metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊2C8, C28.28SD16, C42.247D14, C28.22M4(2), C8⋊2(C7⋊C8), C7⋊1(C8⋊2C8), (C2×C56).8C4, (C4×C8).12D7, C14.1(C4⋊C8), C28.35(C2×C8), (C4×C56).14C2, (C2×C28).45Q8, C28⋊C8.2C2, (C2×C28).397D4, (C2×C4).159D28, (C2×C8).10Dic7, C2.1(C8⋊Dic7), C2.3(C28⋊C8), C14.5(C4.Q8), C4.15(C56⋊C2), (C2×C4).38Dic14, C4.4(C4.Dic7), C2.1(C56.C4), C14.1(C8.C4), (C4×C28).318C22, C22.15(C4⋊Dic7), C4.6(C2×C7⋊C8), (C2×C14).29(C4⋊C4), (C2×C28).291(C2×C4), (C2×C4).67(C2×Dic7), SmallGroup(448,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊2C8
G = < a,b | a56=b8=1, bab-1=a27 >
Smallest permutation representation of C56⋊2C8
►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)
(1 244 310 167 405 91 390 209)(2 271 311 138 406 62 391 180)(3 242 312 165 407 89 392 207)(4 269 313 136 408 60 337 178)(5 240 314 163 409 87 338 205)(6 267 315 134 410 58 339 176)(7 238 316 161 411 85 340 203)(8 265 317 132 412 112 341 174)(9 236 318 159 413 83 342 201)(10 263 319 130 414 110 343 172)(11 234 320 157 415 81 344 199)(12 261 321 128 416 108 345 170)(13 232 322 155 417 79 346 197)(14 259 323 126 418 106 347 224)(15 230 324 153 419 77 348 195)(16 257 325 124 420 104 349 222)(17 228 326 151 421 75 350 193)(18 255 327 122 422 102 351 220)(19 226 328 149 423 73 352 191)(20 253 329 120 424 100 353 218)(21 280 330 147 425 71 354 189)(22 251 331 118 426 98 355 216)(23 278 332 145 427 69 356 187)(24 249 333 116 428 96 357 214)(25 276 334 143 429 67 358 185)(26 247 335 114 430 94 359 212)(27 274 336 141 431 65 360 183)(28 245 281 168 432 92 361 210)(29 272 282 139 433 63 362 181)(30 243 283 166 434 90 363 208)(31 270 284 137 435 61 364 179)(32 241 285 164 436 88 365 206)(33 268 286 135 437 59 366 177)(34 239 287 162 438 86 367 204)(35 266 288 133 439 57 368 175)(36 237 289 160 440 84 369 202)(37 264 290 131 441 111 370 173)(38 235 291 158 442 82 371 200)(39 262 292 129 443 109 372 171)(40 233 293 156 444 80 373 198)(41 260 294 127 445 107 374 169)(42 231 295 154 446 78 375 196)(43 258 296 125 447 105 376 223)(44 229 297 152 448 76 377 194)(45 256 298 123 393 103 378 221)(46 227 299 150 394 74 379 192)(47 254 300 121 395 101 380 219)(48 225 301 148 396 72 381 190)(49 252 302 119 397 99 382 217)(50 279 303 146 398 70 383 188)(51 250 304 117 399 97 384 215)(52 277 305 144 400 68 385 186)(53 248 306 115 401 95 386 213)(54 275 307 142 402 66 387 184)(55 246 308 113 403 93 388 211)(56 273 309 140 404 64 389 182)
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), (1,244,310,167,405,91,390,209)(2,271,311,138,406,62,391,180)(3,242,312,165,407,89,392,207)(4,269,313,136,408,60,337,178)(5,240,314,163,409,87,338,205)(6,267,315,134,410,58,339,176)(7,238,316,161,411,85,340,203)(8,265,317,132,412,112,341,174)(9,236,318,159,413,83,342,201)(10,263,319,130,414,110,343,172)(11,234,320,157,415,81,344,199)(12,261,321,128,416,108,345,170)(13,232,322,155,417,79,346,197)(14,259,323,126,418,106,347,224)(15,230,324,153,419,77,348,195)(16,257,325,124,420,104,349,222)(17,228,326,151,421,75,350,193)(18,255,327,122,422,102,351,220)(19,226,328,149,423,73,352,191)(20,253,329,120,424,100,353,218)(21,280,330,147,425,71,354,189)(22,251,331,118,426,98,355,216)(23,278,332,145,427,69,356,187)(24,249,333,116,428,96,357,214)(25,276,334,143,429,67,358,185)(26,247,335,114,430,94,359,212)(27,274,336,141,431,65,360,183)(28,245,281,168,432,92,361,210)(29,272,282,139,433,63,362,181)(30,243,283,166,434,90,363,208)(31,270,284,137,435,61,364,179)(32,241,285,164,436,88,365,206)(33,268,286,135,437,59,366,177)(34,239,287,162,438,86,367,204)(35,266,288,133,439,57,368,175)(36,237,289,160,440,84,369,202)(37,264,290,131,441,111,370,173)(38,235,291,158,442,82,371,200)(39,262,292,129,443,109,372,171)(40,233,293,156,444,80,373,198)(41,260,294,127,445,107,374,169)(42,231,295,154,446,78,375,196)(43,258,296,125,447,105,376,223)(44,229,297,152,448,76,377,194)(45,256,298,123,393,103,378,221)(46,227,299,150,394,74,379,192)(47,254,300,121,395,101,380,219)(48,225,301,148,396,72,381,190)(49,252,302,119,397,99,382,217)(50,279,303,146,398,70,383,188)(51,250,304,117,399,97,384,215)(52,277,305,144,400,68,385,186)(53,248,306,115,401,95,386,213)(54,275,307,142,402,66,387,184)(55,246,308,113,403,93,388,211)(56,273,309,140,404,64,389,182)>;
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), (1,244,310,167,405,91,390,209)(2,271,311,138,406,62,391,180)(3,242,312,165,407,89,392,207)(4,269,313,136,408,60,337,178)(5,240,314,163,409,87,338,205)(6,267,315,134,410,58,339,176)(7,238,316,161,411,85,340,203)(8,265,317,132,412,112,341,174)(9,236,318,159,413,83,342,201)(10,263,319,130,414,110,343,172)(11,234,320,157,415,81,344,199)(12,261,321,128,416,108,345,170)(13,232,322,155,417,79,346,197)(14,259,323,126,418,106,347,224)(15,230,324,153,419,77,348,195)(16,257,325,124,420,104,349,222)(17,228,326,151,421,75,350,193)(18,255,327,122,422,102,351,220)(19,226,328,149,423,73,352,191)(20,253,329,120,424,100,353,218)(21,280,330,147,425,71,354,189)(22,251,331,118,426,98,355,216)(23,278,332,145,427,69,356,187)(24,249,333,116,428,96,357,214)(25,276,334,143,429,67,358,185)(26,247,335,114,430,94,359,212)(27,274,336,141,431,65,360,183)(28,245,281,168,432,92,361,210)(29,272,282,139,433,63,362,181)(30,243,283,166,434,90,363,208)(31,270,284,137,435,61,364,179)(32,241,285,164,436,88,365,206)(33,268,286,135,437,59,366,177)(34,239,287,162,438,86,367,204)(35,266,288,133,439,57,368,175)(36,237,289,160,440,84,369,202)(37,264,290,131,441,111,370,173)(38,235,291,158,442,82,371,200)(39,262,292,129,443,109,372,171)(40,233,293,156,444,80,373,198)(41,260,294,127,445,107,374,169)(42,231,295,154,446,78,375,196)(43,258,296,125,447,105,376,223)(44,229,297,152,448,76,377,194)(45,256,298,123,393,103,378,221)(46,227,299,150,394,74,379,192)(47,254,300,121,395,101,380,219)(48,225,301,148,396,72,381,190)(49,252,302,119,397,99,382,217)(50,279,303,146,398,70,383,188)(51,250,304,117,399,97,384,215)(52,277,305,144,400,68,385,186)(53,248,306,115,401,95,386,213)(54,275,307,142,402,66,387,184)(55,246,308,113,403,93,388,211)(56,273,309,140,404,64,389,182) );
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)], [(1,244,310,167,405,91,390,209),(2,271,311,138,406,62,391,180),(3,242,312,165,407,89,392,207),(4,269,313,136,408,60,337,178),(5,240,314,163,409,87,338,205),(6,267,315,134,410,58,339,176),(7,238,316,161,411,85,340,203),(8,265,317,132,412,112,341,174),(9,236,318,159,413,83,342,201),(10,263,319,130,414,110,343,172),(11,234,320,157,415,81,344,199),(12,261,321,128,416,108,345,170),(13,232,322,155,417,79,346,197),(14,259,323,126,418,106,347,224),(15,230,324,153,419,77,348,195),(16,257,325,124,420,104,349,222),(17,228,326,151,421,75,350,193),(18,255,327,122,422,102,351,220),(19,226,328,149,423,73,352,191),(20,253,329,120,424,100,353,218),(21,280,330,147,425,71,354,189),(22,251,331,118,426,98,355,216),(23,278,332,145,427,69,356,187),(24,249,333,116,428,96,357,214),(25,276,334,143,429,67,358,185),(26,247,335,114,430,94,359,212),(27,274,336,141,431,65,360,183),(28,245,281,168,432,92,361,210),(29,272,282,139,433,63,362,181),(30,243,283,166,434,90,363,208),(31,270,284,137,435,61,364,179),(32,241,285,164,436,88,365,206),(33,268,286,135,437,59,366,177),(34,239,287,162,438,86,367,204),(35,266,288,133,439,57,368,175),(36,237,289,160,440,84,369,202),(37,264,290,131,441,111,370,173),(38,235,291,158,442,82,371,200),(39,262,292,129,443,109,372,171),(40,233,293,156,444,80,373,198),(41,260,294,127,445,107,374,169),(42,231,295,154,446,78,375,196),(43,258,296,125,447,105,376,223),(44,229,297,152,448,76,377,194),(45,256,298,123,393,103,378,221),(46,227,299,150,394,74,379,192),(47,254,300,121,395,101,380,219),(48,225,301,148,396,72,381,190),(49,252,302,119,397,99,382,217),(50,279,303,146,398,70,383,188),(51,250,304,117,399,97,384,215),(52,277,305,144,400,68,385,186),(53,248,306,115,401,95,386,213),(54,275,307,142,402,66,387,184),(55,246,308,113,403,93,388,211),(56,273,309,140,404,64,389,182)]])
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 |
| type | + | + | + | + | - | + | + | - | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | D7 | M4(2) | SD16 | D14 | Dic7 | C8.C4 | C7⋊C8 | Dic14 | D28 | C56⋊C2 | C4.Dic7 | C56.C4 |
| kernel | C56⋊2C8 | C28⋊C8 | C4×C56 | C2×C56 | C56 | C2×C28 | C2×C28 | C4×C8 | C28 | C28 | C42 | C2×C8 | C14 | C8 | C2×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 3 | 2 | 4 | 3 | 6 | 4 | 12 | 6 | 6 | 24 | 12 | 24 |
Matrix representation of C56⋊2C8 ►in GL4(𝔽113) generated by
| 19 | 96 | 0 | 0 |
| 103 | 9 | 0 | 0 |
| 0 | 0 | 9 | 105 |
| 0 | 0 | 8 | 43 |
| 69 | 0 | 0 | 0 |
| 14 | 44 | 0 | 0 |
| 0 | 0 | 31 | 17 |
| 0 | 0 | 83 | 82 |
G:=sub<GL(4,GF(113))| [19,103,0,0,96,9,0,0,0,0,9,8,0,0,105,43],[69,14,0,0,0,44,0,0,0,0,31,83,0,0,17,82] >;
C56⋊2C8 in GAP, Magma, Sage, TeX
C_{56}\rtimes_2C_8 % in TeX
G:=Group("C56:2C8"); // GroupNames label
G:=SmallGroup(448,14);
// by ID
G=gap.SmallGroup(448,14);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,64,100,1123,136,18822]);
// Polycyclic
G:=Group<a,b|a^56=b^8=1,b*a*b^-1=a^27>;
// generators/relations
Export