direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20⋊2Q8, C60⋊10Q8, C12⋊9Dic10, C12.42D20, C60.172D4, C20⋊2(C3×Q8), (C4×C20).2C6, (C4×C60).8C2, C15⋊11(C4⋊Q8), C4.4(C3×D20), C10.1(C6×D4), C2.4(C6×D20), C10.2(C6×Q8), C4⋊2(C3×Dic10), C6.72(C2×D20), (C4×C12).10D5, C20.27(C3×D4), C4⋊Dic5.4C6, C30.74(C2×Q8), C42.4(C3×D5), C30.274(C2×D4), C2.4(C6×Dic10), (C2×C12).426D10, (C2×Dic10).3C6, C6.42(C2×Dic10), (C2×C60).506C22, (C2×C30).327C23, (C6×Dic10).14C2, (C6×Dic5).150C22, C5⋊1(C3×C4⋊Q8), (C2×C4).74(C6×D5), C22.34(D5×C2×C6), (C2×C20).89(C2×C6), (C3×C4⋊Dic5).18C2, (C2×Dic5).1(C2×C6), (C2×C10).10(C22×C6), (C2×C6).323(C22×D5), SmallGroup(480,662)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20⋊2Q8
G = < a,b,c,d | a3=b20=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 384 in 136 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, C20, C2×C10, C2×C12, C2×C12, C2×C12, C3×Q8, C30, C30, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C4×C12, C3×C4⋊C4, C6×Q8, C3×Dic5, C60, C2×C30, C4⋊Dic5, C4×C20, C2×Dic10, C3×C4⋊Q8, C3×Dic10, C6×Dic5, C2×C60, C2×C60, C20⋊2Q8, C3×C4⋊Dic5, C4×C60, C6×Dic10, C3×C20⋊2Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, D5, C2×C6, C2×D4, C2×Q8, D10, C3×D4, C3×Q8, C22×C6, C3×D5, C4⋊Q8, Dic10, D20, C22×D5, C6×D4, C6×Q8, C6×D5, C2×Dic10, C2×D20, C3×C4⋊Q8, C3×Dic10, C3×D20, D5×C2×C6, C20⋊2Q8, C6×Dic10, C6×D20, C3×C20⋊2Q8
►Regular action on 480 points
Generators in S480
(1 41 318)(2 42 319)(3 43 320)(4 44 301)(5 45 302)(6 46 303)(7 47 304)(8 48 305)(9 49 306)(10 50 307)(11 51 308)(12 52 309)(13 53 310)(14 54 311)(15 55 312)(16 56 313)(17 57 314)(18 58 315)(19 59 316)(20 60 317)(21 171 192)(22 172 193)(23 173 194)(24 174 195)(25 175 196)(26 176 197)(27 177 198)(28 178 199)(29 179 200)(30 180 181)(31 161 182)(32 162 183)(33 163 184)(34 164 185)(35 165 186)(36 166 187)(37 167 188)(38 168 189)(39 169 190)(40 170 191)(61 370 110)(62 371 111)(63 372 112)(64 373 113)(65 374 114)(66 375 115)(67 376 116)(68 377 117)(69 378 118)(70 379 119)(71 380 120)(72 361 101)(73 362 102)(74 363 103)(75 364 104)(76 365 105)(77 366 106)(78 367 107)(79 368 108)(80 369 109)(81 339 263)(82 340 264)(83 321 265)(84 322 266)(85 323 267)(86 324 268)(87 325 269)(88 326 270)(89 327 271)(90 328 272)(91 329 273)(92 330 274)(93 331 275)(94 332 276)(95 333 277)(96 334 278)(97 335 279)(98 336 280)(99 337 261)(100 338 262)(121 409 150)(122 410 151)(123 411 152)(124 412 153)(125 413 154)(126 414 155)(127 415 156)(128 416 157)(129 417 158)(130 418 159)(131 419 160)(132 420 141)(133 401 142)(134 402 143)(135 403 144)(136 404 145)(137 405 146)(138 406 147)(139 407 148)(140 408 149)(201 244 393)(202 245 394)(203 246 395)(204 247 396)(205 248 397)(206 249 398)(207 250 399)(208 251 400)(209 252 381)(210 253 382)(211 254 383)(212 255 384)(213 256 385)(214 257 386)(215 258 387)(216 259 388)(217 260 389)(218 241 390)(219 242 391)(220 243 392)(221 467 290)(222 468 291)(223 469 292)(224 470 293)(225 471 294)(226 472 295)(227 473 296)(228 474 297)(229 475 298)(230 476 299)(231 477 300)(232 478 281)(233 479 282)(234 480 283)(235 461 284)(236 462 285)(237 463 286)(238 464 287)(239 465 288)(240 466 289)(341 451 426)(342 452 427)(343 453 428)(344 454 429)(345 455 430)(346 456 431)(347 457 432)(348 458 433)(349 459 434)(350 460 435)(351 441 436)(352 442 437)(353 443 438)(354 444 439)(355 445 440)(356 446 421)(357 447 422)(358 448 423)(359 449 424)(360 450 425)
(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)
(1 107 168 299)(2 108 169 300)(3 109 170 281)(4 110 171 282)(5 111 172 283)(6 112 173 284)(7 113 174 285)(8 114 175 286)(9 115 176 287)(10 116 177 288)(11 117 178 289)(12 118 179 290)(13 119 180 291)(14 120 161 292)(15 101 162 293)(16 102 163 294)(17 103 164 295)(18 104 165 296)(19 105 166 297)(20 106 167 298)(21 479 301 370)(22 480 302 371)(23 461 303 372)(24 462 304 373)(25 463 305 374)(26 464 306 375)(27 465 307 376)(28 466 308 377)(29 467 309 378)(30 468 310 379)(31 469 311 380)(32 470 312 361)(33 471 313 362)(34 472 314 363)(35 473 315 364)(36 474 316 365)(37 475 317 366)(38 476 318 367)(39 477 319 368)(40 478 320 369)(41 78 189 230)(42 79 190 231)(43 80 191 232)(44 61 192 233)(45 62 193 234)(46 63 194 235)(47 64 195 236)(48 65 196 237)(49 66 197 238)(50 67 198 239)(51 68 199 240)(52 69 200 221)(53 70 181 222)(54 71 182 223)(55 72 183 224)(56 73 184 225)(57 74 185 226)(58 75 186 227)(59 76 187 228)(60 77 188 229)(81 391 348 411)(82 392 349 412)(83 393 350 413)(84 394 351 414)(85 395 352 415)(86 396 353 416)(87 397 354 417)(88 398 355 418)(89 399 356 419)(90 400 357 420)(91 381 358 401)(92 382 359 402)(93 383 360 403)(94 384 341 404)(95 385 342 405)(96 386 343 406)(97 387 344 407)(98 388 345 408)(99 389 346 409)(100 390 347 410)(121 261 260 431)(122 262 241 432)(123 263 242 433)(124 264 243 434)(125 265 244 435)(126 266 245 436)(127 267 246 437)(128 268 247 438)(129 269 248 439)(130 270 249 440)(131 271 250 421)(132 272 251 422)(133 273 252 423)(134 274 253 424)(135 275 254 425)(136 276 255 426)(137 277 256 427)(138 278 257 428)(139 279 258 429)(140 280 259 430)(141 328 208 447)(142 329 209 448)(143 330 210 449)(144 331 211 450)(145 332 212 451)(146 333 213 452)(147 334 214 453)(148 335 215 454)(149 336 216 455)(150 337 217 456)(151 338 218 457)(152 339 219 458)(153 340 220 459)(154 321 201 460)(155 322 202 441)(156 323 203 442)(157 324 204 443)(158 325 205 444)(159 326 206 445)(160 327 207 446)
(1 411 168 391)(2 410 169 390)(3 409 170 389)(4 408 171 388)(5 407 172 387)(6 406 173 386)(7 405 174 385)(8 404 175 384)(9 403 176 383)(10 402 177 382)(11 401 178 381)(12 420 179 400)(13 419 180 399)(14 418 161 398)(15 417 162 397)(16 416 163 396)(17 415 164 395)(18 414 165 394)(19 413 166 393)(20 412 167 392)(21 259 301 140)(22 258 302 139)(23 257 303 138)(24 256 304 137)(25 255 305 136)(26 254 306 135)(27 253 307 134)(28 252 308 133)(29 251 309 132)(30 250 310 131)(31 249 311 130)(32 248 312 129)(33 247 313 128)(34 246 314 127)(35 245 315 126)(36 244 316 125)(37 243 317 124)(38 242 318 123)(39 241 319 122)(40 260 320 121)(41 152 189 219)(42 151 190 218)(43 150 191 217)(44 149 192 216)(45 148 193 215)(46 147 194 214)(47 146 195 213)(48 145 196 212)(49 144 197 211)(50 143 198 210)(51 142 199 209)(52 141 200 208)(53 160 181 207)(54 159 182 206)(55 158 183 205)(56 157 184 204)(57 156 185 203)(58 155 186 202)(59 154 187 201)(60 153 188 220)(61 455 233 336)(62 454 234 335)(63 453 235 334)(64 452 236 333)(65 451 237 332)(66 450 238 331)(67 449 239 330)(68 448 240 329)(69 447 221 328)(70 446 222 327)(71 445 223 326)(72 444 224 325)(73 443 225 324)(74 442 226 323)(75 441 227 322)(76 460 228 321)(77 459 229 340)(78 458 230 339)(79 457 231 338)(80 456 232 337)(81 107 348 299)(82 106 349 298)(83 105 350 297)(84 104 351 296)(85 103 352 295)(86 102 353 294)(87 101 354 293)(88 120 355 292)(89 119 356 291)(90 118 357 290)(91 117 358 289)(92 116 359 288)(93 115 360 287)(94 114 341 286)(95 113 342 285)(96 112 343 284)(97 111 344 283)(98 110 345 282)(99 109 346 281)(100 108 347 300)(261 369 431 478)(262 368 432 477)(263 367 433 476)(264 366 434 475)(265 365 435 474)(266 364 436 473)(267 363 437 472)(268 362 438 471)(269 361 439 470)(270 380 440 469)(271 379 421 468)(272 378 422 467)(273 377 423 466)(274 376 424 465)(275 375 425 464)(276 374 426 463)(277 373 427 462)(278 372 428 461)(279 371 429 480)(280 370 430 479)
G:=sub<Sym(480)| (1,41,318)(2,42,319)(3,43,320)(4,44,301)(5,45,302)(6,46,303)(7,47,304)(8,48,305)(9,49,306)(10,50,307)(11,51,308)(12,52,309)(13,53,310)(14,54,311)(15,55,312)(16,56,313)(17,57,314)(18,58,315)(19,59,316)(20,60,317)(21,171,192)(22,172,193)(23,173,194)(24,174,195)(25,175,196)(26,176,197)(27,177,198)(28,178,199)(29,179,200)(30,180,181)(31,161,182)(32,162,183)(33,163,184)(34,164,185)(35,165,186)(36,166,187)(37,167,188)(38,168,189)(39,169,190)(40,170,191)(61,370,110)(62,371,111)(63,372,112)(64,373,113)(65,374,114)(66,375,115)(67,376,116)(68,377,117)(69,378,118)(70,379,119)(71,380,120)(72,361,101)(73,362,102)(74,363,103)(75,364,104)(76,365,105)(77,366,106)(78,367,107)(79,368,108)(80,369,109)(81,339,263)(82,340,264)(83,321,265)(84,322,266)(85,323,267)(86,324,268)(87,325,269)(88,326,270)(89,327,271)(90,328,272)(91,329,273)(92,330,274)(93,331,275)(94,332,276)(95,333,277)(96,334,278)(97,335,279)(98,336,280)(99,337,261)(100,338,262)(121,409,150)(122,410,151)(123,411,152)(124,412,153)(125,413,154)(126,414,155)(127,415,156)(128,416,157)(129,417,158)(130,418,159)(131,419,160)(132,420,141)(133,401,142)(134,402,143)(135,403,144)(136,404,145)(137,405,146)(138,406,147)(139,407,148)(140,408,149)(201,244,393)(202,245,394)(203,246,395)(204,247,396)(205,248,397)(206,249,398)(207,250,399)(208,251,400)(209,252,381)(210,253,382)(211,254,383)(212,255,384)(213,256,385)(214,257,386)(215,258,387)(216,259,388)(217,260,389)(218,241,390)(219,242,391)(220,243,392)(221,467,290)(222,468,291)(223,469,292)(224,470,293)(225,471,294)(226,472,295)(227,473,296)(228,474,297)(229,475,298)(230,476,299)(231,477,300)(232,478,281)(233,479,282)(234,480,283)(235,461,284)(236,462,285)(237,463,286)(238,464,287)(239,465,288)(240,466,289)(341,451,426)(342,452,427)(343,453,428)(344,454,429)(345,455,430)(346,456,431)(347,457,432)(348,458,433)(349,459,434)(350,460,435)(351,441,436)(352,442,437)(353,443,438)(354,444,439)(355,445,440)(356,446,421)(357,447,422)(358,448,423)(359,449,424)(360,450,425), (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), (1,107,168,299)(2,108,169,300)(3,109,170,281)(4,110,171,282)(5,111,172,283)(6,112,173,284)(7,113,174,285)(8,114,175,286)(9,115,176,287)(10,116,177,288)(11,117,178,289)(12,118,179,290)(13,119,180,291)(14,120,161,292)(15,101,162,293)(16,102,163,294)(17,103,164,295)(18,104,165,296)(19,105,166,297)(20,106,167,298)(21,479,301,370)(22,480,302,371)(23,461,303,372)(24,462,304,373)(25,463,305,374)(26,464,306,375)(27,465,307,376)(28,466,308,377)(29,467,309,378)(30,468,310,379)(31,469,311,380)(32,470,312,361)(33,471,313,362)(34,472,314,363)(35,473,315,364)(36,474,316,365)(37,475,317,366)(38,476,318,367)(39,477,319,368)(40,478,320,369)(41,78,189,230)(42,79,190,231)(43,80,191,232)(44,61,192,233)(45,62,193,234)(46,63,194,235)(47,64,195,236)(48,65,196,237)(49,66,197,238)(50,67,198,239)(51,68,199,240)(52,69,200,221)(53,70,181,222)(54,71,182,223)(55,72,183,224)(56,73,184,225)(57,74,185,226)(58,75,186,227)(59,76,187,228)(60,77,188,229)(81,391,348,411)(82,392,349,412)(83,393,350,413)(84,394,351,414)(85,395,352,415)(86,396,353,416)(87,397,354,417)(88,398,355,418)(89,399,356,419)(90,400,357,420)(91,381,358,401)(92,382,359,402)(93,383,360,403)(94,384,341,404)(95,385,342,405)(96,386,343,406)(97,387,344,407)(98,388,345,408)(99,389,346,409)(100,390,347,410)(121,261,260,431)(122,262,241,432)(123,263,242,433)(124,264,243,434)(125,265,244,435)(126,266,245,436)(127,267,246,437)(128,268,247,438)(129,269,248,439)(130,270,249,440)(131,271,250,421)(132,272,251,422)(133,273,252,423)(134,274,253,424)(135,275,254,425)(136,276,255,426)(137,277,256,427)(138,278,257,428)(139,279,258,429)(140,280,259,430)(141,328,208,447)(142,329,209,448)(143,330,210,449)(144,331,211,450)(145,332,212,451)(146,333,213,452)(147,334,214,453)(148,335,215,454)(149,336,216,455)(150,337,217,456)(151,338,218,457)(152,339,219,458)(153,340,220,459)(154,321,201,460)(155,322,202,441)(156,323,203,442)(157,324,204,443)(158,325,205,444)(159,326,206,445)(160,327,207,446), (1,411,168,391)(2,410,169,390)(3,409,170,389)(4,408,171,388)(5,407,172,387)(6,406,173,386)(7,405,174,385)(8,404,175,384)(9,403,176,383)(10,402,177,382)(11,401,178,381)(12,420,179,400)(13,419,180,399)(14,418,161,398)(15,417,162,397)(16,416,163,396)(17,415,164,395)(18,414,165,394)(19,413,166,393)(20,412,167,392)(21,259,301,140)(22,258,302,139)(23,257,303,138)(24,256,304,137)(25,255,305,136)(26,254,306,135)(27,253,307,134)(28,252,308,133)(29,251,309,132)(30,250,310,131)(31,249,311,130)(32,248,312,129)(33,247,313,128)(34,246,314,127)(35,245,315,126)(36,244,316,125)(37,243,317,124)(38,242,318,123)(39,241,319,122)(40,260,320,121)(41,152,189,219)(42,151,190,218)(43,150,191,217)(44,149,192,216)(45,148,193,215)(46,147,194,214)(47,146,195,213)(48,145,196,212)(49,144,197,211)(50,143,198,210)(51,142,199,209)(52,141,200,208)(53,160,181,207)(54,159,182,206)(55,158,183,205)(56,157,184,204)(57,156,185,203)(58,155,186,202)(59,154,187,201)(60,153,188,220)(61,455,233,336)(62,454,234,335)(63,453,235,334)(64,452,236,333)(65,451,237,332)(66,450,238,331)(67,449,239,330)(68,448,240,329)(69,447,221,328)(70,446,222,327)(71,445,223,326)(72,444,224,325)(73,443,225,324)(74,442,226,323)(75,441,227,322)(76,460,228,321)(77,459,229,340)(78,458,230,339)(79,457,231,338)(80,456,232,337)(81,107,348,299)(82,106,349,298)(83,105,350,297)(84,104,351,296)(85,103,352,295)(86,102,353,294)(87,101,354,293)(88,120,355,292)(89,119,356,291)(90,118,357,290)(91,117,358,289)(92,116,359,288)(93,115,360,287)(94,114,341,286)(95,113,342,285)(96,112,343,284)(97,111,344,283)(98,110,345,282)(99,109,346,281)(100,108,347,300)(261,369,431,478)(262,368,432,477)(263,367,433,476)(264,366,434,475)(265,365,435,474)(266,364,436,473)(267,363,437,472)(268,362,438,471)(269,361,439,470)(270,380,440,469)(271,379,421,468)(272,378,422,467)(273,377,423,466)(274,376,424,465)(275,375,425,464)(276,374,426,463)(277,373,427,462)(278,372,428,461)(279,371,429,480)(280,370,430,479)>;
G:=Group( (1,41,318)(2,42,319)(3,43,320)(4,44,301)(5,45,302)(6,46,303)(7,47,304)(8,48,305)(9,49,306)(10,50,307)(11,51,308)(12,52,309)(13,53,310)(14,54,311)(15,55,312)(16,56,313)(17,57,314)(18,58,315)(19,59,316)(20,60,317)(21,171,192)(22,172,193)(23,173,194)(24,174,195)(25,175,196)(26,176,197)(27,177,198)(28,178,199)(29,179,200)(30,180,181)(31,161,182)(32,162,183)(33,163,184)(34,164,185)(35,165,186)(36,166,187)(37,167,188)(38,168,189)(39,169,190)(40,170,191)(61,370,110)(62,371,111)(63,372,112)(64,373,113)(65,374,114)(66,375,115)(67,376,116)(68,377,117)(69,378,118)(70,379,119)(71,380,120)(72,361,101)(73,362,102)(74,363,103)(75,364,104)(76,365,105)(77,366,106)(78,367,107)(79,368,108)(80,369,109)(81,339,263)(82,340,264)(83,321,265)(84,322,266)(85,323,267)(86,324,268)(87,325,269)(88,326,270)(89,327,271)(90,328,272)(91,329,273)(92,330,274)(93,331,275)(94,332,276)(95,333,277)(96,334,278)(97,335,279)(98,336,280)(99,337,261)(100,338,262)(121,409,150)(122,410,151)(123,411,152)(124,412,153)(125,413,154)(126,414,155)(127,415,156)(128,416,157)(129,417,158)(130,418,159)(131,419,160)(132,420,141)(133,401,142)(134,402,143)(135,403,144)(136,404,145)(137,405,146)(138,406,147)(139,407,148)(140,408,149)(201,244,393)(202,245,394)(203,246,395)(204,247,396)(205,248,397)(206,249,398)(207,250,399)(208,251,400)(209,252,381)(210,253,382)(211,254,383)(212,255,384)(213,256,385)(214,257,386)(215,258,387)(216,259,388)(217,260,389)(218,241,390)(219,242,391)(220,243,392)(221,467,290)(222,468,291)(223,469,292)(224,470,293)(225,471,294)(226,472,295)(227,473,296)(228,474,297)(229,475,298)(230,476,299)(231,477,300)(232,478,281)(233,479,282)(234,480,283)(235,461,284)(236,462,285)(237,463,286)(238,464,287)(239,465,288)(240,466,289)(341,451,426)(342,452,427)(343,453,428)(344,454,429)(345,455,430)(346,456,431)(347,457,432)(348,458,433)(349,459,434)(350,460,435)(351,441,436)(352,442,437)(353,443,438)(354,444,439)(355,445,440)(356,446,421)(357,447,422)(358,448,423)(359,449,424)(360,450,425), (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), (1,107,168,299)(2,108,169,300)(3,109,170,281)(4,110,171,282)(5,111,172,283)(6,112,173,284)(7,113,174,285)(8,114,175,286)(9,115,176,287)(10,116,177,288)(11,117,178,289)(12,118,179,290)(13,119,180,291)(14,120,161,292)(15,101,162,293)(16,102,163,294)(17,103,164,295)(18,104,165,296)(19,105,166,297)(20,106,167,298)(21,479,301,370)(22,480,302,371)(23,461,303,372)(24,462,304,373)(25,463,305,374)(26,464,306,375)(27,465,307,376)(28,466,308,377)(29,467,309,378)(30,468,310,379)(31,469,311,380)(32,470,312,361)(33,471,313,362)(34,472,314,363)(35,473,315,364)(36,474,316,365)(37,475,317,366)(38,476,318,367)(39,477,319,368)(40,478,320,369)(41,78,189,230)(42,79,190,231)(43,80,191,232)(44,61,192,233)(45,62,193,234)(46,63,194,235)(47,64,195,236)(48,65,196,237)(49,66,197,238)(50,67,198,239)(51,68,199,240)(52,69,200,221)(53,70,181,222)(54,71,182,223)(55,72,183,224)(56,73,184,225)(57,74,185,226)(58,75,186,227)(59,76,187,228)(60,77,188,229)(81,391,348,411)(82,392,349,412)(83,393,350,413)(84,394,351,414)(85,395,352,415)(86,396,353,416)(87,397,354,417)(88,398,355,418)(89,399,356,419)(90,400,357,420)(91,381,358,401)(92,382,359,402)(93,383,360,403)(94,384,341,404)(95,385,342,405)(96,386,343,406)(97,387,344,407)(98,388,345,408)(99,389,346,409)(100,390,347,410)(121,261,260,431)(122,262,241,432)(123,263,242,433)(124,264,243,434)(125,265,244,435)(126,266,245,436)(127,267,246,437)(128,268,247,438)(129,269,248,439)(130,270,249,440)(131,271,250,421)(132,272,251,422)(133,273,252,423)(134,274,253,424)(135,275,254,425)(136,276,255,426)(137,277,256,427)(138,278,257,428)(139,279,258,429)(140,280,259,430)(141,328,208,447)(142,329,209,448)(143,330,210,449)(144,331,211,450)(145,332,212,451)(146,333,213,452)(147,334,214,453)(148,335,215,454)(149,336,216,455)(150,337,217,456)(151,338,218,457)(152,339,219,458)(153,340,220,459)(154,321,201,460)(155,322,202,441)(156,323,203,442)(157,324,204,443)(158,325,205,444)(159,326,206,445)(160,327,207,446), (1,411,168,391)(2,410,169,390)(3,409,170,389)(4,408,171,388)(5,407,172,387)(6,406,173,386)(7,405,174,385)(8,404,175,384)(9,403,176,383)(10,402,177,382)(11,401,178,381)(12,420,179,400)(13,419,180,399)(14,418,161,398)(15,417,162,397)(16,416,163,396)(17,415,164,395)(18,414,165,394)(19,413,166,393)(20,412,167,392)(21,259,301,140)(22,258,302,139)(23,257,303,138)(24,256,304,137)(25,255,305,136)(26,254,306,135)(27,253,307,134)(28,252,308,133)(29,251,309,132)(30,250,310,131)(31,249,311,130)(32,248,312,129)(33,247,313,128)(34,246,314,127)(35,245,315,126)(36,244,316,125)(37,243,317,124)(38,242,318,123)(39,241,319,122)(40,260,320,121)(41,152,189,219)(42,151,190,218)(43,150,191,217)(44,149,192,216)(45,148,193,215)(46,147,194,214)(47,146,195,213)(48,145,196,212)(49,144,197,211)(50,143,198,210)(51,142,199,209)(52,141,200,208)(53,160,181,207)(54,159,182,206)(55,158,183,205)(56,157,184,204)(57,156,185,203)(58,155,186,202)(59,154,187,201)(60,153,188,220)(61,455,233,336)(62,454,234,335)(63,453,235,334)(64,452,236,333)(65,451,237,332)(66,450,238,331)(67,449,239,330)(68,448,240,329)(69,447,221,328)(70,446,222,327)(71,445,223,326)(72,444,224,325)(73,443,225,324)(74,442,226,323)(75,441,227,322)(76,460,228,321)(77,459,229,340)(78,458,230,339)(79,457,231,338)(80,456,232,337)(81,107,348,299)(82,106,349,298)(83,105,350,297)(84,104,351,296)(85,103,352,295)(86,102,353,294)(87,101,354,293)(88,120,355,292)(89,119,356,291)(90,118,357,290)(91,117,358,289)(92,116,359,288)(93,115,360,287)(94,114,341,286)(95,113,342,285)(96,112,343,284)(97,111,344,283)(98,110,345,282)(99,109,346,281)(100,108,347,300)(261,369,431,478)(262,368,432,477)(263,367,433,476)(264,366,434,475)(265,365,435,474)(266,364,436,473)(267,363,437,472)(268,362,438,471)(269,361,439,470)(270,380,440,469)(271,379,421,468)(272,378,422,467)(273,377,423,466)(274,376,424,465)(275,375,425,464)(276,374,426,463)(277,373,427,462)(278,372,428,461)(279,371,429,480)(280,370,430,479) );
G=PermutationGroup([[(1,41,318),(2,42,319),(3,43,320),(4,44,301),(5,45,302),(6,46,303),(7,47,304),(8,48,305),(9,49,306),(10,50,307),(11,51,308),(12,52,309),(13,53,310),(14,54,311),(15,55,312),(16,56,313),(17,57,314),(18,58,315),(19,59,316),(20,60,317),(21,171,192),(22,172,193),(23,173,194),(24,174,195),(25,175,196),(26,176,197),(27,177,198),(28,178,199),(29,179,200),(30,180,181),(31,161,182),(32,162,183),(33,163,184),(34,164,185),(35,165,186),(36,166,187),(37,167,188),(38,168,189),(39,169,190),(40,170,191),(61,370,110),(62,371,111),(63,372,112),(64,373,113),(65,374,114),(66,375,115),(67,376,116),(68,377,117),(69,378,118),(70,379,119),(71,380,120),(72,361,101),(73,362,102),(74,363,103),(75,364,104),(76,365,105),(77,366,106),(78,367,107),(79,368,108),(80,369,109),(81,339,263),(82,340,264),(83,321,265),(84,322,266),(85,323,267),(86,324,268),(87,325,269),(88,326,270),(89,327,271),(90,328,272),(91,329,273),(92,330,274),(93,331,275),(94,332,276),(95,333,277),(96,334,278),(97,335,279),(98,336,280),(99,337,261),(100,338,262),(121,409,150),(122,410,151),(123,411,152),(124,412,153),(125,413,154),(126,414,155),(127,415,156),(128,416,157),(129,417,158),(130,418,159),(131,419,160),(132,420,141),(133,401,142),(134,402,143),(135,403,144),(136,404,145),(137,405,146),(138,406,147),(139,407,148),(140,408,149),(201,244,393),(202,245,394),(203,246,395),(204,247,396),(205,248,397),(206,249,398),(207,250,399),(208,251,400),(209,252,381),(210,253,382),(211,254,383),(212,255,384),(213,256,385),(214,257,386),(215,258,387),(216,259,388),(217,260,389),(218,241,390),(219,242,391),(220,243,392),(221,467,290),(222,468,291),(223,469,292),(224,470,293),(225,471,294),(226,472,295),(227,473,296),(228,474,297),(229,475,298),(230,476,299),(231,477,300),(232,478,281),(233,479,282),(234,480,283),(235,461,284),(236,462,285),(237,463,286),(238,464,287),(239,465,288),(240,466,289),(341,451,426),(342,452,427),(343,453,428),(344,454,429),(345,455,430),(346,456,431),(347,457,432),(348,458,433),(349,459,434),(350,460,435),(351,441,436),(352,442,437),(353,443,438),(354,444,439),(355,445,440),(356,446,421),(357,447,422),(358,448,423),(359,449,424),(360,450,425)], [(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)], [(1,107,168,299),(2,108,169,300),(3,109,170,281),(4,110,171,282),(5,111,172,283),(6,112,173,284),(7,113,174,285),(8,114,175,286),(9,115,176,287),(10,116,177,288),(11,117,178,289),(12,118,179,290),(13,119,180,291),(14,120,161,292),(15,101,162,293),(16,102,163,294),(17,103,164,295),(18,104,165,296),(19,105,166,297),(20,106,167,298),(21,479,301,370),(22,480,302,371),(23,461,303,372),(24,462,304,373),(25,463,305,374),(26,464,306,375),(27,465,307,376),(28,466,308,377),(29,467,309,378),(30,468,310,379),(31,469,311,380),(32,470,312,361),(33,471,313,362),(34,472,314,363),(35,473,315,364),(36,474,316,365),(37,475,317,366),(38,476,318,367),(39,477,319,368),(40,478,320,369),(41,78,189,230),(42,79,190,231),(43,80,191,232),(44,61,192,233),(45,62,193,234),(46,63,194,235),(47,64,195,236),(48,65,196,237),(49,66,197,238),(50,67,198,239),(51,68,199,240),(52,69,200,221),(53,70,181,222),(54,71,182,223),(55,72,183,224),(56,73,184,225),(57,74,185,226),(58,75,186,227),(59,76,187,228),(60,77,188,229),(81,391,348,411),(82,392,349,412),(83,393,350,413),(84,394,351,414),(85,395,352,415),(86,396,353,416),(87,397,354,417),(88,398,355,418),(89,399,356,419),(90,400,357,420),(91,381,358,401),(92,382,359,402),(93,383,360,403),(94,384,341,404),(95,385,342,405),(96,386,343,406),(97,387,344,407),(98,388,345,408),(99,389,346,409),(100,390,347,410),(121,261,260,431),(122,262,241,432),(123,263,242,433),(124,264,243,434),(125,265,244,435),(126,266,245,436),(127,267,246,437),(128,268,247,438),(129,269,248,439),(130,270,249,440),(131,271,250,421),(132,272,251,422),(133,273,252,423),(134,274,253,424),(135,275,254,425),(136,276,255,426),(137,277,256,427),(138,278,257,428),(139,279,258,429),(140,280,259,430),(141,328,208,447),(142,329,209,448),(143,330,210,449),(144,331,211,450),(145,332,212,451),(146,333,213,452),(147,334,214,453),(148,335,215,454),(149,336,216,455),(150,337,217,456),(151,338,218,457),(152,339,219,458),(153,340,220,459),(154,321,201,460),(155,322,202,441),(156,323,203,442),(157,324,204,443),(158,325,205,444),(159,326,206,445),(160,327,207,446)], [(1,411,168,391),(2,410,169,390),(3,409,170,389),(4,408,171,388),(5,407,172,387),(6,406,173,386),(7,405,174,385),(8,404,175,384),(9,403,176,383),(10,402,177,382),(11,401,178,381),(12,420,179,400),(13,419,180,399),(14,418,161,398),(15,417,162,397),(16,416,163,396),(17,415,164,395),(18,414,165,394),(19,413,166,393),(20,412,167,392),(21,259,301,140),(22,258,302,139),(23,257,303,138),(24,256,304,137),(25,255,305,136),(26,254,306,135),(27,253,307,134),(28,252,308,133),(29,251,309,132),(30,250,310,131),(31,249,311,130),(32,248,312,129),(33,247,313,128),(34,246,314,127),(35,245,315,126),(36,244,316,125),(37,243,317,124),(38,242,318,123),(39,241,319,122),(40,260,320,121),(41,152,189,219),(42,151,190,218),(43,150,191,217),(44,149,192,216),(45,148,193,215),(46,147,194,214),(47,146,195,213),(48,145,196,212),(49,144,197,211),(50,143,198,210),(51,142,199,209),(52,141,200,208),(53,160,181,207),(54,159,182,206),(55,158,183,205),(56,157,184,204),(57,156,185,203),(58,155,186,202),(59,154,187,201),(60,153,188,220),(61,455,233,336),(62,454,234,335),(63,453,235,334),(64,452,236,333),(65,451,237,332),(66,450,238,331),(67,449,239,330),(68,448,240,329),(69,447,221,328),(70,446,222,327),(71,445,223,326),(72,444,224,325),(73,443,225,324),(74,442,226,323),(75,441,227,322),(76,460,228,321),(77,459,229,340),(78,458,230,339),(79,457,231,338),(80,456,232,337),(81,107,348,299),(82,106,349,298),(83,105,350,297),(84,104,351,296),(85,103,352,295),(86,102,353,294),(87,101,354,293),(88,120,355,292),(89,119,356,291),(90,118,357,290),(91,117,358,289),(92,116,359,288),(93,115,360,287),(94,114,341,286),(95,113,342,285),(96,112,343,284),(97,111,344,283),(98,110,345,282),(99,109,346,281),(100,108,347,300),(261,369,431,478),(262,368,432,477),(263,367,433,476),(264,366,434,475),(265,365,435,474),(266,364,436,473),(267,363,437,472),(268,362,438,471),(269,361,439,470),(270,380,440,469),(271,379,421,468),(272,378,422,467),(273,377,423,466),(274,376,424,465),(275,375,425,464),(276,374,426,463),(277,373,427,462),(278,372,428,461),(279,371,429,480),(280,370,430,479)]])
138 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | ··· | 6F | 10A | ··· | 10F | 12A | ··· | 12L | 12M | ··· | 12T | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 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 | 1 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
138 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | Q8 | D5 | D10 | C3×D4 | C3×Q8 | C3×D5 | Dic10 | D20 | C6×D5 | C3×Dic10 | C3×D20 |
| kernel | C3×C20⋊2Q8 | C3×C4⋊Dic5 | C4×C60 | C6×Dic10 | C20⋊2Q8 | C4⋊Dic5 | C4×C20 | C2×Dic10 | C60 | C60 | C4×C12 | C2×C12 | C20 | C20 | C42 | C12 | C12 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 6 | 4 | 8 | 4 | 16 | 8 | 12 | 32 | 16 |
Matrix representation of C3×C20⋊2Q8 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 | 0 |
| 0 | 0 | 0 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 16 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 60 | 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 | 27 | 2 |
| 0 | 0 | 0 | 0 | 1 | 34 |
| 16 | 32 | 0 | 0 | 0 | 0 |
| 32 | 45 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 28 | 0 | 0 |
| 0 | 0 | 35 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 57 |
| 0 | 0 | 0 | 0 | 32 | 35 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[0,1,0,0,0,0,60,0,0,0,0,0,0,0,60,16,0,0,0,0,1,44,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,1,0,0,0,0,2,34],[16,32,0,0,0,0,32,45,0,0,0,0,0,0,27,35,0,0,0,0,28,34,0,0,0,0,0,0,26,32,0,0,0,0,57,35] >;
C3×C20⋊2Q8 in GAP, Magma, Sage, TeX
C_3\times C_{20}\rtimes_2Q_8 % in TeX
G:=Group("C3xC20:2Q8"); // GroupNames label
G:=SmallGroup(480,662);
// by ID
G=gap.SmallGroup(480,662);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,701,344,590,142,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations