direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C59, C4.C118, C236.3C2, C118.7C22, C2.2(C2×C118), SmallGroup(472,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C59
G = < a,b,c | a59=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C59
►Regular action on 472 points
Generators in S472
(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)
(1 366 109 341)(2 367 110 342)(3 368 111 343)(4 369 112 344)(5 370 113 345)(6 371 114 346)(7 372 115 347)(8 373 116 348)(9 374 117 349)(10 375 118 350)(11 376 60 351)(12 377 61 352)(13 378 62 353)(14 379 63 354)(15 380 64 296)(16 381 65 297)(17 382 66 298)(18 383 67 299)(19 384 68 300)(20 385 69 301)(21 386 70 302)(22 387 71 303)(23 388 72 304)(24 389 73 305)(25 390 74 306)(26 391 75 307)(27 392 76 308)(28 393 77 309)(29 394 78 310)(30 395 79 311)(31 396 80 312)(32 397 81 313)(33 398 82 314)(34 399 83 315)(35 400 84 316)(36 401 85 317)(37 402 86 318)(38 403 87 319)(39 404 88 320)(40 405 89 321)(41 406 90 322)(42 407 91 323)(43 408 92 324)(44 409 93 325)(45 410 94 326)(46 411 95 327)(47 412 96 328)(48 413 97 329)(49 355 98 330)(50 356 99 331)(51 357 100 332)(52 358 101 333)(53 359 102 334)(54 360 103 335)(55 361 104 336)(56 362 105 337)(57 363 106 338)(58 364 107 339)(59 365 108 340)(119 182 417 265)(120 183 418 266)(121 184 419 267)(122 185 420 268)(123 186 421 269)(124 187 422 270)(125 188 423 271)(126 189 424 272)(127 190 425 273)(128 191 426 274)(129 192 427 275)(130 193 428 276)(131 194 429 277)(132 195 430 278)(133 196 431 279)(134 197 432 280)(135 198 433 281)(136 199 434 282)(137 200 435 283)(138 201 436 284)(139 202 437 285)(140 203 438 286)(141 204 439 287)(142 205 440 288)(143 206 441 289)(144 207 442 290)(145 208 443 291)(146 209 444 292)(147 210 445 293)(148 211 446 294)(149 212 447 295)(150 213 448 237)(151 214 449 238)(152 215 450 239)(153 216 451 240)(154 217 452 241)(155 218 453 242)(156 219 454 243)(157 220 455 244)(158 221 456 245)(159 222 457 246)(160 223 458 247)(161 224 459 248)(162 225 460 249)(163 226 461 250)(164 227 462 251)(165 228 463 252)(166 229 464 253)(167 230 465 254)(168 231 466 255)(169 232 467 256)(170 233 468 257)(171 234 469 258)(172 235 470 259)(173 236 471 260)(174 178 472 261)(175 179 414 262)(176 180 415 263)(177 181 416 264)
(1 268 109 185)(2 269 110 186)(3 270 111 187)(4 271 112 188)(5 272 113 189)(6 273 114 190)(7 274 115 191)(8 275 116 192)(9 276 117 193)(10 277 118 194)(11 278 60 195)(12 279 61 196)(13 280 62 197)(14 281 63 198)(15 282 64 199)(16 283 65 200)(17 284 66 201)(18 285 67 202)(19 286 68 203)(20 287 69 204)(21 288 70 205)(22 289 71 206)(23 290 72 207)(24 291 73 208)(25 292 74 209)(26 293 75 210)(27 294 76 211)(28 295 77 212)(29 237 78 213)(30 238 79 214)(31 239 80 215)(32 240 81 216)(33 241 82 217)(34 242 83 218)(35 243 84 219)(36 244 85 220)(37 245 86 221)(38 246 87 222)(39 247 88 223)(40 248 89 224)(41 249 90 225)(42 250 91 226)(43 251 92 227)(44 252 93 228)(45 253 94 229)(46 254 95 230)(47 255 96 231)(48 256 97 232)(49 257 98 233)(50 258 99 234)(51 259 100 235)(52 260 101 236)(53 261 102 178)(54 262 103 179)(55 263 104 180)(56 264 105 181)(57 265 106 182)(58 266 107 183)(59 267 108 184)(119 363 417 338)(120 364 418 339)(121 365 419 340)(122 366 420 341)(123 367 421 342)(124 368 422 343)(125 369 423 344)(126 370 424 345)(127 371 425 346)(128 372 426 347)(129 373 427 348)(130 374 428 349)(131 375 429 350)(132 376 430 351)(133 377 431 352)(134 378 432 353)(135 379 433 354)(136 380 434 296)(137 381 435 297)(138 382 436 298)(139 383 437 299)(140 384 438 300)(141 385 439 301)(142 386 440 302)(143 387 441 303)(144 388 442 304)(145 389 443 305)(146 390 444 306)(147 391 445 307)(148 392 446 308)(149 393 447 309)(150 394 448 310)(151 395 449 311)(152 396 450 312)(153 397 451 313)(154 398 452 314)(155 399 453 315)(156 400 454 316)(157 401 455 317)(158 402 456 318)(159 403 457 319)(160 404 458 320)(161 405 459 321)(162 406 460 322)(163 407 461 323)(164 408 462 324)(165 409 463 325)(166 410 464 326)(167 411 465 327)(168 412 466 328)(169 413 467 329)(170 355 468 330)(171 356 469 331)(172 357 470 332)(173 358 471 333)(174 359 472 334)(175 360 414 335)(176 361 415 336)(177 362 416 337)
G:=sub<Sym(472)| (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), (1,366,109,341)(2,367,110,342)(3,368,111,343)(4,369,112,344)(5,370,113,345)(6,371,114,346)(7,372,115,347)(8,373,116,348)(9,374,117,349)(10,375,118,350)(11,376,60,351)(12,377,61,352)(13,378,62,353)(14,379,63,354)(15,380,64,296)(16,381,65,297)(17,382,66,298)(18,383,67,299)(19,384,68,300)(20,385,69,301)(21,386,70,302)(22,387,71,303)(23,388,72,304)(24,389,73,305)(25,390,74,306)(26,391,75,307)(27,392,76,308)(28,393,77,309)(29,394,78,310)(30,395,79,311)(31,396,80,312)(32,397,81,313)(33,398,82,314)(34,399,83,315)(35,400,84,316)(36,401,85,317)(37,402,86,318)(38,403,87,319)(39,404,88,320)(40,405,89,321)(41,406,90,322)(42,407,91,323)(43,408,92,324)(44,409,93,325)(45,410,94,326)(46,411,95,327)(47,412,96,328)(48,413,97,329)(49,355,98,330)(50,356,99,331)(51,357,100,332)(52,358,101,333)(53,359,102,334)(54,360,103,335)(55,361,104,336)(56,362,105,337)(57,363,106,338)(58,364,107,339)(59,365,108,340)(119,182,417,265)(120,183,418,266)(121,184,419,267)(122,185,420,268)(123,186,421,269)(124,187,422,270)(125,188,423,271)(126,189,424,272)(127,190,425,273)(128,191,426,274)(129,192,427,275)(130,193,428,276)(131,194,429,277)(132,195,430,278)(133,196,431,279)(134,197,432,280)(135,198,433,281)(136,199,434,282)(137,200,435,283)(138,201,436,284)(139,202,437,285)(140,203,438,286)(141,204,439,287)(142,205,440,288)(143,206,441,289)(144,207,442,290)(145,208,443,291)(146,209,444,292)(147,210,445,293)(148,211,446,294)(149,212,447,295)(150,213,448,237)(151,214,449,238)(152,215,450,239)(153,216,451,240)(154,217,452,241)(155,218,453,242)(156,219,454,243)(157,220,455,244)(158,221,456,245)(159,222,457,246)(160,223,458,247)(161,224,459,248)(162,225,460,249)(163,226,461,250)(164,227,462,251)(165,228,463,252)(166,229,464,253)(167,230,465,254)(168,231,466,255)(169,232,467,256)(170,233,468,257)(171,234,469,258)(172,235,470,259)(173,236,471,260)(174,178,472,261)(175,179,414,262)(176,180,415,263)(177,181,416,264), (1,268,109,185)(2,269,110,186)(3,270,111,187)(4,271,112,188)(5,272,113,189)(6,273,114,190)(7,274,115,191)(8,275,116,192)(9,276,117,193)(10,277,118,194)(11,278,60,195)(12,279,61,196)(13,280,62,197)(14,281,63,198)(15,282,64,199)(16,283,65,200)(17,284,66,201)(18,285,67,202)(19,286,68,203)(20,287,69,204)(21,288,70,205)(22,289,71,206)(23,290,72,207)(24,291,73,208)(25,292,74,209)(26,293,75,210)(27,294,76,211)(28,295,77,212)(29,237,78,213)(30,238,79,214)(31,239,80,215)(32,240,81,216)(33,241,82,217)(34,242,83,218)(35,243,84,219)(36,244,85,220)(37,245,86,221)(38,246,87,222)(39,247,88,223)(40,248,89,224)(41,249,90,225)(42,250,91,226)(43,251,92,227)(44,252,93,228)(45,253,94,229)(46,254,95,230)(47,255,96,231)(48,256,97,232)(49,257,98,233)(50,258,99,234)(51,259,100,235)(52,260,101,236)(53,261,102,178)(54,262,103,179)(55,263,104,180)(56,264,105,181)(57,265,106,182)(58,266,107,183)(59,267,108,184)(119,363,417,338)(120,364,418,339)(121,365,419,340)(122,366,420,341)(123,367,421,342)(124,368,422,343)(125,369,423,344)(126,370,424,345)(127,371,425,346)(128,372,426,347)(129,373,427,348)(130,374,428,349)(131,375,429,350)(132,376,430,351)(133,377,431,352)(134,378,432,353)(135,379,433,354)(136,380,434,296)(137,381,435,297)(138,382,436,298)(139,383,437,299)(140,384,438,300)(141,385,439,301)(142,386,440,302)(143,387,441,303)(144,388,442,304)(145,389,443,305)(146,390,444,306)(147,391,445,307)(148,392,446,308)(149,393,447,309)(150,394,448,310)(151,395,449,311)(152,396,450,312)(153,397,451,313)(154,398,452,314)(155,399,453,315)(156,400,454,316)(157,401,455,317)(158,402,456,318)(159,403,457,319)(160,404,458,320)(161,405,459,321)(162,406,460,322)(163,407,461,323)(164,408,462,324)(165,409,463,325)(166,410,464,326)(167,411,465,327)(168,412,466,328)(169,413,467,329)(170,355,468,330)(171,356,469,331)(172,357,470,332)(173,358,471,333)(174,359,472,334)(175,360,414,335)(176,361,415,336)(177,362,416,337)>;
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), (1,366,109,341)(2,367,110,342)(3,368,111,343)(4,369,112,344)(5,370,113,345)(6,371,114,346)(7,372,115,347)(8,373,116,348)(9,374,117,349)(10,375,118,350)(11,376,60,351)(12,377,61,352)(13,378,62,353)(14,379,63,354)(15,380,64,296)(16,381,65,297)(17,382,66,298)(18,383,67,299)(19,384,68,300)(20,385,69,301)(21,386,70,302)(22,387,71,303)(23,388,72,304)(24,389,73,305)(25,390,74,306)(26,391,75,307)(27,392,76,308)(28,393,77,309)(29,394,78,310)(30,395,79,311)(31,396,80,312)(32,397,81,313)(33,398,82,314)(34,399,83,315)(35,400,84,316)(36,401,85,317)(37,402,86,318)(38,403,87,319)(39,404,88,320)(40,405,89,321)(41,406,90,322)(42,407,91,323)(43,408,92,324)(44,409,93,325)(45,410,94,326)(46,411,95,327)(47,412,96,328)(48,413,97,329)(49,355,98,330)(50,356,99,331)(51,357,100,332)(52,358,101,333)(53,359,102,334)(54,360,103,335)(55,361,104,336)(56,362,105,337)(57,363,106,338)(58,364,107,339)(59,365,108,340)(119,182,417,265)(120,183,418,266)(121,184,419,267)(122,185,420,268)(123,186,421,269)(124,187,422,270)(125,188,423,271)(126,189,424,272)(127,190,425,273)(128,191,426,274)(129,192,427,275)(130,193,428,276)(131,194,429,277)(132,195,430,278)(133,196,431,279)(134,197,432,280)(135,198,433,281)(136,199,434,282)(137,200,435,283)(138,201,436,284)(139,202,437,285)(140,203,438,286)(141,204,439,287)(142,205,440,288)(143,206,441,289)(144,207,442,290)(145,208,443,291)(146,209,444,292)(147,210,445,293)(148,211,446,294)(149,212,447,295)(150,213,448,237)(151,214,449,238)(152,215,450,239)(153,216,451,240)(154,217,452,241)(155,218,453,242)(156,219,454,243)(157,220,455,244)(158,221,456,245)(159,222,457,246)(160,223,458,247)(161,224,459,248)(162,225,460,249)(163,226,461,250)(164,227,462,251)(165,228,463,252)(166,229,464,253)(167,230,465,254)(168,231,466,255)(169,232,467,256)(170,233,468,257)(171,234,469,258)(172,235,470,259)(173,236,471,260)(174,178,472,261)(175,179,414,262)(176,180,415,263)(177,181,416,264), (1,268,109,185)(2,269,110,186)(3,270,111,187)(4,271,112,188)(5,272,113,189)(6,273,114,190)(7,274,115,191)(8,275,116,192)(9,276,117,193)(10,277,118,194)(11,278,60,195)(12,279,61,196)(13,280,62,197)(14,281,63,198)(15,282,64,199)(16,283,65,200)(17,284,66,201)(18,285,67,202)(19,286,68,203)(20,287,69,204)(21,288,70,205)(22,289,71,206)(23,290,72,207)(24,291,73,208)(25,292,74,209)(26,293,75,210)(27,294,76,211)(28,295,77,212)(29,237,78,213)(30,238,79,214)(31,239,80,215)(32,240,81,216)(33,241,82,217)(34,242,83,218)(35,243,84,219)(36,244,85,220)(37,245,86,221)(38,246,87,222)(39,247,88,223)(40,248,89,224)(41,249,90,225)(42,250,91,226)(43,251,92,227)(44,252,93,228)(45,253,94,229)(46,254,95,230)(47,255,96,231)(48,256,97,232)(49,257,98,233)(50,258,99,234)(51,259,100,235)(52,260,101,236)(53,261,102,178)(54,262,103,179)(55,263,104,180)(56,264,105,181)(57,265,106,182)(58,266,107,183)(59,267,108,184)(119,363,417,338)(120,364,418,339)(121,365,419,340)(122,366,420,341)(123,367,421,342)(124,368,422,343)(125,369,423,344)(126,370,424,345)(127,371,425,346)(128,372,426,347)(129,373,427,348)(130,374,428,349)(131,375,429,350)(132,376,430,351)(133,377,431,352)(134,378,432,353)(135,379,433,354)(136,380,434,296)(137,381,435,297)(138,382,436,298)(139,383,437,299)(140,384,438,300)(141,385,439,301)(142,386,440,302)(143,387,441,303)(144,388,442,304)(145,389,443,305)(146,390,444,306)(147,391,445,307)(148,392,446,308)(149,393,447,309)(150,394,448,310)(151,395,449,311)(152,396,450,312)(153,397,451,313)(154,398,452,314)(155,399,453,315)(156,400,454,316)(157,401,455,317)(158,402,456,318)(159,403,457,319)(160,404,458,320)(161,405,459,321)(162,406,460,322)(163,407,461,323)(164,408,462,324)(165,409,463,325)(166,410,464,326)(167,411,465,327)(168,412,466,328)(169,413,467,329)(170,355,468,330)(171,356,469,331)(172,357,470,332)(173,358,471,333)(174,359,472,334)(175,360,414,335)(176,361,415,336)(177,362,416,337) );
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)], [(1,366,109,341),(2,367,110,342),(3,368,111,343),(4,369,112,344),(5,370,113,345),(6,371,114,346),(7,372,115,347),(8,373,116,348),(9,374,117,349),(10,375,118,350),(11,376,60,351),(12,377,61,352),(13,378,62,353),(14,379,63,354),(15,380,64,296),(16,381,65,297),(17,382,66,298),(18,383,67,299),(19,384,68,300),(20,385,69,301),(21,386,70,302),(22,387,71,303),(23,388,72,304),(24,389,73,305),(25,390,74,306),(26,391,75,307),(27,392,76,308),(28,393,77,309),(29,394,78,310),(30,395,79,311),(31,396,80,312),(32,397,81,313),(33,398,82,314),(34,399,83,315),(35,400,84,316),(36,401,85,317),(37,402,86,318),(38,403,87,319),(39,404,88,320),(40,405,89,321),(41,406,90,322),(42,407,91,323),(43,408,92,324),(44,409,93,325),(45,410,94,326),(46,411,95,327),(47,412,96,328),(48,413,97,329),(49,355,98,330),(50,356,99,331),(51,357,100,332),(52,358,101,333),(53,359,102,334),(54,360,103,335),(55,361,104,336),(56,362,105,337),(57,363,106,338),(58,364,107,339),(59,365,108,340),(119,182,417,265),(120,183,418,266),(121,184,419,267),(122,185,420,268),(123,186,421,269),(124,187,422,270),(125,188,423,271),(126,189,424,272),(127,190,425,273),(128,191,426,274),(129,192,427,275),(130,193,428,276),(131,194,429,277),(132,195,430,278),(133,196,431,279),(134,197,432,280),(135,198,433,281),(136,199,434,282),(137,200,435,283),(138,201,436,284),(139,202,437,285),(140,203,438,286),(141,204,439,287),(142,205,440,288),(143,206,441,289),(144,207,442,290),(145,208,443,291),(146,209,444,292),(147,210,445,293),(148,211,446,294),(149,212,447,295),(150,213,448,237),(151,214,449,238),(152,215,450,239),(153,216,451,240),(154,217,452,241),(155,218,453,242),(156,219,454,243),(157,220,455,244),(158,221,456,245),(159,222,457,246),(160,223,458,247),(161,224,459,248),(162,225,460,249),(163,226,461,250),(164,227,462,251),(165,228,463,252),(166,229,464,253),(167,230,465,254),(168,231,466,255),(169,232,467,256),(170,233,468,257),(171,234,469,258),(172,235,470,259),(173,236,471,260),(174,178,472,261),(175,179,414,262),(176,180,415,263),(177,181,416,264)], [(1,268,109,185),(2,269,110,186),(3,270,111,187),(4,271,112,188),(5,272,113,189),(6,273,114,190),(7,274,115,191),(8,275,116,192),(9,276,117,193),(10,277,118,194),(11,278,60,195),(12,279,61,196),(13,280,62,197),(14,281,63,198),(15,282,64,199),(16,283,65,200),(17,284,66,201),(18,285,67,202),(19,286,68,203),(20,287,69,204),(21,288,70,205),(22,289,71,206),(23,290,72,207),(24,291,73,208),(25,292,74,209),(26,293,75,210),(27,294,76,211),(28,295,77,212),(29,237,78,213),(30,238,79,214),(31,239,80,215),(32,240,81,216),(33,241,82,217),(34,242,83,218),(35,243,84,219),(36,244,85,220),(37,245,86,221),(38,246,87,222),(39,247,88,223),(40,248,89,224),(41,249,90,225),(42,250,91,226),(43,251,92,227),(44,252,93,228),(45,253,94,229),(46,254,95,230),(47,255,96,231),(48,256,97,232),(49,257,98,233),(50,258,99,234),(51,259,100,235),(52,260,101,236),(53,261,102,178),(54,262,103,179),(55,263,104,180),(56,264,105,181),(57,265,106,182),(58,266,107,183),(59,267,108,184),(119,363,417,338),(120,364,418,339),(121,365,419,340),(122,366,420,341),(123,367,421,342),(124,368,422,343),(125,369,423,344),(126,370,424,345),(127,371,425,346),(128,372,426,347),(129,373,427,348),(130,374,428,349),(131,375,429,350),(132,376,430,351),(133,377,431,352),(134,378,432,353),(135,379,433,354),(136,380,434,296),(137,381,435,297),(138,382,436,298),(139,383,437,299),(140,384,438,300),(141,385,439,301),(142,386,440,302),(143,387,441,303),(144,388,442,304),(145,389,443,305),(146,390,444,306),(147,391,445,307),(148,392,446,308),(149,393,447,309),(150,394,448,310),(151,395,449,311),(152,396,450,312),(153,397,451,313),(154,398,452,314),(155,399,453,315),(156,400,454,316),(157,401,455,317),(158,402,456,318),(159,403,457,319),(160,404,458,320),(161,405,459,321),(162,406,460,322),(163,407,461,323),(164,408,462,324),(165,409,463,325),(166,410,464,326),(167,411,465,327),(168,412,466,328),(169,413,467,329),(170,355,468,330),(171,356,469,331),(172,357,470,332),(173,358,471,333),(174,359,472,334),(175,360,414,335),(176,361,415,336),(177,362,416,337)]])
295 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 59A | ··· | 59BF | 118A | ··· | 118BF | 236A | ··· | 236FR |
| order | 1 | 2 | 4 | 4 | 4 | 59 | ··· | 59 | 118 | ··· | 118 | 236 | ··· | 236 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
295 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C59 | C118 | Q8 | Q8×C59 |
| kernel | Q8×C59 | C236 | Q8 | C4 | C59 | C1 |
| # reps | 1 | 3 | 58 | 174 | 1 | 58 |
Matrix representation of Q8×C59 ►in GL2(𝔽709) generated by
| 20 | 0 |
| 0 | 20 |
| 1 | 707 |
| 1 | 708 |
| 417 | 489 |
| 307 | 292 |
G:=sub<GL(2,GF(709))| [20,0,0,20],[1,1,707,708],[417,307,489,292] >;
Q8×C59 in GAP, Magma, Sage, TeX
Q_8\times C_{59} % in TeX
G:=Group("Q8xC59"); // GroupNames label
G:=SmallGroup(472,10);
// by ID
G=gap.SmallGroup(472,10);
# by ID
G:=PCGroup([4,-2,-2,-59,-2,944,1905,949]);
// Polycyclic
G:=Group<a,b,c|a^59=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export