direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C6×Dic10, C30.69C24, C60.266C23, C10⋊1(C6×Q8), C30⋊6(C2×Q8), (C2×C30)⋊12Q8, C15⋊7(C22×Q8), C10.1(C23×C6), C23.38(C6×D5), C6.69(C23×D5), (C2×C12).437D10, (C22×C20).12C6, (C22×C60).19C2, C20.33(C22×C6), (C22×C12).18D5, (C2×C60).515C22, (C2×C30).379C23, C12.239(C22×D5), (C22×C6).135D10, (C22×Dic5).8C6, Dic5.1(C22×C6), (C3×Dic5).53C23, (C22×C30).164C22, (C6×Dic5).256C22, C5⋊1(Q8×C2×C6), C4.32(D5×C2×C6), (C2×C10)⋊6(C3×Q8), C2.3(D5×C22×C6), (C2×C4).87(C6×D5), C22.28(D5×C2×C6), (C2×C20).98(C2×C6), (C2×C6×Dic5).14C2, (C22×C4).10(C3×D5), (C22×C10).51(C2×C6), (C2×C10).62(C22×C6), (C2×Dic5).44(C2×C6), (C2×C6).375(C22×D5), SmallGroup(480,1135)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 720 in 312 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C4 [×8], C22 [×7], C5, C6, C6 [×6], C2×C4 [×6], C2×C4 [×12], Q8 [×16], C23, C10, C10 [×6], C12 [×4], C12 [×8], C2×C6 [×7], C15, C22×C4, C22×C4 [×2], C2×Q8 [×12], Dic5 [×8], C20 [×4], C2×C10 [×7], C2×C12 [×6], C2×C12 [×12], C3×Q8 [×16], C22×C6, C30, C30 [×6], C22×Q8, Dic10 [×16], C2×Dic5 [×12], C2×C20 [×6], C22×C10, C22×C12, C22×C12 [×2], C6×Q8 [×12], C3×Dic5 [×8], C60 [×4], C2×C30 [×7], C2×Dic10 [×12], C22×Dic5 [×2], C22×C20, Q8×C2×C6, C3×Dic10 [×16], C6×Dic5 [×12], C2×C60 [×6], C22×C30, C22×Dic10, C6×Dic10 [×12], C2×C6×Dic5 [×2], C22×C60, C2×C6×Dic10
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], Q8 [×4], C23 [×15], D5, C2×C6 [×35], C2×Q8 [×6], C24, D10 [×7], C3×Q8 [×4], C22×C6 [×15], C3×D5, C22×Q8, Dic10 [×4], C22×D5 [×7], C6×Q8 [×6], C23×C6, C6×D5 [×7], C2×Dic10 [×6], C23×D5, Q8×C2×C6, C3×Dic10 [×4], D5×C2×C6 [×7], C22×Dic10, C6×Dic10 [×6], D5×C22×C6, C2×C6×Dic10
Generators and relations
G = < a,b,c,d | a2=b6=c20=1, d2=c10, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 480 points
Generators in S480
(1 159)(2 160)(3 141)(4 142)(5 143)(6 144)(7 145)(8 146)(9 147)(10 148)(11 149)(12 150)(13 151)(14 152)(15 153)(16 154)(17 155)(18 156)(19 157)(20 158)(21 242)(22 243)(23 244)(24 245)(25 246)(26 247)(27 248)(28 249)(29 250)(30 251)(31 252)(32 253)(33 254)(34 255)(35 256)(36 257)(37 258)(38 259)(39 260)(40 241)(41 329)(42 330)(43 331)(44 332)(45 333)(46 334)(47 335)(48 336)(49 337)(50 338)(51 339)(52 340)(53 321)(54 322)(55 323)(56 324)(57 325)(58 326)(59 327)(60 328)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 101)(80 102)(81 286)(82 287)(83 288)(84 289)(85 290)(86 291)(87 292)(88 293)(89 294)(90 295)(91 296)(92 297)(93 298)(94 299)(95 300)(96 281)(97 282)(98 283)(99 284)(100 285)(121 301)(122 302)(123 303)(124 304)(125 305)(126 306)(127 307)(128 308)(129 309)(130 310)(131 311)(132 312)(133 313)(134 314)(135 315)(136 316)(137 317)(138 318)(139 319)(140 320)(161 468)(162 469)(163 470)(164 471)(165 472)(166 473)(167 474)(168 475)(169 476)(170 477)(171 478)(172 479)(173 480)(174 461)(175 462)(176 463)(177 464)(178 465)(179 466)(180 467)(181 220)(182 201)(183 202)(184 203)(185 204)(186 205)(187 206)(188 207)(189 208)(190 209)(191 210)(192 211)(193 212)(194 213)(195 214)(196 215)(197 216)(198 217)(199 218)(200 219)(221 457)(222 458)(223 459)(224 460)(225 441)(226 442)(227 443)(228 444)(229 445)(230 446)(231 447)(232 448)(233 449)(234 450)(235 451)(236 452)(237 453)(238 454)(239 455)(240 456)(261 406)(262 407)(263 408)(264 409)(265 410)(266 411)(267 412)(268 413)(269 414)(270 415)(271 416)(272 417)(273 418)(274 419)(275 420)(276 401)(277 402)(278 403)(279 404)(280 405)(341 387)(342 388)(343 389)(344 390)(345 391)(346 392)(347 393)(348 394)(349 395)(350 396)(351 397)(352 398)(353 399)(354 400)(355 381)(356 382)(357 383)(358 384)(359 385)(360 386)(361 438)(362 439)(363 440)(364 421)(365 422)(366 423)(367 424)(368 425)(369 426)(370 427)(371 428)(372 429)(373 430)(374 431)(375 432)(376 433)(377 434)(378 435)(379 436)(380 437)
(1 128 367 418 392 211)(2 129 368 419 393 212)(3 130 369 420 394 213)(4 131 370 401 395 214)(5 132 371 402 396 215)(6 133 372 403 397 216)(7 134 373 404 398 217)(8 135 374 405 399 218)(9 136 375 406 400 219)(10 137 376 407 381 220)(11 138 377 408 382 201)(12 139 378 409 383 202)(13 140 379 410 384 203)(14 121 380 411 385 204)(15 122 361 412 386 205)(16 123 362 413 387 206)(17 124 363 414 388 207)(18 125 364 415 389 208)(19 126 365 416 390 209)(20 127 366 417 391 210)(21 167 335 231 116 281)(22 168 336 232 117 282)(23 169 337 233 118 283)(24 170 338 234 119 284)(25 171 339 235 120 285)(26 172 340 236 101 286)(27 173 321 237 102 287)(28 174 322 238 103 288)(29 175 323 239 104 289)(30 176 324 240 105 290)(31 177 325 221 106 291)(32 178 326 222 107 292)(33 179 327 223 108 293)(34 180 328 224 109 294)(35 161 329 225 110 295)(36 162 330 226 111 296)(37 163 331 227 112 297)(38 164 332 228 113 298)(39 165 333 229 114 299)(40 166 334 230 115 300)(41 441 68 90 256 468)(42 442 69 91 257 469)(43 443 70 92 258 470)(44 444 71 93 259 471)(45 445 72 94 260 472)(46 446 73 95 241 473)(47 447 74 96 242 474)(48 448 75 97 243 475)(49 449 76 98 244 476)(50 450 77 99 245 477)(51 451 78 100 246 478)(52 452 79 81 247 479)(53 453 80 82 248 480)(54 454 61 83 249 461)(55 455 62 84 250 462)(56 456 63 85 251 463)(57 457 64 86 252 464)(58 458 65 87 253 465)(59 459 66 88 254 466)(60 460 67 89 255 467)(141 310 426 275 348 194)(142 311 427 276 349 195)(143 312 428 277 350 196)(144 313 429 278 351 197)(145 314 430 279 352 198)(146 315 431 280 353 199)(147 316 432 261 354 200)(148 317 433 262 355 181)(149 318 434 263 356 182)(150 319 435 264 357 183)(151 320 436 265 358 184)(152 301 437 266 359 185)(153 302 438 267 360 186)(154 303 439 268 341 187)(155 304 440 269 342 188)(156 305 421 270 343 189)(157 306 422 271 344 190)(158 307 423 272 345 191)(159 308 424 273 346 192)(160 309 425 274 347 193)
(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 79 11 69)(2 78 12 68)(3 77 13 67)(4 76 14 66)(5 75 15 65)(6 74 16 64)(7 73 17 63)(8 72 18 62)(9 71 19 61)(10 70 20 80)(21 439 31 429)(22 438 32 428)(23 437 33 427)(24 436 34 426)(25 435 35 425)(26 434 36 424)(27 433 37 423)(28 432 38 422)(29 431 39 421)(30 430 40 440)(41 393 51 383)(42 392 52 382)(43 391 53 381)(44 390 54 400)(45 389 55 399)(46 388 56 398)(47 387 57 397)(48 386 58 396)(49 385 59 395)(50 384 60 394)(81 138 91 128)(82 137 92 127)(83 136 93 126)(84 135 94 125)(85 134 95 124)(86 133 96 123)(87 132 97 122)(88 131 98 121)(89 130 99 140)(90 129 100 139)(101 149 111 159)(102 148 112 158)(103 147 113 157)(104 146 114 156)(105 145 115 155)(106 144 116 154)(107 143 117 153)(108 142 118 152)(109 141 119 151)(110 160 120 150)(161 274 171 264)(162 273 172 263)(163 272 173 262)(164 271 174 261)(165 270 175 280)(166 269 176 279)(167 268 177 278)(168 267 178 277)(169 266 179 276)(170 265 180 275)(181 227 191 237)(182 226 192 236)(183 225 193 235)(184 224 194 234)(185 223 195 233)(186 222 196 232)(187 221 197 231)(188 240 198 230)(189 239 199 229)(190 238 200 228)(201 442 211 452)(202 441 212 451)(203 460 213 450)(204 459 214 449)(205 458 215 448)(206 457 216 447)(207 456 217 446)(208 455 218 445)(209 454 219 444)(210 453 220 443)(241 363 251 373)(242 362 252 372)(243 361 253 371)(244 380 254 370)(245 379 255 369)(246 378 256 368)(247 377 257 367)(248 376 258 366)(249 375 259 365)(250 374 260 364)(281 303 291 313)(282 302 292 312)(283 301 293 311)(284 320 294 310)(285 319 295 309)(286 318 296 308)(287 317 297 307)(288 316 298 306)(289 315 299 305)(290 314 300 304)(321 355 331 345)(322 354 332 344)(323 353 333 343)(324 352 334 342)(325 351 335 341)(326 350 336 360)(327 349 337 359)(328 348 338 358)(329 347 339 357)(330 346 340 356)(401 476 411 466)(402 475 412 465)(403 474 413 464)(404 473 414 463)(405 472 415 462)(406 471 416 461)(407 470 417 480)(408 469 418 479)(409 468 419 478)(410 467 420 477)
G:=sub<Sym(480)| (1,159)(2,160)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(21,242)(22,243)(23,244)(24,245)(25,246)(26,247)(27,248)(28,249)(29,250)(30,251)(31,252)(32,253)(33,254)(34,255)(35,256)(36,257)(37,258)(38,259)(39,260)(40,241)(41,329)(42,330)(43,331)(44,332)(45,333)(46,334)(47,335)(48,336)(49,337)(50,338)(51,339)(52,340)(53,321)(54,322)(55,323)(56,324)(57,325)(58,326)(59,327)(60,328)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,113)(72,114)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,101)(80,102)(81,286)(82,287)(83,288)(84,289)(85,290)(86,291)(87,292)(88,293)(89,294)(90,295)(91,296)(92,297)(93,298)(94,299)(95,300)(96,281)(97,282)(98,283)(99,284)(100,285)(121,301)(122,302)(123,303)(124,304)(125,305)(126,306)(127,307)(128,308)(129,309)(130,310)(131,311)(132,312)(133,313)(134,314)(135,315)(136,316)(137,317)(138,318)(139,319)(140,320)(161,468)(162,469)(163,470)(164,471)(165,472)(166,473)(167,474)(168,475)(169,476)(170,477)(171,478)(172,479)(173,480)(174,461)(175,462)(176,463)(177,464)(178,465)(179,466)(180,467)(181,220)(182,201)(183,202)(184,203)(185,204)(186,205)(187,206)(188,207)(189,208)(190,209)(191,210)(192,211)(193,212)(194,213)(195,214)(196,215)(197,216)(198,217)(199,218)(200,219)(221,457)(222,458)(223,459)(224,460)(225,441)(226,442)(227,443)(228,444)(229,445)(230,446)(231,447)(232,448)(233,449)(234,450)(235,451)(236,452)(237,453)(238,454)(239,455)(240,456)(261,406)(262,407)(263,408)(264,409)(265,410)(266,411)(267,412)(268,413)(269,414)(270,415)(271,416)(272,417)(273,418)(274,419)(275,420)(276,401)(277,402)(278,403)(279,404)(280,405)(341,387)(342,388)(343,389)(344,390)(345,391)(346,392)(347,393)(348,394)(349,395)(350,396)(351,397)(352,398)(353,399)(354,400)(355,381)(356,382)(357,383)(358,384)(359,385)(360,386)(361,438)(362,439)(363,440)(364,421)(365,422)(366,423)(367,424)(368,425)(369,426)(370,427)(371,428)(372,429)(373,430)(374,431)(375,432)(376,433)(377,434)(378,435)(379,436)(380,437), (1,128,367,418,392,211)(2,129,368,419,393,212)(3,130,369,420,394,213)(4,131,370,401,395,214)(5,132,371,402,396,215)(6,133,372,403,397,216)(7,134,373,404,398,217)(8,135,374,405,399,218)(9,136,375,406,400,219)(10,137,376,407,381,220)(11,138,377,408,382,201)(12,139,378,409,383,202)(13,140,379,410,384,203)(14,121,380,411,385,204)(15,122,361,412,386,205)(16,123,362,413,387,206)(17,124,363,414,388,207)(18,125,364,415,389,208)(19,126,365,416,390,209)(20,127,366,417,391,210)(21,167,335,231,116,281)(22,168,336,232,117,282)(23,169,337,233,118,283)(24,170,338,234,119,284)(25,171,339,235,120,285)(26,172,340,236,101,286)(27,173,321,237,102,287)(28,174,322,238,103,288)(29,175,323,239,104,289)(30,176,324,240,105,290)(31,177,325,221,106,291)(32,178,326,222,107,292)(33,179,327,223,108,293)(34,180,328,224,109,294)(35,161,329,225,110,295)(36,162,330,226,111,296)(37,163,331,227,112,297)(38,164,332,228,113,298)(39,165,333,229,114,299)(40,166,334,230,115,300)(41,441,68,90,256,468)(42,442,69,91,257,469)(43,443,70,92,258,470)(44,444,71,93,259,471)(45,445,72,94,260,472)(46,446,73,95,241,473)(47,447,74,96,242,474)(48,448,75,97,243,475)(49,449,76,98,244,476)(50,450,77,99,245,477)(51,451,78,100,246,478)(52,452,79,81,247,479)(53,453,80,82,248,480)(54,454,61,83,249,461)(55,455,62,84,250,462)(56,456,63,85,251,463)(57,457,64,86,252,464)(58,458,65,87,253,465)(59,459,66,88,254,466)(60,460,67,89,255,467)(141,310,426,275,348,194)(142,311,427,276,349,195)(143,312,428,277,350,196)(144,313,429,278,351,197)(145,314,430,279,352,198)(146,315,431,280,353,199)(147,316,432,261,354,200)(148,317,433,262,355,181)(149,318,434,263,356,182)(150,319,435,264,357,183)(151,320,436,265,358,184)(152,301,437,266,359,185)(153,302,438,267,360,186)(154,303,439,268,341,187)(155,304,440,269,342,188)(156,305,421,270,343,189)(157,306,422,271,344,190)(158,307,423,272,345,191)(159,308,424,273,346,192)(160,309,425,274,347,193), (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), (1,79,11,69)(2,78,12,68)(3,77,13,67)(4,76,14,66)(5,75,15,65)(6,74,16,64)(7,73,17,63)(8,72,18,62)(9,71,19,61)(10,70,20,80)(21,439,31,429)(22,438,32,428)(23,437,33,427)(24,436,34,426)(25,435,35,425)(26,434,36,424)(27,433,37,423)(28,432,38,422)(29,431,39,421)(30,430,40,440)(41,393,51,383)(42,392,52,382)(43,391,53,381)(44,390,54,400)(45,389,55,399)(46,388,56,398)(47,387,57,397)(48,386,58,396)(49,385,59,395)(50,384,60,394)(81,138,91,128)(82,137,92,127)(83,136,93,126)(84,135,94,125)(85,134,95,124)(86,133,96,123)(87,132,97,122)(88,131,98,121)(89,130,99,140)(90,129,100,139)(101,149,111,159)(102,148,112,158)(103,147,113,157)(104,146,114,156)(105,145,115,155)(106,144,116,154)(107,143,117,153)(108,142,118,152)(109,141,119,151)(110,160,120,150)(161,274,171,264)(162,273,172,263)(163,272,173,262)(164,271,174,261)(165,270,175,280)(166,269,176,279)(167,268,177,278)(168,267,178,277)(169,266,179,276)(170,265,180,275)(181,227,191,237)(182,226,192,236)(183,225,193,235)(184,224,194,234)(185,223,195,233)(186,222,196,232)(187,221,197,231)(188,240,198,230)(189,239,199,229)(190,238,200,228)(201,442,211,452)(202,441,212,451)(203,460,213,450)(204,459,214,449)(205,458,215,448)(206,457,216,447)(207,456,217,446)(208,455,218,445)(209,454,219,444)(210,453,220,443)(241,363,251,373)(242,362,252,372)(243,361,253,371)(244,380,254,370)(245,379,255,369)(246,378,256,368)(247,377,257,367)(248,376,258,366)(249,375,259,365)(250,374,260,364)(281,303,291,313)(282,302,292,312)(283,301,293,311)(284,320,294,310)(285,319,295,309)(286,318,296,308)(287,317,297,307)(288,316,298,306)(289,315,299,305)(290,314,300,304)(321,355,331,345)(322,354,332,344)(323,353,333,343)(324,352,334,342)(325,351,335,341)(326,350,336,360)(327,349,337,359)(328,348,338,358)(329,347,339,357)(330,346,340,356)(401,476,411,466)(402,475,412,465)(403,474,413,464)(404,473,414,463)(405,472,415,462)(406,471,416,461)(407,470,417,480)(408,469,418,479)(409,468,419,478)(410,467,420,477)>;
G:=Group( (1,159)(2,160)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(21,242)(22,243)(23,244)(24,245)(25,246)(26,247)(27,248)(28,249)(29,250)(30,251)(31,252)(32,253)(33,254)(34,255)(35,256)(36,257)(37,258)(38,259)(39,260)(40,241)(41,329)(42,330)(43,331)(44,332)(45,333)(46,334)(47,335)(48,336)(49,337)(50,338)(51,339)(52,340)(53,321)(54,322)(55,323)(56,324)(57,325)(58,326)(59,327)(60,328)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,113)(72,114)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,101)(80,102)(81,286)(82,287)(83,288)(84,289)(85,290)(86,291)(87,292)(88,293)(89,294)(90,295)(91,296)(92,297)(93,298)(94,299)(95,300)(96,281)(97,282)(98,283)(99,284)(100,285)(121,301)(122,302)(123,303)(124,304)(125,305)(126,306)(127,307)(128,308)(129,309)(130,310)(131,311)(132,312)(133,313)(134,314)(135,315)(136,316)(137,317)(138,318)(139,319)(140,320)(161,468)(162,469)(163,470)(164,471)(165,472)(166,473)(167,474)(168,475)(169,476)(170,477)(171,478)(172,479)(173,480)(174,461)(175,462)(176,463)(177,464)(178,465)(179,466)(180,467)(181,220)(182,201)(183,202)(184,203)(185,204)(186,205)(187,206)(188,207)(189,208)(190,209)(191,210)(192,211)(193,212)(194,213)(195,214)(196,215)(197,216)(198,217)(199,218)(200,219)(221,457)(222,458)(223,459)(224,460)(225,441)(226,442)(227,443)(228,444)(229,445)(230,446)(231,447)(232,448)(233,449)(234,450)(235,451)(236,452)(237,453)(238,454)(239,455)(240,456)(261,406)(262,407)(263,408)(264,409)(265,410)(266,411)(267,412)(268,413)(269,414)(270,415)(271,416)(272,417)(273,418)(274,419)(275,420)(276,401)(277,402)(278,403)(279,404)(280,405)(341,387)(342,388)(343,389)(344,390)(345,391)(346,392)(347,393)(348,394)(349,395)(350,396)(351,397)(352,398)(353,399)(354,400)(355,381)(356,382)(357,383)(358,384)(359,385)(360,386)(361,438)(362,439)(363,440)(364,421)(365,422)(366,423)(367,424)(368,425)(369,426)(370,427)(371,428)(372,429)(373,430)(374,431)(375,432)(376,433)(377,434)(378,435)(379,436)(380,437), (1,128,367,418,392,211)(2,129,368,419,393,212)(3,130,369,420,394,213)(4,131,370,401,395,214)(5,132,371,402,396,215)(6,133,372,403,397,216)(7,134,373,404,398,217)(8,135,374,405,399,218)(9,136,375,406,400,219)(10,137,376,407,381,220)(11,138,377,408,382,201)(12,139,378,409,383,202)(13,140,379,410,384,203)(14,121,380,411,385,204)(15,122,361,412,386,205)(16,123,362,413,387,206)(17,124,363,414,388,207)(18,125,364,415,389,208)(19,126,365,416,390,209)(20,127,366,417,391,210)(21,167,335,231,116,281)(22,168,336,232,117,282)(23,169,337,233,118,283)(24,170,338,234,119,284)(25,171,339,235,120,285)(26,172,340,236,101,286)(27,173,321,237,102,287)(28,174,322,238,103,288)(29,175,323,239,104,289)(30,176,324,240,105,290)(31,177,325,221,106,291)(32,178,326,222,107,292)(33,179,327,223,108,293)(34,180,328,224,109,294)(35,161,329,225,110,295)(36,162,330,226,111,296)(37,163,331,227,112,297)(38,164,332,228,113,298)(39,165,333,229,114,299)(40,166,334,230,115,300)(41,441,68,90,256,468)(42,442,69,91,257,469)(43,443,70,92,258,470)(44,444,71,93,259,471)(45,445,72,94,260,472)(46,446,73,95,241,473)(47,447,74,96,242,474)(48,448,75,97,243,475)(49,449,76,98,244,476)(50,450,77,99,245,477)(51,451,78,100,246,478)(52,452,79,81,247,479)(53,453,80,82,248,480)(54,454,61,83,249,461)(55,455,62,84,250,462)(56,456,63,85,251,463)(57,457,64,86,252,464)(58,458,65,87,253,465)(59,459,66,88,254,466)(60,460,67,89,255,467)(141,310,426,275,348,194)(142,311,427,276,349,195)(143,312,428,277,350,196)(144,313,429,278,351,197)(145,314,430,279,352,198)(146,315,431,280,353,199)(147,316,432,261,354,200)(148,317,433,262,355,181)(149,318,434,263,356,182)(150,319,435,264,357,183)(151,320,436,265,358,184)(152,301,437,266,359,185)(153,302,438,267,360,186)(154,303,439,268,341,187)(155,304,440,269,342,188)(156,305,421,270,343,189)(157,306,422,271,344,190)(158,307,423,272,345,191)(159,308,424,273,346,192)(160,309,425,274,347,193), (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), (1,79,11,69)(2,78,12,68)(3,77,13,67)(4,76,14,66)(5,75,15,65)(6,74,16,64)(7,73,17,63)(8,72,18,62)(9,71,19,61)(10,70,20,80)(21,439,31,429)(22,438,32,428)(23,437,33,427)(24,436,34,426)(25,435,35,425)(26,434,36,424)(27,433,37,423)(28,432,38,422)(29,431,39,421)(30,430,40,440)(41,393,51,383)(42,392,52,382)(43,391,53,381)(44,390,54,400)(45,389,55,399)(46,388,56,398)(47,387,57,397)(48,386,58,396)(49,385,59,395)(50,384,60,394)(81,138,91,128)(82,137,92,127)(83,136,93,126)(84,135,94,125)(85,134,95,124)(86,133,96,123)(87,132,97,122)(88,131,98,121)(89,130,99,140)(90,129,100,139)(101,149,111,159)(102,148,112,158)(103,147,113,157)(104,146,114,156)(105,145,115,155)(106,144,116,154)(107,143,117,153)(108,142,118,152)(109,141,119,151)(110,160,120,150)(161,274,171,264)(162,273,172,263)(163,272,173,262)(164,271,174,261)(165,270,175,280)(166,269,176,279)(167,268,177,278)(168,267,178,277)(169,266,179,276)(170,265,180,275)(181,227,191,237)(182,226,192,236)(183,225,193,235)(184,224,194,234)(185,223,195,233)(186,222,196,232)(187,221,197,231)(188,240,198,230)(189,239,199,229)(190,238,200,228)(201,442,211,452)(202,441,212,451)(203,460,213,450)(204,459,214,449)(205,458,215,448)(206,457,216,447)(207,456,217,446)(208,455,218,445)(209,454,219,444)(210,453,220,443)(241,363,251,373)(242,362,252,372)(243,361,253,371)(244,380,254,370)(245,379,255,369)(246,378,256,368)(247,377,257,367)(248,376,258,366)(249,375,259,365)(250,374,260,364)(281,303,291,313)(282,302,292,312)(283,301,293,311)(284,320,294,310)(285,319,295,309)(286,318,296,308)(287,317,297,307)(288,316,298,306)(289,315,299,305)(290,314,300,304)(321,355,331,345)(322,354,332,344)(323,353,333,343)(324,352,334,342)(325,351,335,341)(326,350,336,360)(327,349,337,359)(328,348,338,358)(329,347,339,357)(330,346,340,356)(401,476,411,466)(402,475,412,465)(403,474,413,464)(404,473,414,463)(405,472,415,462)(406,471,416,461)(407,470,417,480)(408,469,418,479)(409,468,419,478)(410,467,420,477) );
G=PermutationGroup([(1,159),(2,160),(3,141),(4,142),(5,143),(6,144),(7,145),(8,146),(9,147),(10,148),(11,149),(12,150),(13,151),(14,152),(15,153),(16,154),(17,155),(18,156),(19,157),(20,158),(21,242),(22,243),(23,244),(24,245),(25,246),(26,247),(27,248),(28,249),(29,250),(30,251),(31,252),(32,253),(33,254),(34,255),(35,256),(36,257),(37,258),(38,259),(39,260),(40,241),(41,329),(42,330),(43,331),(44,332),(45,333),(46,334),(47,335),(48,336),(49,337),(50,338),(51,339),(52,340),(53,321),(54,322),(55,323),(56,324),(57,325),(58,326),(59,327),(60,328),(61,103),(62,104),(63,105),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,113),(72,114),(73,115),(74,116),(75,117),(76,118),(77,119),(78,120),(79,101),(80,102),(81,286),(82,287),(83,288),(84,289),(85,290),(86,291),(87,292),(88,293),(89,294),(90,295),(91,296),(92,297),(93,298),(94,299),(95,300),(96,281),(97,282),(98,283),(99,284),(100,285),(121,301),(122,302),(123,303),(124,304),(125,305),(126,306),(127,307),(128,308),(129,309),(130,310),(131,311),(132,312),(133,313),(134,314),(135,315),(136,316),(137,317),(138,318),(139,319),(140,320),(161,468),(162,469),(163,470),(164,471),(165,472),(166,473),(167,474),(168,475),(169,476),(170,477),(171,478),(172,479),(173,480),(174,461),(175,462),(176,463),(177,464),(178,465),(179,466),(180,467),(181,220),(182,201),(183,202),(184,203),(185,204),(186,205),(187,206),(188,207),(189,208),(190,209),(191,210),(192,211),(193,212),(194,213),(195,214),(196,215),(197,216),(198,217),(199,218),(200,219),(221,457),(222,458),(223,459),(224,460),(225,441),(226,442),(227,443),(228,444),(229,445),(230,446),(231,447),(232,448),(233,449),(234,450),(235,451),(236,452),(237,453),(238,454),(239,455),(240,456),(261,406),(262,407),(263,408),(264,409),(265,410),(266,411),(267,412),(268,413),(269,414),(270,415),(271,416),(272,417),(273,418),(274,419),(275,420),(276,401),(277,402),(278,403),(279,404),(280,405),(341,387),(342,388),(343,389),(344,390),(345,391),(346,392),(347,393),(348,394),(349,395),(350,396),(351,397),(352,398),(353,399),(354,400),(355,381),(356,382),(357,383),(358,384),(359,385),(360,386),(361,438),(362,439),(363,440),(364,421),(365,422),(366,423),(367,424),(368,425),(369,426),(370,427),(371,428),(372,429),(373,430),(374,431),(375,432),(376,433),(377,434),(378,435),(379,436),(380,437)], [(1,128,367,418,392,211),(2,129,368,419,393,212),(3,130,369,420,394,213),(4,131,370,401,395,214),(5,132,371,402,396,215),(6,133,372,403,397,216),(7,134,373,404,398,217),(8,135,374,405,399,218),(9,136,375,406,400,219),(10,137,376,407,381,220),(11,138,377,408,382,201),(12,139,378,409,383,202),(13,140,379,410,384,203),(14,121,380,411,385,204),(15,122,361,412,386,205),(16,123,362,413,387,206),(17,124,363,414,388,207),(18,125,364,415,389,208),(19,126,365,416,390,209),(20,127,366,417,391,210),(21,167,335,231,116,281),(22,168,336,232,117,282),(23,169,337,233,118,283),(24,170,338,234,119,284),(25,171,339,235,120,285),(26,172,340,236,101,286),(27,173,321,237,102,287),(28,174,322,238,103,288),(29,175,323,239,104,289),(30,176,324,240,105,290),(31,177,325,221,106,291),(32,178,326,222,107,292),(33,179,327,223,108,293),(34,180,328,224,109,294),(35,161,329,225,110,295),(36,162,330,226,111,296),(37,163,331,227,112,297),(38,164,332,228,113,298),(39,165,333,229,114,299),(40,166,334,230,115,300),(41,441,68,90,256,468),(42,442,69,91,257,469),(43,443,70,92,258,470),(44,444,71,93,259,471),(45,445,72,94,260,472),(46,446,73,95,241,473),(47,447,74,96,242,474),(48,448,75,97,243,475),(49,449,76,98,244,476),(50,450,77,99,245,477),(51,451,78,100,246,478),(52,452,79,81,247,479),(53,453,80,82,248,480),(54,454,61,83,249,461),(55,455,62,84,250,462),(56,456,63,85,251,463),(57,457,64,86,252,464),(58,458,65,87,253,465),(59,459,66,88,254,466),(60,460,67,89,255,467),(141,310,426,275,348,194),(142,311,427,276,349,195),(143,312,428,277,350,196),(144,313,429,278,351,197),(145,314,430,279,352,198),(146,315,431,280,353,199),(147,316,432,261,354,200),(148,317,433,262,355,181),(149,318,434,263,356,182),(150,319,435,264,357,183),(151,320,436,265,358,184),(152,301,437,266,359,185),(153,302,438,267,360,186),(154,303,439,268,341,187),(155,304,440,269,342,188),(156,305,421,270,343,189),(157,306,422,271,344,190),(158,307,423,272,345,191),(159,308,424,273,346,192),(160,309,425,274,347,193)], [(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)], [(1,79,11,69),(2,78,12,68),(3,77,13,67),(4,76,14,66),(5,75,15,65),(6,74,16,64),(7,73,17,63),(8,72,18,62),(9,71,19,61),(10,70,20,80),(21,439,31,429),(22,438,32,428),(23,437,33,427),(24,436,34,426),(25,435,35,425),(26,434,36,424),(27,433,37,423),(28,432,38,422),(29,431,39,421),(30,430,40,440),(41,393,51,383),(42,392,52,382),(43,391,53,381),(44,390,54,400),(45,389,55,399),(46,388,56,398),(47,387,57,397),(48,386,58,396),(49,385,59,395),(50,384,60,394),(81,138,91,128),(82,137,92,127),(83,136,93,126),(84,135,94,125),(85,134,95,124),(86,133,96,123),(87,132,97,122),(88,131,98,121),(89,130,99,140),(90,129,100,139),(101,149,111,159),(102,148,112,158),(103,147,113,157),(104,146,114,156),(105,145,115,155),(106,144,116,154),(107,143,117,153),(108,142,118,152),(109,141,119,151),(110,160,120,150),(161,274,171,264),(162,273,172,263),(163,272,173,262),(164,271,174,261),(165,270,175,280),(166,269,176,279),(167,268,177,278),(168,267,178,277),(169,266,179,276),(170,265,180,275),(181,227,191,237),(182,226,192,236),(183,225,193,235),(184,224,194,234),(185,223,195,233),(186,222,196,232),(187,221,197,231),(188,240,198,230),(189,239,199,229),(190,238,200,228),(201,442,211,452),(202,441,212,451),(203,460,213,450),(204,459,214,449),(205,458,215,448),(206,457,216,447),(207,456,217,446),(208,455,218,445),(209,454,219,444),(210,453,220,443),(241,363,251,373),(242,362,252,372),(243,361,253,371),(244,380,254,370),(245,379,255,369),(246,378,256,368),(247,377,257,367),(248,376,258,366),(249,375,259,365),(250,374,260,364),(281,303,291,313),(282,302,292,312),(283,301,293,311),(284,320,294,310),(285,319,295,309),(286,318,296,308),(287,317,297,307),(288,316,298,306),(289,315,299,305),(290,314,300,304),(321,355,331,345),(322,354,332,344),(323,353,333,343),(324,352,334,342),(325,351,335,341),(326,350,336,360),(327,349,337,359),(328,348,338,358),(329,347,339,357),(330,346,340,356),(401,476,411,466),(402,475,412,465),(403,474,413,464),(404,473,414,463),(405,472,415,462),(406,471,416,461),(407,470,417,480),(408,469,418,479),(409,468,419,478),(410,467,420,477)])
Matrix representation ►G ⊆ GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 48 | 0 |
| 0 | 0 | 0 | 0 | 48 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 60 | 17 | 0 | 0 |
| 0 | 0 | 0 | 2 | 36 |
| 0 | 0 | 0 | 27 | 29 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 53 | 58 |
| 0 | 0 | 0 | 42 | 8 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,48,0,0,0,0,0,48],[1,0,0,0,0,0,0,60,0,0,0,1,17,0,0,0,0,0,2,27,0,0,0,36,29],[60,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,53,42,0,0,0,58,8] >;
156 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 6A | ··· | 6N | 10A | ··· | 10N | 12A | ··· | 12H | 12I | ··· | 12X | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | Q8 | D5 | D10 | D10 | C3×Q8 | C3×D5 | Dic10 | C6×D5 | C6×D5 | C3×Dic10 |
| kernel | C2×C6×Dic10 | C6×Dic10 | C2×C6×Dic5 | C22×C60 | C22×Dic10 | C2×Dic10 | C22×Dic5 | C22×C20 | C2×C30 | C22×C12 | C2×C12 | C22×C6 | C2×C10 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 2 | 1 | 2 | 24 | 4 | 2 | 4 | 2 | 12 | 2 | 8 | 4 | 16 | 24 | 4 | 32 |
In GAP, Magma, Sage, TeX
C_2\times C_6\times Dic_{10} % in TeX
G:=Group("C2xC6xDic10"); // GroupNames label
G:=SmallGroup(480,1135);
// by ID
G=gap.SmallGroup(480,1135);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,336,1571,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^20=1,d^2=c^10,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
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ℤ
𝔽
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