direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C61, C4.C122, C244.3C2, C122.7C22, C2.2(C2×C122), SmallGroup(488,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C61
G = < a,b,c | a61=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C61
►Regular action on 488 points
Generators in S488
(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)(428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488)
(1 171 470 421)(2 172 471 422)(3 173 472 423)(4 174 473 424)(5 175 474 425)(6 176 475 426)(7 177 476 427)(8 178 477 367)(9 179 478 368)(10 180 479 369)(11 181 480 370)(12 182 481 371)(13 183 482 372)(14 123 483 373)(15 124 484 374)(16 125 485 375)(17 126 486 376)(18 127 487 377)(19 128 488 378)(20 129 428 379)(21 130 429 380)(22 131 430 381)(23 132 431 382)(24 133 432 383)(25 134 433 384)(26 135 434 385)(27 136 435 386)(28 137 436 387)(29 138 437 388)(30 139 438 389)(31 140 439 390)(32 141 440 391)(33 142 441 392)(34 143 442 393)(35 144 443 394)(36 145 444 395)(37 146 445 396)(38 147 446 397)(39 148 447 398)(40 149 448 399)(41 150 449 400)(42 151 450 401)(43 152 451 402)(44 153 452 403)(45 154 453 404)(46 155 454 405)(47 156 455 406)(48 157 456 407)(49 158 457 408)(50 159 458 409)(51 160 459 410)(52 161 460 411)(53 162 461 412)(54 163 462 413)(55 164 463 414)(56 165 464 415)(57 166 465 416)(58 167 466 417)(59 168 467 418)(60 169 468 419)(61 170 469 420)(62 238 333 247)(63 239 334 248)(64 240 335 249)(65 241 336 250)(66 242 337 251)(67 243 338 252)(68 244 339 253)(69 184 340 254)(70 185 341 255)(71 186 342 256)(72 187 343 257)(73 188 344 258)(74 189 345 259)(75 190 346 260)(76 191 347 261)(77 192 348 262)(78 193 349 263)(79 194 350 264)(80 195 351 265)(81 196 352 266)(82 197 353 267)(83 198 354 268)(84 199 355 269)(85 200 356 270)(86 201 357 271)(87 202 358 272)(88 203 359 273)(89 204 360 274)(90 205 361 275)(91 206 362 276)(92 207 363 277)(93 208 364 278)(94 209 365 279)(95 210 366 280)(96 211 306 281)(97 212 307 282)(98 213 308 283)(99 214 309 284)(100 215 310 285)(101 216 311 286)(102 217 312 287)(103 218 313 288)(104 219 314 289)(105 220 315 290)(106 221 316 291)(107 222 317 292)(108 223 318 293)(109 224 319 294)(110 225 320 295)(111 226 321 296)(112 227 322 297)(113 228 323 298)(114 229 324 299)(115 230 325 300)(116 231 326 301)(117 232 327 302)(118 233 328 303)(119 234 329 304)(120 235 330 305)(121 236 331 245)(122 237 332 246)
(1 316 470 106)(2 317 471 107)(3 318 472 108)(4 319 473 109)(5 320 474 110)(6 321 475 111)(7 322 476 112)(8 323 477 113)(9 324 478 114)(10 325 479 115)(11 326 480 116)(12 327 481 117)(13 328 482 118)(14 329 483 119)(15 330 484 120)(16 331 485 121)(17 332 486 122)(18 333 487 62)(19 334 488 63)(20 335 428 64)(21 336 429 65)(22 337 430 66)(23 338 431 67)(24 339 432 68)(25 340 433 69)(26 341 434 70)(27 342 435 71)(28 343 436 72)(29 344 437 73)(30 345 438 74)(31 346 439 75)(32 347 440 76)(33 348 441 77)(34 349 442 78)(35 350 443 79)(36 351 444 80)(37 352 445 81)(38 353 446 82)(39 354 447 83)(40 355 448 84)(41 356 449 85)(42 357 450 86)(43 358 451 87)(44 359 452 88)(45 360 453 89)(46 361 454 90)(47 362 455 91)(48 363 456 92)(49 364 457 93)(50 365 458 94)(51 366 459 95)(52 306 460 96)(53 307 461 97)(54 308 462 98)(55 309 463 99)(56 310 464 100)(57 311 465 101)(58 312 466 102)(59 313 467 103)(60 314 468 104)(61 315 469 105)(123 234 373 304)(124 235 374 305)(125 236 375 245)(126 237 376 246)(127 238 377 247)(128 239 378 248)(129 240 379 249)(130 241 380 250)(131 242 381 251)(132 243 382 252)(133 244 383 253)(134 184 384 254)(135 185 385 255)(136 186 386 256)(137 187 387 257)(138 188 388 258)(139 189 389 259)(140 190 390 260)(141 191 391 261)(142 192 392 262)(143 193 393 263)(144 194 394 264)(145 195 395 265)(146 196 396 266)(147 197 397 267)(148 198 398 268)(149 199 399 269)(150 200 400 270)(151 201 401 271)(152 202 402 272)(153 203 403 273)(154 204 404 274)(155 205 405 275)(156 206 406 276)(157 207 407 277)(158 208 408 278)(159 209 409 279)(160 210 410 280)(161 211 411 281)(162 212 412 282)(163 213 413 283)(164 214 414 284)(165 215 415 285)(166 216 416 286)(167 217 417 287)(168 218 418 288)(169 219 419 289)(170 220 420 290)(171 221 421 291)(172 222 422 292)(173 223 423 293)(174 224 424 294)(175 225 425 295)(176 226 426 296)(177 227 427 297)(178 228 367 298)(179 229 368 299)(180 230 369 300)(181 231 370 301)(182 232 371 302)(183 233 372 303)
G:=sub<Sym(488)| (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)(428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488), (1,171,470,421)(2,172,471,422)(3,173,472,423)(4,174,473,424)(5,175,474,425)(6,176,475,426)(7,177,476,427)(8,178,477,367)(9,179,478,368)(10,180,479,369)(11,181,480,370)(12,182,481,371)(13,183,482,372)(14,123,483,373)(15,124,484,374)(16,125,485,375)(17,126,486,376)(18,127,487,377)(19,128,488,378)(20,129,428,379)(21,130,429,380)(22,131,430,381)(23,132,431,382)(24,133,432,383)(25,134,433,384)(26,135,434,385)(27,136,435,386)(28,137,436,387)(29,138,437,388)(30,139,438,389)(31,140,439,390)(32,141,440,391)(33,142,441,392)(34,143,442,393)(35,144,443,394)(36,145,444,395)(37,146,445,396)(38,147,446,397)(39,148,447,398)(40,149,448,399)(41,150,449,400)(42,151,450,401)(43,152,451,402)(44,153,452,403)(45,154,453,404)(46,155,454,405)(47,156,455,406)(48,157,456,407)(49,158,457,408)(50,159,458,409)(51,160,459,410)(52,161,460,411)(53,162,461,412)(54,163,462,413)(55,164,463,414)(56,165,464,415)(57,166,465,416)(58,167,466,417)(59,168,467,418)(60,169,468,419)(61,170,469,420)(62,238,333,247)(63,239,334,248)(64,240,335,249)(65,241,336,250)(66,242,337,251)(67,243,338,252)(68,244,339,253)(69,184,340,254)(70,185,341,255)(71,186,342,256)(72,187,343,257)(73,188,344,258)(74,189,345,259)(75,190,346,260)(76,191,347,261)(77,192,348,262)(78,193,349,263)(79,194,350,264)(80,195,351,265)(81,196,352,266)(82,197,353,267)(83,198,354,268)(84,199,355,269)(85,200,356,270)(86,201,357,271)(87,202,358,272)(88,203,359,273)(89,204,360,274)(90,205,361,275)(91,206,362,276)(92,207,363,277)(93,208,364,278)(94,209,365,279)(95,210,366,280)(96,211,306,281)(97,212,307,282)(98,213,308,283)(99,214,309,284)(100,215,310,285)(101,216,311,286)(102,217,312,287)(103,218,313,288)(104,219,314,289)(105,220,315,290)(106,221,316,291)(107,222,317,292)(108,223,318,293)(109,224,319,294)(110,225,320,295)(111,226,321,296)(112,227,322,297)(113,228,323,298)(114,229,324,299)(115,230,325,300)(116,231,326,301)(117,232,327,302)(118,233,328,303)(119,234,329,304)(120,235,330,305)(121,236,331,245)(122,237,332,246), (1,316,470,106)(2,317,471,107)(3,318,472,108)(4,319,473,109)(5,320,474,110)(6,321,475,111)(7,322,476,112)(8,323,477,113)(9,324,478,114)(10,325,479,115)(11,326,480,116)(12,327,481,117)(13,328,482,118)(14,329,483,119)(15,330,484,120)(16,331,485,121)(17,332,486,122)(18,333,487,62)(19,334,488,63)(20,335,428,64)(21,336,429,65)(22,337,430,66)(23,338,431,67)(24,339,432,68)(25,340,433,69)(26,341,434,70)(27,342,435,71)(28,343,436,72)(29,344,437,73)(30,345,438,74)(31,346,439,75)(32,347,440,76)(33,348,441,77)(34,349,442,78)(35,350,443,79)(36,351,444,80)(37,352,445,81)(38,353,446,82)(39,354,447,83)(40,355,448,84)(41,356,449,85)(42,357,450,86)(43,358,451,87)(44,359,452,88)(45,360,453,89)(46,361,454,90)(47,362,455,91)(48,363,456,92)(49,364,457,93)(50,365,458,94)(51,366,459,95)(52,306,460,96)(53,307,461,97)(54,308,462,98)(55,309,463,99)(56,310,464,100)(57,311,465,101)(58,312,466,102)(59,313,467,103)(60,314,468,104)(61,315,469,105)(123,234,373,304)(124,235,374,305)(125,236,375,245)(126,237,376,246)(127,238,377,247)(128,239,378,248)(129,240,379,249)(130,241,380,250)(131,242,381,251)(132,243,382,252)(133,244,383,253)(134,184,384,254)(135,185,385,255)(136,186,386,256)(137,187,387,257)(138,188,388,258)(139,189,389,259)(140,190,390,260)(141,191,391,261)(142,192,392,262)(143,193,393,263)(144,194,394,264)(145,195,395,265)(146,196,396,266)(147,197,397,267)(148,198,398,268)(149,199,399,269)(150,200,400,270)(151,201,401,271)(152,202,402,272)(153,203,403,273)(154,204,404,274)(155,205,405,275)(156,206,406,276)(157,207,407,277)(158,208,408,278)(159,209,409,279)(160,210,410,280)(161,211,411,281)(162,212,412,282)(163,213,413,283)(164,214,414,284)(165,215,415,285)(166,216,416,286)(167,217,417,287)(168,218,418,288)(169,219,419,289)(170,220,420,290)(171,221,421,291)(172,222,422,292)(173,223,423,293)(174,224,424,294)(175,225,425,295)(176,226,426,296)(177,227,427,297)(178,228,367,298)(179,229,368,299)(180,230,369,300)(181,231,370,301)(182,232,371,302)(183,233,372,303)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61)(62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122)(123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183)(184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244)(245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305)(306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366)(367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427)(428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488), (1,171,470,421)(2,172,471,422)(3,173,472,423)(4,174,473,424)(5,175,474,425)(6,176,475,426)(7,177,476,427)(8,178,477,367)(9,179,478,368)(10,180,479,369)(11,181,480,370)(12,182,481,371)(13,183,482,372)(14,123,483,373)(15,124,484,374)(16,125,485,375)(17,126,486,376)(18,127,487,377)(19,128,488,378)(20,129,428,379)(21,130,429,380)(22,131,430,381)(23,132,431,382)(24,133,432,383)(25,134,433,384)(26,135,434,385)(27,136,435,386)(28,137,436,387)(29,138,437,388)(30,139,438,389)(31,140,439,390)(32,141,440,391)(33,142,441,392)(34,143,442,393)(35,144,443,394)(36,145,444,395)(37,146,445,396)(38,147,446,397)(39,148,447,398)(40,149,448,399)(41,150,449,400)(42,151,450,401)(43,152,451,402)(44,153,452,403)(45,154,453,404)(46,155,454,405)(47,156,455,406)(48,157,456,407)(49,158,457,408)(50,159,458,409)(51,160,459,410)(52,161,460,411)(53,162,461,412)(54,163,462,413)(55,164,463,414)(56,165,464,415)(57,166,465,416)(58,167,466,417)(59,168,467,418)(60,169,468,419)(61,170,469,420)(62,238,333,247)(63,239,334,248)(64,240,335,249)(65,241,336,250)(66,242,337,251)(67,243,338,252)(68,244,339,253)(69,184,340,254)(70,185,341,255)(71,186,342,256)(72,187,343,257)(73,188,344,258)(74,189,345,259)(75,190,346,260)(76,191,347,261)(77,192,348,262)(78,193,349,263)(79,194,350,264)(80,195,351,265)(81,196,352,266)(82,197,353,267)(83,198,354,268)(84,199,355,269)(85,200,356,270)(86,201,357,271)(87,202,358,272)(88,203,359,273)(89,204,360,274)(90,205,361,275)(91,206,362,276)(92,207,363,277)(93,208,364,278)(94,209,365,279)(95,210,366,280)(96,211,306,281)(97,212,307,282)(98,213,308,283)(99,214,309,284)(100,215,310,285)(101,216,311,286)(102,217,312,287)(103,218,313,288)(104,219,314,289)(105,220,315,290)(106,221,316,291)(107,222,317,292)(108,223,318,293)(109,224,319,294)(110,225,320,295)(111,226,321,296)(112,227,322,297)(113,228,323,298)(114,229,324,299)(115,230,325,300)(116,231,326,301)(117,232,327,302)(118,233,328,303)(119,234,329,304)(120,235,330,305)(121,236,331,245)(122,237,332,246), (1,316,470,106)(2,317,471,107)(3,318,472,108)(4,319,473,109)(5,320,474,110)(6,321,475,111)(7,322,476,112)(8,323,477,113)(9,324,478,114)(10,325,479,115)(11,326,480,116)(12,327,481,117)(13,328,482,118)(14,329,483,119)(15,330,484,120)(16,331,485,121)(17,332,486,122)(18,333,487,62)(19,334,488,63)(20,335,428,64)(21,336,429,65)(22,337,430,66)(23,338,431,67)(24,339,432,68)(25,340,433,69)(26,341,434,70)(27,342,435,71)(28,343,436,72)(29,344,437,73)(30,345,438,74)(31,346,439,75)(32,347,440,76)(33,348,441,77)(34,349,442,78)(35,350,443,79)(36,351,444,80)(37,352,445,81)(38,353,446,82)(39,354,447,83)(40,355,448,84)(41,356,449,85)(42,357,450,86)(43,358,451,87)(44,359,452,88)(45,360,453,89)(46,361,454,90)(47,362,455,91)(48,363,456,92)(49,364,457,93)(50,365,458,94)(51,366,459,95)(52,306,460,96)(53,307,461,97)(54,308,462,98)(55,309,463,99)(56,310,464,100)(57,311,465,101)(58,312,466,102)(59,313,467,103)(60,314,468,104)(61,315,469,105)(123,234,373,304)(124,235,374,305)(125,236,375,245)(126,237,376,246)(127,238,377,247)(128,239,378,248)(129,240,379,249)(130,241,380,250)(131,242,381,251)(132,243,382,252)(133,244,383,253)(134,184,384,254)(135,185,385,255)(136,186,386,256)(137,187,387,257)(138,188,388,258)(139,189,389,259)(140,190,390,260)(141,191,391,261)(142,192,392,262)(143,193,393,263)(144,194,394,264)(145,195,395,265)(146,196,396,266)(147,197,397,267)(148,198,398,268)(149,199,399,269)(150,200,400,270)(151,201,401,271)(152,202,402,272)(153,203,403,273)(154,204,404,274)(155,205,405,275)(156,206,406,276)(157,207,407,277)(158,208,408,278)(159,209,409,279)(160,210,410,280)(161,211,411,281)(162,212,412,282)(163,213,413,283)(164,214,414,284)(165,215,415,285)(166,216,416,286)(167,217,417,287)(168,218,418,288)(169,219,419,289)(170,220,420,290)(171,221,421,291)(172,222,422,292)(173,223,423,293)(174,224,424,294)(175,225,425,295)(176,226,426,296)(177,227,427,297)(178,228,367,298)(179,229,368,299)(180,230,369,300)(181,231,370,301)(182,232,371,302)(183,233,372,303) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61),(62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122),(123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183),(184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244),(245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305),(306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366),(367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427),(428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488)], [(1,171,470,421),(2,172,471,422),(3,173,472,423),(4,174,473,424),(5,175,474,425),(6,176,475,426),(7,177,476,427),(8,178,477,367),(9,179,478,368),(10,180,479,369),(11,181,480,370),(12,182,481,371),(13,183,482,372),(14,123,483,373),(15,124,484,374),(16,125,485,375),(17,126,486,376),(18,127,487,377),(19,128,488,378),(20,129,428,379),(21,130,429,380),(22,131,430,381),(23,132,431,382),(24,133,432,383),(25,134,433,384),(26,135,434,385),(27,136,435,386),(28,137,436,387),(29,138,437,388),(30,139,438,389),(31,140,439,390),(32,141,440,391),(33,142,441,392),(34,143,442,393),(35,144,443,394),(36,145,444,395),(37,146,445,396),(38,147,446,397),(39,148,447,398),(40,149,448,399),(41,150,449,400),(42,151,450,401),(43,152,451,402),(44,153,452,403),(45,154,453,404),(46,155,454,405),(47,156,455,406),(48,157,456,407),(49,158,457,408),(50,159,458,409),(51,160,459,410),(52,161,460,411),(53,162,461,412),(54,163,462,413),(55,164,463,414),(56,165,464,415),(57,166,465,416),(58,167,466,417),(59,168,467,418),(60,169,468,419),(61,170,469,420),(62,238,333,247),(63,239,334,248),(64,240,335,249),(65,241,336,250),(66,242,337,251),(67,243,338,252),(68,244,339,253),(69,184,340,254),(70,185,341,255),(71,186,342,256),(72,187,343,257),(73,188,344,258),(74,189,345,259),(75,190,346,260),(76,191,347,261),(77,192,348,262),(78,193,349,263),(79,194,350,264),(80,195,351,265),(81,196,352,266),(82,197,353,267),(83,198,354,268),(84,199,355,269),(85,200,356,270),(86,201,357,271),(87,202,358,272),(88,203,359,273),(89,204,360,274),(90,205,361,275),(91,206,362,276),(92,207,363,277),(93,208,364,278),(94,209,365,279),(95,210,366,280),(96,211,306,281),(97,212,307,282),(98,213,308,283),(99,214,309,284),(100,215,310,285),(101,216,311,286),(102,217,312,287),(103,218,313,288),(104,219,314,289),(105,220,315,290),(106,221,316,291),(107,222,317,292),(108,223,318,293),(109,224,319,294),(110,225,320,295),(111,226,321,296),(112,227,322,297),(113,228,323,298),(114,229,324,299),(115,230,325,300),(116,231,326,301),(117,232,327,302),(118,233,328,303),(119,234,329,304),(120,235,330,305),(121,236,331,245),(122,237,332,246)], [(1,316,470,106),(2,317,471,107),(3,318,472,108),(4,319,473,109),(5,320,474,110),(6,321,475,111),(7,322,476,112),(8,323,477,113),(9,324,478,114),(10,325,479,115),(11,326,480,116),(12,327,481,117),(13,328,482,118),(14,329,483,119),(15,330,484,120),(16,331,485,121),(17,332,486,122),(18,333,487,62),(19,334,488,63),(20,335,428,64),(21,336,429,65),(22,337,430,66),(23,338,431,67),(24,339,432,68),(25,340,433,69),(26,341,434,70),(27,342,435,71),(28,343,436,72),(29,344,437,73),(30,345,438,74),(31,346,439,75),(32,347,440,76),(33,348,441,77),(34,349,442,78),(35,350,443,79),(36,351,444,80),(37,352,445,81),(38,353,446,82),(39,354,447,83),(40,355,448,84),(41,356,449,85),(42,357,450,86),(43,358,451,87),(44,359,452,88),(45,360,453,89),(46,361,454,90),(47,362,455,91),(48,363,456,92),(49,364,457,93),(50,365,458,94),(51,366,459,95),(52,306,460,96),(53,307,461,97),(54,308,462,98),(55,309,463,99),(56,310,464,100),(57,311,465,101),(58,312,466,102),(59,313,467,103),(60,314,468,104),(61,315,469,105),(123,234,373,304),(124,235,374,305),(125,236,375,245),(126,237,376,246),(127,238,377,247),(128,239,378,248),(129,240,379,249),(130,241,380,250),(131,242,381,251),(132,243,382,252),(133,244,383,253),(134,184,384,254),(135,185,385,255),(136,186,386,256),(137,187,387,257),(138,188,388,258),(139,189,389,259),(140,190,390,260),(141,191,391,261),(142,192,392,262),(143,193,393,263),(144,194,394,264),(145,195,395,265),(146,196,396,266),(147,197,397,267),(148,198,398,268),(149,199,399,269),(150,200,400,270),(151,201,401,271),(152,202,402,272),(153,203,403,273),(154,204,404,274),(155,205,405,275),(156,206,406,276),(157,207,407,277),(158,208,408,278),(159,209,409,279),(160,210,410,280),(161,211,411,281),(162,212,412,282),(163,213,413,283),(164,214,414,284),(165,215,415,285),(166,216,416,286),(167,217,417,287),(168,218,418,288),(169,219,419,289),(170,220,420,290),(171,221,421,291),(172,222,422,292),(173,223,423,293),(174,224,424,294),(175,225,425,295),(176,226,426,296),(177,227,427,297),(178,228,367,298),(179,229,368,299),(180,230,369,300),(181,231,370,301),(182,232,371,302),(183,233,372,303)]])
305 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 61A | ··· | 61BH | 122A | ··· | 122BH | 244A | ··· | 244FX |
| order | 1 | 2 | 4 | 4 | 4 | 61 | ··· | 61 | 122 | ··· | 122 | 244 | ··· | 244 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
305 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C61 | C122 | Q8 | Q8×C61 |
| kernel | Q8×C61 | C244 | Q8 | C4 | C61 | C1 |
| # reps | 1 | 3 | 60 | 180 | 1 | 60 |
Matrix representation of Q8×C61 ►in GL2(𝔽733) generated by
| 429 | 0 |
| 0 | 429 |
| 174 | 731 |
| 112 | 559 |
| 64 | 233 |
| 728 | 669 |
G:=sub<GL(2,GF(733))| [429,0,0,429],[174,112,731,559],[64,728,233,669] >;
Q8×C61 in GAP, Magma, Sage, TeX
Q_8\times C_{61} % in TeX
G:=Group("Q8xC61"); // GroupNames label
G:=SmallGroup(488,11);
// by ID
G=gap.SmallGroup(488,11);
# by ID
G:=PCGroup([4,-2,-2,-61,-2,976,1969,981]);
// Polycyclic
G:=Group<a,b,c|a^61=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export