322 268)(24 309 269)(25 310 270)(26 311 271)(27 312 272)(28 313 273)(29 151 291)(30 152 292)(31 153 293)(32 154 294)(33 141 281)(34 142 282)(35 143 283)(36 144 284)(37 145 285)(38 146 286)(39 147 287)(40 148 288)(41 149 289)(42 150 290)(43 171 304)(44 172 305)(45 173 306)(46 174 307)(47 175 308)(48 176 295)(49 177 296)(50 178 297)(51 179 298)(52 180 299)(53 181 300)(54 182 301)(55 169 302)(56 170 303)(57 199 238)(58 200 225)(59 201 226)(60 202 227)(61 203 228)(62 204 229)(63 205 230)(64 206 231)(65 207 232)(66 208 233)(67 209 234)(68 210 235)(69 197 236)(70 198 237)(71 185 331)(72 186 332)(73 187 333)(74 188 334)(75 189 335)(76 190 336)(77 191 323)(78 192 324)(79 193 325)(80 194 326)(81 195 327)(82 196 328)(83 183 329)(84 184 330)(85 253 159)(86 254 160)(87 255 161)(88 256 162)(89 257 163)(90 258 164)(91 259 165)(92 260 166)(93 261 167)(94 262 168)(95 263 155)(96 264 156)(97 265 157)(98 266 158)(113 246 127)(114 247 128)(115 248 129)(116 249 130)(117 250 131)(118 251 132)(119 252 133)(120 239 134)(121 240 135)(122 241 136)(123 242 137)(124 243 138)(125 244 139)(126 245 140)
(1 137 334 145 307 266 316 208)(2 138 335 146 308 253 317 209)(3 139 336 147 295 254 318 210)(4 140 323 148 296 255 319 197)(5 127 324 149 297 256 320 198)(6 128 325 150 298 257 321 199)(7 129 326 151 299 258 322 200)(8 130 327 152 300 259 309 201)(9 131 328 153 301 260 310 202)(10 132 329 154 302 261 311 203)(11 133 330 141 303 262 312 204)(12 134 331 142 304 263 313 205)(13 135 332 143 305 264 314 206)(14 136 333 144 306 265 315 207)(15 231 222 121 186 283 172 156)(16 232 223 122 187 284 173 157)(17 233 224 123 188 285 174 158)(18 234 211 124 189 286 175 159)(19 235 212 125 190 287 176 160)(20 236 213 126 191 288 177 161)(21 237 214 113 192 289 178 162)(22 238 215 114 193 290 179 163)(23 225 216 115 194 291 180 164)(24 226 217 116 195 292 181 165)(25 227 218 117 196 293 182 166)(26 228 219 118 183 294 169 167)(27 229 220 119 184 281 170 168)(28 230 221 120 185 282 171 155)(29 52 90 268 58 101 248 80)(30 53 91 269 59 102 249 81)(31 54 92 270 60 103 250 82)(32 55 93 271 61 104 251 83)(33 56 94 272 62 105 252 84)(34 43 95 273 63 106 239 71)(35 44 96 274 64 107 240 72)(36 45 97 275 65 108 241 73)(37 46 98 276 66 109 242 74)(38 47 85 277 67 110 243 75)(39 48 86 278 68 111 244 76)(40 49 87 279 69 112 245 77)(41 50 88 280 70 99 246 78)(42 51 89 267 57 100 247 79)

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

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

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

168 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 7A ··· 7F 8A ··· 8H 12A 12B 12C 12D 14A ··· 14R 21A ··· 21F 28A ··· 28X 42A ··· 42R 56A ··· 56AV 84A ··· 84X order 1 2 2 2 3 4 4 4 4 6 6 6 7 ··· 7 8 ··· 8 12 12 12 12 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 56 ··· 56 84 ··· 84 size 1 1 1 1 2 1 1 1 1 2 2 2 1 ··· 1 3 ··· 3 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 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C7 C8 C14 C14 C28 C28 C56 S3 Dic3 D6 Dic3 C3⋊C8 S3×C7 C7×Dic3 S3×C14 C7×Dic3 C7×C3⋊C8 kernel C14×C3⋊C8 C7×C3⋊C8 C2×C84 C84 C2×C42 C2×C3⋊C8 C42 C3⋊C8 C2×C12 C12 C2×C6 C6 C2×C28 C28 C28 C2×C14 C14 C2×C4 C4 C4 C22 C2 # reps 1 2 1 2 2 6 8 12 6 12 12 48 1 1 1 1 4 6 6 6 6 24

Matrix representation of C14×C3⋊C8 in GL4(𝔽337) generated by

 336 0 0 0 0 336 0 0 0 0 79 0 0 0 0 79
,
 1 0 0 0 0 1 0 0 0 0 0 336 0 0 1 336
,
 85 0 0 0 0 1 0 0 0 0 137 158 0 0 295 200
G:=sub<GL(4,GF(337))| [336,0,0,0,0,336,0,0,0,0,79,0,0,0,0,79],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,336,336],[85,0,0,0,0,1,0,0,0,0,137,295,0,0,158,200] >;

C14×C3⋊C8 in GAP, Magma, Sage, TeX

C_{14}\times C_3\rtimes C_8
% in TeX

G:=Group("C14xC3:C8");
// GroupNames label

G:=SmallGroup(336,79);
// by ID

G=gap.SmallGroup(336,79);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,168,69,8069]);
// Polycyclic

G:=Group<a,b,c|a^14=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

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