metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56.4C8, C28.14C42, C14.4M5(2), C42.4Dic7, C7⋊C16⋊7C4, C8.2(C7⋊C8), C14.4(C4×C8), (C2×C28).3C8, (C4×C8).15D7, C8.39(C4×D7), C7⋊1(C16⋊5C4), C56.56(C2×C4), (C2×C56).23C4, (C4×C28).18C4, (C4×C56).19C2, C28.41(C2×C8), (C2×C8).330D14, (C2×C8).13Dic7, C4.14(C4×Dic7), C2.1(C28.C8), (C2×C56).395C22, C2.4(C4×C7⋊C8), C4.13(C2×C7⋊C8), (C2×C7⋊C16).7C2, (C2×C4).2(C7⋊C8), C22.8(C2×C7⋊C8), (C2×C14).26(C2×C8), (C2×C28).310(C2×C4), (C2×C4).91(C2×Dic7), SmallGroup(448,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.C8
G = < a,b | a56=1, b8=a28, bab-1=a13 >
Smallest permutation representation of C56.C8
►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)
(1 280 302 442 69 358 212 114 29 252 330 414 97 386 184 142)(2 237 303 399 70 371 213 127 30 265 331 427 98 343 185 155)(3 250 304 412 71 384 214 140 31 278 332 440 99 356 186 168)(4 263 305 425 72 341 215 153 32 235 333 397 100 369 187 125)(5 276 306 438 73 354 216 166 33 248 334 410 101 382 188 138)(6 233 307 395 74 367 217 123 34 261 335 423 102 339 189 151)(7 246 308 408 75 380 218 136 35 274 336 436 103 352 190 164)(8 259 309 421 76 337 219 149 36 231 281 393 104 365 191 121)(9 272 310 434 77 350 220 162 37 244 282 406 105 378 192 134)(10 229 311 447 78 363 221 119 38 257 283 419 106 391 193 147)(11 242 312 404 79 376 222 132 39 270 284 432 107 348 194 160)(12 255 313 417 80 389 223 145 40 227 285 445 108 361 195 117)(13 268 314 430 81 346 224 158 41 240 286 402 109 374 196 130)(14 225 315 443 82 359 169 115 42 253 287 415 110 387 197 143)(15 238 316 400 83 372 170 128 43 266 288 428 111 344 198 156)(16 251 317 413 84 385 171 141 44 279 289 441 112 357 199 113)(17 264 318 426 85 342 172 154 45 236 290 398 57 370 200 126)(18 277 319 439 86 355 173 167 46 249 291 411 58 383 201 139)(19 234 320 396 87 368 174 124 47 262 292 424 59 340 202 152)(20 247 321 409 88 381 175 137 48 275 293 437 60 353 203 165)(21 260 322 422 89 338 176 150 49 232 294 394 61 366 204 122)(22 273 323 435 90 351 177 163 50 245 295 407 62 379 205 135)(23 230 324 448 91 364 178 120 51 258 296 420 63 392 206 148)(24 243 325 405 92 377 179 133 52 271 297 433 64 349 207 161)(25 256 326 418 93 390 180 146 53 228 298 446 65 362 208 118)(26 269 327 431 94 347 181 159 54 241 299 403 66 375 209 131)(27 226 328 444 95 360 182 116 55 254 300 416 67 388 210 144)(28 239 329 401 96 373 183 129 56 267 301 429 68 345 211 157)
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), (1,280,302,442,69,358,212,114,29,252,330,414,97,386,184,142)(2,237,303,399,70,371,213,127,30,265,331,427,98,343,185,155)(3,250,304,412,71,384,214,140,31,278,332,440,99,356,186,168)(4,263,305,425,72,341,215,153,32,235,333,397,100,369,187,125)(5,276,306,438,73,354,216,166,33,248,334,410,101,382,188,138)(6,233,307,395,74,367,217,123,34,261,335,423,102,339,189,151)(7,246,308,408,75,380,218,136,35,274,336,436,103,352,190,164)(8,259,309,421,76,337,219,149,36,231,281,393,104,365,191,121)(9,272,310,434,77,350,220,162,37,244,282,406,105,378,192,134)(10,229,311,447,78,363,221,119,38,257,283,419,106,391,193,147)(11,242,312,404,79,376,222,132,39,270,284,432,107,348,194,160)(12,255,313,417,80,389,223,145,40,227,285,445,108,361,195,117)(13,268,314,430,81,346,224,158,41,240,286,402,109,374,196,130)(14,225,315,443,82,359,169,115,42,253,287,415,110,387,197,143)(15,238,316,400,83,372,170,128,43,266,288,428,111,344,198,156)(16,251,317,413,84,385,171,141,44,279,289,441,112,357,199,113)(17,264,318,426,85,342,172,154,45,236,290,398,57,370,200,126)(18,277,319,439,86,355,173,167,46,249,291,411,58,383,201,139)(19,234,320,396,87,368,174,124,47,262,292,424,59,340,202,152)(20,247,321,409,88,381,175,137,48,275,293,437,60,353,203,165)(21,260,322,422,89,338,176,150,49,232,294,394,61,366,204,122)(22,273,323,435,90,351,177,163,50,245,295,407,62,379,205,135)(23,230,324,448,91,364,178,120,51,258,296,420,63,392,206,148)(24,243,325,405,92,377,179,133,52,271,297,433,64,349,207,161)(25,256,326,418,93,390,180,146,53,228,298,446,65,362,208,118)(26,269,327,431,94,347,181,159,54,241,299,403,66,375,209,131)(27,226,328,444,95,360,182,116,55,254,300,416,67,388,210,144)(28,239,329,401,96,373,183,129,56,267,301,429,68,345,211,157)>;
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), (1,280,302,442,69,358,212,114,29,252,330,414,97,386,184,142)(2,237,303,399,70,371,213,127,30,265,331,427,98,343,185,155)(3,250,304,412,71,384,214,140,31,278,332,440,99,356,186,168)(4,263,305,425,72,341,215,153,32,235,333,397,100,369,187,125)(5,276,306,438,73,354,216,166,33,248,334,410,101,382,188,138)(6,233,307,395,74,367,217,123,34,261,335,423,102,339,189,151)(7,246,308,408,75,380,218,136,35,274,336,436,103,352,190,164)(8,259,309,421,76,337,219,149,36,231,281,393,104,365,191,121)(9,272,310,434,77,350,220,162,37,244,282,406,105,378,192,134)(10,229,311,447,78,363,221,119,38,257,283,419,106,391,193,147)(11,242,312,404,79,376,222,132,39,270,284,432,107,348,194,160)(12,255,313,417,80,389,223,145,40,227,285,445,108,361,195,117)(13,268,314,430,81,346,224,158,41,240,286,402,109,374,196,130)(14,225,315,443,82,359,169,115,42,253,287,415,110,387,197,143)(15,238,316,400,83,372,170,128,43,266,288,428,111,344,198,156)(16,251,317,413,84,385,171,141,44,279,289,441,112,357,199,113)(17,264,318,426,85,342,172,154,45,236,290,398,57,370,200,126)(18,277,319,439,86,355,173,167,46,249,291,411,58,383,201,139)(19,234,320,396,87,368,174,124,47,262,292,424,59,340,202,152)(20,247,321,409,88,381,175,137,48,275,293,437,60,353,203,165)(21,260,322,422,89,338,176,150,49,232,294,394,61,366,204,122)(22,273,323,435,90,351,177,163,50,245,295,407,62,379,205,135)(23,230,324,448,91,364,178,120,51,258,296,420,63,392,206,148)(24,243,325,405,92,377,179,133,52,271,297,433,64,349,207,161)(25,256,326,418,93,390,180,146,53,228,298,446,65,362,208,118)(26,269,327,431,94,347,181,159,54,241,299,403,66,375,209,131)(27,226,328,444,95,360,182,116,55,254,300,416,67,388,210,144)(28,239,329,401,96,373,183,129,56,267,301,429,68,345,211,157) );
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)], [(1,280,302,442,69,358,212,114,29,252,330,414,97,386,184,142),(2,237,303,399,70,371,213,127,30,265,331,427,98,343,185,155),(3,250,304,412,71,384,214,140,31,278,332,440,99,356,186,168),(4,263,305,425,72,341,215,153,32,235,333,397,100,369,187,125),(5,276,306,438,73,354,216,166,33,248,334,410,101,382,188,138),(6,233,307,395,74,367,217,123,34,261,335,423,102,339,189,151),(7,246,308,408,75,380,218,136,35,274,336,436,103,352,190,164),(8,259,309,421,76,337,219,149,36,231,281,393,104,365,191,121),(9,272,310,434,77,350,220,162,37,244,282,406,105,378,192,134),(10,229,311,447,78,363,221,119,38,257,283,419,106,391,193,147),(11,242,312,404,79,376,222,132,39,270,284,432,107,348,194,160),(12,255,313,417,80,389,223,145,40,227,285,445,108,361,195,117),(13,268,314,430,81,346,224,158,41,240,286,402,109,374,196,130),(14,225,315,443,82,359,169,115,42,253,287,415,110,387,197,143),(15,238,316,400,83,372,170,128,43,266,288,428,111,344,198,156),(16,251,317,413,84,385,171,141,44,279,289,441,112,357,199,113),(17,264,318,426,85,342,172,154,45,236,290,398,57,370,200,126),(18,277,319,439,86,355,173,167,46,249,291,411,58,383,201,139),(19,234,320,396,87,368,174,124,47,262,292,424,59,340,202,152),(20,247,321,409,88,381,175,137,48,275,293,437,60,353,203,165),(21,260,322,422,89,338,176,150,49,232,294,394,61,366,204,122),(22,273,323,435,90,351,177,163,50,245,295,407,62,379,205,135),(23,230,324,448,91,364,178,120,51,258,296,420,63,392,206,148),(24,243,325,405,92,377,179,133,52,271,297,433,64,349,207,161),(25,256,326,418,93,390,180,146,53,228,298,446,65,362,208,118),(26,269,327,431,94,347,181,159,54,241,299,403,66,375,209,131),(27,226,328,444,95,360,182,116,55,254,300,416,67,388,210,144),(28,239,329,401,96,373,183,129,56,267,301,429,68,345,211,157)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 16A | ··· | 16P | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | D7 | Dic7 | Dic7 | D14 | M5(2) | C7⋊C8 | C4×D7 | C7⋊C8 | C28.C8 |
| kernel | C56.C8 | C2×C7⋊C16 | C4×C56 | C7⋊C16 | C4×C28 | C2×C56 | C56 | C2×C28 | C4×C8 | C42 | C2×C8 | C2×C8 | C14 | C8 | C8 | C2×C4 | C2 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 8 | 8 | 3 | 3 | 3 | 3 | 8 | 12 | 12 | 12 | 48 |
Matrix representation of C56.C8 ►in GL3(𝔽113) generated by
| 98 | 0 | 0 |
| 0 | 13 | 70 |
| 0 | 43 | 111 |
| 98 | 0 | 0 |
| 0 | 55 | 46 |
| 0 | 10 | 58 |
G:=sub<GL(3,GF(113))| [98,0,0,0,13,43,0,70,111],[98,0,0,0,55,10,0,46,58] >;
C56.C8 in GAP, Magma, Sage, TeX
C_{56}.C_8 % in TeX
G:=Group("C56.C8"); // GroupNames label
G:=SmallGroup(448,18);
// by ID
G=gap.SmallGroup(448,18);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,477,64,100,102,18822]);
// Polycyclic
G:=Group<a,b|a^56=1,b^8=a^28,b*a*b^-1=a^13>;
// generators/relations
Export