metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊4C8, C42.280D14, C28.15M4(2), C8⋊3(C7⋊C8), C7⋊2(C8⋊C8), C14.2(C4×C8), (C4×C8).14D7, C28.39(C2×C8), (C4×C56).17C2, (C2×C56).21C4, (C2×C8).12Dic7, C2.2(C56⋊C4), C14.4(C8⋊C4), C4.13(C8⋊D7), (C2×C14).15C42, (C4×C28).336C22, C22.15(C4×Dic7), C2.3(C4×C7⋊C8), C4.11(C2×C7⋊C8), (C4×C7⋊C8).17C2, (C2×C7⋊C8).11C4, (C2×C4).165(C4×D7), (C2×C28).239(C2×C4), (C2×C4).89(C2×Dic7), SmallGroup(448,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊C8
G = < a,b | a56=b8=1, bab-1=a13 >
Subgroups: 164 in 66 conjugacy classes, 51 normal (13 characteristic)
C1, C2, C2, C4, C22, C7, C8, C8, C2×C4, C2×C4, C14, C14, C42, C2×C8, C2×C8, C28, C2×C14, C4×C8, C4×C8, C7⋊C8, C56, C2×C28, C2×C28, C8⋊C8, C2×C7⋊C8, C4×C28, C2×C56, C4×C7⋊C8, C4×C56, C56⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D7, C42, C2×C8, M4(2), Dic7, D14, C4×C8, C8⋊C4, C7⋊C8, C4×D7, C2×Dic7, C8⋊C8, C8⋊D7, C2×C7⋊C8, C4×Dic7, C4×C7⋊C8, C56⋊C4, 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)
(1 325 249 167 224 421 102 356)(2 282 250 124 169 434 103 369)(3 295 251 137 170 447 104 382)(4 308 252 150 171 404 105 339)(5 321 253 163 172 417 106 352)(6 334 254 120 173 430 107 365)(7 291 255 133 174 443 108 378)(8 304 256 146 175 400 109 391)(9 317 257 159 176 413 110 348)(10 330 258 116 177 426 111 361)(11 287 259 129 178 439 112 374)(12 300 260 142 179 396 57 387)(13 313 261 155 180 409 58 344)(14 326 262 168 181 422 59 357)(15 283 263 125 182 435 60 370)(16 296 264 138 183 448 61 383)(17 309 265 151 184 405 62 340)(18 322 266 164 185 418 63 353)(19 335 267 121 186 431 64 366)(20 292 268 134 187 444 65 379)(21 305 269 147 188 401 66 392)(22 318 270 160 189 414 67 349)(23 331 271 117 190 427 68 362)(24 288 272 130 191 440 69 375)(25 301 273 143 192 397 70 388)(26 314 274 156 193 410 71 345)(27 327 275 113 194 423 72 358)(28 284 276 126 195 436 73 371)(29 297 277 139 196 393 74 384)(30 310 278 152 197 406 75 341)(31 323 279 165 198 419 76 354)(32 336 280 122 199 432 77 367)(33 293 225 135 200 445 78 380)(34 306 226 148 201 402 79 337)(35 319 227 161 202 415 80 350)(36 332 228 118 203 428 81 363)(37 289 229 131 204 441 82 376)(38 302 230 144 205 398 83 389)(39 315 231 157 206 411 84 346)(40 328 232 114 207 424 85 359)(41 285 233 127 208 437 86 372)(42 298 234 140 209 394 87 385)(43 311 235 153 210 407 88 342)(44 324 236 166 211 420 89 355)(45 281 237 123 212 433 90 368)(46 294 238 136 213 446 91 381)(47 307 239 149 214 403 92 338)(48 320 240 162 215 416 93 351)(49 333 241 119 216 429 94 364)(50 290 242 132 217 442 95 377)(51 303 243 145 218 399 96 390)(52 316 244 158 219 412 97 347)(53 329 245 115 220 425 98 360)(54 286 246 128 221 438 99 373)(55 299 247 141 222 395 100 386)(56 312 248 154 223 408 101 343)
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), (1,325,249,167,224,421,102,356)(2,282,250,124,169,434,103,369)(3,295,251,137,170,447,104,382)(4,308,252,150,171,404,105,339)(5,321,253,163,172,417,106,352)(6,334,254,120,173,430,107,365)(7,291,255,133,174,443,108,378)(8,304,256,146,175,400,109,391)(9,317,257,159,176,413,110,348)(10,330,258,116,177,426,111,361)(11,287,259,129,178,439,112,374)(12,300,260,142,179,396,57,387)(13,313,261,155,180,409,58,344)(14,326,262,168,181,422,59,357)(15,283,263,125,182,435,60,370)(16,296,264,138,183,448,61,383)(17,309,265,151,184,405,62,340)(18,322,266,164,185,418,63,353)(19,335,267,121,186,431,64,366)(20,292,268,134,187,444,65,379)(21,305,269,147,188,401,66,392)(22,318,270,160,189,414,67,349)(23,331,271,117,190,427,68,362)(24,288,272,130,191,440,69,375)(25,301,273,143,192,397,70,388)(26,314,274,156,193,410,71,345)(27,327,275,113,194,423,72,358)(28,284,276,126,195,436,73,371)(29,297,277,139,196,393,74,384)(30,310,278,152,197,406,75,341)(31,323,279,165,198,419,76,354)(32,336,280,122,199,432,77,367)(33,293,225,135,200,445,78,380)(34,306,226,148,201,402,79,337)(35,319,227,161,202,415,80,350)(36,332,228,118,203,428,81,363)(37,289,229,131,204,441,82,376)(38,302,230,144,205,398,83,389)(39,315,231,157,206,411,84,346)(40,328,232,114,207,424,85,359)(41,285,233,127,208,437,86,372)(42,298,234,140,209,394,87,385)(43,311,235,153,210,407,88,342)(44,324,236,166,211,420,89,355)(45,281,237,123,212,433,90,368)(46,294,238,136,213,446,91,381)(47,307,239,149,214,403,92,338)(48,320,240,162,215,416,93,351)(49,333,241,119,216,429,94,364)(50,290,242,132,217,442,95,377)(51,303,243,145,218,399,96,390)(52,316,244,158,219,412,97,347)(53,329,245,115,220,425,98,360)(54,286,246,128,221,438,99,373)(55,299,247,141,222,395,100,386)(56,312,248,154,223,408,101,343)>;
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), (1,325,249,167,224,421,102,356)(2,282,250,124,169,434,103,369)(3,295,251,137,170,447,104,382)(4,308,252,150,171,404,105,339)(5,321,253,163,172,417,106,352)(6,334,254,120,173,430,107,365)(7,291,255,133,174,443,108,378)(8,304,256,146,175,400,109,391)(9,317,257,159,176,413,110,348)(10,330,258,116,177,426,111,361)(11,287,259,129,178,439,112,374)(12,300,260,142,179,396,57,387)(13,313,261,155,180,409,58,344)(14,326,262,168,181,422,59,357)(15,283,263,125,182,435,60,370)(16,296,264,138,183,448,61,383)(17,309,265,151,184,405,62,340)(18,322,266,164,185,418,63,353)(19,335,267,121,186,431,64,366)(20,292,268,134,187,444,65,379)(21,305,269,147,188,401,66,392)(22,318,270,160,189,414,67,349)(23,331,271,117,190,427,68,362)(24,288,272,130,191,440,69,375)(25,301,273,143,192,397,70,388)(26,314,274,156,193,410,71,345)(27,327,275,113,194,423,72,358)(28,284,276,126,195,436,73,371)(29,297,277,139,196,393,74,384)(30,310,278,152,197,406,75,341)(31,323,279,165,198,419,76,354)(32,336,280,122,199,432,77,367)(33,293,225,135,200,445,78,380)(34,306,226,148,201,402,79,337)(35,319,227,161,202,415,80,350)(36,332,228,118,203,428,81,363)(37,289,229,131,204,441,82,376)(38,302,230,144,205,398,83,389)(39,315,231,157,206,411,84,346)(40,328,232,114,207,424,85,359)(41,285,233,127,208,437,86,372)(42,298,234,140,209,394,87,385)(43,311,235,153,210,407,88,342)(44,324,236,166,211,420,89,355)(45,281,237,123,212,433,90,368)(46,294,238,136,213,446,91,381)(47,307,239,149,214,403,92,338)(48,320,240,162,215,416,93,351)(49,333,241,119,216,429,94,364)(50,290,242,132,217,442,95,377)(51,303,243,145,218,399,96,390)(52,316,244,158,219,412,97,347)(53,329,245,115,220,425,98,360)(54,286,246,128,221,438,99,373)(55,299,247,141,222,395,100,386)(56,312,248,154,223,408,101,343) );
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)], [(1,325,249,167,224,421,102,356),(2,282,250,124,169,434,103,369),(3,295,251,137,170,447,104,382),(4,308,252,150,171,404,105,339),(5,321,253,163,172,417,106,352),(6,334,254,120,173,430,107,365),(7,291,255,133,174,443,108,378),(8,304,256,146,175,400,109,391),(9,317,257,159,176,413,110,348),(10,330,258,116,177,426,111,361),(11,287,259,129,178,439,112,374),(12,300,260,142,179,396,57,387),(13,313,261,155,180,409,58,344),(14,326,262,168,181,422,59,357),(15,283,263,125,182,435,60,370),(16,296,264,138,183,448,61,383),(17,309,265,151,184,405,62,340),(18,322,266,164,185,418,63,353),(19,335,267,121,186,431,64,366),(20,292,268,134,187,444,65,379),(21,305,269,147,188,401,66,392),(22,318,270,160,189,414,67,349),(23,331,271,117,190,427,68,362),(24,288,272,130,191,440,69,375),(25,301,273,143,192,397,70,388),(26,314,274,156,193,410,71,345),(27,327,275,113,194,423,72,358),(28,284,276,126,195,436,73,371),(29,297,277,139,196,393,74,384),(30,310,278,152,197,406,75,341),(31,323,279,165,198,419,76,354),(32,336,280,122,199,432,77,367),(33,293,225,135,200,445,78,380),(34,306,226,148,201,402,79,337),(35,319,227,161,202,415,80,350),(36,332,228,118,203,428,81,363),(37,289,229,131,204,441,82,376),(38,302,230,144,205,398,83,389),(39,315,231,157,206,411,84,346),(40,328,232,114,207,424,85,359),(41,285,233,127,208,437,86,372),(42,298,234,140,209,394,87,385),(43,311,235,153,210,407,88,342),(44,324,236,166,211,420,89,355),(45,281,237,123,212,433,90,368),(46,294,238,136,213,446,91,381),(47,307,239,149,214,403,92,338),(48,320,240,162,215,416,93,351),(49,333,241,119,216,429,94,364),(50,290,242,132,217,442,95,377),(51,303,243,145,218,399,96,390),(52,316,244,158,219,412,97,347),(53,329,245,115,220,425,98,360),(54,286,246,128,221,438,99,373),(55,299,247,141,222,395,100,386),(56,312,248,154,223,408,101,343)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8X | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D7 | M4(2) | D14 | Dic7 | C7⋊C8 | C4×D7 | C8⋊D7 |
| kernel | C56⋊C8 | C4×C7⋊C8 | C4×C56 | C2×C7⋊C8 | C2×C56 | C56 | C4×C8 | C28 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 3 | 8 | 3 | 6 | 24 | 12 | 48 |
Matrix representation of C56⋊C8 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 13 | 70 |
| 0 | 43 | 111 |
| 69 | 0 | 0 |
| 0 | 17 | 31 |
| 0 | 100 | 96 |
G:=sub<GL(3,GF(113))| [112,0,0,0,13,43,0,70,111],[69,0,0,0,17,100,0,31,96] >;
C56⋊C8 in GAP, Magma, Sage, TeX
C_{56}\rtimes C_8 % in TeX
G:=Group("C56:C8"); // GroupNames label
G:=SmallGroup(448,12);
// by ID
G=gap.SmallGroup(448,12);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,477,64,100,136,18822]);
// Polycyclic
G:=Group<a,b|a^56=b^8=1,b*a*b^-1=a^13>;
// generators/relations