direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C57, C4.C114, C76.7C6, C228.7C2, C12.3C38, C114.24C22, C6.7(C2×C38), C2.2(C2×C114), C38.15(C2×C6), SmallGroup(456,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C57
G = < a,b,c | a57=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C57
►Regular action on 456 points
Generators in S456
(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)
(1 320 282 417)(2 321 283 418)(3 322 284 419)(4 323 285 420)(5 324 229 421)(6 325 230 422)(7 326 231 423)(8 327 232 424)(9 328 233 425)(10 329 234 426)(11 330 235 427)(12 331 236 428)(13 332 237 429)(14 333 238 430)(15 334 239 431)(16 335 240 432)(17 336 241 433)(18 337 242 434)(19 338 243 435)(20 339 244 436)(21 340 245 437)(22 341 246 438)(23 342 247 439)(24 286 248 440)(25 287 249 441)(26 288 250 442)(27 289 251 443)(28 290 252 444)(29 291 253 445)(30 292 254 446)(31 293 255 447)(32 294 256 448)(33 295 257 449)(34 296 258 450)(35 297 259 451)(36 298 260 452)(37 299 261 453)(38 300 262 454)(39 301 263 455)(40 302 264 456)(41 303 265 400)(42 304 266 401)(43 305 267 402)(44 306 268 403)(45 307 269 404)(46 308 270 405)(47 309 271 406)(48 310 272 407)(49 311 273 408)(50 312 274 409)(51 313 275 410)(52 314 276 411)(53 315 277 412)(54 316 278 413)(55 317 279 414)(56 318 280 415)(57 319 281 416)(58 181 369 148)(59 182 370 149)(60 183 371 150)(61 184 372 151)(62 185 373 152)(63 186 374 153)(64 187 375 154)(65 188 376 155)(66 189 377 156)(67 190 378 157)(68 191 379 158)(69 192 380 159)(70 193 381 160)(71 194 382 161)(72 195 383 162)(73 196 384 163)(74 197 385 164)(75 198 386 165)(76 199 387 166)(77 200 388 167)(78 201 389 168)(79 202 390 169)(80 203 391 170)(81 204 392 171)(82 205 393 115)(83 206 394 116)(84 207 395 117)(85 208 396 118)(86 209 397 119)(87 210 398 120)(88 211 399 121)(89 212 343 122)(90 213 344 123)(91 214 345 124)(92 215 346 125)(93 216 347 126)(94 217 348 127)(95 218 349 128)(96 219 350 129)(97 220 351 130)(98 221 352 131)(99 222 353 132)(100 223 354 133)(101 224 355 134)(102 225 356 135)(103 226 357 136)(104 227 358 137)(105 228 359 138)(106 172 360 139)(107 173 361 140)(108 174 362 141)(109 175 363 142)(110 176 364 143)(111 177 365 144)(112 178 366 145)(113 179 367 146)(114 180 368 147)
(1 59 282 370)(2 60 283 371)(3 61 284 372)(4 62 285 373)(5 63 229 374)(6 64 230 375)(7 65 231 376)(8 66 232 377)(9 67 233 378)(10 68 234 379)(11 69 235 380)(12 70 236 381)(13 71 237 382)(14 72 238 383)(15 73 239 384)(16 74 240 385)(17 75 241 386)(18 76 242 387)(19 77 243 388)(20 78 244 389)(21 79 245 390)(22 80 246 391)(23 81 247 392)(24 82 248 393)(25 83 249 394)(26 84 250 395)(27 85 251 396)(28 86 252 397)(29 87 253 398)(30 88 254 399)(31 89 255 343)(32 90 256 344)(33 91 257 345)(34 92 258 346)(35 93 259 347)(36 94 260 348)(37 95 261 349)(38 96 262 350)(39 97 263 351)(40 98 264 352)(41 99 265 353)(42 100 266 354)(43 101 267 355)(44 102 268 356)(45 103 269 357)(46 104 270 358)(47 105 271 359)(48 106 272 360)(49 107 273 361)(50 108 274 362)(51 109 275 363)(52 110 276 364)(53 111 277 365)(54 112 278 366)(55 113 279 367)(56 114 280 368)(57 58 281 369)(115 440 205 286)(116 441 206 287)(117 442 207 288)(118 443 208 289)(119 444 209 290)(120 445 210 291)(121 446 211 292)(122 447 212 293)(123 448 213 294)(124 449 214 295)(125 450 215 296)(126 451 216 297)(127 452 217 298)(128 453 218 299)(129 454 219 300)(130 455 220 301)(131 456 221 302)(132 400 222 303)(133 401 223 304)(134 402 224 305)(135 403 225 306)(136 404 226 307)(137 405 227 308)(138 406 228 309)(139 407 172 310)(140 408 173 311)(141 409 174 312)(142 410 175 313)(143 411 176 314)(144 412 177 315)(145 413 178 316)(146 414 179 317)(147 415 180 318)(148 416 181 319)(149 417 182 320)(150 418 183 321)(151 419 184 322)(152 420 185 323)(153 421 186 324)(154 422 187 325)(155 423 188 326)(156 424 189 327)(157 425 190 328)(158 426 191 329)(159 427 192 330)(160 428 193 331)(161 429 194 332)(162 430 195 333)(163 431 196 334)(164 432 197 335)(165 433 198 336)(166 434 199 337)(167 435 200 338)(168 436 201 339)(169 437 202 340)(170 438 203 341)(171 439 204 342)
G:=sub<Sym(456)| (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), (1,320,282,417)(2,321,283,418)(3,322,284,419)(4,323,285,420)(5,324,229,421)(6,325,230,422)(7,326,231,423)(8,327,232,424)(9,328,233,425)(10,329,234,426)(11,330,235,427)(12,331,236,428)(13,332,237,429)(14,333,238,430)(15,334,239,431)(16,335,240,432)(17,336,241,433)(18,337,242,434)(19,338,243,435)(20,339,244,436)(21,340,245,437)(22,341,246,438)(23,342,247,439)(24,286,248,440)(25,287,249,441)(26,288,250,442)(27,289,251,443)(28,290,252,444)(29,291,253,445)(30,292,254,446)(31,293,255,447)(32,294,256,448)(33,295,257,449)(34,296,258,450)(35,297,259,451)(36,298,260,452)(37,299,261,453)(38,300,262,454)(39,301,263,455)(40,302,264,456)(41,303,265,400)(42,304,266,401)(43,305,267,402)(44,306,268,403)(45,307,269,404)(46,308,270,405)(47,309,271,406)(48,310,272,407)(49,311,273,408)(50,312,274,409)(51,313,275,410)(52,314,276,411)(53,315,277,412)(54,316,278,413)(55,317,279,414)(56,318,280,415)(57,319,281,416)(58,181,369,148)(59,182,370,149)(60,183,371,150)(61,184,372,151)(62,185,373,152)(63,186,374,153)(64,187,375,154)(65,188,376,155)(66,189,377,156)(67,190,378,157)(68,191,379,158)(69,192,380,159)(70,193,381,160)(71,194,382,161)(72,195,383,162)(73,196,384,163)(74,197,385,164)(75,198,386,165)(76,199,387,166)(77,200,388,167)(78,201,389,168)(79,202,390,169)(80,203,391,170)(81,204,392,171)(82,205,393,115)(83,206,394,116)(84,207,395,117)(85,208,396,118)(86,209,397,119)(87,210,398,120)(88,211,399,121)(89,212,343,122)(90,213,344,123)(91,214,345,124)(92,215,346,125)(93,216,347,126)(94,217,348,127)(95,218,349,128)(96,219,350,129)(97,220,351,130)(98,221,352,131)(99,222,353,132)(100,223,354,133)(101,224,355,134)(102,225,356,135)(103,226,357,136)(104,227,358,137)(105,228,359,138)(106,172,360,139)(107,173,361,140)(108,174,362,141)(109,175,363,142)(110,176,364,143)(111,177,365,144)(112,178,366,145)(113,179,367,146)(114,180,368,147), (1,59,282,370)(2,60,283,371)(3,61,284,372)(4,62,285,373)(5,63,229,374)(6,64,230,375)(7,65,231,376)(8,66,232,377)(9,67,233,378)(10,68,234,379)(11,69,235,380)(12,70,236,381)(13,71,237,382)(14,72,238,383)(15,73,239,384)(16,74,240,385)(17,75,241,386)(18,76,242,387)(19,77,243,388)(20,78,244,389)(21,79,245,390)(22,80,246,391)(23,81,247,392)(24,82,248,393)(25,83,249,394)(26,84,250,395)(27,85,251,396)(28,86,252,397)(29,87,253,398)(30,88,254,399)(31,89,255,343)(32,90,256,344)(33,91,257,345)(34,92,258,346)(35,93,259,347)(36,94,260,348)(37,95,261,349)(38,96,262,350)(39,97,263,351)(40,98,264,352)(41,99,265,353)(42,100,266,354)(43,101,267,355)(44,102,268,356)(45,103,269,357)(46,104,270,358)(47,105,271,359)(48,106,272,360)(49,107,273,361)(50,108,274,362)(51,109,275,363)(52,110,276,364)(53,111,277,365)(54,112,278,366)(55,113,279,367)(56,114,280,368)(57,58,281,369)(115,440,205,286)(116,441,206,287)(117,442,207,288)(118,443,208,289)(119,444,209,290)(120,445,210,291)(121,446,211,292)(122,447,212,293)(123,448,213,294)(124,449,214,295)(125,450,215,296)(126,451,216,297)(127,452,217,298)(128,453,218,299)(129,454,219,300)(130,455,220,301)(131,456,221,302)(132,400,222,303)(133,401,223,304)(134,402,224,305)(135,403,225,306)(136,404,226,307)(137,405,227,308)(138,406,228,309)(139,407,172,310)(140,408,173,311)(141,409,174,312)(142,410,175,313)(143,411,176,314)(144,412,177,315)(145,413,178,316)(146,414,179,317)(147,415,180,318)(148,416,181,319)(149,417,182,320)(150,418,183,321)(151,419,184,322)(152,420,185,323)(153,421,186,324)(154,422,187,325)(155,423,188,326)(156,424,189,327)(157,425,190,328)(158,426,191,329)(159,427,192,330)(160,428,193,331)(161,429,194,332)(162,430,195,333)(163,431,196,334)(164,432,197,335)(165,433,198,336)(166,434,199,337)(167,435,200,338)(168,436,201,339)(169,437,202,340)(170,438,203,341)(171,439,204,342)>;
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), (1,320,282,417)(2,321,283,418)(3,322,284,419)(4,323,285,420)(5,324,229,421)(6,325,230,422)(7,326,231,423)(8,327,232,424)(9,328,233,425)(10,329,234,426)(11,330,235,427)(12,331,236,428)(13,332,237,429)(14,333,238,430)(15,334,239,431)(16,335,240,432)(17,336,241,433)(18,337,242,434)(19,338,243,435)(20,339,244,436)(21,340,245,437)(22,341,246,438)(23,342,247,439)(24,286,248,440)(25,287,249,441)(26,288,250,442)(27,289,251,443)(28,290,252,444)(29,291,253,445)(30,292,254,446)(31,293,255,447)(32,294,256,448)(33,295,257,449)(34,296,258,450)(35,297,259,451)(36,298,260,452)(37,299,261,453)(38,300,262,454)(39,301,263,455)(40,302,264,456)(41,303,265,400)(42,304,266,401)(43,305,267,402)(44,306,268,403)(45,307,269,404)(46,308,270,405)(47,309,271,406)(48,310,272,407)(49,311,273,408)(50,312,274,409)(51,313,275,410)(52,314,276,411)(53,315,277,412)(54,316,278,413)(55,317,279,414)(56,318,280,415)(57,319,281,416)(58,181,369,148)(59,182,370,149)(60,183,371,150)(61,184,372,151)(62,185,373,152)(63,186,374,153)(64,187,375,154)(65,188,376,155)(66,189,377,156)(67,190,378,157)(68,191,379,158)(69,192,380,159)(70,193,381,160)(71,194,382,161)(72,195,383,162)(73,196,384,163)(74,197,385,164)(75,198,386,165)(76,199,387,166)(77,200,388,167)(78,201,389,168)(79,202,390,169)(80,203,391,170)(81,204,392,171)(82,205,393,115)(83,206,394,116)(84,207,395,117)(85,208,396,118)(86,209,397,119)(87,210,398,120)(88,211,399,121)(89,212,343,122)(90,213,344,123)(91,214,345,124)(92,215,346,125)(93,216,347,126)(94,217,348,127)(95,218,349,128)(96,219,350,129)(97,220,351,130)(98,221,352,131)(99,222,353,132)(100,223,354,133)(101,224,355,134)(102,225,356,135)(103,226,357,136)(104,227,358,137)(105,228,359,138)(106,172,360,139)(107,173,361,140)(108,174,362,141)(109,175,363,142)(110,176,364,143)(111,177,365,144)(112,178,366,145)(113,179,367,146)(114,180,368,147), (1,59,282,370)(2,60,283,371)(3,61,284,372)(4,62,285,373)(5,63,229,374)(6,64,230,375)(7,65,231,376)(8,66,232,377)(9,67,233,378)(10,68,234,379)(11,69,235,380)(12,70,236,381)(13,71,237,382)(14,72,238,383)(15,73,239,384)(16,74,240,385)(17,75,241,386)(18,76,242,387)(19,77,243,388)(20,78,244,389)(21,79,245,390)(22,80,246,391)(23,81,247,392)(24,82,248,393)(25,83,249,394)(26,84,250,395)(27,85,251,396)(28,86,252,397)(29,87,253,398)(30,88,254,399)(31,89,255,343)(32,90,256,344)(33,91,257,345)(34,92,258,346)(35,93,259,347)(36,94,260,348)(37,95,261,349)(38,96,262,350)(39,97,263,351)(40,98,264,352)(41,99,265,353)(42,100,266,354)(43,101,267,355)(44,102,268,356)(45,103,269,357)(46,104,270,358)(47,105,271,359)(48,106,272,360)(49,107,273,361)(50,108,274,362)(51,109,275,363)(52,110,276,364)(53,111,277,365)(54,112,278,366)(55,113,279,367)(56,114,280,368)(57,58,281,369)(115,440,205,286)(116,441,206,287)(117,442,207,288)(118,443,208,289)(119,444,209,290)(120,445,210,291)(121,446,211,292)(122,447,212,293)(123,448,213,294)(124,449,214,295)(125,450,215,296)(126,451,216,297)(127,452,217,298)(128,453,218,299)(129,454,219,300)(130,455,220,301)(131,456,221,302)(132,400,222,303)(133,401,223,304)(134,402,224,305)(135,403,225,306)(136,404,226,307)(137,405,227,308)(138,406,228,309)(139,407,172,310)(140,408,173,311)(141,409,174,312)(142,410,175,313)(143,411,176,314)(144,412,177,315)(145,413,178,316)(146,414,179,317)(147,415,180,318)(148,416,181,319)(149,417,182,320)(150,418,183,321)(151,419,184,322)(152,420,185,323)(153,421,186,324)(154,422,187,325)(155,423,188,326)(156,424,189,327)(157,425,190,328)(158,426,191,329)(159,427,192,330)(160,428,193,331)(161,429,194,332)(162,430,195,333)(163,431,196,334)(164,432,197,335)(165,433,198,336)(166,434,199,337)(167,435,200,338)(168,436,201,339)(169,437,202,340)(170,438,203,341)(171,439,204,342) );
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)], [(1,320,282,417),(2,321,283,418),(3,322,284,419),(4,323,285,420),(5,324,229,421),(6,325,230,422),(7,326,231,423),(8,327,232,424),(9,328,233,425),(10,329,234,426),(11,330,235,427),(12,331,236,428),(13,332,237,429),(14,333,238,430),(15,334,239,431),(16,335,240,432),(17,336,241,433),(18,337,242,434),(19,338,243,435),(20,339,244,436),(21,340,245,437),(22,341,246,438),(23,342,247,439),(24,286,248,440),(25,287,249,441),(26,288,250,442),(27,289,251,443),(28,290,252,444),(29,291,253,445),(30,292,254,446),(31,293,255,447),(32,294,256,448),(33,295,257,449),(34,296,258,450),(35,297,259,451),(36,298,260,452),(37,299,261,453),(38,300,262,454),(39,301,263,455),(40,302,264,456),(41,303,265,400),(42,304,266,401),(43,305,267,402),(44,306,268,403),(45,307,269,404),(46,308,270,405),(47,309,271,406),(48,310,272,407),(49,311,273,408),(50,312,274,409),(51,313,275,410),(52,314,276,411),(53,315,277,412),(54,316,278,413),(55,317,279,414),(56,318,280,415),(57,319,281,416),(58,181,369,148),(59,182,370,149),(60,183,371,150),(61,184,372,151),(62,185,373,152),(63,186,374,153),(64,187,375,154),(65,188,376,155),(66,189,377,156),(67,190,378,157),(68,191,379,158),(69,192,380,159),(70,193,381,160),(71,194,382,161),(72,195,383,162),(73,196,384,163),(74,197,385,164),(75,198,386,165),(76,199,387,166),(77,200,388,167),(78,201,389,168),(79,202,390,169),(80,203,391,170),(81,204,392,171),(82,205,393,115),(83,206,394,116),(84,207,395,117),(85,208,396,118),(86,209,397,119),(87,210,398,120),(88,211,399,121),(89,212,343,122),(90,213,344,123),(91,214,345,124),(92,215,346,125),(93,216,347,126),(94,217,348,127),(95,218,349,128),(96,219,350,129),(97,220,351,130),(98,221,352,131),(99,222,353,132),(100,223,354,133),(101,224,355,134),(102,225,356,135),(103,226,357,136),(104,227,358,137),(105,228,359,138),(106,172,360,139),(107,173,361,140),(108,174,362,141),(109,175,363,142),(110,176,364,143),(111,177,365,144),(112,178,366,145),(113,179,367,146),(114,180,368,147)], [(1,59,282,370),(2,60,283,371),(3,61,284,372),(4,62,285,373),(5,63,229,374),(6,64,230,375),(7,65,231,376),(8,66,232,377),(9,67,233,378),(10,68,234,379),(11,69,235,380),(12,70,236,381),(13,71,237,382),(14,72,238,383),(15,73,239,384),(16,74,240,385),(17,75,241,386),(18,76,242,387),(19,77,243,388),(20,78,244,389),(21,79,245,390),(22,80,246,391),(23,81,247,392),(24,82,248,393),(25,83,249,394),(26,84,250,395),(27,85,251,396),(28,86,252,397),(29,87,253,398),(30,88,254,399),(31,89,255,343),(32,90,256,344),(33,91,257,345),(34,92,258,346),(35,93,259,347),(36,94,260,348),(37,95,261,349),(38,96,262,350),(39,97,263,351),(40,98,264,352),(41,99,265,353),(42,100,266,354),(43,101,267,355),(44,102,268,356),(45,103,269,357),(46,104,270,358),(47,105,271,359),(48,106,272,360),(49,107,273,361),(50,108,274,362),(51,109,275,363),(52,110,276,364),(53,111,277,365),(54,112,278,366),(55,113,279,367),(56,114,280,368),(57,58,281,369),(115,440,205,286),(116,441,206,287),(117,442,207,288),(118,443,208,289),(119,444,209,290),(120,445,210,291),(121,446,211,292),(122,447,212,293),(123,448,213,294),(124,449,214,295),(125,450,215,296),(126,451,216,297),(127,452,217,298),(128,453,218,299),(129,454,219,300),(130,455,220,301),(131,456,221,302),(132,400,222,303),(133,401,223,304),(134,402,224,305),(135,403,225,306),(136,404,226,307),(137,405,227,308),(138,406,228,309),(139,407,172,310),(140,408,173,311),(141,409,174,312),(142,410,175,313),(143,411,176,314),(144,412,177,315),(145,413,178,316),(146,414,179,317),(147,415,180,318),(148,416,181,319),(149,417,182,320),(150,418,183,321),(151,419,184,322),(152,420,185,323),(153,421,186,324),(154,422,187,325),(155,423,188,326),(156,424,189,327),(157,425,190,328),(158,426,191,329),(159,427,192,330),(160,428,193,331),(161,429,194,332),(162,430,195,333),(163,431,196,334),(164,432,197,335),(165,433,198,336),(166,434,199,337),(167,435,200,338),(168,436,201,339),(169,437,202,340),(170,438,203,341),(171,439,204,342)]])
285 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 12A | ··· | 12F | 19A | ··· | 19R | 38A | ··· | 38R | 57A | ··· | 57AJ | 76A | ··· | 76BB | 114A | ··· | 114AJ | 228A | ··· | 228DD |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
285 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | - | |||||||||
| image | C1 | C2 | C3 | C6 | C19 | C38 | C57 | C114 | Q8 | C3×Q8 | Q8×C19 | Q8×C57 |
| kernel | Q8×C57 | C228 | Q8×C19 | C76 | C3×Q8 | C12 | Q8 | C4 | C57 | C19 | C3 | C1 |
| # reps | 1 | 3 | 2 | 6 | 18 | 54 | 36 | 108 | 1 | 2 | 18 | 36 |
Matrix representation of Q8×C57 ►in GL3(𝔽229) generated by
| 94 | 0 | 0 |
| 0 | 44 | 0 |
| 0 | 0 | 44 |
| 1 | 0 | 0 |
| 0 | 228 | 227 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 143 | 87 |
| 0 | 15 | 86 |
G:=sub<GL(3,GF(229))| [94,0,0,0,44,0,0,0,44],[1,0,0,0,228,1,0,227,1],[1,0,0,0,143,15,0,87,86] >;
Q8×C57 in GAP, Magma, Sage, TeX
Q_8\times C_{57} % in TeX
G:=Group("Q8xC57"); // GroupNames label
G:=SmallGroup(456,41);
// by ID
G=gap.SmallGroup(456,41);
# by ID
G:=PCGroup([5,-2,-2,-3,-19,-2,1140,2301,1146]);
// Polycyclic
G:=Group<a,b,c|a^57=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export