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G = Q8×C57order 456 = 23·3·19

Direct product of C57 and Q8

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

C1C2 — Q8×C57
C1C2C38C114C228 — Q8×C57
C1C2 — Q8×C57
C1C114 — Q8×C57

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

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

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

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

285 conjugacy classes

class 1  2 3A3B4A4B4C6A6B12A···12F19A···19R38A···38R57A···57AJ76A···76BB114A···114AJ228A···228DD
order12334446612···1219···1938···3857···5776···76114···114228···228
size1111222112···21···11···11···12···21···12···2

285 irreducible representations

dim111111112222
type++-
imageC1C2C3C6C19C38C57C114Q8C3×Q8Q8×C19Q8×C57
kernelQ8×C57C228Q8×C19C76C3×Q8C12Q8C4C57C19C3C1
# reps1326185436108121836

Matrix representation of Q8×C57 in GL3(𝔽229) generated by

9400
0440
0044
,
100
0228227
011
,
100
014387
01586
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

Subgroup lattice of Q8×C57 in TeX

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