metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C57⋊1C8, C76.2S3, C4.2D57, C2.Dic57, C38.Dic3, C6.Dic19, C228.2C2, C114.1C4, C12.2D19, C19⋊(C3⋊C8), C3⋊(C19⋊C8), SmallGroup(456,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C57 — C57⋊C8 |
Generators and relations for C57⋊C8
G = < a,b | a57=b8=1, bab-1=a-1 >
Smallest permutation representation of C57⋊C8
►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)
(1 441 205 311 113 390 165 250)(2 440 206 310 114 389 166 249)(3 439 207 309 58 388 167 248)(4 438 208 308 59 387 168 247)(5 437 209 307 60 386 169 246)(6 436 210 306 61 385 170 245)(7 435 211 305 62 384 171 244)(8 434 212 304 63 383 115 243)(9 433 213 303 64 382 116 242)(10 432 214 302 65 381 117 241)(11 431 215 301 66 380 118 240)(12 430 216 300 67 379 119 239)(13 429 217 299 68 378 120 238)(14 428 218 298 69 377 121 237)(15 427 219 297 70 376 122 236)(16 426 220 296 71 375 123 235)(17 425 221 295 72 374 124 234)(18 424 222 294 73 373 125 233)(19 423 223 293 74 372 126 232)(20 422 224 292 75 371 127 231)(21 421 225 291 76 370 128 230)(22 420 226 290 77 369 129 229)(23 419 227 289 78 368 130 285)(24 418 228 288 79 367 131 284)(25 417 172 287 80 366 132 283)(26 416 173 286 81 365 133 282)(27 415 174 342 82 364 134 281)(28 414 175 341 83 363 135 280)(29 413 176 340 84 362 136 279)(30 412 177 339 85 361 137 278)(31 411 178 338 86 360 138 277)(32 410 179 337 87 359 139 276)(33 409 180 336 88 358 140 275)(34 408 181 335 89 357 141 274)(35 407 182 334 90 356 142 273)(36 406 183 333 91 355 143 272)(37 405 184 332 92 354 144 271)(38 404 185 331 93 353 145 270)(39 403 186 330 94 352 146 269)(40 402 187 329 95 351 147 268)(41 401 188 328 96 350 148 267)(42 400 189 327 97 349 149 266)(43 456 190 326 98 348 150 265)(44 455 191 325 99 347 151 264)(45 454 192 324 100 346 152 263)(46 453 193 323 101 345 153 262)(47 452 194 322 102 344 154 261)(48 451 195 321 103 343 155 260)(49 450 196 320 104 399 156 259)(50 449 197 319 105 398 157 258)(51 448 198 318 106 397 158 257)(52 447 199 317 107 396 159 256)(53 446 200 316 108 395 160 255)(54 445 201 315 109 394 161 254)(55 444 202 314 110 393 162 253)(56 443 203 313 111 392 163 252)(57 442 204 312 112 391 164 251)
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), (1,441,205,311,113,390,165,250)(2,440,206,310,114,389,166,249)(3,439,207,309,58,388,167,248)(4,438,208,308,59,387,168,247)(5,437,209,307,60,386,169,246)(6,436,210,306,61,385,170,245)(7,435,211,305,62,384,171,244)(8,434,212,304,63,383,115,243)(9,433,213,303,64,382,116,242)(10,432,214,302,65,381,117,241)(11,431,215,301,66,380,118,240)(12,430,216,300,67,379,119,239)(13,429,217,299,68,378,120,238)(14,428,218,298,69,377,121,237)(15,427,219,297,70,376,122,236)(16,426,220,296,71,375,123,235)(17,425,221,295,72,374,124,234)(18,424,222,294,73,373,125,233)(19,423,223,293,74,372,126,232)(20,422,224,292,75,371,127,231)(21,421,225,291,76,370,128,230)(22,420,226,290,77,369,129,229)(23,419,227,289,78,368,130,285)(24,418,228,288,79,367,131,284)(25,417,172,287,80,366,132,283)(26,416,173,286,81,365,133,282)(27,415,174,342,82,364,134,281)(28,414,175,341,83,363,135,280)(29,413,176,340,84,362,136,279)(30,412,177,339,85,361,137,278)(31,411,178,338,86,360,138,277)(32,410,179,337,87,359,139,276)(33,409,180,336,88,358,140,275)(34,408,181,335,89,357,141,274)(35,407,182,334,90,356,142,273)(36,406,183,333,91,355,143,272)(37,405,184,332,92,354,144,271)(38,404,185,331,93,353,145,270)(39,403,186,330,94,352,146,269)(40,402,187,329,95,351,147,268)(41,401,188,328,96,350,148,267)(42,400,189,327,97,349,149,266)(43,456,190,326,98,348,150,265)(44,455,191,325,99,347,151,264)(45,454,192,324,100,346,152,263)(46,453,193,323,101,345,153,262)(47,452,194,322,102,344,154,261)(48,451,195,321,103,343,155,260)(49,450,196,320,104,399,156,259)(50,449,197,319,105,398,157,258)(51,448,198,318,106,397,158,257)(52,447,199,317,107,396,159,256)(53,446,200,316,108,395,160,255)(54,445,201,315,109,394,161,254)(55,444,202,314,110,393,162,253)(56,443,203,313,111,392,163,252)(57,442,204,312,112,391,164,251)>;
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), (1,441,205,311,113,390,165,250)(2,440,206,310,114,389,166,249)(3,439,207,309,58,388,167,248)(4,438,208,308,59,387,168,247)(5,437,209,307,60,386,169,246)(6,436,210,306,61,385,170,245)(7,435,211,305,62,384,171,244)(8,434,212,304,63,383,115,243)(9,433,213,303,64,382,116,242)(10,432,214,302,65,381,117,241)(11,431,215,301,66,380,118,240)(12,430,216,300,67,379,119,239)(13,429,217,299,68,378,120,238)(14,428,218,298,69,377,121,237)(15,427,219,297,70,376,122,236)(16,426,220,296,71,375,123,235)(17,425,221,295,72,374,124,234)(18,424,222,294,73,373,125,233)(19,423,223,293,74,372,126,232)(20,422,224,292,75,371,127,231)(21,421,225,291,76,370,128,230)(22,420,226,290,77,369,129,229)(23,419,227,289,78,368,130,285)(24,418,228,288,79,367,131,284)(25,417,172,287,80,366,132,283)(26,416,173,286,81,365,133,282)(27,415,174,342,82,364,134,281)(28,414,175,341,83,363,135,280)(29,413,176,340,84,362,136,279)(30,412,177,339,85,361,137,278)(31,411,178,338,86,360,138,277)(32,410,179,337,87,359,139,276)(33,409,180,336,88,358,140,275)(34,408,181,335,89,357,141,274)(35,407,182,334,90,356,142,273)(36,406,183,333,91,355,143,272)(37,405,184,332,92,354,144,271)(38,404,185,331,93,353,145,270)(39,403,186,330,94,352,146,269)(40,402,187,329,95,351,147,268)(41,401,188,328,96,350,148,267)(42,400,189,327,97,349,149,266)(43,456,190,326,98,348,150,265)(44,455,191,325,99,347,151,264)(45,454,192,324,100,346,152,263)(46,453,193,323,101,345,153,262)(47,452,194,322,102,344,154,261)(48,451,195,321,103,343,155,260)(49,450,196,320,104,399,156,259)(50,449,197,319,105,398,157,258)(51,448,198,318,106,397,158,257)(52,447,199,317,107,396,159,256)(53,446,200,316,108,395,160,255)(54,445,201,315,109,394,161,254)(55,444,202,314,110,393,162,253)(56,443,203,313,111,392,163,252)(57,442,204,312,112,391,164,251) );
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)], [(1,441,205,311,113,390,165,250),(2,440,206,310,114,389,166,249),(3,439,207,309,58,388,167,248),(4,438,208,308,59,387,168,247),(5,437,209,307,60,386,169,246),(6,436,210,306,61,385,170,245),(7,435,211,305,62,384,171,244),(8,434,212,304,63,383,115,243),(9,433,213,303,64,382,116,242),(10,432,214,302,65,381,117,241),(11,431,215,301,66,380,118,240),(12,430,216,300,67,379,119,239),(13,429,217,299,68,378,120,238),(14,428,218,298,69,377,121,237),(15,427,219,297,70,376,122,236),(16,426,220,296,71,375,123,235),(17,425,221,295,72,374,124,234),(18,424,222,294,73,373,125,233),(19,423,223,293,74,372,126,232),(20,422,224,292,75,371,127,231),(21,421,225,291,76,370,128,230),(22,420,226,290,77,369,129,229),(23,419,227,289,78,368,130,285),(24,418,228,288,79,367,131,284),(25,417,172,287,80,366,132,283),(26,416,173,286,81,365,133,282),(27,415,174,342,82,364,134,281),(28,414,175,341,83,363,135,280),(29,413,176,340,84,362,136,279),(30,412,177,339,85,361,137,278),(31,411,178,338,86,360,138,277),(32,410,179,337,87,359,139,276),(33,409,180,336,88,358,140,275),(34,408,181,335,89,357,141,274),(35,407,182,334,90,356,142,273),(36,406,183,333,91,355,143,272),(37,405,184,332,92,354,144,271),(38,404,185,331,93,353,145,270),(39,403,186,330,94,352,146,269),(40,402,187,329,95,351,147,268),(41,401,188,328,96,350,148,267),(42,400,189,327,97,349,149,266),(43,456,190,326,98,348,150,265),(44,455,191,325,99,347,151,264),(45,454,192,324,100,346,152,263),(46,453,193,323,101,345,153,262),(47,452,194,322,102,344,154,261),(48,451,195,321,103,343,155,260),(49,450,196,320,104,399,156,259),(50,449,197,319,105,398,157,258),(51,448,198,318,106,397,158,257),(52,447,199,317,107,396,159,256),(53,446,200,316,108,395,160,255),(54,445,201,315,109,394,161,254),(55,444,202,314,110,393,162,253),(56,443,203,313,111,392,163,252),(57,442,204,312,112,391,164,251)]])
120 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57R | 76A | ··· | 76R | 114A | ··· | 114R | 228A | ··· | 228AJ |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 57 | 57 | 57 | 57 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | S3 | Dic3 | C3⋊C8 | D19 | Dic19 | D57 | C19⋊C8 | Dic57 | C57⋊C8 |
| kernel | C57⋊C8 | C228 | C114 | C57 | C76 | C38 | C19 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 9 | 9 | 18 | 18 | 18 | 36 |
Matrix representation of C57⋊C8 ►in GL2(𝔽457) generated by
| 196 | 299 |
| 354 | 132 |
| 237 | 27 |
| 116 | 220 |
G:=sub<GL(2,GF(457))| [196,354,299,132],[237,116,27,220] >;
C57⋊C8 in GAP, Magma, Sage, TeX
C_{57}\rtimes C_8 % in TeX
G:=Group("C57:C8"); // GroupNames label
G:=SmallGroup(456,5);
// by ID
G=gap.SmallGroup(456,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,10,26,323,10804]);
// Polycyclic
G:=Group<a,b|a^57=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export