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G = C1209C4order 480 = 25·3·5

1st semidirect product of C120 and C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C1209C4, C6.4D40, C241Dic5, C81Dic15, C405Dic3, C30.25D8, C2.1D120, C10.4D24, C60.23Q8, C30.11Q16, C2.2Dic60, C4.5Dic30, C6.2Dic20, C22.9D60, C20.20Dic6, C10.2Dic12, C12.20Dic10, (C2×C40).5S3, (C2×C8).3D15, (C2×C24).5D5, C53(C241C4), C32(C405C4), (C2×C120).9C2, (C2×C4).72D30, (C2×C6).15D20, C1510(C2.D8), C30.40(C4⋊C4), C605C4.3C2, C60.231(C2×C4), (C2×C20).386D6, (C2×C30).101D4, (C2×C10).15D12, C6.8(C4⋊Dic5), C4.7(C2×Dic15), C2.4(C605C4), (C2×C12).388D10, C12.36(C2×Dic5), C20.57(C2×Dic3), (C2×C60).473C22, C10.15(C4⋊Dic3), SmallGroup(480,178)

Series: Derived Chief Lower central Upper central

C1C60 — C1209C4
C1C5C15C30C2×C30C2×C60C605C4 — C1209C4
C15C30C60 — C1209C4
C1C22C2×C4C2×C8

Generators and relations for C1209C4
 G = < a,b | a120=b4=1, bab-1=a-1 >

Subgroups: 404 in 72 conjugacy classes, 47 normal (37 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3 [×2], C12 [×2], C2×C6, C15, C4⋊C4 [×2], C2×C8, Dic5 [×2], C20 [×2], C2×C10, C24 [×2], C2×Dic3 [×2], C2×C12, C30 [×3], C2.D8, C40 [×2], C2×Dic5 [×2], C2×C20, C4⋊Dic3 [×2], C2×C24, Dic15 [×2], C60 [×2], C2×C30, C4⋊Dic5 [×2], C2×C40, C241C4, C120 [×2], C2×Dic15 [×2], C2×C60, C405C4, C605C4 [×2], C2×C120, C1209C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, Dic3 [×2], D6, C4⋊C4, D8, Q16, Dic5 [×2], D10, Dic6, D12, C2×Dic3, D15, C2.D8, Dic10, D20, C2×Dic5, D24, Dic12, C4⋊Dic3, Dic15 [×2], D30, D40, Dic20, C4⋊Dic5, C241C4, Dic30, D60, C2×Dic15, C405C4, D120, Dic60, C605C4, C1209C4

Smallest permutation representation of C1209C4
Regular action on 480 points
Generators in S480
(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)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480)
(1 345 122 427)(2 344 123 426)(3 343 124 425)(4 342 125 424)(5 341 126 423)(6 340 127 422)(7 339 128 421)(8 338 129 420)(9 337 130 419)(10 336 131 418)(11 335 132 417)(12 334 133 416)(13 333 134 415)(14 332 135 414)(15 331 136 413)(16 330 137 412)(17 329 138 411)(18 328 139 410)(19 327 140 409)(20 326 141 408)(21 325 142 407)(22 324 143 406)(23 323 144 405)(24 322 145 404)(25 321 146 403)(26 320 147 402)(27 319 148 401)(28 318 149 400)(29 317 150 399)(30 316 151 398)(31 315 152 397)(32 314 153 396)(33 313 154 395)(34 312 155 394)(35 311 156 393)(36 310 157 392)(37 309 158 391)(38 308 159 390)(39 307 160 389)(40 306 161 388)(41 305 162 387)(42 304 163 386)(43 303 164 385)(44 302 165 384)(45 301 166 383)(46 300 167 382)(47 299 168 381)(48 298 169 380)(49 297 170 379)(50 296 171 378)(51 295 172 377)(52 294 173 376)(53 293 174 375)(54 292 175 374)(55 291 176 373)(56 290 177 372)(57 289 178 371)(58 288 179 370)(59 287 180 369)(60 286 181 368)(61 285 182 367)(62 284 183 366)(63 283 184 365)(64 282 185 364)(65 281 186 363)(66 280 187 362)(67 279 188 361)(68 278 189 480)(69 277 190 479)(70 276 191 478)(71 275 192 477)(72 274 193 476)(73 273 194 475)(74 272 195 474)(75 271 196 473)(76 270 197 472)(77 269 198 471)(78 268 199 470)(79 267 200 469)(80 266 201 468)(81 265 202 467)(82 264 203 466)(83 263 204 465)(84 262 205 464)(85 261 206 463)(86 260 207 462)(87 259 208 461)(88 258 209 460)(89 257 210 459)(90 256 211 458)(91 255 212 457)(92 254 213 456)(93 253 214 455)(94 252 215 454)(95 251 216 453)(96 250 217 452)(97 249 218 451)(98 248 219 450)(99 247 220 449)(100 246 221 448)(101 245 222 447)(102 244 223 446)(103 243 224 445)(104 242 225 444)(105 241 226 443)(106 360 227 442)(107 359 228 441)(108 358 229 440)(109 357 230 439)(110 356 231 438)(111 355 232 437)(112 354 233 436)(113 353 234 435)(114 352 235 434)(115 351 236 433)(116 350 237 432)(117 349 238 431)(118 348 239 430)(119 347 240 429)(120 346 121 428)

G:=sub<Sym(480)| (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)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480), (1,345,122,427)(2,344,123,426)(3,343,124,425)(4,342,125,424)(5,341,126,423)(6,340,127,422)(7,339,128,421)(8,338,129,420)(9,337,130,419)(10,336,131,418)(11,335,132,417)(12,334,133,416)(13,333,134,415)(14,332,135,414)(15,331,136,413)(16,330,137,412)(17,329,138,411)(18,328,139,410)(19,327,140,409)(20,326,141,408)(21,325,142,407)(22,324,143,406)(23,323,144,405)(24,322,145,404)(25,321,146,403)(26,320,147,402)(27,319,148,401)(28,318,149,400)(29,317,150,399)(30,316,151,398)(31,315,152,397)(32,314,153,396)(33,313,154,395)(34,312,155,394)(35,311,156,393)(36,310,157,392)(37,309,158,391)(38,308,159,390)(39,307,160,389)(40,306,161,388)(41,305,162,387)(42,304,163,386)(43,303,164,385)(44,302,165,384)(45,301,166,383)(46,300,167,382)(47,299,168,381)(48,298,169,380)(49,297,170,379)(50,296,171,378)(51,295,172,377)(52,294,173,376)(53,293,174,375)(54,292,175,374)(55,291,176,373)(56,290,177,372)(57,289,178,371)(58,288,179,370)(59,287,180,369)(60,286,181,368)(61,285,182,367)(62,284,183,366)(63,283,184,365)(64,282,185,364)(65,281,186,363)(66,280,187,362)(67,279,188,361)(68,278,189,480)(69,277,190,479)(70,276,191,478)(71,275,192,477)(72,274,193,476)(73,273,194,475)(74,272,195,474)(75,271,196,473)(76,270,197,472)(77,269,198,471)(78,268,199,470)(79,267,200,469)(80,266,201,468)(81,265,202,467)(82,264,203,466)(83,263,204,465)(84,262,205,464)(85,261,206,463)(86,260,207,462)(87,259,208,461)(88,258,209,460)(89,257,210,459)(90,256,211,458)(91,255,212,457)(92,254,213,456)(93,253,214,455)(94,252,215,454)(95,251,216,453)(96,250,217,452)(97,249,218,451)(98,248,219,450)(99,247,220,449)(100,246,221,448)(101,245,222,447)(102,244,223,446)(103,243,224,445)(104,242,225,444)(105,241,226,443)(106,360,227,442)(107,359,228,441)(108,358,229,440)(109,357,230,439)(110,356,231,438)(111,355,232,437)(112,354,233,436)(113,353,234,435)(114,352,235,434)(115,351,236,433)(116,350,237,432)(117,349,238,431)(118,348,239,430)(119,347,240,429)(120,346,121,428)>;

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)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480), (1,345,122,427)(2,344,123,426)(3,343,124,425)(4,342,125,424)(5,341,126,423)(6,340,127,422)(7,339,128,421)(8,338,129,420)(9,337,130,419)(10,336,131,418)(11,335,132,417)(12,334,133,416)(13,333,134,415)(14,332,135,414)(15,331,136,413)(16,330,137,412)(17,329,138,411)(18,328,139,410)(19,327,140,409)(20,326,141,408)(21,325,142,407)(22,324,143,406)(23,323,144,405)(24,322,145,404)(25,321,146,403)(26,320,147,402)(27,319,148,401)(28,318,149,400)(29,317,150,399)(30,316,151,398)(31,315,152,397)(32,314,153,396)(33,313,154,395)(34,312,155,394)(35,311,156,393)(36,310,157,392)(37,309,158,391)(38,308,159,390)(39,307,160,389)(40,306,161,388)(41,305,162,387)(42,304,163,386)(43,303,164,385)(44,302,165,384)(45,301,166,383)(46,300,167,382)(47,299,168,381)(48,298,169,380)(49,297,170,379)(50,296,171,378)(51,295,172,377)(52,294,173,376)(53,293,174,375)(54,292,175,374)(55,291,176,373)(56,290,177,372)(57,289,178,371)(58,288,179,370)(59,287,180,369)(60,286,181,368)(61,285,182,367)(62,284,183,366)(63,283,184,365)(64,282,185,364)(65,281,186,363)(66,280,187,362)(67,279,188,361)(68,278,189,480)(69,277,190,479)(70,276,191,478)(71,275,192,477)(72,274,193,476)(73,273,194,475)(74,272,195,474)(75,271,196,473)(76,270,197,472)(77,269,198,471)(78,268,199,470)(79,267,200,469)(80,266,201,468)(81,265,202,467)(82,264,203,466)(83,263,204,465)(84,262,205,464)(85,261,206,463)(86,260,207,462)(87,259,208,461)(88,258,209,460)(89,257,210,459)(90,256,211,458)(91,255,212,457)(92,254,213,456)(93,253,214,455)(94,252,215,454)(95,251,216,453)(96,250,217,452)(97,249,218,451)(98,248,219,450)(99,247,220,449)(100,246,221,448)(101,245,222,447)(102,244,223,446)(103,243,224,445)(104,242,225,444)(105,241,226,443)(106,360,227,442)(107,359,228,441)(108,358,229,440)(109,357,230,439)(110,356,231,438)(111,355,232,437)(112,354,233,436)(113,353,234,435)(114,352,235,434)(115,351,236,433)(116,350,237,432)(117,349,238,431)(118,348,239,430)(119,347,240,429)(120,346,121,428) );

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),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360),(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480)], [(1,345,122,427),(2,344,123,426),(3,343,124,425),(4,342,125,424),(5,341,126,423),(6,340,127,422),(7,339,128,421),(8,338,129,420),(9,337,130,419),(10,336,131,418),(11,335,132,417),(12,334,133,416),(13,333,134,415),(14,332,135,414),(15,331,136,413),(16,330,137,412),(17,329,138,411),(18,328,139,410),(19,327,140,409),(20,326,141,408),(21,325,142,407),(22,324,143,406),(23,323,144,405),(24,322,145,404),(25,321,146,403),(26,320,147,402),(27,319,148,401),(28,318,149,400),(29,317,150,399),(30,316,151,398),(31,315,152,397),(32,314,153,396),(33,313,154,395),(34,312,155,394),(35,311,156,393),(36,310,157,392),(37,309,158,391),(38,308,159,390),(39,307,160,389),(40,306,161,388),(41,305,162,387),(42,304,163,386),(43,303,164,385),(44,302,165,384),(45,301,166,383),(46,300,167,382),(47,299,168,381),(48,298,169,380),(49,297,170,379),(50,296,171,378),(51,295,172,377),(52,294,173,376),(53,293,174,375),(54,292,175,374),(55,291,176,373),(56,290,177,372),(57,289,178,371),(58,288,179,370),(59,287,180,369),(60,286,181,368),(61,285,182,367),(62,284,183,366),(63,283,184,365),(64,282,185,364),(65,281,186,363),(66,280,187,362),(67,279,188,361),(68,278,189,480),(69,277,190,479),(70,276,191,478),(71,275,192,477),(72,274,193,476),(73,273,194,475),(74,272,195,474),(75,271,196,473),(76,270,197,472),(77,269,198,471),(78,268,199,470),(79,267,200,469),(80,266,201,468),(81,265,202,467),(82,264,203,466),(83,263,204,465),(84,262,205,464),(85,261,206,463),(86,260,207,462),(87,259,208,461),(88,258,209,460),(89,257,210,459),(90,256,211,458),(91,255,212,457),(92,254,213,456),(93,253,214,455),(94,252,215,454),(95,251,216,453),(96,250,217,452),(97,249,218,451),(98,248,219,450),(99,247,220,449),(100,246,221,448),(101,245,222,447),(102,244,223,446),(103,243,224,445),(104,242,225,444),(105,241,226,443),(106,360,227,442),(107,359,228,441),(108,358,229,440),(109,357,230,439),(110,356,231,438),(111,355,232,437),(112,354,233,436),(113,353,234,435),(114,352,235,434),(115,351,236,433),(116,350,237,432),(117,349,238,431),(118,348,239,430),(119,347,240,429),(120,346,121,428)])

126 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D15A15B15C15D20A···20H24A···24H30A···30L40A···40P60A···60P120A···120AF
order1222344444455666888810···10121212121515151520···2024···2430···3040···4060···60120···120
size1111222606060602222222222···2222222222···22···22···22···22···22···2

126 irreducible representations

dim11112222222222222222222222222
type++++-++-++--+-++-++--++--++-
imageC1C2C2C4S3Q8D4D5Dic3D6D8Q16Dic5D10Dic6D12D15Dic10D20D24Dic12Dic15D30D40Dic20Dic30D60D120Dic60
kernelC1209C4C605C4C2×C120C120C2×C40C60C2×C30C2×C24C40C2×C20C30C30C24C2×C12C20C2×C10C2×C8C12C2×C6C10C10C8C2×C4C6C6C4C22C2C2
# reps1214111221224222444448488881616

Matrix representation of C1209C4 in GL5(𝔽241)

10000
01736300
01783000
00020228
000207227
,
640000
02116400
01783000
00019249
00013349

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,173,178,0,0,0,63,30,0,0,0,0,0,20,207,0,0,0,228,227],[64,0,0,0,0,0,211,178,0,0,0,64,30,0,0,0,0,0,192,133,0,0,0,49,49] >;

C1209C4 in GAP, Magma, Sage, TeX

C_{120}\rtimes_9C_4
% in TeX

G:=Group("C120:9C4");
// GroupNames label

G:=SmallGroup(480,178);
// by ID

G=gap.SmallGroup(480,178);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,675,80,2693,18822]);
// Polycyclic

G:=Group<a,b|a^120=b^4=1,b*a*b^-1=a^-1>;
// generators/relations

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