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## G = C12×Dic7order 336 = 24·3·7

### Direct product of C12 and Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C12×Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C2×C42 — C6×Dic7 — C12×Dic7
 Lower central C7 — C12×Dic7
 Upper central C1 — C2×C12

Generators and relations for C12×Dic7
G = < a,b,c | a12=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 144 in 60 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C7, C2×C4, C2×C4, C12, C12, C2×C6, C14, C14, C42, C21, C2×C12, C2×C12, Dic7, C28, C2×C14, C42, C42, C4×C12, C2×Dic7, C2×C28, C3×Dic7, C84, C2×C42, C4×Dic7, C6×Dic7, C2×C84, C12×Dic7
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, D7, C42, C2×C12, Dic7, D14, C3×D7, C4×C12, C4×D7, C2×Dic7, C3×Dic7, C6×D7, C4×Dic7, C12×D7, C6×Dic7, C12×Dic7

Smallest permutation representation of C12×Dic7
Regular action on 336 points
Generators in S336
(1 210 41 192 242 108 233 213 145 311 330 168)(2 197 42 193 243 109 234 214 146 312 331 155)(3 198 29 194 244 110 235 215 147 313 332 156)(4 199 30 195 245 111 236 216 148 314 333 157)(5 200 31 196 246 112 237 217 149 315 334 158)(6 201 32 183 247 99 238 218 150 316 335 159)(7 202 33 184 248 100 225 219 151 317 336 160)(8 203 34 185 249 101 226 220 152 318 323 161)(9 204 35 186 250 102 227 221 153 319 324 162)(10 205 36 187 251 103 228 222 154 320 325 163)(11 206 37 188 252 104 229 223 141 321 326 164)(12 207 38 189 239 105 230 224 142 322 327 165)(13 208 39 190 240 106 231 211 143 309 328 166)(14 209 40 191 241 107 232 212 144 310 329 167)(15 95 78 70 277 290 297 176 47 133 114 262)(16 96 79 57 278 291 298 177 48 134 115 263)(17 97 80 58 279 292 299 178 49 135 116 264)(18 98 81 59 280 293 300 179 50 136 117 265)(19 85 82 60 267 294 301 180 51 137 118 266)(20 86 83 61 268 281 302 181 52 138 119 253)(21 87 84 62 269 282 303 182 53 139 120 254)(22 88 71 63 270 283 304 169 54 140 121 255)(23 89 72 64 271 284 305 170 55 127 122 256)(24 90 73 65 272 285 306 171 56 128 123 257)(25 91 74 66 273 286 307 172 43 129 124 258)(26 92 75 67 274 287 308 173 44 130 125 259)(27 93 76 68 275 288 295 174 45 131 126 260)(28 94 77 69 276 289 296 175 46 132 113 261)
(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)
(1 267 8 274)(2 280 9 273)(3 279 10 272)(4 278 11 271)(5 277 12 270)(6 276 13 269)(7 275 14 268)(15 142 22 149)(16 141 23 148)(17 154 24 147)(18 153 25 146)(19 152 26 145)(20 151 27 144)(21 150 28 143)(29 299 36 306)(30 298 37 305)(31 297 38 304)(32 296 39 303)(33 295 40 302)(34 308 41 301)(35 307 42 300)(43 243 50 250)(44 242 51 249)(45 241 52 248)(46 240 53 247)(47 239 54 246)(48 252 55 245)(49 251 56 244)(57 164 64 157)(58 163 65 156)(59 162 66 155)(60 161 67 168)(61 160 68 167)(62 159 69 166)(63 158 70 165)(71 334 78 327)(72 333 79 326)(73 332 80 325)(74 331 81 324)(75 330 82 323)(76 329 83 336)(77 328 84 335)(85 318 92 311)(86 317 93 310)(87 316 94 309)(88 315 95 322)(89 314 96 321)(90 313 97 320)(91 312 98 319)(99 132 106 139)(100 131 107 138)(101 130 108 137)(102 129 109 136)(103 128 110 135)(104 127 111 134)(105 140 112 133)(113 231 120 238)(114 230 121 237)(115 229 122 236)(116 228 123 235)(117 227 124 234)(118 226 125 233)(119 225 126 232)(169 196 176 189)(170 195 177 188)(171 194 178 187)(172 193 179 186)(173 192 180 185)(174 191 181 184)(175 190 182 183)(197 293 204 286)(198 292 205 285)(199 291 206 284)(200 290 207 283)(201 289 208 282)(202 288 209 281)(203 287 210 294)(211 254 218 261)(212 253 219 260)(213 266 220 259)(214 265 221 258)(215 264 222 257)(216 263 223 256)(217 262 224 255)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 7A 7B 7C 12A ··· 12H 12I ··· 12X 14A ··· 14I 21A ··· 21F 28A ··· 28L 42A ··· 42R 84A ··· 84X order 1 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 7 7 7 12 ··· 12 12 ··· 12 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 1 1 1 1 1 1 1 1 7 ··· 7 1 ··· 1 2 2 2 1 ··· 1 7 ··· 7 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

120 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 C2 C3 C4 C4 C6 C6 C12 C12 D7 Dic7 D14 C3×D7 C4×D7 C3×Dic7 C6×D7 C12×D7 kernel C12×Dic7 C6×Dic7 C2×C84 C4×Dic7 C3×Dic7 C84 C2×Dic7 C2×C28 Dic7 C28 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 2 1 2 8 4 4 2 16 8 3 6 3 6 12 12 6 24

Matrix representation of C12×Dic7 in GL3(𝔽337) generated by

 1 0 0 0 265 0 0 0 265
,
 336 0 0 0 0 1 0 336 303
,
 148 0 0 0 97 80 0 152 240
G:=sub<GL(3,GF(337))| [1,0,0,0,265,0,0,0,265],[336,0,0,0,0,336,0,1,303],[148,0,0,0,97,152,0,80,240] >;

C12×Dic7 in GAP, Magma, Sage, TeX

C_{12}\times {\rm Dic}_7
% in TeX

G:=Group("C12xDic7");
// GroupNames label

G:=SmallGroup(336,65);
// by ID

G=gap.SmallGroup(336,65);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,151,10373]);
// Polycyclic

G:=Group<a,b,c|a^12=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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