direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3×C5⋊C8, Dic15⋊1C8, C30.6C42, C15⋊3(C4×C8), C15⋊C8⋊3C4, C5⋊3(C8×Dic3), C10.7(S3×C8), C30.5(C2×C8), C6.11(C4×F5), C6.3(D5⋊C8), (C5×Dic3)⋊1C8, C2.3(Dic3×F5), C2.2(D15⋊C8), (C2×Dic3).8F5, C10.6(C4×Dic3), C22.11(S3×F5), (C2×Dic15).3C4, (C10×Dic3).2C4, Dic5.11(C4×S3), (C2×Dic5).141D6, Dic5.9(C2×Dic3), (Dic3×Dic5).19C2, (C6×Dic5).134C22, C3⋊2(C4×C5⋊C8), (C3×C5⋊C8)⋊3C4, C6.3(C2×C5⋊C8), C2.2(S3×C5⋊C8), (C6×C5⋊C8).1C2, (C2×C5⋊C8).4S3, (C2×C10).4(C4×S3), (C2×C30).2(C2×C4), (C2×C6).12(C2×F5), (C2×C15⋊C8).1C2, (C3×Dic5).17(C2×C4), SmallGroup(480,244)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — Dic3×C5⋊C8 |
Generators and relations for Dic3×C5⋊C8
G = < a,b,c,d | a6=c5=d8=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 308 in 88 conjugacy classes, 46 normal (34 characteristic)
C1, C2, C3, C4, C22, C5, C6, C8, C2×C4, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C2×C8, Dic5, Dic5, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C30, C4×C8, C5⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, C5×Dic3, C3×Dic5, Dic15, C2×C30, C4×Dic5, C2×C5⋊C8, C2×C5⋊C8, C8×Dic3, C3×C5⋊C8, C15⋊C8, C6×Dic5, C10×Dic3, C2×Dic15, C4×C5⋊C8, Dic3×Dic5, C6×C5⋊C8, C2×C15⋊C8, Dic3×C5⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, F5, C4×S3, C2×Dic3, C4×C8, C5⋊C8, C2×F5, S3×C8, C4×Dic3, D5⋊C8, C4×F5, C2×C5⋊C8, C8×Dic3, S3×F5, C4×C5⋊C8, Dic3×F5, S3×C5⋊C8, D15⋊C8, Dic3×C5⋊C8
►Regular action on 480 points
Generators in S480
(1 445 286 110 335 460)(2 446 287 111 336 461)(3 447 288 112 329 462)(4 448 281 105 330 463)(5 441 282 106 331 464)(6 442 283 107 332 457)(7 443 284 108 333 458)(8 444 285 109 334 459)(9 404 228 362 190 259)(10 405 229 363 191 260)(11 406 230 364 192 261)(12 407 231 365 185 262)(13 408 232 366 186 263)(14 401 225 367 187 264)(15 402 226 368 188 257)(16 403 227 361 189 258)(17 390 452 412 84 193)(18 391 453 413 85 194)(19 392 454 414 86 195)(20 385 455 415 87 196)(21 386 456 416 88 197)(22 387 449 409 81 198)(23 388 450 410 82 199)(24 389 451 411 83 200)(25 326 343 152 208 174)(26 327 344 145 201 175)(27 328 337 146 202 176)(28 321 338 147 203 169)(29 322 339 148 204 170)(30 323 340 149 205 171)(31 324 341 150 206 172)(32 325 342 151 207 173)(33 353 58 302 164 276)(34 354 59 303 165 277)(35 355 60 304 166 278)(36 356 61 297 167 279)(37 357 62 298 168 280)(38 358 63 299 161 273)(39 359 64 300 162 274)(40 360 57 301 163 275)(41 296 240 180 247 427)(42 289 233 181 248 428)(43 290 234 182 241 429)(44 291 235 183 242 430)(45 292 236 184 243 431)(46 293 237 177 244 432)(47 294 238 178 245 425)(48 295 239 179 246 426)(49 92 376 218 398 117)(50 93 369 219 399 118)(51 94 370 220 400 119)(52 95 371 221 393 120)(53 96 372 222 394 113)(54 89 373 223 395 114)(55 90 374 224 396 115)(56 91 375 217 397 116)(65 123 465 74 478 435)(66 124 466 75 479 436)(67 125 467 76 480 437)(68 126 468 77 473 438)(69 127 469 78 474 439)(70 128 470 79 475 440)(71 121 471 80 476 433)(72 122 472 73 477 434)(97 347 270 216 157 135)(98 348 271 209 158 136)(99 349 272 210 159 129)(100 350 265 211 160 130)(101 351 266 212 153 131)(102 352 267 213 154 132)(103 345 268 214 155 133)(104 346 269 215 156 134)(137 378 311 318 250 424)(138 379 312 319 251 417)(139 380 305 320 252 418)(140 381 306 313 253 419)(141 382 307 314 254 420)(142 383 308 315 255 421)(143 384 309 316 256 422)(144 377 310 317 249 423)
(1 42 110 181)(2 43 111 182)(3 44 112 183)(4 45 105 184)(5 46 106 177)(6 47 107 178)(7 48 108 179)(8 41 109 180)(9 169 362 338)(10 170 363 339)(11 171 364 340)(12 172 365 341)(13 173 366 342)(14 174 367 343)(15 175 368 344)(16 176 361 337)(17 57 412 275)(18 58 413 276)(19 59 414 277)(20 60 415 278)(21 61 416 279)(22 62 409 280)(23 63 410 273)(24 64 411 274)(25 225 152 264)(26 226 145 257)(27 227 146 258)(28 228 147 259)(29 229 148 260)(30 230 149 261)(31 231 150 262)(32 232 151 263)(33 194 302 453)(34 195 303 454)(35 196 304 455)(36 197 297 456)(37 198 298 449)(38 199 299 450)(39 200 300 451)(40 193 301 452)(49 253 218 381)(50 254 219 382)(51 255 220 383)(52 256 221 384)(53 249 222 377)(54 250 223 378)(55 251 224 379)(56 252 217 380)(65 349 74 159)(66 350 75 160)(67 351 76 153)(68 352 77 154)(69 345 78 155)(70 346 79 156)(71 347 80 157)(72 348 73 158)(81 168 387 357)(82 161 388 358)(83 162 389 359)(84 163 390 360)(85 164 391 353)(86 165 392 354)(87 166 385 355)(88 167 386 356)(89 318 395 137)(90 319 396 138)(91 320 397 139)(92 313 398 140)(93 314 399 141)(94 315 400 142)(95 316 393 143)(96 317 394 144)(97 476 216 121)(98 477 209 122)(99 478 210 123)(100 479 211 124)(101 480 212 125)(102 473 213 126)(103 474 214 127)(104 475 215 128)(113 423 372 310)(114 424 373 311)(115 417 374 312)(116 418 375 305)(117 419 376 306)(118 420 369 307)(119 421 370 308)(120 422 371 309)(129 435 272 465)(130 436 265 466)(131 437 266 467)(132 438 267 468)(133 439 268 469)(134 440 269 470)(135 433 270 471)(136 434 271 472)(185 324 407 206)(186 325 408 207)(187 326 401 208)(188 327 402 201)(189 328 403 202)(190 321 404 203)(191 322 405 204)(192 323 406 205)(233 445 428 335)(234 446 429 336)(235 447 430 329)(236 448 431 330)(237 441 432 331)(238 442 425 332)(239 443 426 333)(240 444 427 334)(241 461 290 287)(242 462 291 288)(243 463 292 281)(244 464 293 282)(245 457 294 283)(246 458 295 284)(247 459 296 285)(248 460 289 286)
(1 83 323 211 224)(2 212 84 217 324)(3 218 213 325 85)(4 326 219 86 214)(5 87 327 215 220)(6 216 88 221 328)(7 222 209 321 81)(8 322 223 82 210)(9 298 426 144 72)(10 137 299 65 427)(11 66 138 428 300)(12 429 67 301 139)(13 302 430 140 68)(14 141 303 69 431)(15 70 142 432 304)(16 425 71 297 143)(17 116 150 287 131)(18 288 117 132 151)(19 133 281 152 118)(20 145 134 119 282)(21 120 146 283 135)(22 284 113 136 147)(23 129 285 148 114)(24 149 130 115 286)(25 369 414 268 463)(26 269 370 464 415)(27 457 270 416 371)(28 409 458 372 271)(29 373 410 272 459)(30 265 374 460 411)(31 461 266 412 375)(32 413 462 376 267)(33 235 313 77 366)(34 78 236 367 314)(35 368 79 315 237)(36 316 361 238 80)(37 239 317 73 362)(38 74 240 363 318)(39 364 75 319 233)(40 320 365 234 76)(41 405 378 161 123)(42 162 406 124 379)(43 125 163 380 407)(44 381 126 408 164)(45 401 382 165 127)(46 166 402 128 383)(47 121 167 384 403)(48 377 122 404 168)(49 102 207 391 112)(50 392 103 105 208)(51 106 385 201 104)(52 202 107 97 386)(53 98 203 387 108)(54 388 99 109 204)(55 110 389 205 100)(56 206 111 101 390)(57 418 262 241 437)(58 242 419 438 263)(59 439 243 264 420)(60 257 440 421 244)(61 422 258 245 433)(62 246 423 434 259)(63 435 247 260 424)(64 261 436 417 248)(89 450 349 334 170)(90 335 451 171 350)(91 172 336 351 452)(92 352 173 453 329)(93 454 345 330 174)(94 331 455 175 346)(95 176 332 347 456)(96 348 169 449 333)(153 193 397 341 446)(154 342 194 447 398)(155 448 343 399 195)(156 400 441 196 344)(157 197 393 337 442)(158 338 198 443 394)(159 444 339 395 199)(160 396 445 200 340)(177 355 188 475 255)(178 476 356 256 189)(179 249 477 190 357)(180 191 250 358 478)(181 359 192 479 251)(182 480 360 252 185)(183 253 473 186 353)(184 187 254 354 474)(225 307 277 469 292)(226 470 308 293 278)(227 294 471 279 309)(228 280 295 310 472)(229 311 273 465 296)(230 466 312 289 274)(231 290 467 275 305)(232 276 291 306 468)
(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)
G:=sub<Sym(480)| (1,445,286,110,335,460)(2,446,287,111,336,461)(3,447,288,112,329,462)(4,448,281,105,330,463)(5,441,282,106,331,464)(6,442,283,107,332,457)(7,443,284,108,333,458)(8,444,285,109,334,459)(9,404,228,362,190,259)(10,405,229,363,191,260)(11,406,230,364,192,261)(12,407,231,365,185,262)(13,408,232,366,186,263)(14,401,225,367,187,264)(15,402,226,368,188,257)(16,403,227,361,189,258)(17,390,452,412,84,193)(18,391,453,413,85,194)(19,392,454,414,86,195)(20,385,455,415,87,196)(21,386,456,416,88,197)(22,387,449,409,81,198)(23,388,450,410,82,199)(24,389,451,411,83,200)(25,326,343,152,208,174)(26,327,344,145,201,175)(27,328,337,146,202,176)(28,321,338,147,203,169)(29,322,339,148,204,170)(30,323,340,149,205,171)(31,324,341,150,206,172)(32,325,342,151,207,173)(33,353,58,302,164,276)(34,354,59,303,165,277)(35,355,60,304,166,278)(36,356,61,297,167,279)(37,357,62,298,168,280)(38,358,63,299,161,273)(39,359,64,300,162,274)(40,360,57,301,163,275)(41,296,240,180,247,427)(42,289,233,181,248,428)(43,290,234,182,241,429)(44,291,235,183,242,430)(45,292,236,184,243,431)(46,293,237,177,244,432)(47,294,238,178,245,425)(48,295,239,179,246,426)(49,92,376,218,398,117)(50,93,369,219,399,118)(51,94,370,220,400,119)(52,95,371,221,393,120)(53,96,372,222,394,113)(54,89,373,223,395,114)(55,90,374,224,396,115)(56,91,375,217,397,116)(65,123,465,74,478,435)(66,124,466,75,479,436)(67,125,467,76,480,437)(68,126,468,77,473,438)(69,127,469,78,474,439)(70,128,470,79,475,440)(71,121,471,80,476,433)(72,122,472,73,477,434)(97,347,270,216,157,135)(98,348,271,209,158,136)(99,349,272,210,159,129)(100,350,265,211,160,130)(101,351,266,212,153,131)(102,352,267,213,154,132)(103,345,268,214,155,133)(104,346,269,215,156,134)(137,378,311,318,250,424)(138,379,312,319,251,417)(139,380,305,320,252,418)(140,381,306,313,253,419)(141,382,307,314,254,420)(142,383,308,315,255,421)(143,384,309,316,256,422)(144,377,310,317,249,423), (1,42,110,181)(2,43,111,182)(3,44,112,183)(4,45,105,184)(5,46,106,177)(6,47,107,178)(7,48,108,179)(8,41,109,180)(9,169,362,338)(10,170,363,339)(11,171,364,340)(12,172,365,341)(13,173,366,342)(14,174,367,343)(15,175,368,344)(16,176,361,337)(17,57,412,275)(18,58,413,276)(19,59,414,277)(20,60,415,278)(21,61,416,279)(22,62,409,280)(23,63,410,273)(24,64,411,274)(25,225,152,264)(26,226,145,257)(27,227,146,258)(28,228,147,259)(29,229,148,260)(30,230,149,261)(31,231,150,262)(32,232,151,263)(33,194,302,453)(34,195,303,454)(35,196,304,455)(36,197,297,456)(37,198,298,449)(38,199,299,450)(39,200,300,451)(40,193,301,452)(49,253,218,381)(50,254,219,382)(51,255,220,383)(52,256,221,384)(53,249,222,377)(54,250,223,378)(55,251,224,379)(56,252,217,380)(65,349,74,159)(66,350,75,160)(67,351,76,153)(68,352,77,154)(69,345,78,155)(70,346,79,156)(71,347,80,157)(72,348,73,158)(81,168,387,357)(82,161,388,358)(83,162,389,359)(84,163,390,360)(85,164,391,353)(86,165,392,354)(87,166,385,355)(88,167,386,356)(89,318,395,137)(90,319,396,138)(91,320,397,139)(92,313,398,140)(93,314,399,141)(94,315,400,142)(95,316,393,143)(96,317,394,144)(97,476,216,121)(98,477,209,122)(99,478,210,123)(100,479,211,124)(101,480,212,125)(102,473,213,126)(103,474,214,127)(104,475,215,128)(113,423,372,310)(114,424,373,311)(115,417,374,312)(116,418,375,305)(117,419,376,306)(118,420,369,307)(119,421,370,308)(120,422,371,309)(129,435,272,465)(130,436,265,466)(131,437,266,467)(132,438,267,468)(133,439,268,469)(134,440,269,470)(135,433,270,471)(136,434,271,472)(185,324,407,206)(186,325,408,207)(187,326,401,208)(188,327,402,201)(189,328,403,202)(190,321,404,203)(191,322,405,204)(192,323,406,205)(233,445,428,335)(234,446,429,336)(235,447,430,329)(236,448,431,330)(237,441,432,331)(238,442,425,332)(239,443,426,333)(240,444,427,334)(241,461,290,287)(242,462,291,288)(243,463,292,281)(244,464,293,282)(245,457,294,283)(246,458,295,284)(247,459,296,285)(248,460,289,286), (1,83,323,211,224)(2,212,84,217,324)(3,218,213,325,85)(4,326,219,86,214)(5,87,327,215,220)(6,216,88,221,328)(7,222,209,321,81)(8,322,223,82,210)(9,298,426,144,72)(10,137,299,65,427)(11,66,138,428,300)(12,429,67,301,139)(13,302,430,140,68)(14,141,303,69,431)(15,70,142,432,304)(16,425,71,297,143)(17,116,150,287,131)(18,288,117,132,151)(19,133,281,152,118)(20,145,134,119,282)(21,120,146,283,135)(22,284,113,136,147)(23,129,285,148,114)(24,149,130,115,286)(25,369,414,268,463)(26,269,370,464,415)(27,457,270,416,371)(28,409,458,372,271)(29,373,410,272,459)(30,265,374,460,411)(31,461,266,412,375)(32,413,462,376,267)(33,235,313,77,366)(34,78,236,367,314)(35,368,79,315,237)(36,316,361,238,80)(37,239,317,73,362)(38,74,240,363,318)(39,364,75,319,233)(40,320,365,234,76)(41,405,378,161,123)(42,162,406,124,379)(43,125,163,380,407)(44,381,126,408,164)(45,401,382,165,127)(46,166,402,128,383)(47,121,167,384,403)(48,377,122,404,168)(49,102,207,391,112)(50,392,103,105,208)(51,106,385,201,104)(52,202,107,97,386)(53,98,203,387,108)(54,388,99,109,204)(55,110,389,205,100)(56,206,111,101,390)(57,418,262,241,437)(58,242,419,438,263)(59,439,243,264,420)(60,257,440,421,244)(61,422,258,245,433)(62,246,423,434,259)(63,435,247,260,424)(64,261,436,417,248)(89,450,349,334,170)(90,335,451,171,350)(91,172,336,351,452)(92,352,173,453,329)(93,454,345,330,174)(94,331,455,175,346)(95,176,332,347,456)(96,348,169,449,333)(153,193,397,341,446)(154,342,194,447,398)(155,448,343,399,195)(156,400,441,196,344)(157,197,393,337,442)(158,338,198,443,394)(159,444,339,395,199)(160,396,445,200,340)(177,355,188,475,255)(178,476,356,256,189)(179,249,477,190,357)(180,191,250,358,478)(181,359,192,479,251)(182,480,360,252,185)(183,253,473,186,353)(184,187,254,354,474)(225,307,277,469,292)(226,470,308,293,278)(227,294,471,279,309)(228,280,295,310,472)(229,311,273,465,296)(230,466,312,289,274)(231,290,467,275,305)(232,276,291,306,468), (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)>;
G:=Group( (1,445,286,110,335,460)(2,446,287,111,336,461)(3,447,288,112,329,462)(4,448,281,105,330,463)(5,441,282,106,331,464)(6,442,283,107,332,457)(7,443,284,108,333,458)(8,444,285,109,334,459)(9,404,228,362,190,259)(10,405,229,363,191,260)(11,406,230,364,192,261)(12,407,231,365,185,262)(13,408,232,366,186,263)(14,401,225,367,187,264)(15,402,226,368,188,257)(16,403,227,361,189,258)(17,390,452,412,84,193)(18,391,453,413,85,194)(19,392,454,414,86,195)(20,385,455,415,87,196)(21,386,456,416,88,197)(22,387,449,409,81,198)(23,388,450,410,82,199)(24,389,451,411,83,200)(25,326,343,152,208,174)(26,327,344,145,201,175)(27,328,337,146,202,176)(28,321,338,147,203,169)(29,322,339,148,204,170)(30,323,340,149,205,171)(31,324,341,150,206,172)(32,325,342,151,207,173)(33,353,58,302,164,276)(34,354,59,303,165,277)(35,355,60,304,166,278)(36,356,61,297,167,279)(37,357,62,298,168,280)(38,358,63,299,161,273)(39,359,64,300,162,274)(40,360,57,301,163,275)(41,296,240,180,247,427)(42,289,233,181,248,428)(43,290,234,182,241,429)(44,291,235,183,242,430)(45,292,236,184,243,431)(46,293,237,177,244,432)(47,294,238,178,245,425)(48,295,239,179,246,426)(49,92,376,218,398,117)(50,93,369,219,399,118)(51,94,370,220,400,119)(52,95,371,221,393,120)(53,96,372,222,394,113)(54,89,373,223,395,114)(55,90,374,224,396,115)(56,91,375,217,397,116)(65,123,465,74,478,435)(66,124,466,75,479,436)(67,125,467,76,480,437)(68,126,468,77,473,438)(69,127,469,78,474,439)(70,128,470,79,475,440)(71,121,471,80,476,433)(72,122,472,73,477,434)(97,347,270,216,157,135)(98,348,271,209,158,136)(99,349,272,210,159,129)(100,350,265,211,160,130)(101,351,266,212,153,131)(102,352,267,213,154,132)(103,345,268,214,155,133)(104,346,269,215,156,134)(137,378,311,318,250,424)(138,379,312,319,251,417)(139,380,305,320,252,418)(140,381,306,313,253,419)(141,382,307,314,254,420)(142,383,308,315,255,421)(143,384,309,316,256,422)(144,377,310,317,249,423), (1,42,110,181)(2,43,111,182)(3,44,112,183)(4,45,105,184)(5,46,106,177)(6,47,107,178)(7,48,108,179)(8,41,109,180)(9,169,362,338)(10,170,363,339)(11,171,364,340)(12,172,365,341)(13,173,366,342)(14,174,367,343)(15,175,368,344)(16,176,361,337)(17,57,412,275)(18,58,413,276)(19,59,414,277)(20,60,415,278)(21,61,416,279)(22,62,409,280)(23,63,410,273)(24,64,411,274)(25,225,152,264)(26,226,145,257)(27,227,146,258)(28,228,147,259)(29,229,148,260)(30,230,149,261)(31,231,150,262)(32,232,151,263)(33,194,302,453)(34,195,303,454)(35,196,304,455)(36,197,297,456)(37,198,298,449)(38,199,299,450)(39,200,300,451)(40,193,301,452)(49,253,218,381)(50,254,219,382)(51,255,220,383)(52,256,221,384)(53,249,222,377)(54,250,223,378)(55,251,224,379)(56,252,217,380)(65,349,74,159)(66,350,75,160)(67,351,76,153)(68,352,77,154)(69,345,78,155)(70,346,79,156)(71,347,80,157)(72,348,73,158)(81,168,387,357)(82,161,388,358)(83,162,389,359)(84,163,390,360)(85,164,391,353)(86,165,392,354)(87,166,385,355)(88,167,386,356)(89,318,395,137)(90,319,396,138)(91,320,397,139)(92,313,398,140)(93,314,399,141)(94,315,400,142)(95,316,393,143)(96,317,394,144)(97,476,216,121)(98,477,209,122)(99,478,210,123)(100,479,211,124)(101,480,212,125)(102,473,213,126)(103,474,214,127)(104,475,215,128)(113,423,372,310)(114,424,373,311)(115,417,374,312)(116,418,375,305)(117,419,376,306)(118,420,369,307)(119,421,370,308)(120,422,371,309)(129,435,272,465)(130,436,265,466)(131,437,266,467)(132,438,267,468)(133,439,268,469)(134,440,269,470)(135,433,270,471)(136,434,271,472)(185,324,407,206)(186,325,408,207)(187,326,401,208)(188,327,402,201)(189,328,403,202)(190,321,404,203)(191,322,405,204)(192,323,406,205)(233,445,428,335)(234,446,429,336)(235,447,430,329)(236,448,431,330)(237,441,432,331)(238,442,425,332)(239,443,426,333)(240,444,427,334)(241,461,290,287)(242,462,291,288)(243,463,292,281)(244,464,293,282)(245,457,294,283)(246,458,295,284)(247,459,296,285)(248,460,289,286), (1,83,323,211,224)(2,212,84,217,324)(3,218,213,325,85)(4,326,219,86,214)(5,87,327,215,220)(6,216,88,221,328)(7,222,209,321,81)(8,322,223,82,210)(9,298,426,144,72)(10,137,299,65,427)(11,66,138,428,300)(12,429,67,301,139)(13,302,430,140,68)(14,141,303,69,431)(15,70,142,432,304)(16,425,71,297,143)(17,116,150,287,131)(18,288,117,132,151)(19,133,281,152,118)(20,145,134,119,282)(21,120,146,283,135)(22,284,113,136,147)(23,129,285,148,114)(24,149,130,115,286)(25,369,414,268,463)(26,269,370,464,415)(27,457,270,416,371)(28,409,458,372,271)(29,373,410,272,459)(30,265,374,460,411)(31,461,266,412,375)(32,413,462,376,267)(33,235,313,77,366)(34,78,236,367,314)(35,368,79,315,237)(36,316,361,238,80)(37,239,317,73,362)(38,74,240,363,318)(39,364,75,319,233)(40,320,365,234,76)(41,405,378,161,123)(42,162,406,124,379)(43,125,163,380,407)(44,381,126,408,164)(45,401,382,165,127)(46,166,402,128,383)(47,121,167,384,403)(48,377,122,404,168)(49,102,207,391,112)(50,392,103,105,208)(51,106,385,201,104)(52,202,107,97,386)(53,98,203,387,108)(54,388,99,109,204)(55,110,389,205,100)(56,206,111,101,390)(57,418,262,241,437)(58,242,419,438,263)(59,439,243,264,420)(60,257,440,421,244)(61,422,258,245,433)(62,246,423,434,259)(63,435,247,260,424)(64,261,436,417,248)(89,450,349,334,170)(90,335,451,171,350)(91,172,336,351,452)(92,352,173,453,329)(93,454,345,330,174)(94,331,455,175,346)(95,176,332,347,456)(96,348,169,449,333)(153,193,397,341,446)(154,342,194,447,398)(155,448,343,399,195)(156,400,441,196,344)(157,197,393,337,442)(158,338,198,443,394)(159,444,339,395,199)(160,396,445,200,340)(177,355,188,475,255)(178,476,356,256,189)(179,249,477,190,357)(180,191,250,358,478)(181,359,192,479,251)(182,480,360,252,185)(183,253,473,186,353)(184,187,254,354,474)(225,307,277,469,292)(226,470,308,293,278)(227,294,471,279,309)(228,280,295,310,472)(229,311,273,465,296)(230,466,312,289,274)(231,290,467,275,305)(232,276,291,306,468), (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) );
G=PermutationGroup([[(1,445,286,110,335,460),(2,446,287,111,336,461),(3,447,288,112,329,462),(4,448,281,105,330,463),(5,441,282,106,331,464),(6,442,283,107,332,457),(7,443,284,108,333,458),(8,444,285,109,334,459),(9,404,228,362,190,259),(10,405,229,363,191,260),(11,406,230,364,192,261),(12,407,231,365,185,262),(13,408,232,366,186,263),(14,401,225,367,187,264),(15,402,226,368,188,257),(16,403,227,361,189,258),(17,390,452,412,84,193),(18,391,453,413,85,194),(19,392,454,414,86,195),(20,385,455,415,87,196),(21,386,456,416,88,197),(22,387,449,409,81,198),(23,388,450,410,82,199),(24,389,451,411,83,200),(25,326,343,152,208,174),(26,327,344,145,201,175),(27,328,337,146,202,176),(28,321,338,147,203,169),(29,322,339,148,204,170),(30,323,340,149,205,171),(31,324,341,150,206,172),(32,325,342,151,207,173),(33,353,58,302,164,276),(34,354,59,303,165,277),(35,355,60,304,166,278),(36,356,61,297,167,279),(37,357,62,298,168,280),(38,358,63,299,161,273),(39,359,64,300,162,274),(40,360,57,301,163,275),(41,296,240,180,247,427),(42,289,233,181,248,428),(43,290,234,182,241,429),(44,291,235,183,242,430),(45,292,236,184,243,431),(46,293,237,177,244,432),(47,294,238,178,245,425),(48,295,239,179,246,426),(49,92,376,218,398,117),(50,93,369,219,399,118),(51,94,370,220,400,119),(52,95,371,221,393,120),(53,96,372,222,394,113),(54,89,373,223,395,114),(55,90,374,224,396,115),(56,91,375,217,397,116),(65,123,465,74,478,435),(66,124,466,75,479,436),(67,125,467,76,480,437),(68,126,468,77,473,438),(69,127,469,78,474,439),(70,128,470,79,475,440),(71,121,471,80,476,433),(72,122,472,73,477,434),(97,347,270,216,157,135),(98,348,271,209,158,136),(99,349,272,210,159,129),(100,350,265,211,160,130),(101,351,266,212,153,131),(102,352,267,213,154,132),(103,345,268,214,155,133),(104,346,269,215,156,134),(137,378,311,318,250,424),(138,379,312,319,251,417),(139,380,305,320,252,418),(140,381,306,313,253,419),(141,382,307,314,254,420),(142,383,308,315,255,421),(143,384,309,316,256,422),(144,377,310,317,249,423)], [(1,42,110,181),(2,43,111,182),(3,44,112,183),(4,45,105,184),(5,46,106,177),(6,47,107,178),(7,48,108,179),(8,41,109,180),(9,169,362,338),(10,170,363,339),(11,171,364,340),(12,172,365,341),(13,173,366,342),(14,174,367,343),(15,175,368,344),(16,176,361,337),(17,57,412,275),(18,58,413,276),(19,59,414,277),(20,60,415,278),(21,61,416,279),(22,62,409,280),(23,63,410,273),(24,64,411,274),(25,225,152,264),(26,226,145,257),(27,227,146,258),(28,228,147,259),(29,229,148,260),(30,230,149,261),(31,231,150,262),(32,232,151,263),(33,194,302,453),(34,195,303,454),(35,196,304,455),(36,197,297,456),(37,198,298,449),(38,199,299,450),(39,200,300,451),(40,193,301,452),(49,253,218,381),(50,254,219,382),(51,255,220,383),(52,256,221,384),(53,249,222,377),(54,250,223,378),(55,251,224,379),(56,252,217,380),(65,349,74,159),(66,350,75,160),(67,351,76,153),(68,352,77,154),(69,345,78,155),(70,346,79,156),(71,347,80,157),(72,348,73,158),(81,168,387,357),(82,161,388,358),(83,162,389,359),(84,163,390,360),(85,164,391,353),(86,165,392,354),(87,166,385,355),(88,167,386,356),(89,318,395,137),(90,319,396,138),(91,320,397,139),(92,313,398,140),(93,314,399,141),(94,315,400,142),(95,316,393,143),(96,317,394,144),(97,476,216,121),(98,477,209,122),(99,478,210,123),(100,479,211,124),(101,480,212,125),(102,473,213,126),(103,474,214,127),(104,475,215,128),(113,423,372,310),(114,424,373,311),(115,417,374,312),(116,418,375,305),(117,419,376,306),(118,420,369,307),(119,421,370,308),(120,422,371,309),(129,435,272,465),(130,436,265,466),(131,437,266,467),(132,438,267,468),(133,439,268,469),(134,440,269,470),(135,433,270,471),(136,434,271,472),(185,324,407,206),(186,325,408,207),(187,326,401,208),(188,327,402,201),(189,328,403,202),(190,321,404,203),(191,322,405,204),(192,323,406,205),(233,445,428,335),(234,446,429,336),(235,447,430,329),(236,448,431,330),(237,441,432,331),(238,442,425,332),(239,443,426,333),(240,444,427,334),(241,461,290,287),(242,462,291,288),(243,463,292,281),(244,464,293,282),(245,457,294,283),(246,458,295,284),(247,459,296,285),(248,460,289,286)], [(1,83,323,211,224),(2,212,84,217,324),(3,218,213,325,85),(4,326,219,86,214),(5,87,327,215,220),(6,216,88,221,328),(7,222,209,321,81),(8,322,223,82,210),(9,298,426,144,72),(10,137,299,65,427),(11,66,138,428,300),(12,429,67,301,139),(13,302,430,140,68),(14,141,303,69,431),(15,70,142,432,304),(16,425,71,297,143),(17,116,150,287,131),(18,288,117,132,151),(19,133,281,152,118),(20,145,134,119,282),(21,120,146,283,135),(22,284,113,136,147),(23,129,285,148,114),(24,149,130,115,286),(25,369,414,268,463),(26,269,370,464,415),(27,457,270,416,371),(28,409,458,372,271),(29,373,410,272,459),(30,265,374,460,411),(31,461,266,412,375),(32,413,462,376,267),(33,235,313,77,366),(34,78,236,367,314),(35,368,79,315,237),(36,316,361,238,80),(37,239,317,73,362),(38,74,240,363,318),(39,364,75,319,233),(40,320,365,234,76),(41,405,378,161,123),(42,162,406,124,379),(43,125,163,380,407),(44,381,126,408,164),(45,401,382,165,127),(46,166,402,128,383),(47,121,167,384,403),(48,377,122,404,168),(49,102,207,391,112),(50,392,103,105,208),(51,106,385,201,104),(52,202,107,97,386),(53,98,203,387,108),(54,388,99,109,204),(55,110,389,205,100),(56,206,111,101,390),(57,418,262,241,437),(58,242,419,438,263),(59,439,243,264,420),(60,257,440,421,244),(61,422,258,245,433),(62,246,423,434,259),(63,435,247,260,424),(64,261,436,417,248),(89,450,349,334,170),(90,335,451,171,350),(91,172,336,351,452),(92,352,173,453,329),(93,454,345,330,174),(94,331,455,175,346),(95,176,332,347,456),(96,348,169,449,333),(153,193,397,341,446),(154,342,194,447,398),(155,448,343,399,195),(156,400,441,196,344),(157,197,393,337,442),(158,338,198,443,394),(159,444,339,395,199),(160,396,445,200,340),(177,355,188,475,255),(178,476,356,256,189),(179,249,477,190,357),(180,191,250,358,478),(181,359,192,479,251),(182,480,360,252,185),(183,253,473,186,353),(184,187,254,354,474),(225,307,277,469,292),(226,470,308,293,278),(227,294,471,279,309),(228,280,295,310,472),(229,311,273,465,296),(230,466,312,289,274),(231,290,467,275,305),(232,276,291,306,468)], [(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)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5 | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15 | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | 30 | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 15 | 15 | 15 | 15 | 4 | 2 | 2 | 2 | 5 | ··· | 5 | 15 | ··· | 15 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 8 | 12 | 12 | 12 | 12 | 10 | ··· | 10 | 8 | 8 | 8 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | - | + | + | - | + | + | - | - | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8 | S3 | Dic3 | D6 | C4×S3 | C4×S3 | S3×C8 | F5 | C5⋊C8 | C2×F5 | D5⋊C8 | C4×F5 | S3×F5 | Dic3×F5 | S3×C5⋊C8 | D15⋊C8 |
| kernel | Dic3×C5⋊C8 | Dic3×Dic5 | C6×C5⋊C8 | C2×C15⋊C8 | C3×C5⋊C8 | C15⋊C8 | C10×Dic3 | C2×Dic15 | C5×Dic3 | Dic15 | C2×C5⋊C8 | C5⋊C8 | C2×Dic5 | Dic5 | C2×C10 | C10 | C2×Dic3 | Dic3 | C2×C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 |
Matrix representation of Dic3×C5⋊C8 ►in GL6(𝔽241)
| 1 | 240 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 177 | 64 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 192 | 101 | 198 | 152 |
| 0 | 0 | 149 | 12 | 37 | 103 |
| 0 | 0 | 229 | 204 | 138 | 60 |
| 0 | 0 | 89 | 161 | 49 | 140 |
G:=sub<GL(6,GF(241))| [1,1,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[177,0,0,0,0,0,64,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,192,149,229,89,0,0,101,12,204,161,0,0,198,37,138,49,0,0,152,103,60,140] >;
Dic3×C5⋊C8 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_5\rtimes C_8 % in TeX
G:=Group("Dic3xC5:C8"); // GroupNames label
G:=SmallGroup(480,244);
// by ID
G=gap.SmallGroup(480,244);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,100,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^5=d^8=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations