metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7⋊C64, C14.C32, C56.2C8, C32.2D7, C112.2C4, C224.3C2, C28.2C16, C16.3Dic7, C2.(C7⋊C32), C8.4(C7⋊C8), C4.2(C7⋊C16), SmallGroup(448,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C64 |
Generators and relations for C7⋊C64
G = < a,b | a7=b64=1, bab-1=a-1 >
Smallest permutation representation of C7⋊C64
►Regular action on 448 points
Generators in S448
(1 380 194 151 397 290 102)(2 103 291 398 152 195 381)(3 382 196 153 399 292 104)(4 105 293 400 154 197 383)(5 384 198 155 401 294 106)(6 107 295 402 156 199 321)(7 322 200 157 403 296 108)(8 109 297 404 158 201 323)(9 324 202 159 405 298 110)(10 111 299 406 160 203 325)(11 326 204 161 407 300 112)(12 113 301 408 162 205 327)(13 328 206 163 409 302 114)(14 115 303 410 164 207 329)(15 330 208 165 411 304 116)(16 117 305 412 166 209 331)(17 332 210 167 413 306 118)(18 119 307 414 168 211 333)(19 334 212 169 415 308 120)(20 121 309 416 170 213 335)(21 336 214 171 417 310 122)(22 123 311 418 172 215 337)(23 338 216 173 419 312 124)(24 125 313 420 174 217 339)(25 340 218 175 421 314 126)(26 127 315 422 176 219 341)(27 342 220 177 423 316 128)(28 65 317 424 178 221 343)(29 344 222 179 425 318 66)(30 67 319 426 180 223 345)(31 346 224 181 427 320 68)(32 69 257 428 182 225 347)(33 348 226 183 429 258 70)(34 71 259 430 184 227 349)(35 350 228 185 431 260 72)(36 73 261 432 186 229 351)(37 352 230 187 433 262 74)(38 75 263 434 188 231 353)(39 354 232 189 435 264 76)(40 77 265 436 190 233 355)(41 356 234 191 437 266 78)(42 79 267 438 192 235 357)(43 358 236 129 439 268 80)(44 81 269 440 130 237 359)(45 360 238 131 441 270 82)(46 83 271 442 132 239 361)(47 362 240 133 443 272 84)(48 85 273 444 134 241 363)(49 364 242 135 445 274 86)(50 87 275 446 136 243 365)(51 366 244 137 447 276 88)(52 89 277 448 138 245 367)(53 368 246 139 385 278 90)(54 91 279 386 140 247 369)(55 370 248 141 387 280 92)(56 93 281 388 142 249 371)(57 372 250 143 389 282 94)(58 95 283 390 144 251 373)(59 374 252 145 391 284 96)(60 97 285 392 146 253 375)(61 376 254 147 393 286 98)(62 99 287 394 148 255 377)(63 378 256 149 395 288 100)(64 101 289 396 150 193 379)
(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)| (1,380,194,151,397,290,102)(2,103,291,398,152,195,381)(3,382,196,153,399,292,104)(4,105,293,400,154,197,383)(5,384,198,155,401,294,106)(6,107,295,402,156,199,321)(7,322,200,157,403,296,108)(8,109,297,404,158,201,323)(9,324,202,159,405,298,110)(10,111,299,406,160,203,325)(11,326,204,161,407,300,112)(12,113,301,408,162,205,327)(13,328,206,163,409,302,114)(14,115,303,410,164,207,329)(15,330,208,165,411,304,116)(16,117,305,412,166,209,331)(17,332,210,167,413,306,118)(18,119,307,414,168,211,333)(19,334,212,169,415,308,120)(20,121,309,416,170,213,335)(21,336,214,171,417,310,122)(22,123,311,418,172,215,337)(23,338,216,173,419,312,124)(24,125,313,420,174,217,339)(25,340,218,175,421,314,126)(26,127,315,422,176,219,341)(27,342,220,177,423,316,128)(28,65,317,424,178,221,343)(29,344,222,179,425,318,66)(30,67,319,426,180,223,345)(31,346,224,181,427,320,68)(32,69,257,428,182,225,347)(33,348,226,183,429,258,70)(34,71,259,430,184,227,349)(35,350,228,185,431,260,72)(36,73,261,432,186,229,351)(37,352,230,187,433,262,74)(38,75,263,434,188,231,353)(39,354,232,189,435,264,76)(40,77,265,436,190,233,355)(41,356,234,191,437,266,78)(42,79,267,438,192,235,357)(43,358,236,129,439,268,80)(44,81,269,440,130,237,359)(45,360,238,131,441,270,82)(46,83,271,442,132,239,361)(47,362,240,133,443,272,84)(48,85,273,444,134,241,363)(49,364,242,135,445,274,86)(50,87,275,446,136,243,365)(51,366,244,137,447,276,88)(52,89,277,448,138,245,367)(53,368,246,139,385,278,90)(54,91,279,386,140,247,369)(55,370,248,141,387,280,92)(56,93,281,388,142,249,371)(57,372,250,143,389,282,94)(58,95,283,390,144,251,373)(59,374,252,145,391,284,96)(60,97,285,392,146,253,375)(61,376,254,147,393,286,98)(62,99,287,394,148,255,377)(63,378,256,149,395,288,100)(64,101,289,396,150,193,379), (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( (1,380,194,151,397,290,102)(2,103,291,398,152,195,381)(3,382,196,153,399,292,104)(4,105,293,400,154,197,383)(5,384,198,155,401,294,106)(6,107,295,402,156,199,321)(7,322,200,157,403,296,108)(8,109,297,404,158,201,323)(9,324,202,159,405,298,110)(10,111,299,406,160,203,325)(11,326,204,161,407,300,112)(12,113,301,408,162,205,327)(13,328,206,163,409,302,114)(14,115,303,410,164,207,329)(15,330,208,165,411,304,116)(16,117,305,412,166,209,331)(17,332,210,167,413,306,118)(18,119,307,414,168,211,333)(19,334,212,169,415,308,120)(20,121,309,416,170,213,335)(21,336,214,171,417,310,122)(22,123,311,418,172,215,337)(23,338,216,173,419,312,124)(24,125,313,420,174,217,339)(25,340,218,175,421,314,126)(26,127,315,422,176,219,341)(27,342,220,177,423,316,128)(28,65,317,424,178,221,343)(29,344,222,179,425,318,66)(30,67,319,426,180,223,345)(31,346,224,181,427,320,68)(32,69,257,428,182,225,347)(33,348,226,183,429,258,70)(34,71,259,430,184,227,349)(35,350,228,185,431,260,72)(36,73,261,432,186,229,351)(37,352,230,187,433,262,74)(38,75,263,434,188,231,353)(39,354,232,189,435,264,76)(40,77,265,436,190,233,355)(41,356,234,191,437,266,78)(42,79,267,438,192,235,357)(43,358,236,129,439,268,80)(44,81,269,440,130,237,359)(45,360,238,131,441,270,82)(46,83,271,442,132,239,361)(47,362,240,133,443,272,84)(48,85,273,444,134,241,363)(49,364,242,135,445,274,86)(50,87,275,446,136,243,365)(51,366,244,137,447,276,88)(52,89,277,448,138,245,367)(53,368,246,139,385,278,90)(54,91,279,386,140,247,369)(55,370,248,141,387,280,92)(56,93,281,388,142,249,371)(57,372,250,143,389,282,94)(58,95,283,390,144,251,373)(59,374,252,145,391,284,96)(60,97,285,392,146,253,375)(61,376,254,147,393,286,98)(62,99,287,394,148,255,377)(63,378,256,149,395,288,100)(64,101,289,396,150,193,379), (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([[(1,380,194,151,397,290,102),(2,103,291,398,152,195,381),(3,382,196,153,399,292,104),(4,105,293,400,154,197,383),(5,384,198,155,401,294,106),(6,107,295,402,156,199,321),(7,322,200,157,403,296,108),(8,109,297,404,158,201,323),(9,324,202,159,405,298,110),(10,111,299,406,160,203,325),(11,326,204,161,407,300,112),(12,113,301,408,162,205,327),(13,328,206,163,409,302,114),(14,115,303,410,164,207,329),(15,330,208,165,411,304,116),(16,117,305,412,166,209,331),(17,332,210,167,413,306,118),(18,119,307,414,168,211,333),(19,334,212,169,415,308,120),(20,121,309,416,170,213,335),(21,336,214,171,417,310,122),(22,123,311,418,172,215,337),(23,338,216,173,419,312,124),(24,125,313,420,174,217,339),(25,340,218,175,421,314,126),(26,127,315,422,176,219,341),(27,342,220,177,423,316,128),(28,65,317,424,178,221,343),(29,344,222,179,425,318,66),(30,67,319,426,180,223,345),(31,346,224,181,427,320,68),(32,69,257,428,182,225,347),(33,348,226,183,429,258,70),(34,71,259,430,184,227,349),(35,350,228,185,431,260,72),(36,73,261,432,186,229,351),(37,352,230,187,433,262,74),(38,75,263,434,188,231,353),(39,354,232,189,435,264,76),(40,77,265,436,190,233,355),(41,356,234,191,437,266,78),(42,79,267,438,192,235,357),(43,358,236,129,439,268,80),(44,81,269,440,130,237,359),(45,360,238,131,441,270,82),(46,83,271,442,132,239,361),(47,362,240,133,443,272,84),(48,85,273,444,134,241,363),(49,364,242,135,445,274,86),(50,87,275,446,136,243,365),(51,366,244,137,447,276,88),(52,89,277,448,138,245,367),(53,368,246,139,385,278,90),(54,91,279,386,140,247,369),(55,370,248,141,387,280,92),(56,93,281,388,142,249,371),(57,372,250,143,389,282,94),(58,95,283,390,144,251,373),(59,374,252,145,391,284,96),(60,97,285,392,146,253,375),(61,376,254,147,393,286,98),(62,99,287,394,148,255,377),(63,378,256,149,395,288,100),(64,101,289,396,150,193,379)], [(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)]])
160 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 16A | ··· | 16H | 28A | ··· | 28F | 32A | ··· | 32P | 56A | ··· | 56L | 64A | ··· | 64AF | 112A | ··· | 112X | 224A | ··· | 224AV |
| order | 1 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 64 | ··· | 64 | 112 | ··· | 112 | 224 | ··· | 224 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||
| image | C1 | C2 | C4 | C8 | C16 | C32 | C64 | D7 | Dic7 | C7⋊C8 | C7⋊C16 | C7⋊C32 | C7⋊C64 |
| kernel | C7⋊C64 | C224 | C112 | C56 | C28 | C14 | C7 | C32 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 16 | 32 | 3 | 3 | 6 | 12 | 24 | 48 |
Matrix representation of C7⋊C64 ►in GL3(𝔽449) generated by
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 448 | 354 |
| 188 | 0 | 0 |
| 0 | 381 | 135 |
| 0 | 309 | 68 |
G:=sub<GL(3,GF(449))| [1,0,0,0,0,448,0,1,354],[188,0,0,0,381,309,0,135,68] >;
C7⋊C64 in GAP, Magma, Sage, TeX
C_7\rtimes C_{64} % in TeX
G:=Group("C7:C64"); // GroupNames label
G:=SmallGroup(448,1);
// by ID
G=gap.SmallGroup(448,1);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,14,36,58,80,102,18822]);
// Polycyclic
G:=Group<a,b|a^7=b^64=1,b*a*b^-1=a^-1>;
// generators/relations
Export