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G = C57⋊Q8order 456 = 23·3·19

The semidirect product of C57 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C57⋊Q8, C38.7D6, C6.7D38, C31Dic38, C191Dic6, 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

C1C114 — C57⋊Q8
C1C19C57C114C3×Dic19 — C57⋊Q8
C57C114 — C57⋊Q8
C1C2

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 >

3C4
19C4
57C4
57Q8
19C12
19Dic3
3C76
3Dic19
19Dic6
3Dic38

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

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

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

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

63 conjugacy classes

class 1  2  3 4A4B4C 6 12A12B19A···19I38A···38I57A···57I76A···76R114A···114I
order1234446121219···1938···3857···5776···76114···114
size112638114238382···22···24···46···64···4

63 irreducible representations

dim1111222222244
type+++++-+-++-+-
imageC1C2C2C2S3Q8D6Dic6D19D38Dic38S3×D19C57⋊Q8
kernelC57⋊Q8Dic3×C19C3×Dic19Dic57Dic19C57C38C19Dic3C6C3C2C1
# reps11111112991899

Matrix representation of C57⋊Q8 in GL4(𝔽229) generated by

1393900
663500
0022757
00121
,
1000
0100
0050227
00220179
,
21914900
271000
00158179
0011071
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

Export

Subgroup lattice of C57⋊Q8 in TeX

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