direct product, abelian, monomial, 2-elementary
Aliases: C8×C56, SmallGroup(448,125)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C8×C56 |
| C1 — C8×C56 |
| C1 — C8×C56 |
Generators and relations for C8×C56
G = < a,b | a8=b56=1, ab=ba >
Subgroups: 74, all normal (8 characteristic)
C1, C2, C4, C22, C7, C8, C2×C4, C14, C42, C2×C8, C28, C2×C14, C4×C8, C56, C2×C28, C82, C4×C28, C2×C56, C4×C56, C8×C56
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, C14, C42, C2×C8, C28, C2×C14, C4×C8, C56, C2×C28, C82, C4×C28, C2×C56, C4×C56, C8×C56
►Regular action on 448 points
Generators in S448
(1 201 251 308 392 107 434 127)(2 202 252 309 337 108 435 128)(3 203 253 310 338 109 436 129)(4 204 254 311 339 110 437 130)(5 205 255 312 340 111 438 131)(6 206 256 313 341 112 439 132)(7 207 257 314 342 57 440 133)(8 208 258 315 343 58 441 134)(9 209 259 316 344 59 442 135)(10 210 260 317 345 60 443 136)(11 211 261 318 346 61 444 137)(12 212 262 319 347 62 445 138)(13 213 263 320 348 63 446 139)(14 214 264 321 349 64 447 140)(15 215 265 322 350 65 448 141)(16 216 266 323 351 66 393 142)(17 217 267 324 352 67 394 143)(18 218 268 325 353 68 395 144)(19 219 269 326 354 69 396 145)(20 220 270 327 355 70 397 146)(21 221 271 328 356 71 398 147)(22 222 272 329 357 72 399 148)(23 223 273 330 358 73 400 149)(24 224 274 331 359 74 401 150)(25 169 275 332 360 75 402 151)(26 170 276 333 361 76 403 152)(27 171 277 334 362 77 404 153)(28 172 278 335 363 78 405 154)(29 173 279 336 364 79 406 155)(30 174 280 281 365 80 407 156)(31 175 225 282 366 81 408 157)(32 176 226 283 367 82 409 158)(33 177 227 284 368 83 410 159)(34 178 228 285 369 84 411 160)(35 179 229 286 370 85 412 161)(36 180 230 287 371 86 413 162)(37 181 231 288 372 87 414 163)(38 182 232 289 373 88 415 164)(39 183 233 290 374 89 416 165)(40 184 234 291 375 90 417 166)(41 185 235 292 376 91 418 167)(42 186 236 293 377 92 419 168)(43 187 237 294 378 93 420 113)(44 188 238 295 379 94 421 114)(45 189 239 296 380 95 422 115)(46 190 240 297 381 96 423 116)(47 191 241 298 382 97 424 117)(48 192 242 299 383 98 425 118)(49 193 243 300 384 99 426 119)(50 194 244 301 385 100 427 120)(51 195 245 302 386 101 428 121)(52 196 246 303 387 102 429 122)(53 197 247 304 388 103 430 123)(54 198 248 305 389 104 431 124)(55 199 249 306 390 105 432 125)(56 200 250 307 391 106 433 126)
(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)
G:=sub<Sym(448)| (1,201,251,308,392,107,434,127)(2,202,252,309,337,108,435,128)(3,203,253,310,338,109,436,129)(4,204,254,311,339,110,437,130)(5,205,255,312,340,111,438,131)(6,206,256,313,341,112,439,132)(7,207,257,314,342,57,440,133)(8,208,258,315,343,58,441,134)(9,209,259,316,344,59,442,135)(10,210,260,317,345,60,443,136)(11,211,261,318,346,61,444,137)(12,212,262,319,347,62,445,138)(13,213,263,320,348,63,446,139)(14,214,264,321,349,64,447,140)(15,215,265,322,350,65,448,141)(16,216,266,323,351,66,393,142)(17,217,267,324,352,67,394,143)(18,218,268,325,353,68,395,144)(19,219,269,326,354,69,396,145)(20,220,270,327,355,70,397,146)(21,221,271,328,356,71,398,147)(22,222,272,329,357,72,399,148)(23,223,273,330,358,73,400,149)(24,224,274,331,359,74,401,150)(25,169,275,332,360,75,402,151)(26,170,276,333,361,76,403,152)(27,171,277,334,362,77,404,153)(28,172,278,335,363,78,405,154)(29,173,279,336,364,79,406,155)(30,174,280,281,365,80,407,156)(31,175,225,282,366,81,408,157)(32,176,226,283,367,82,409,158)(33,177,227,284,368,83,410,159)(34,178,228,285,369,84,411,160)(35,179,229,286,370,85,412,161)(36,180,230,287,371,86,413,162)(37,181,231,288,372,87,414,163)(38,182,232,289,373,88,415,164)(39,183,233,290,374,89,416,165)(40,184,234,291,375,90,417,166)(41,185,235,292,376,91,418,167)(42,186,236,293,377,92,419,168)(43,187,237,294,378,93,420,113)(44,188,238,295,379,94,421,114)(45,189,239,296,380,95,422,115)(46,190,240,297,381,96,423,116)(47,191,241,298,382,97,424,117)(48,192,242,299,383,98,425,118)(49,193,243,300,384,99,426,119)(50,194,244,301,385,100,427,120)(51,195,245,302,386,101,428,121)(52,196,246,303,387,102,429,122)(53,197,247,304,388,103,430,123)(54,198,248,305,389,104,431,124)(55,199,249,306,390,105,432,125)(56,200,250,307,391,106,433,126), (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)>;
G:=Group( (1,201,251,308,392,107,434,127)(2,202,252,309,337,108,435,128)(3,203,253,310,338,109,436,129)(4,204,254,311,339,110,437,130)(5,205,255,312,340,111,438,131)(6,206,256,313,341,112,439,132)(7,207,257,314,342,57,440,133)(8,208,258,315,343,58,441,134)(9,209,259,316,344,59,442,135)(10,210,260,317,345,60,443,136)(11,211,261,318,346,61,444,137)(12,212,262,319,347,62,445,138)(13,213,263,320,348,63,446,139)(14,214,264,321,349,64,447,140)(15,215,265,322,350,65,448,141)(16,216,266,323,351,66,393,142)(17,217,267,324,352,67,394,143)(18,218,268,325,353,68,395,144)(19,219,269,326,354,69,396,145)(20,220,270,327,355,70,397,146)(21,221,271,328,356,71,398,147)(22,222,272,329,357,72,399,148)(23,223,273,330,358,73,400,149)(24,224,274,331,359,74,401,150)(25,169,275,332,360,75,402,151)(26,170,276,333,361,76,403,152)(27,171,277,334,362,77,404,153)(28,172,278,335,363,78,405,154)(29,173,279,336,364,79,406,155)(30,174,280,281,365,80,407,156)(31,175,225,282,366,81,408,157)(32,176,226,283,367,82,409,158)(33,177,227,284,368,83,410,159)(34,178,228,285,369,84,411,160)(35,179,229,286,370,85,412,161)(36,180,230,287,371,86,413,162)(37,181,231,288,372,87,414,163)(38,182,232,289,373,88,415,164)(39,183,233,290,374,89,416,165)(40,184,234,291,375,90,417,166)(41,185,235,292,376,91,418,167)(42,186,236,293,377,92,419,168)(43,187,237,294,378,93,420,113)(44,188,238,295,379,94,421,114)(45,189,239,296,380,95,422,115)(46,190,240,297,381,96,423,116)(47,191,241,298,382,97,424,117)(48,192,242,299,383,98,425,118)(49,193,243,300,384,99,426,119)(50,194,244,301,385,100,427,120)(51,195,245,302,386,101,428,121)(52,196,246,303,387,102,429,122)(53,197,247,304,388,103,430,123)(54,198,248,305,389,104,431,124)(55,199,249,306,390,105,432,125)(56,200,250,307,391,106,433,126), (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) );
G=PermutationGroup([[(1,201,251,308,392,107,434,127),(2,202,252,309,337,108,435,128),(3,203,253,310,338,109,436,129),(4,204,254,311,339,110,437,130),(5,205,255,312,340,111,438,131),(6,206,256,313,341,112,439,132),(7,207,257,314,342,57,440,133),(8,208,258,315,343,58,441,134),(9,209,259,316,344,59,442,135),(10,210,260,317,345,60,443,136),(11,211,261,318,346,61,444,137),(12,212,262,319,347,62,445,138),(13,213,263,320,348,63,446,139),(14,214,264,321,349,64,447,140),(15,215,265,322,350,65,448,141),(16,216,266,323,351,66,393,142),(17,217,267,324,352,67,394,143),(18,218,268,325,353,68,395,144),(19,219,269,326,354,69,396,145),(20,220,270,327,355,70,397,146),(21,221,271,328,356,71,398,147),(22,222,272,329,357,72,399,148),(23,223,273,330,358,73,400,149),(24,224,274,331,359,74,401,150),(25,169,275,332,360,75,402,151),(26,170,276,333,361,76,403,152),(27,171,277,334,362,77,404,153),(28,172,278,335,363,78,405,154),(29,173,279,336,364,79,406,155),(30,174,280,281,365,80,407,156),(31,175,225,282,366,81,408,157),(32,176,226,283,367,82,409,158),(33,177,227,284,368,83,410,159),(34,178,228,285,369,84,411,160),(35,179,229,286,370,85,412,161),(36,180,230,287,371,86,413,162),(37,181,231,288,372,87,414,163),(38,182,232,289,373,88,415,164),(39,183,233,290,374,89,416,165),(40,184,234,291,375,90,417,166),(41,185,235,292,376,91,418,167),(42,186,236,293,377,92,419,168),(43,187,237,294,378,93,420,113),(44,188,238,295,379,94,421,114),(45,189,239,296,380,95,422,115),(46,190,240,297,381,96,423,116),(47,191,241,298,382,97,424,117),(48,192,242,299,383,98,425,118),(49,193,243,300,384,99,426,119),(50,194,244,301,385,100,427,120),(51,195,245,302,386,101,428,121),(52,196,246,303,387,102,429,122),(53,197,247,304,388,103,430,123),(54,198,248,305,389,104,431,124),(55,199,249,306,390,105,432,125),(56,200,250,307,391,106,433,126)], [(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)]])
448 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 7A | ··· | 7F | 8A | ··· | 8AV | 14A | ··· | 14R | 28A | ··· | 28BT | 56A | ··· | 56KB |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
448 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C4 | C7 | C8 | C14 | C28 | C56 |
| kernel | C8×C56 | C4×C56 | C2×C56 | C82 | C56 | C4×C8 | C2×C8 | C8 |
| # reps | 1 | 3 | 12 | 6 | 48 | 18 | 72 | 288 |
Matrix representation of C8×C56 ►in GL2(𝔽113) generated by
| 95 | 0 |
| 0 | 1 |
| 72 | 0 |
| 0 | 69 |
G:=sub<GL(2,GF(113))| [95,0,0,1],[72,0,0,69] >;
C8×C56 in GAP, Magma, Sage, TeX
C_8\times C_{56} % in TeX
G:=Group("C8xC56"); // GroupNames label
G:=SmallGroup(448,125);
// by ID
G=gap.SmallGroup(448,125);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,196,400,136,172]);
// Polycyclic
G:=Group<a,b|a^8=b^56=1,a*b=b*a>;
// generators/relations