metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊1C8, C4.16D56, C28.34D8, C28.17Q16, C4.9Dic28, C42.248D14, C28.23M4(2), C8⋊1(C7⋊C8), (C4×C8).6D7, C7⋊1(C8⋊1C8), (C2×C56).9C4, (C4×C56).8C2, C14.2(C4⋊C8), C28.36(C2×C8), (C2×C28).46Q8, C28⋊C8.3C2, (C2×C8).8Dic7, (C2×C4).160D28, (C2×C28).398D4, C14.6(C2.D8), C2.4(C28⋊C8), C2.1(C56⋊1C4), (C2×C4).39Dic14, C4.5(C4.Dic7), C14.2(C8.C4), C2.2(C56.C4), (C4×C28).319C22, C22.16(C4⋊Dic7), C4.7(C2×C7⋊C8), (C2×C14).30(C4⋊C4), (C2×C28).292(C2×C4), (C2×C4).68(C2×Dic7), SmallGroup(448,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊1C8
G = < a,b | a56=b8=1, bab-1=a-1 >
Smallest permutation representation of C56⋊1C8
►Regular action on 448 points
Generators in S448
(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)
(1 199 428 344 120 333 84 242)(2 198 429 343 121 332 85 241)(3 197 430 342 122 331 86 240)(4 196 431 341 123 330 87 239)(5 195 432 340 124 329 88 238)(6 194 433 339 125 328 89 237)(7 193 434 338 126 327 90 236)(8 192 435 337 127 326 91 235)(9 191 436 392 128 325 92 234)(10 190 437 391 129 324 93 233)(11 189 438 390 130 323 94 232)(12 188 439 389 131 322 95 231)(13 187 440 388 132 321 96 230)(14 186 441 387 133 320 97 229)(15 185 442 386 134 319 98 228)(16 184 443 385 135 318 99 227)(17 183 444 384 136 317 100 226)(18 182 445 383 137 316 101 225)(19 181 446 382 138 315 102 280)(20 180 447 381 139 314 103 279)(21 179 448 380 140 313 104 278)(22 178 393 379 141 312 105 277)(23 177 394 378 142 311 106 276)(24 176 395 377 143 310 107 275)(25 175 396 376 144 309 108 274)(26 174 397 375 145 308 109 273)(27 173 398 374 146 307 110 272)(28 172 399 373 147 306 111 271)(29 171 400 372 148 305 112 270)(30 170 401 371 149 304 57 269)(31 169 402 370 150 303 58 268)(32 224 403 369 151 302 59 267)(33 223 404 368 152 301 60 266)(34 222 405 367 153 300 61 265)(35 221 406 366 154 299 62 264)(36 220 407 365 155 298 63 263)(37 219 408 364 156 297 64 262)(38 218 409 363 157 296 65 261)(39 217 410 362 158 295 66 260)(40 216 411 361 159 294 67 259)(41 215 412 360 160 293 68 258)(42 214 413 359 161 292 69 257)(43 213 414 358 162 291 70 256)(44 212 415 357 163 290 71 255)(45 211 416 356 164 289 72 254)(46 210 417 355 165 288 73 253)(47 209 418 354 166 287 74 252)(48 208 419 353 167 286 75 251)(49 207 420 352 168 285 76 250)(50 206 421 351 113 284 77 249)(51 205 422 350 114 283 78 248)(52 204 423 349 115 282 79 247)(53 203 424 348 116 281 80 246)(54 202 425 347 117 336 81 245)(55 201 426 346 118 335 82 244)(56 200 427 345 119 334 83 243)
G:=sub<Sym(448)| (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), (1,199,428,344,120,333,84,242)(2,198,429,343,121,332,85,241)(3,197,430,342,122,331,86,240)(4,196,431,341,123,330,87,239)(5,195,432,340,124,329,88,238)(6,194,433,339,125,328,89,237)(7,193,434,338,126,327,90,236)(8,192,435,337,127,326,91,235)(9,191,436,392,128,325,92,234)(10,190,437,391,129,324,93,233)(11,189,438,390,130,323,94,232)(12,188,439,389,131,322,95,231)(13,187,440,388,132,321,96,230)(14,186,441,387,133,320,97,229)(15,185,442,386,134,319,98,228)(16,184,443,385,135,318,99,227)(17,183,444,384,136,317,100,226)(18,182,445,383,137,316,101,225)(19,181,446,382,138,315,102,280)(20,180,447,381,139,314,103,279)(21,179,448,380,140,313,104,278)(22,178,393,379,141,312,105,277)(23,177,394,378,142,311,106,276)(24,176,395,377,143,310,107,275)(25,175,396,376,144,309,108,274)(26,174,397,375,145,308,109,273)(27,173,398,374,146,307,110,272)(28,172,399,373,147,306,111,271)(29,171,400,372,148,305,112,270)(30,170,401,371,149,304,57,269)(31,169,402,370,150,303,58,268)(32,224,403,369,151,302,59,267)(33,223,404,368,152,301,60,266)(34,222,405,367,153,300,61,265)(35,221,406,366,154,299,62,264)(36,220,407,365,155,298,63,263)(37,219,408,364,156,297,64,262)(38,218,409,363,157,296,65,261)(39,217,410,362,158,295,66,260)(40,216,411,361,159,294,67,259)(41,215,412,360,160,293,68,258)(42,214,413,359,161,292,69,257)(43,213,414,358,162,291,70,256)(44,212,415,357,163,290,71,255)(45,211,416,356,164,289,72,254)(46,210,417,355,165,288,73,253)(47,209,418,354,166,287,74,252)(48,208,419,353,167,286,75,251)(49,207,420,352,168,285,76,250)(50,206,421,351,113,284,77,249)(51,205,422,350,114,283,78,248)(52,204,423,349,115,282,79,247)(53,203,424,348,116,281,80,246)(54,202,425,347,117,336,81,245)(55,201,426,346,118,335,82,244)(56,200,427,345,119,334,83,243)>;
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), (1,199,428,344,120,333,84,242)(2,198,429,343,121,332,85,241)(3,197,430,342,122,331,86,240)(4,196,431,341,123,330,87,239)(5,195,432,340,124,329,88,238)(6,194,433,339,125,328,89,237)(7,193,434,338,126,327,90,236)(8,192,435,337,127,326,91,235)(9,191,436,392,128,325,92,234)(10,190,437,391,129,324,93,233)(11,189,438,390,130,323,94,232)(12,188,439,389,131,322,95,231)(13,187,440,388,132,321,96,230)(14,186,441,387,133,320,97,229)(15,185,442,386,134,319,98,228)(16,184,443,385,135,318,99,227)(17,183,444,384,136,317,100,226)(18,182,445,383,137,316,101,225)(19,181,446,382,138,315,102,280)(20,180,447,381,139,314,103,279)(21,179,448,380,140,313,104,278)(22,178,393,379,141,312,105,277)(23,177,394,378,142,311,106,276)(24,176,395,377,143,310,107,275)(25,175,396,376,144,309,108,274)(26,174,397,375,145,308,109,273)(27,173,398,374,146,307,110,272)(28,172,399,373,147,306,111,271)(29,171,400,372,148,305,112,270)(30,170,401,371,149,304,57,269)(31,169,402,370,150,303,58,268)(32,224,403,369,151,302,59,267)(33,223,404,368,152,301,60,266)(34,222,405,367,153,300,61,265)(35,221,406,366,154,299,62,264)(36,220,407,365,155,298,63,263)(37,219,408,364,156,297,64,262)(38,218,409,363,157,296,65,261)(39,217,410,362,158,295,66,260)(40,216,411,361,159,294,67,259)(41,215,412,360,160,293,68,258)(42,214,413,359,161,292,69,257)(43,213,414,358,162,291,70,256)(44,212,415,357,163,290,71,255)(45,211,416,356,164,289,72,254)(46,210,417,355,165,288,73,253)(47,209,418,354,166,287,74,252)(48,208,419,353,167,286,75,251)(49,207,420,352,168,285,76,250)(50,206,421,351,113,284,77,249)(51,205,422,350,114,283,78,248)(52,204,423,349,115,282,79,247)(53,203,424,348,116,281,80,246)(54,202,425,347,117,336,81,245)(55,201,426,346,118,335,82,244)(56,200,427,345,119,334,83,243) );
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)], [(1,199,428,344,120,333,84,242),(2,198,429,343,121,332,85,241),(3,197,430,342,122,331,86,240),(4,196,431,341,123,330,87,239),(5,195,432,340,124,329,88,238),(6,194,433,339,125,328,89,237),(7,193,434,338,126,327,90,236),(8,192,435,337,127,326,91,235),(9,191,436,392,128,325,92,234),(10,190,437,391,129,324,93,233),(11,189,438,390,130,323,94,232),(12,188,439,389,131,322,95,231),(13,187,440,388,132,321,96,230),(14,186,441,387,133,320,97,229),(15,185,442,386,134,319,98,228),(16,184,443,385,135,318,99,227),(17,183,444,384,136,317,100,226),(18,182,445,383,137,316,101,225),(19,181,446,382,138,315,102,280),(20,180,447,381,139,314,103,279),(21,179,448,380,140,313,104,278),(22,178,393,379,141,312,105,277),(23,177,394,378,142,311,106,276),(24,176,395,377,143,310,107,275),(25,175,396,376,144,309,108,274),(26,174,397,375,145,308,109,273),(27,173,398,374,146,307,110,272),(28,172,399,373,147,306,111,271),(29,171,400,372,148,305,112,270),(30,170,401,371,149,304,57,269),(31,169,402,370,150,303,58,268),(32,224,403,369,151,302,59,267),(33,223,404,368,152,301,60,266),(34,222,405,367,153,300,61,265),(35,221,406,366,154,299,62,264),(36,220,407,365,155,298,63,263),(37,219,408,364,156,297,64,262),(38,218,409,363,157,296,65,261),(39,217,410,362,158,295,66,260),(40,216,411,361,159,294,67,259),(41,215,412,360,160,293,68,258),(42,214,413,359,161,292,69,257),(43,213,414,358,162,291,70,256),(44,212,415,357,163,290,71,255),(45,211,416,356,164,289,72,254),(46,210,417,355,165,288,73,253),(47,209,418,354,166,287,74,252),(48,208,419,353,167,286,75,251),(49,207,420,352,168,285,76,250),(50,206,421,351,113,284,77,249),(51,205,422,350,114,283,78,248),(52,204,423,349,115,282,79,247),(53,203,424,348,116,281,80,246),(54,202,425,347,117,336,81,245),(55,201,426,346,118,335,82,244),(56,200,427,345,119,334,83,243)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 28 | ··· | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | - | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | D7 | M4(2) | D8 | Q16 | D14 | Dic7 | C8.C4 | C7⋊C8 | Dic14 | D28 | D56 | Dic28 | C4.Dic7 | C56.C4 |
| kernel | C56⋊1C8 | C28⋊C8 | C4×C56 | C2×C56 | C56 | C2×C28 | C2×C28 | C4×C8 | C28 | C28 | C28 | C42 | C2×C8 | C14 | C8 | C2×C4 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 3 | 2 | 2 | 2 | 3 | 6 | 4 | 12 | 6 | 6 | 12 | 12 | 12 | 24 |
Matrix representation of C56⋊1C8 ►in GL4(𝔽113) generated by
| 8 | 17 | 0 | 0 |
| 96 | 77 | 0 | 0 |
| 0 | 0 | 80 | 22 |
| 0 | 0 | 60 | 49 |
| 71 | 79 | 0 | 0 |
| 88 | 42 | 0 | 0 |
| 0 | 0 | 28 | 33 |
| 0 | 0 | 76 | 85 |
G:=sub<GL(4,GF(113))| [8,96,0,0,17,77,0,0,0,0,80,60,0,0,22,49],[71,88,0,0,79,42,0,0,0,0,28,76,0,0,33,85] >;
C56⋊1C8 in GAP, Magma, Sage, TeX
C_{56}\rtimes_1C_8 % in TeX
G:=Group("C56:1C8"); // GroupNames label
G:=SmallGroup(448,15);
// by ID
G=gap.SmallGroup(448,15);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,288,100,1123,136,18822]);
// Polycyclic
G:=Group<a,b|a^56=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export