direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C62, C62.12C23, C124.20C22, C4.4(C2×C62), (C2×C4).3C62, (C2×C124).9C2, C22.4(C2×C62), C2.2(C22×C62), (C2×C62).15C22, SmallGroup(496,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C62
G = < a,b,c | a62=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C62
►Regular action on 496 points
Generators in S496
(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)(435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496)
(1 132 381 191)(2 133 382 192)(3 134 383 193)(4 135 384 194)(5 136 385 195)(6 137 386 196)(7 138 387 197)(8 139 388 198)(9 140 389 199)(10 141 390 200)(11 142 391 201)(12 143 392 202)(13 144 393 203)(14 145 394 204)(15 146 395 205)(16 147 396 206)(17 148 397 207)(18 149 398 208)(19 150 399 209)(20 151 400 210)(21 152 401 211)(22 153 402 212)(23 154 403 213)(24 155 404 214)(25 156 405 215)(26 157 406 216)(27 158 407 217)(28 159 408 218)(29 160 409 219)(30 161 410 220)(31 162 411 221)(32 163 412 222)(33 164 413 223)(34 165 414 224)(35 166 415 225)(36 167 416 226)(37 168 417 227)(38 169 418 228)(39 170 419 229)(40 171 420 230)(41 172 421 231)(42 173 422 232)(43 174 423 233)(44 175 424 234)(45 176 425 235)(46 177 426 236)(47 178 427 237)(48 179 428 238)(49 180 429 239)(50 181 430 240)(51 182 431 241)(52 183 432 242)(53 184 433 243)(54 185 434 244)(55 186 373 245)(56 125 374 246)(57 126 375 247)(58 127 376 248)(59 128 377 187)(60 129 378 188)(61 130 379 189)(62 131 380 190)(63 338 300 467)(64 339 301 468)(65 340 302 469)(66 341 303 470)(67 342 304 471)(68 343 305 472)(69 344 306 473)(70 345 307 474)(71 346 308 475)(72 347 309 476)(73 348 310 477)(74 349 249 478)(75 350 250 479)(76 351 251 480)(77 352 252 481)(78 353 253 482)(79 354 254 483)(80 355 255 484)(81 356 256 485)(82 357 257 486)(83 358 258 487)(84 359 259 488)(85 360 260 489)(86 361 261 490)(87 362 262 491)(88 363 263 492)(89 364 264 493)(90 365 265 494)(91 366 266 495)(92 367 267 496)(93 368 268 435)(94 369 269 436)(95 370 270 437)(96 371 271 438)(97 372 272 439)(98 311 273 440)(99 312 274 441)(100 313 275 442)(101 314 276 443)(102 315 277 444)(103 316 278 445)(104 317 279 446)(105 318 280 447)(106 319 281 448)(107 320 282 449)(108 321 283 450)(109 322 284 451)(110 323 285 452)(111 324 286 453)(112 325 287 454)(113 326 288 455)(114 327 289 456)(115 328 290 457)(116 329 291 458)(117 330 292 459)(118 331 293 460)(119 332 294 461)(120 333 295 462)(121 334 296 463)(122 335 297 464)(123 336 298 465)(124 337 299 466)
(1 495 381 366)(2 496 382 367)(3 435 383 368)(4 436 384 369)(5 437 385 370)(6 438 386 371)(7 439 387 372)(8 440 388 311)(9 441 389 312)(10 442 390 313)(11 443 391 314)(12 444 392 315)(13 445 393 316)(14 446 394 317)(15 447 395 318)(16 448 396 319)(17 449 397 320)(18 450 398 321)(19 451 399 322)(20 452 400 323)(21 453 401 324)(22 454 402 325)(23 455 403 326)(24 456 404 327)(25 457 405 328)(26 458 406 329)(27 459 407 330)(28 460 408 331)(29 461 409 332)(30 462 410 333)(31 463 411 334)(32 464 412 335)(33 465 413 336)(34 466 414 337)(35 467 415 338)(36 468 416 339)(37 469 417 340)(38 470 418 341)(39 471 419 342)(40 472 420 343)(41 473 421 344)(42 474 422 345)(43 475 423 346)(44 476 424 347)(45 477 425 348)(46 478 426 349)(47 479 427 350)(48 480 428 351)(49 481 429 352)(50 482 430 353)(51 483 431 354)(52 484 432 355)(53 485 433 356)(54 486 434 357)(55 487 373 358)(56 488 374 359)(57 489 375 360)(58 490 376 361)(59 491 377 362)(60 492 378 363)(61 493 379 364)(62 494 380 365)(63 166 300 225)(64 167 301 226)(65 168 302 227)(66 169 303 228)(67 170 304 229)(68 171 305 230)(69 172 306 231)(70 173 307 232)(71 174 308 233)(72 175 309 234)(73 176 310 235)(74 177 249 236)(75 178 250 237)(76 179 251 238)(77 180 252 239)(78 181 253 240)(79 182 254 241)(80 183 255 242)(81 184 256 243)(82 185 257 244)(83 186 258 245)(84 125 259 246)(85 126 260 247)(86 127 261 248)(87 128 262 187)(88 129 263 188)(89 130 264 189)(90 131 265 190)(91 132 266 191)(92 133 267 192)(93 134 268 193)(94 135 269 194)(95 136 270 195)(96 137 271 196)(97 138 272 197)(98 139 273 198)(99 140 274 199)(100 141 275 200)(101 142 276 201)(102 143 277 202)(103 144 278 203)(104 145 279 204)(105 146 280 205)(106 147 281 206)(107 148 282 207)(108 149 283 208)(109 150 284 209)(110 151 285 210)(111 152 286 211)(112 153 287 212)(113 154 288 213)(114 155 289 214)(115 156 290 215)(116 157 291 216)(117 158 292 217)(118 159 293 218)(119 160 294 219)(120 161 295 220)(121 162 296 221)(122 163 297 222)(123 164 298 223)(124 165 299 224)
G:=sub<Sym(496)| (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)(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,132,381,191)(2,133,382,192)(3,134,383,193)(4,135,384,194)(5,136,385,195)(6,137,386,196)(7,138,387,197)(8,139,388,198)(9,140,389,199)(10,141,390,200)(11,142,391,201)(12,143,392,202)(13,144,393,203)(14,145,394,204)(15,146,395,205)(16,147,396,206)(17,148,397,207)(18,149,398,208)(19,150,399,209)(20,151,400,210)(21,152,401,211)(22,153,402,212)(23,154,403,213)(24,155,404,214)(25,156,405,215)(26,157,406,216)(27,158,407,217)(28,159,408,218)(29,160,409,219)(30,161,410,220)(31,162,411,221)(32,163,412,222)(33,164,413,223)(34,165,414,224)(35,166,415,225)(36,167,416,226)(37,168,417,227)(38,169,418,228)(39,170,419,229)(40,171,420,230)(41,172,421,231)(42,173,422,232)(43,174,423,233)(44,175,424,234)(45,176,425,235)(46,177,426,236)(47,178,427,237)(48,179,428,238)(49,180,429,239)(50,181,430,240)(51,182,431,241)(52,183,432,242)(53,184,433,243)(54,185,434,244)(55,186,373,245)(56,125,374,246)(57,126,375,247)(58,127,376,248)(59,128,377,187)(60,129,378,188)(61,130,379,189)(62,131,380,190)(63,338,300,467)(64,339,301,468)(65,340,302,469)(66,341,303,470)(67,342,304,471)(68,343,305,472)(69,344,306,473)(70,345,307,474)(71,346,308,475)(72,347,309,476)(73,348,310,477)(74,349,249,478)(75,350,250,479)(76,351,251,480)(77,352,252,481)(78,353,253,482)(79,354,254,483)(80,355,255,484)(81,356,256,485)(82,357,257,486)(83,358,258,487)(84,359,259,488)(85,360,260,489)(86,361,261,490)(87,362,262,491)(88,363,263,492)(89,364,264,493)(90,365,265,494)(91,366,266,495)(92,367,267,496)(93,368,268,435)(94,369,269,436)(95,370,270,437)(96,371,271,438)(97,372,272,439)(98,311,273,440)(99,312,274,441)(100,313,275,442)(101,314,276,443)(102,315,277,444)(103,316,278,445)(104,317,279,446)(105,318,280,447)(106,319,281,448)(107,320,282,449)(108,321,283,450)(109,322,284,451)(110,323,285,452)(111,324,286,453)(112,325,287,454)(113,326,288,455)(114,327,289,456)(115,328,290,457)(116,329,291,458)(117,330,292,459)(118,331,293,460)(119,332,294,461)(120,333,295,462)(121,334,296,463)(122,335,297,464)(123,336,298,465)(124,337,299,466), (1,495,381,366)(2,496,382,367)(3,435,383,368)(4,436,384,369)(5,437,385,370)(6,438,386,371)(7,439,387,372)(8,440,388,311)(9,441,389,312)(10,442,390,313)(11,443,391,314)(12,444,392,315)(13,445,393,316)(14,446,394,317)(15,447,395,318)(16,448,396,319)(17,449,397,320)(18,450,398,321)(19,451,399,322)(20,452,400,323)(21,453,401,324)(22,454,402,325)(23,455,403,326)(24,456,404,327)(25,457,405,328)(26,458,406,329)(27,459,407,330)(28,460,408,331)(29,461,409,332)(30,462,410,333)(31,463,411,334)(32,464,412,335)(33,465,413,336)(34,466,414,337)(35,467,415,338)(36,468,416,339)(37,469,417,340)(38,470,418,341)(39,471,419,342)(40,472,420,343)(41,473,421,344)(42,474,422,345)(43,475,423,346)(44,476,424,347)(45,477,425,348)(46,478,426,349)(47,479,427,350)(48,480,428,351)(49,481,429,352)(50,482,430,353)(51,483,431,354)(52,484,432,355)(53,485,433,356)(54,486,434,357)(55,487,373,358)(56,488,374,359)(57,489,375,360)(58,490,376,361)(59,491,377,362)(60,492,378,363)(61,493,379,364)(62,494,380,365)(63,166,300,225)(64,167,301,226)(65,168,302,227)(66,169,303,228)(67,170,304,229)(68,171,305,230)(69,172,306,231)(70,173,307,232)(71,174,308,233)(72,175,309,234)(73,176,310,235)(74,177,249,236)(75,178,250,237)(76,179,251,238)(77,180,252,239)(78,181,253,240)(79,182,254,241)(80,183,255,242)(81,184,256,243)(82,185,257,244)(83,186,258,245)(84,125,259,246)(85,126,260,247)(86,127,261,248)(87,128,262,187)(88,129,263,188)(89,130,264,189)(90,131,265,190)(91,132,266,191)(92,133,267,192)(93,134,268,193)(94,135,269,194)(95,136,270,195)(96,137,271,196)(97,138,272,197)(98,139,273,198)(99,140,274,199)(100,141,275,200)(101,142,276,201)(102,143,277,202)(103,144,278,203)(104,145,279,204)(105,146,280,205)(106,147,281,206)(107,148,282,207)(108,149,283,208)(109,150,284,209)(110,151,285,210)(111,152,286,211)(112,153,287,212)(113,154,288,213)(114,155,289,214)(115,156,290,215)(116,157,291,216)(117,158,292,217)(118,159,293,218)(119,160,294,219)(120,161,295,220)(121,162,296,221)(122,163,297,222)(123,164,298,223)(124,165,299,224)>;
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)(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,132,381,191)(2,133,382,192)(3,134,383,193)(4,135,384,194)(5,136,385,195)(6,137,386,196)(7,138,387,197)(8,139,388,198)(9,140,389,199)(10,141,390,200)(11,142,391,201)(12,143,392,202)(13,144,393,203)(14,145,394,204)(15,146,395,205)(16,147,396,206)(17,148,397,207)(18,149,398,208)(19,150,399,209)(20,151,400,210)(21,152,401,211)(22,153,402,212)(23,154,403,213)(24,155,404,214)(25,156,405,215)(26,157,406,216)(27,158,407,217)(28,159,408,218)(29,160,409,219)(30,161,410,220)(31,162,411,221)(32,163,412,222)(33,164,413,223)(34,165,414,224)(35,166,415,225)(36,167,416,226)(37,168,417,227)(38,169,418,228)(39,170,419,229)(40,171,420,230)(41,172,421,231)(42,173,422,232)(43,174,423,233)(44,175,424,234)(45,176,425,235)(46,177,426,236)(47,178,427,237)(48,179,428,238)(49,180,429,239)(50,181,430,240)(51,182,431,241)(52,183,432,242)(53,184,433,243)(54,185,434,244)(55,186,373,245)(56,125,374,246)(57,126,375,247)(58,127,376,248)(59,128,377,187)(60,129,378,188)(61,130,379,189)(62,131,380,190)(63,338,300,467)(64,339,301,468)(65,340,302,469)(66,341,303,470)(67,342,304,471)(68,343,305,472)(69,344,306,473)(70,345,307,474)(71,346,308,475)(72,347,309,476)(73,348,310,477)(74,349,249,478)(75,350,250,479)(76,351,251,480)(77,352,252,481)(78,353,253,482)(79,354,254,483)(80,355,255,484)(81,356,256,485)(82,357,257,486)(83,358,258,487)(84,359,259,488)(85,360,260,489)(86,361,261,490)(87,362,262,491)(88,363,263,492)(89,364,264,493)(90,365,265,494)(91,366,266,495)(92,367,267,496)(93,368,268,435)(94,369,269,436)(95,370,270,437)(96,371,271,438)(97,372,272,439)(98,311,273,440)(99,312,274,441)(100,313,275,442)(101,314,276,443)(102,315,277,444)(103,316,278,445)(104,317,279,446)(105,318,280,447)(106,319,281,448)(107,320,282,449)(108,321,283,450)(109,322,284,451)(110,323,285,452)(111,324,286,453)(112,325,287,454)(113,326,288,455)(114,327,289,456)(115,328,290,457)(116,329,291,458)(117,330,292,459)(118,331,293,460)(119,332,294,461)(120,333,295,462)(121,334,296,463)(122,335,297,464)(123,336,298,465)(124,337,299,466), (1,495,381,366)(2,496,382,367)(3,435,383,368)(4,436,384,369)(5,437,385,370)(6,438,386,371)(7,439,387,372)(8,440,388,311)(9,441,389,312)(10,442,390,313)(11,443,391,314)(12,444,392,315)(13,445,393,316)(14,446,394,317)(15,447,395,318)(16,448,396,319)(17,449,397,320)(18,450,398,321)(19,451,399,322)(20,452,400,323)(21,453,401,324)(22,454,402,325)(23,455,403,326)(24,456,404,327)(25,457,405,328)(26,458,406,329)(27,459,407,330)(28,460,408,331)(29,461,409,332)(30,462,410,333)(31,463,411,334)(32,464,412,335)(33,465,413,336)(34,466,414,337)(35,467,415,338)(36,468,416,339)(37,469,417,340)(38,470,418,341)(39,471,419,342)(40,472,420,343)(41,473,421,344)(42,474,422,345)(43,475,423,346)(44,476,424,347)(45,477,425,348)(46,478,426,349)(47,479,427,350)(48,480,428,351)(49,481,429,352)(50,482,430,353)(51,483,431,354)(52,484,432,355)(53,485,433,356)(54,486,434,357)(55,487,373,358)(56,488,374,359)(57,489,375,360)(58,490,376,361)(59,491,377,362)(60,492,378,363)(61,493,379,364)(62,494,380,365)(63,166,300,225)(64,167,301,226)(65,168,302,227)(66,169,303,228)(67,170,304,229)(68,171,305,230)(69,172,306,231)(70,173,307,232)(71,174,308,233)(72,175,309,234)(73,176,310,235)(74,177,249,236)(75,178,250,237)(76,179,251,238)(77,180,252,239)(78,181,253,240)(79,182,254,241)(80,183,255,242)(81,184,256,243)(82,185,257,244)(83,186,258,245)(84,125,259,246)(85,126,260,247)(86,127,261,248)(87,128,262,187)(88,129,263,188)(89,130,264,189)(90,131,265,190)(91,132,266,191)(92,133,267,192)(93,134,268,193)(94,135,269,194)(95,136,270,195)(96,137,271,196)(97,138,272,197)(98,139,273,198)(99,140,274,199)(100,141,275,200)(101,142,276,201)(102,143,277,202)(103,144,278,203)(104,145,279,204)(105,146,280,205)(106,147,281,206)(107,148,282,207)(108,149,283,208)(109,150,284,209)(110,151,285,210)(111,152,286,211)(112,153,287,212)(113,154,288,213)(114,155,289,214)(115,156,290,215)(116,157,291,216)(117,158,292,217)(118,159,293,218)(119,160,294,219)(120,161,295,220)(121,162,296,221)(122,163,297,222)(123,164,298,223)(124,165,299,224) );
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),(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496)], [(1,132,381,191),(2,133,382,192),(3,134,383,193),(4,135,384,194),(5,136,385,195),(6,137,386,196),(7,138,387,197),(8,139,388,198),(9,140,389,199),(10,141,390,200),(11,142,391,201),(12,143,392,202),(13,144,393,203),(14,145,394,204),(15,146,395,205),(16,147,396,206),(17,148,397,207),(18,149,398,208),(19,150,399,209),(20,151,400,210),(21,152,401,211),(22,153,402,212),(23,154,403,213),(24,155,404,214),(25,156,405,215),(26,157,406,216),(27,158,407,217),(28,159,408,218),(29,160,409,219),(30,161,410,220),(31,162,411,221),(32,163,412,222),(33,164,413,223),(34,165,414,224),(35,166,415,225),(36,167,416,226),(37,168,417,227),(38,169,418,228),(39,170,419,229),(40,171,420,230),(41,172,421,231),(42,173,422,232),(43,174,423,233),(44,175,424,234),(45,176,425,235),(46,177,426,236),(47,178,427,237),(48,179,428,238),(49,180,429,239),(50,181,430,240),(51,182,431,241),(52,183,432,242),(53,184,433,243),(54,185,434,244),(55,186,373,245),(56,125,374,246),(57,126,375,247),(58,127,376,248),(59,128,377,187),(60,129,378,188),(61,130,379,189),(62,131,380,190),(63,338,300,467),(64,339,301,468),(65,340,302,469),(66,341,303,470),(67,342,304,471),(68,343,305,472),(69,344,306,473),(70,345,307,474),(71,346,308,475),(72,347,309,476),(73,348,310,477),(74,349,249,478),(75,350,250,479),(76,351,251,480),(77,352,252,481),(78,353,253,482),(79,354,254,483),(80,355,255,484),(81,356,256,485),(82,357,257,486),(83,358,258,487),(84,359,259,488),(85,360,260,489),(86,361,261,490),(87,362,262,491),(88,363,263,492),(89,364,264,493),(90,365,265,494),(91,366,266,495),(92,367,267,496),(93,368,268,435),(94,369,269,436),(95,370,270,437),(96,371,271,438),(97,372,272,439),(98,311,273,440),(99,312,274,441),(100,313,275,442),(101,314,276,443),(102,315,277,444),(103,316,278,445),(104,317,279,446),(105,318,280,447),(106,319,281,448),(107,320,282,449),(108,321,283,450),(109,322,284,451),(110,323,285,452),(111,324,286,453),(112,325,287,454),(113,326,288,455),(114,327,289,456),(115,328,290,457),(116,329,291,458),(117,330,292,459),(118,331,293,460),(119,332,294,461),(120,333,295,462),(121,334,296,463),(122,335,297,464),(123,336,298,465),(124,337,299,466)], [(1,495,381,366),(2,496,382,367),(3,435,383,368),(4,436,384,369),(5,437,385,370),(6,438,386,371),(7,439,387,372),(8,440,388,311),(9,441,389,312),(10,442,390,313),(11,443,391,314),(12,444,392,315),(13,445,393,316),(14,446,394,317),(15,447,395,318),(16,448,396,319),(17,449,397,320),(18,450,398,321),(19,451,399,322),(20,452,400,323),(21,453,401,324),(22,454,402,325),(23,455,403,326),(24,456,404,327),(25,457,405,328),(26,458,406,329),(27,459,407,330),(28,460,408,331),(29,461,409,332),(30,462,410,333),(31,463,411,334),(32,464,412,335),(33,465,413,336),(34,466,414,337),(35,467,415,338),(36,468,416,339),(37,469,417,340),(38,470,418,341),(39,471,419,342),(40,472,420,343),(41,473,421,344),(42,474,422,345),(43,475,423,346),(44,476,424,347),(45,477,425,348),(46,478,426,349),(47,479,427,350),(48,480,428,351),(49,481,429,352),(50,482,430,353),(51,483,431,354),(52,484,432,355),(53,485,433,356),(54,486,434,357),(55,487,373,358),(56,488,374,359),(57,489,375,360),(58,490,376,361),(59,491,377,362),(60,492,378,363),(61,493,379,364),(62,494,380,365),(63,166,300,225),(64,167,301,226),(65,168,302,227),(66,169,303,228),(67,170,304,229),(68,171,305,230),(69,172,306,231),(70,173,307,232),(71,174,308,233),(72,175,309,234),(73,176,310,235),(74,177,249,236),(75,178,250,237),(76,179,251,238),(77,180,252,239),(78,181,253,240),(79,182,254,241),(80,183,255,242),(81,184,256,243),(82,185,257,244),(83,186,258,245),(84,125,259,246),(85,126,260,247),(86,127,261,248),(87,128,262,187),(88,129,263,188),(89,130,264,189),(90,131,265,190),(91,132,266,191),(92,133,267,192),(93,134,268,193),(94,135,269,194),(95,136,270,195),(96,137,271,196),(97,138,272,197),(98,139,273,198),(99,140,274,199),(100,141,275,200),(101,142,276,201),(102,143,277,202),(103,144,278,203),(104,145,279,204),(105,146,280,205),(106,147,281,206),(107,148,282,207),(108,149,283,208),(109,150,284,209),(110,151,285,210),(111,152,286,211),(112,153,287,212),(113,154,288,213),(114,155,289,214),(115,156,290,215),(116,157,291,216),(117,158,292,217),(118,159,293,218),(119,160,294,219),(120,161,295,220),(121,162,296,221),(122,163,297,222),(123,164,298,223),(124,165,299,224)]])
310 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 31A | ··· | 31AD | 62A | ··· | 62CL | 124A | ··· | 124FX |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
310 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | ||||
| image | C1 | C2 | C2 | C31 | C62 | C62 | Q8 | Q8×C31 |
| kernel | Q8×C62 | C2×C124 | Q8×C31 | C2×Q8 | C2×C4 | Q8 | C62 | C2 |
| # reps | 1 | 3 | 4 | 30 | 90 | 120 | 2 | 60 |
Matrix representation of Q8×C62 ►in GL3(𝔽373) generated by
| 372 | 0 | 0 |
| 0 | 17 | 0 |
| 0 | 0 | 17 |
| 372 | 0 | 0 |
| 0 | 372 | 371 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 104 | 208 |
| 0 | 0 | 269 |
G:=sub<GL(3,GF(373))| [372,0,0,0,17,0,0,0,17],[372,0,0,0,372,1,0,371,1],[1,0,0,0,104,0,0,208,269] >;
Q8×C62 in GAP, Magma, Sage, TeX
Q_8\times C_{62} % in TeX
G:=Group("Q8xC62"); // GroupNames label
G:=SmallGroup(496,39);
// by ID
G=gap.SmallGroup(496,39);
# by ID
G:=PCGroup([5,-2,-2,-2,-31,-2,1240,2501,1246]);
// Polycyclic
G:=Group<a,b,c|a^62=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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