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G = Q8×C61order 488 = 23·61

Direct product of C61 and Q8

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

C1C2 — Q8×C61
C1C2C122C244 — Q8×C61
C1C2 — Q8×C61
C1C122 — Q8×C61

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)
(1 182 458 66)(2 183 459 67)(3 123 460 68)(4 124 461 69)(5 125 462 70)(6 126 463 71)(7 127 464 72)(8 128 465 73)(9 129 466 74)(10 130 467 75)(11 131 468 76)(12 132 469 77)(13 133 470 78)(14 134 471 79)(15 135 472 80)(16 136 473 81)(17 137 474 82)(18 138 475 83)(19 139 476 84)(20 140 477 85)(21 141 478 86)(22 142 479 87)(23 143 480 88)(24 144 481 89)(25 145 482 90)(26 146 483 91)(27 147 484 92)(28 148 485 93)(29 149 486 94)(30 150 487 95)(31 151 488 96)(32 152 428 97)(33 153 429 98)(34 154 430 99)(35 155 431 100)(36 156 432 101)(37 157 433 102)(38 158 434 103)(39 159 435 104)(40 160 436 105)(41 161 437 106)(42 162 438 107)(43 163 439 108)(44 164 440 109)(45 165 441 110)(46 166 442 111)(47 167 443 112)(48 168 444 113)(49 169 445 114)(50 170 446 115)(51 171 447 116)(52 172 448 117)(53 173 449 118)(54 174 450 119)(55 175 451 120)(56 176 452 121)(57 177 453 122)(58 178 454 62)(59 179 455 63)(60 180 456 64)(61 181 457 65)(184 386 350 289)(185 387 351 290)(186 388 352 291)(187 389 353 292)(188 390 354 293)(189 391 355 294)(190 392 356 295)(191 393 357 296)(192 394 358 297)(193 395 359 298)(194 396 360 299)(195 397 361 300)(196 398 362 301)(197 399 363 302)(198 400 364 303)(199 401 365 304)(200 402 366 305)(201 403 306 245)(202 404 307 246)(203 405 308 247)(204 406 309 248)(205 407 310 249)(206 408 311 250)(207 409 312 251)(208 410 313 252)(209 411 314 253)(210 412 315 254)(211 413 316 255)(212 414 317 256)(213 415 318 257)(214 416 319 258)(215 417 320 259)(216 418 321 260)(217 419 322 261)(218 420 323 262)(219 421 324 263)(220 422 325 264)(221 423 326 265)(222 424 327 266)(223 425 328 267)(224 426 329 268)(225 427 330 269)(226 367 331 270)(227 368 332 271)(228 369 333 272)(229 370 334 273)(230 371 335 274)(231 372 336 275)(232 373 337 276)(233 374 338 277)(234 375 339 278)(235 376 340 279)(236 377 341 280)(237 378 342 281)(238 379 343 282)(239 380 344 283)(240 381 345 284)(241 382 346 285)(242 383 347 286)(243 384 348 287)(244 385 349 288)
(1 271 458 368)(2 272 459 369)(3 273 460 370)(4 274 461 371)(5 275 462 372)(6 276 463 373)(7 277 464 374)(8 278 465 375)(9 279 466 376)(10 280 467 377)(11 281 468 378)(12 282 469 379)(13 283 470 380)(14 284 471 381)(15 285 472 382)(16 286 473 383)(17 287 474 384)(18 288 475 385)(19 289 476 386)(20 290 477 387)(21 291 478 388)(22 292 479 389)(23 293 480 390)(24 294 481 391)(25 295 482 392)(26 296 483 393)(27 297 484 394)(28 298 485 395)(29 299 486 396)(30 300 487 397)(31 301 488 398)(32 302 428 399)(33 303 429 400)(34 304 430 401)(35 305 431 402)(36 245 432 403)(37 246 433 404)(38 247 434 405)(39 248 435 406)(40 249 436 407)(41 250 437 408)(42 251 438 409)(43 252 439 410)(44 253 440 411)(45 254 441 412)(46 255 442 413)(47 256 443 414)(48 257 444 415)(49 258 445 416)(50 259 446 417)(51 260 447 418)(52 261 448 419)(53 262 449 420)(54 263 450 421)(55 264 451 422)(56 265 452 423)(57 266 453 424)(58 267 454 425)(59 268 455 426)(60 269 456 427)(61 270 457 367)(62 223 178 328)(63 224 179 329)(64 225 180 330)(65 226 181 331)(66 227 182 332)(67 228 183 333)(68 229 123 334)(69 230 124 335)(70 231 125 336)(71 232 126 337)(72 233 127 338)(73 234 128 339)(74 235 129 340)(75 236 130 341)(76 237 131 342)(77 238 132 343)(78 239 133 344)(79 240 134 345)(80 241 135 346)(81 242 136 347)(82 243 137 348)(83 244 138 349)(84 184 139 350)(85 185 140 351)(86 186 141 352)(87 187 142 353)(88 188 143 354)(89 189 144 355)(90 190 145 356)(91 191 146 357)(92 192 147 358)(93 193 148 359)(94 194 149 360)(95 195 150 361)(96 196 151 362)(97 197 152 363)(98 198 153 364)(99 199 154 365)(100 200 155 366)(101 201 156 306)(102 202 157 307)(103 203 158 308)(104 204 159 309)(105 205 160 310)(106 206 161 311)(107 207 162 312)(108 208 163 313)(109 209 164 314)(110 210 165 315)(111 211 166 316)(112 212 167 317)(113 213 168 318)(114 214 169 319)(115 215 170 320)(116 216 171 321)(117 217 172 322)(118 218 173 323)(119 219 174 324)(120 220 175 325)(121 221 176 326)(122 222 177 327)

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

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), (1,182,458,66)(2,183,459,67)(3,123,460,68)(4,124,461,69)(5,125,462,70)(6,126,463,71)(7,127,464,72)(8,128,465,73)(9,129,466,74)(10,130,467,75)(11,131,468,76)(12,132,469,77)(13,133,470,78)(14,134,471,79)(15,135,472,80)(16,136,473,81)(17,137,474,82)(18,138,475,83)(19,139,476,84)(20,140,477,85)(21,141,478,86)(22,142,479,87)(23,143,480,88)(24,144,481,89)(25,145,482,90)(26,146,483,91)(27,147,484,92)(28,148,485,93)(29,149,486,94)(30,150,487,95)(31,151,488,96)(32,152,428,97)(33,153,429,98)(34,154,430,99)(35,155,431,100)(36,156,432,101)(37,157,433,102)(38,158,434,103)(39,159,435,104)(40,160,436,105)(41,161,437,106)(42,162,438,107)(43,163,439,108)(44,164,440,109)(45,165,441,110)(46,166,442,111)(47,167,443,112)(48,168,444,113)(49,169,445,114)(50,170,446,115)(51,171,447,116)(52,172,448,117)(53,173,449,118)(54,174,450,119)(55,175,451,120)(56,176,452,121)(57,177,453,122)(58,178,454,62)(59,179,455,63)(60,180,456,64)(61,181,457,65)(184,386,350,289)(185,387,351,290)(186,388,352,291)(187,389,353,292)(188,390,354,293)(189,391,355,294)(190,392,356,295)(191,393,357,296)(192,394,358,297)(193,395,359,298)(194,396,360,299)(195,397,361,300)(196,398,362,301)(197,399,363,302)(198,400,364,303)(199,401,365,304)(200,402,366,305)(201,403,306,245)(202,404,307,246)(203,405,308,247)(204,406,309,248)(205,407,310,249)(206,408,311,250)(207,409,312,251)(208,410,313,252)(209,411,314,253)(210,412,315,254)(211,413,316,255)(212,414,317,256)(213,415,318,257)(214,416,319,258)(215,417,320,259)(216,418,321,260)(217,419,322,261)(218,420,323,262)(219,421,324,263)(220,422,325,264)(221,423,326,265)(222,424,327,266)(223,425,328,267)(224,426,329,268)(225,427,330,269)(226,367,331,270)(227,368,332,271)(228,369,333,272)(229,370,334,273)(230,371,335,274)(231,372,336,275)(232,373,337,276)(233,374,338,277)(234,375,339,278)(235,376,340,279)(236,377,341,280)(237,378,342,281)(238,379,343,282)(239,380,344,283)(240,381,345,284)(241,382,346,285)(242,383,347,286)(243,384,348,287)(244,385,349,288), (1,271,458,368)(2,272,459,369)(3,273,460,370)(4,274,461,371)(5,275,462,372)(6,276,463,373)(7,277,464,374)(8,278,465,375)(9,279,466,376)(10,280,467,377)(11,281,468,378)(12,282,469,379)(13,283,470,380)(14,284,471,381)(15,285,472,382)(16,286,473,383)(17,287,474,384)(18,288,475,385)(19,289,476,386)(20,290,477,387)(21,291,478,388)(22,292,479,389)(23,293,480,390)(24,294,481,391)(25,295,482,392)(26,296,483,393)(27,297,484,394)(28,298,485,395)(29,299,486,396)(30,300,487,397)(31,301,488,398)(32,302,428,399)(33,303,429,400)(34,304,430,401)(35,305,431,402)(36,245,432,403)(37,246,433,404)(38,247,434,405)(39,248,435,406)(40,249,436,407)(41,250,437,408)(42,251,438,409)(43,252,439,410)(44,253,440,411)(45,254,441,412)(46,255,442,413)(47,256,443,414)(48,257,444,415)(49,258,445,416)(50,259,446,417)(51,260,447,418)(52,261,448,419)(53,262,449,420)(54,263,450,421)(55,264,451,422)(56,265,452,423)(57,266,453,424)(58,267,454,425)(59,268,455,426)(60,269,456,427)(61,270,457,367)(62,223,178,328)(63,224,179,329)(64,225,180,330)(65,226,181,331)(66,227,182,332)(67,228,183,333)(68,229,123,334)(69,230,124,335)(70,231,125,336)(71,232,126,337)(72,233,127,338)(73,234,128,339)(74,235,129,340)(75,236,130,341)(76,237,131,342)(77,238,132,343)(78,239,133,344)(79,240,134,345)(80,241,135,346)(81,242,136,347)(82,243,137,348)(83,244,138,349)(84,184,139,350)(85,185,140,351)(86,186,141,352)(87,187,142,353)(88,188,143,354)(89,189,144,355)(90,190,145,356)(91,191,146,357)(92,192,147,358)(93,193,148,359)(94,194,149,360)(95,195,150,361)(96,196,151,362)(97,197,152,363)(98,198,153,364)(99,199,154,365)(100,200,155,366)(101,201,156,306)(102,202,157,307)(103,203,158,308)(104,204,159,309)(105,205,160,310)(106,206,161,311)(107,207,162,312)(108,208,163,313)(109,209,164,314)(110,210,165,315)(111,211,166,316)(112,212,167,317)(113,213,168,318)(114,214,169,319)(115,215,170,320)(116,216,171,321)(117,217,172,322)(118,218,173,323)(119,219,174,324)(120,220,175,325)(121,221,176,326)(122,222,177,327) );

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

305 conjugacy classes

class 1  2 4A4B4C61A···61BH122A···122BH244A···244FX
order1244461···61122···122244···244
size112221···11···12···2

305 irreducible representations

dim111122
type++-
imageC1C2C61C122Q8Q8×C61
kernelQ8×C61C244Q8C4C61C1
# reps1360180160

Matrix representation of Q8×C61 in GL2(𝔽733) generated by

4290
0429
,
174731
112559
,
64233
728669
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

Subgroup lattice of Q8×C61 in TeX

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