metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C57⋊Q8, C38.7D6, C6.7D38, C3⋊1Dic38, C19⋊1Dic6, Dic19.S3, Dic3.D19, C114.7C22, Dic57.2C2, C2.7(S3×D19), (C3×Dic19).1C2, (Dic3×C19).1C2, SmallGroup(456,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C57⋊Q8
G = < a,b,c | a57=b4=1, c2=b2, bab-1=a20, cac-1=a37, cbc-1=b-1 >
Smallest permutation representation of C57⋊Q8
►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)
(1 172 112 147)(2 192 113 167)(3 212 114 130)(4 175 58 150)(5 195 59 170)(6 215 60 133)(7 178 61 153)(8 198 62 116)(9 218 63 136)(10 181 64 156)(11 201 65 119)(12 221 66 139)(13 184 67 159)(14 204 68 122)(15 224 69 142)(16 187 70 162)(17 207 71 125)(18 227 72 145)(19 190 73 165)(20 210 74 128)(21 173 75 148)(22 193 76 168)(23 213 77 131)(24 176 78 151)(25 196 79 171)(26 216 80 134)(27 179 81 154)(28 199 82 117)(29 219 83 137)(30 182 84 157)(31 202 85 120)(32 222 86 140)(33 185 87 160)(34 205 88 123)(35 225 89 143)(36 188 90 163)(37 208 91 126)(38 228 92 146)(39 191 93 166)(40 211 94 129)(41 174 95 149)(42 194 96 169)(43 214 97 132)(44 177 98 152)(45 197 99 115)(46 217 100 135)(47 180 101 155)(48 200 102 118)(49 220 103 138)(50 183 104 158)(51 203 105 121)(52 223 106 141)(53 186 107 161)(54 206 108 124)(55 226 109 144)(56 189 110 164)(57 209 111 127)(229 357 337 428)(230 377 338 448)(231 397 339 411)(232 360 340 431)(233 380 341 451)(234 343 342 414)(235 363 286 434)(236 383 287 454)(237 346 288 417)(238 366 289 437)(239 386 290 400)(240 349 291 420)(241 369 292 440)(242 389 293 403)(243 352 294 423)(244 372 295 443)(245 392 296 406)(246 355 297 426)(247 375 298 446)(248 395 299 409)(249 358 300 429)(250 378 301 449)(251 398 302 412)(252 361 303 432)(253 381 304 452)(254 344 305 415)(255 364 306 435)(256 384 307 455)(257 347 308 418)(258 367 309 438)(259 387 310 401)(260 350 311 421)(261 370 312 441)(262 390 313 404)(263 353 314 424)(264 373 315 444)(265 393 316 407)(266 356 317 427)(267 376 318 447)(268 396 319 410)(269 359 320 430)(270 379 321 450)(271 399 322 413)(272 362 323 433)(273 382 324 453)(274 345 325 416)(275 365 326 436)(276 385 327 456)(277 348 328 419)(278 368 329 439)(279 388 330 402)(280 351 331 422)(281 371 332 442)(282 391 333 405)(283 354 334 425)(284 374 335 445)(285 394 336 408)
(1 318 112 267)(2 298 113 247)(3 335 114 284)(4 315 58 264)(5 295 59 244)(6 332 60 281)(7 312 61 261)(8 292 62 241)(9 329 63 278)(10 309 64 258)(11 289 65 238)(12 326 66 275)(13 306 67 255)(14 286 68 235)(15 323 69 272)(16 303 70 252)(17 340 71 232)(18 320 72 269)(19 300 73 249)(20 337 74 229)(21 317 75 266)(22 297 76 246)(23 334 77 283)(24 314 78 263)(25 294 79 243)(26 331 80 280)(27 311 81 260)(28 291 82 240)(29 328 83 277)(30 308 84 257)(31 288 85 237)(32 325 86 274)(33 305 87 254)(34 342 88 234)(35 322 89 271)(36 302 90 251)(37 339 91 231)(38 319 92 268)(39 299 93 248)(40 336 94 285)(41 316 95 265)(42 296 96 245)(43 333 97 282)(44 313 98 262)(45 293 99 242)(46 330 100 279)(47 310 101 259)(48 290 102 239)(49 327 103 276)(50 307 104 256)(51 287 105 236)(52 324 106 273)(53 304 107 253)(54 341 108 233)(55 321 109 270)(56 301 110 250)(57 338 111 230)(115 403 197 389)(116 440 198 369)(117 420 199 349)(118 400 200 386)(119 437 201 366)(120 417 202 346)(121 454 203 383)(122 434 204 363)(123 414 205 343)(124 451 206 380)(125 431 207 360)(126 411 208 397)(127 448 209 377)(128 428 210 357)(129 408 211 394)(130 445 212 374)(131 425 213 354)(132 405 214 391)(133 442 215 371)(134 422 216 351)(135 402 217 388)(136 439 218 368)(137 419 219 348)(138 456 220 385)(139 436 221 365)(140 416 222 345)(141 453 223 382)(142 433 224 362)(143 413 225 399)(144 450 226 379)(145 430 227 359)(146 410 228 396)(147 447 172 376)(148 427 173 356)(149 407 174 393)(150 444 175 373)(151 424 176 353)(152 404 177 390)(153 441 178 370)(154 421 179 350)(155 401 180 387)(156 438 181 367)(157 418 182 347)(158 455 183 384)(159 435 184 364)(160 415 185 344)(161 452 186 381)(162 432 187 361)(163 412 188 398)(164 449 189 378)(165 429 190 358)(166 409 191 395)(167 446 192 375)(168 426 193 355)(169 406 194 392)(170 443 195 372)(171 423 196 352)
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), (1,172,112,147)(2,192,113,167)(3,212,114,130)(4,175,58,150)(5,195,59,170)(6,215,60,133)(7,178,61,153)(8,198,62,116)(9,218,63,136)(10,181,64,156)(11,201,65,119)(12,221,66,139)(13,184,67,159)(14,204,68,122)(15,224,69,142)(16,187,70,162)(17,207,71,125)(18,227,72,145)(19,190,73,165)(20,210,74,128)(21,173,75,148)(22,193,76,168)(23,213,77,131)(24,176,78,151)(25,196,79,171)(26,216,80,134)(27,179,81,154)(28,199,82,117)(29,219,83,137)(30,182,84,157)(31,202,85,120)(32,222,86,140)(33,185,87,160)(34,205,88,123)(35,225,89,143)(36,188,90,163)(37,208,91,126)(38,228,92,146)(39,191,93,166)(40,211,94,129)(41,174,95,149)(42,194,96,169)(43,214,97,132)(44,177,98,152)(45,197,99,115)(46,217,100,135)(47,180,101,155)(48,200,102,118)(49,220,103,138)(50,183,104,158)(51,203,105,121)(52,223,106,141)(53,186,107,161)(54,206,108,124)(55,226,109,144)(56,189,110,164)(57,209,111,127)(229,357,337,428)(230,377,338,448)(231,397,339,411)(232,360,340,431)(233,380,341,451)(234,343,342,414)(235,363,286,434)(236,383,287,454)(237,346,288,417)(238,366,289,437)(239,386,290,400)(240,349,291,420)(241,369,292,440)(242,389,293,403)(243,352,294,423)(244,372,295,443)(245,392,296,406)(246,355,297,426)(247,375,298,446)(248,395,299,409)(249,358,300,429)(250,378,301,449)(251,398,302,412)(252,361,303,432)(253,381,304,452)(254,344,305,415)(255,364,306,435)(256,384,307,455)(257,347,308,418)(258,367,309,438)(259,387,310,401)(260,350,311,421)(261,370,312,441)(262,390,313,404)(263,353,314,424)(264,373,315,444)(265,393,316,407)(266,356,317,427)(267,376,318,447)(268,396,319,410)(269,359,320,430)(270,379,321,450)(271,399,322,413)(272,362,323,433)(273,382,324,453)(274,345,325,416)(275,365,326,436)(276,385,327,456)(277,348,328,419)(278,368,329,439)(279,388,330,402)(280,351,331,422)(281,371,332,442)(282,391,333,405)(283,354,334,425)(284,374,335,445)(285,394,336,408), (1,318,112,267)(2,298,113,247)(3,335,114,284)(4,315,58,264)(5,295,59,244)(6,332,60,281)(7,312,61,261)(8,292,62,241)(9,329,63,278)(10,309,64,258)(11,289,65,238)(12,326,66,275)(13,306,67,255)(14,286,68,235)(15,323,69,272)(16,303,70,252)(17,340,71,232)(18,320,72,269)(19,300,73,249)(20,337,74,229)(21,317,75,266)(22,297,76,246)(23,334,77,283)(24,314,78,263)(25,294,79,243)(26,331,80,280)(27,311,81,260)(28,291,82,240)(29,328,83,277)(30,308,84,257)(31,288,85,237)(32,325,86,274)(33,305,87,254)(34,342,88,234)(35,322,89,271)(36,302,90,251)(37,339,91,231)(38,319,92,268)(39,299,93,248)(40,336,94,285)(41,316,95,265)(42,296,96,245)(43,333,97,282)(44,313,98,262)(45,293,99,242)(46,330,100,279)(47,310,101,259)(48,290,102,239)(49,327,103,276)(50,307,104,256)(51,287,105,236)(52,324,106,273)(53,304,107,253)(54,341,108,233)(55,321,109,270)(56,301,110,250)(57,338,111,230)(115,403,197,389)(116,440,198,369)(117,420,199,349)(118,400,200,386)(119,437,201,366)(120,417,202,346)(121,454,203,383)(122,434,204,363)(123,414,205,343)(124,451,206,380)(125,431,207,360)(126,411,208,397)(127,448,209,377)(128,428,210,357)(129,408,211,394)(130,445,212,374)(131,425,213,354)(132,405,214,391)(133,442,215,371)(134,422,216,351)(135,402,217,388)(136,439,218,368)(137,419,219,348)(138,456,220,385)(139,436,221,365)(140,416,222,345)(141,453,223,382)(142,433,224,362)(143,413,225,399)(144,450,226,379)(145,430,227,359)(146,410,228,396)(147,447,172,376)(148,427,173,356)(149,407,174,393)(150,444,175,373)(151,424,176,353)(152,404,177,390)(153,441,178,370)(154,421,179,350)(155,401,180,387)(156,438,181,367)(157,418,182,347)(158,455,183,384)(159,435,184,364)(160,415,185,344)(161,452,186,381)(162,432,187,361)(163,412,188,398)(164,449,189,378)(165,429,190,358)(166,409,191,395)(167,446,192,375)(168,426,193,355)(169,406,194,392)(170,443,195,372)(171,423,196,352)>;
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), (1,172,112,147)(2,192,113,167)(3,212,114,130)(4,175,58,150)(5,195,59,170)(6,215,60,133)(7,178,61,153)(8,198,62,116)(9,218,63,136)(10,181,64,156)(11,201,65,119)(12,221,66,139)(13,184,67,159)(14,204,68,122)(15,224,69,142)(16,187,70,162)(17,207,71,125)(18,227,72,145)(19,190,73,165)(20,210,74,128)(21,173,75,148)(22,193,76,168)(23,213,77,131)(24,176,78,151)(25,196,79,171)(26,216,80,134)(27,179,81,154)(28,199,82,117)(29,219,83,137)(30,182,84,157)(31,202,85,120)(32,222,86,140)(33,185,87,160)(34,205,88,123)(35,225,89,143)(36,188,90,163)(37,208,91,126)(38,228,92,146)(39,191,93,166)(40,211,94,129)(41,174,95,149)(42,194,96,169)(43,214,97,132)(44,177,98,152)(45,197,99,115)(46,217,100,135)(47,180,101,155)(48,200,102,118)(49,220,103,138)(50,183,104,158)(51,203,105,121)(52,223,106,141)(53,186,107,161)(54,206,108,124)(55,226,109,144)(56,189,110,164)(57,209,111,127)(229,357,337,428)(230,377,338,448)(231,397,339,411)(232,360,340,431)(233,380,341,451)(234,343,342,414)(235,363,286,434)(236,383,287,454)(237,346,288,417)(238,366,289,437)(239,386,290,400)(240,349,291,420)(241,369,292,440)(242,389,293,403)(243,352,294,423)(244,372,295,443)(245,392,296,406)(246,355,297,426)(247,375,298,446)(248,395,299,409)(249,358,300,429)(250,378,301,449)(251,398,302,412)(252,361,303,432)(253,381,304,452)(254,344,305,415)(255,364,306,435)(256,384,307,455)(257,347,308,418)(258,367,309,438)(259,387,310,401)(260,350,311,421)(261,370,312,441)(262,390,313,404)(263,353,314,424)(264,373,315,444)(265,393,316,407)(266,356,317,427)(267,376,318,447)(268,396,319,410)(269,359,320,430)(270,379,321,450)(271,399,322,413)(272,362,323,433)(273,382,324,453)(274,345,325,416)(275,365,326,436)(276,385,327,456)(277,348,328,419)(278,368,329,439)(279,388,330,402)(280,351,331,422)(281,371,332,442)(282,391,333,405)(283,354,334,425)(284,374,335,445)(285,394,336,408), (1,318,112,267)(2,298,113,247)(3,335,114,284)(4,315,58,264)(5,295,59,244)(6,332,60,281)(7,312,61,261)(8,292,62,241)(9,329,63,278)(10,309,64,258)(11,289,65,238)(12,326,66,275)(13,306,67,255)(14,286,68,235)(15,323,69,272)(16,303,70,252)(17,340,71,232)(18,320,72,269)(19,300,73,249)(20,337,74,229)(21,317,75,266)(22,297,76,246)(23,334,77,283)(24,314,78,263)(25,294,79,243)(26,331,80,280)(27,311,81,260)(28,291,82,240)(29,328,83,277)(30,308,84,257)(31,288,85,237)(32,325,86,274)(33,305,87,254)(34,342,88,234)(35,322,89,271)(36,302,90,251)(37,339,91,231)(38,319,92,268)(39,299,93,248)(40,336,94,285)(41,316,95,265)(42,296,96,245)(43,333,97,282)(44,313,98,262)(45,293,99,242)(46,330,100,279)(47,310,101,259)(48,290,102,239)(49,327,103,276)(50,307,104,256)(51,287,105,236)(52,324,106,273)(53,304,107,253)(54,341,108,233)(55,321,109,270)(56,301,110,250)(57,338,111,230)(115,403,197,389)(116,440,198,369)(117,420,199,349)(118,400,200,386)(119,437,201,366)(120,417,202,346)(121,454,203,383)(122,434,204,363)(123,414,205,343)(124,451,206,380)(125,431,207,360)(126,411,208,397)(127,448,209,377)(128,428,210,357)(129,408,211,394)(130,445,212,374)(131,425,213,354)(132,405,214,391)(133,442,215,371)(134,422,216,351)(135,402,217,388)(136,439,218,368)(137,419,219,348)(138,456,220,385)(139,436,221,365)(140,416,222,345)(141,453,223,382)(142,433,224,362)(143,413,225,399)(144,450,226,379)(145,430,227,359)(146,410,228,396)(147,447,172,376)(148,427,173,356)(149,407,174,393)(150,444,175,373)(151,424,176,353)(152,404,177,390)(153,441,178,370)(154,421,179,350)(155,401,180,387)(156,438,181,367)(157,418,182,347)(158,455,183,384)(159,435,184,364)(160,415,185,344)(161,452,186,381)(162,432,187,361)(163,412,188,398)(164,449,189,378)(165,429,190,358)(166,409,191,395)(167,446,192,375)(168,426,193,355)(169,406,194,392)(170,443,195,372)(171,423,196,352) );
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)], [(1,172,112,147),(2,192,113,167),(3,212,114,130),(4,175,58,150),(5,195,59,170),(6,215,60,133),(7,178,61,153),(8,198,62,116),(9,218,63,136),(10,181,64,156),(11,201,65,119),(12,221,66,139),(13,184,67,159),(14,204,68,122),(15,224,69,142),(16,187,70,162),(17,207,71,125),(18,227,72,145),(19,190,73,165),(20,210,74,128),(21,173,75,148),(22,193,76,168),(23,213,77,131),(24,176,78,151),(25,196,79,171),(26,216,80,134),(27,179,81,154),(28,199,82,117),(29,219,83,137),(30,182,84,157),(31,202,85,120),(32,222,86,140),(33,185,87,160),(34,205,88,123),(35,225,89,143),(36,188,90,163),(37,208,91,126),(38,228,92,146),(39,191,93,166),(40,211,94,129),(41,174,95,149),(42,194,96,169),(43,214,97,132),(44,177,98,152),(45,197,99,115),(46,217,100,135),(47,180,101,155),(48,200,102,118),(49,220,103,138),(50,183,104,158),(51,203,105,121),(52,223,106,141),(53,186,107,161),(54,206,108,124),(55,226,109,144),(56,189,110,164),(57,209,111,127),(229,357,337,428),(230,377,338,448),(231,397,339,411),(232,360,340,431),(233,380,341,451),(234,343,342,414),(235,363,286,434),(236,383,287,454),(237,346,288,417),(238,366,289,437),(239,386,290,400),(240,349,291,420),(241,369,292,440),(242,389,293,403),(243,352,294,423),(244,372,295,443),(245,392,296,406),(246,355,297,426),(247,375,298,446),(248,395,299,409),(249,358,300,429),(250,378,301,449),(251,398,302,412),(252,361,303,432),(253,381,304,452),(254,344,305,415),(255,364,306,435),(256,384,307,455),(257,347,308,418),(258,367,309,438),(259,387,310,401),(260,350,311,421),(261,370,312,441),(262,390,313,404),(263,353,314,424),(264,373,315,444),(265,393,316,407),(266,356,317,427),(267,376,318,447),(268,396,319,410),(269,359,320,430),(270,379,321,450),(271,399,322,413),(272,362,323,433),(273,382,324,453),(274,345,325,416),(275,365,326,436),(276,385,327,456),(277,348,328,419),(278,368,329,439),(279,388,330,402),(280,351,331,422),(281,371,332,442),(282,391,333,405),(283,354,334,425),(284,374,335,445),(285,394,336,408)], [(1,318,112,267),(2,298,113,247),(3,335,114,284),(4,315,58,264),(5,295,59,244),(6,332,60,281),(7,312,61,261),(8,292,62,241),(9,329,63,278),(10,309,64,258),(11,289,65,238),(12,326,66,275),(13,306,67,255),(14,286,68,235),(15,323,69,272),(16,303,70,252),(17,340,71,232),(18,320,72,269),(19,300,73,249),(20,337,74,229),(21,317,75,266),(22,297,76,246),(23,334,77,283),(24,314,78,263),(25,294,79,243),(26,331,80,280),(27,311,81,260),(28,291,82,240),(29,328,83,277),(30,308,84,257),(31,288,85,237),(32,325,86,274),(33,305,87,254),(34,342,88,234),(35,322,89,271),(36,302,90,251),(37,339,91,231),(38,319,92,268),(39,299,93,248),(40,336,94,285),(41,316,95,265),(42,296,96,245),(43,333,97,282),(44,313,98,262),(45,293,99,242),(46,330,100,279),(47,310,101,259),(48,290,102,239),(49,327,103,276),(50,307,104,256),(51,287,105,236),(52,324,106,273),(53,304,107,253),(54,341,108,233),(55,321,109,270),(56,301,110,250),(57,338,111,230),(115,403,197,389),(116,440,198,369),(117,420,199,349),(118,400,200,386),(119,437,201,366),(120,417,202,346),(121,454,203,383),(122,434,204,363),(123,414,205,343),(124,451,206,380),(125,431,207,360),(126,411,208,397),(127,448,209,377),(128,428,210,357),(129,408,211,394),(130,445,212,374),(131,425,213,354),(132,405,214,391),(133,442,215,371),(134,422,216,351),(135,402,217,388),(136,439,218,368),(137,419,219,348),(138,456,220,385),(139,436,221,365),(140,416,222,345),(141,453,223,382),(142,433,224,362),(143,413,225,399),(144,450,226,379),(145,430,227,359),(146,410,228,396),(147,447,172,376),(148,427,173,356),(149,407,174,393),(150,444,175,373),(151,424,176,353),(152,404,177,390),(153,441,178,370),(154,421,179,350),(155,401,180,387),(156,438,181,367),(157,418,182,347),(158,455,183,384),(159,435,184,364),(160,415,185,344),(161,452,186,381),(162,432,187,361),(163,412,188,398),(164,449,189,378),(165,429,190,358),(166,409,191,395),(167,446,192,375),(168,426,193,355),(169,406,194,392),(170,443,195,372),(171,423,196,352)]])
63 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 12A | 12B | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57I | 76A | ··· | 76R | 114A | ··· | 114I |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 |
| size | 1 | 1 | 2 | 6 | 38 | 114 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | Dic6 | D19 | D38 | Dic38 | S3×D19 | C57⋊Q8 |
| kernel | C57⋊Q8 | Dic3×C19 | C3×Dic19 | Dic57 | Dic19 | C57 | C38 | C19 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 18 | 9 | 9 |
Matrix representation of C57⋊Q8 ►in GL4(𝔽229) generated by
| 139 | 39 | 0 | 0 |
| 66 | 35 | 0 | 0 |
| 0 | 0 | 227 | 57 |
| 0 | 0 | 12 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 50 | 227 |
| 0 | 0 | 220 | 179 |
| 219 | 149 | 0 | 0 |
| 27 | 10 | 0 | 0 |
| 0 | 0 | 158 | 179 |
| 0 | 0 | 110 | 71 |
G:=sub<GL(4,GF(229))| [139,66,0,0,39,35,0,0,0,0,227,12,0,0,57,1],[1,0,0,0,0,1,0,0,0,0,50,220,0,0,227,179],[219,27,0,0,149,10,0,0,0,0,158,110,0,0,179,71] >;
C57⋊Q8 in GAP, Magma, Sage, TeX
C_{57}\rtimes Q_8 % in TeX
G:=Group("C57:Q8"); // GroupNames label
G:=SmallGroup(456,18);
// by ID
G=gap.SmallGroup(456,18);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,20,61,26,168,10804]);
// Polycyclic
G:=Group<a,b,c|a^57=b^4=1,c^2=b^2,b*a*b^-1=a^20,c*a*c^-1=a^37,c*b*c^-1=b^-1>;
// generators/relations
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