direct product, abelian, monomial, 7-elementary
Aliases: C7×C63, SmallGroup(441,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C7×C63 |
| C1 — C7×C63 |
| C1 — C7×C63 |
Generators and relations for C7×C63
G = < a,b | a7=b63=1, ab=ba >
Smallest permutation representation of C7×C63
►Regular action on 441 points
Generators in S441
(1 414 290 100 206 375 152)(2 415 291 101 207 376 153)(3 416 292 102 208 377 154)(4 417 293 103 209 378 155)(5 418 294 104 210 316 156)(6 419 295 105 211 317 157)(7 420 296 106 212 318 158)(8 421 297 107 213 319 159)(9 422 298 108 214 320 160)(10 423 299 109 215 321 161)(11 424 300 110 216 322 162)(12 425 301 111 217 323 163)(13 426 302 112 218 324 164)(14 427 303 113 219 325 165)(15 428 304 114 220 326 166)(16 429 305 115 221 327 167)(17 430 306 116 222 328 168)(18 431 307 117 223 329 169)(19 432 308 118 224 330 170)(20 433 309 119 225 331 171)(21 434 310 120 226 332 172)(22 435 311 121 227 333 173)(23 436 312 122 228 334 174)(24 437 313 123 229 335 175)(25 438 314 124 230 336 176)(26 439 315 125 231 337 177)(27 440 253 126 232 338 178)(28 441 254 64 233 339 179)(29 379 255 65 234 340 180)(30 380 256 66 235 341 181)(31 381 257 67 236 342 182)(32 382 258 68 237 343 183)(33 383 259 69 238 344 184)(34 384 260 70 239 345 185)(35 385 261 71 240 346 186)(36 386 262 72 241 347 187)(37 387 263 73 242 348 188)(38 388 264 74 243 349 189)(39 389 265 75 244 350 127)(40 390 266 76 245 351 128)(41 391 267 77 246 352 129)(42 392 268 78 247 353 130)(43 393 269 79 248 354 131)(44 394 270 80 249 355 132)(45 395 271 81 250 356 133)(46 396 272 82 251 357 134)(47 397 273 83 252 358 135)(48 398 274 84 190 359 136)(49 399 275 85 191 360 137)(50 400 276 86 192 361 138)(51 401 277 87 193 362 139)(52 402 278 88 194 363 140)(53 403 279 89 195 364 141)(54 404 280 90 196 365 142)(55 405 281 91 197 366 143)(56 406 282 92 198 367 144)(57 407 283 93 199 368 145)(58 408 284 94 200 369 146)(59 409 285 95 201 370 147)(60 410 286 96 202 371 148)(61 411 287 97 203 372 149)(62 412 288 98 204 373 150)(63 413 289 99 205 374 151)
(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)
G:=sub<Sym(441)| (1,414,290,100,206,375,152)(2,415,291,101,207,376,153)(3,416,292,102,208,377,154)(4,417,293,103,209,378,155)(5,418,294,104,210,316,156)(6,419,295,105,211,317,157)(7,420,296,106,212,318,158)(8,421,297,107,213,319,159)(9,422,298,108,214,320,160)(10,423,299,109,215,321,161)(11,424,300,110,216,322,162)(12,425,301,111,217,323,163)(13,426,302,112,218,324,164)(14,427,303,113,219,325,165)(15,428,304,114,220,326,166)(16,429,305,115,221,327,167)(17,430,306,116,222,328,168)(18,431,307,117,223,329,169)(19,432,308,118,224,330,170)(20,433,309,119,225,331,171)(21,434,310,120,226,332,172)(22,435,311,121,227,333,173)(23,436,312,122,228,334,174)(24,437,313,123,229,335,175)(25,438,314,124,230,336,176)(26,439,315,125,231,337,177)(27,440,253,126,232,338,178)(28,441,254,64,233,339,179)(29,379,255,65,234,340,180)(30,380,256,66,235,341,181)(31,381,257,67,236,342,182)(32,382,258,68,237,343,183)(33,383,259,69,238,344,184)(34,384,260,70,239,345,185)(35,385,261,71,240,346,186)(36,386,262,72,241,347,187)(37,387,263,73,242,348,188)(38,388,264,74,243,349,189)(39,389,265,75,244,350,127)(40,390,266,76,245,351,128)(41,391,267,77,246,352,129)(42,392,268,78,247,353,130)(43,393,269,79,248,354,131)(44,394,270,80,249,355,132)(45,395,271,81,250,356,133)(46,396,272,82,251,357,134)(47,397,273,83,252,358,135)(48,398,274,84,190,359,136)(49,399,275,85,191,360,137)(50,400,276,86,192,361,138)(51,401,277,87,193,362,139)(52,402,278,88,194,363,140)(53,403,279,89,195,364,141)(54,404,280,90,196,365,142)(55,405,281,91,197,366,143)(56,406,282,92,198,367,144)(57,407,283,93,199,368,145)(58,408,284,94,200,369,146)(59,409,285,95,201,370,147)(60,410,286,96,202,371,148)(61,411,287,97,203,372,149)(62,412,288,98,204,373,150)(63,413,289,99,205,374,151), (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)>;
G:=Group( (1,414,290,100,206,375,152)(2,415,291,101,207,376,153)(3,416,292,102,208,377,154)(4,417,293,103,209,378,155)(5,418,294,104,210,316,156)(6,419,295,105,211,317,157)(7,420,296,106,212,318,158)(8,421,297,107,213,319,159)(9,422,298,108,214,320,160)(10,423,299,109,215,321,161)(11,424,300,110,216,322,162)(12,425,301,111,217,323,163)(13,426,302,112,218,324,164)(14,427,303,113,219,325,165)(15,428,304,114,220,326,166)(16,429,305,115,221,327,167)(17,430,306,116,222,328,168)(18,431,307,117,223,329,169)(19,432,308,118,224,330,170)(20,433,309,119,225,331,171)(21,434,310,120,226,332,172)(22,435,311,121,227,333,173)(23,436,312,122,228,334,174)(24,437,313,123,229,335,175)(25,438,314,124,230,336,176)(26,439,315,125,231,337,177)(27,440,253,126,232,338,178)(28,441,254,64,233,339,179)(29,379,255,65,234,340,180)(30,380,256,66,235,341,181)(31,381,257,67,236,342,182)(32,382,258,68,237,343,183)(33,383,259,69,238,344,184)(34,384,260,70,239,345,185)(35,385,261,71,240,346,186)(36,386,262,72,241,347,187)(37,387,263,73,242,348,188)(38,388,264,74,243,349,189)(39,389,265,75,244,350,127)(40,390,266,76,245,351,128)(41,391,267,77,246,352,129)(42,392,268,78,247,353,130)(43,393,269,79,248,354,131)(44,394,270,80,249,355,132)(45,395,271,81,250,356,133)(46,396,272,82,251,357,134)(47,397,273,83,252,358,135)(48,398,274,84,190,359,136)(49,399,275,85,191,360,137)(50,400,276,86,192,361,138)(51,401,277,87,193,362,139)(52,402,278,88,194,363,140)(53,403,279,89,195,364,141)(54,404,280,90,196,365,142)(55,405,281,91,197,366,143)(56,406,282,92,198,367,144)(57,407,283,93,199,368,145)(58,408,284,94,200,369,146)(59,409,285,95,201,370,147)(60,410,286,96,202,371,148)(61,411,287,97,203,372,149)(62,412,288,98,204,373,150)(63,413,289,99,205,374,151), (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) );
G=PermutationGroup([[(1,414,290,100,206,375,152),(2,415,291,101,207,376,153),(3,416,292,102,208,377,154),(4,417,293,103,209,378,155),(5,418,294,104,210,316,156),(6,419,295,105,211,317,157),(7,420,296,106,212,318,158),(8,421,297,107,213,319,159),(9,422,298,108,214,320,160),(10,423,299,109,215,321,161),(11,424,300,110,216,322,162),(12,425,301,111,217,323,163),(13,426,302,112,218,324,164),(14,427,303,113,219,325,165),(15,428,304,114,220,326,166),(16,429,305,115,221,327,167),(17,430,306,116,222,328,168),(18,431,307,117,223,329,169),(19,432,308,118,224,330,170),(20,433,309,119,225,331,171),(21,434,310,120,226,332,172),(22,435,311,121,227,333,173),(23,436,312,122,228,334,174),(24,437,313,123,229,335,175),(25,438,314,124,230,336,176),(26,439,315,125,231,337,177),(27,440,253,126,232,338,178),(28,441,254,64,233,339,179),(29,379,255,65,234,340,180),(30,380,256,66,235,341,181),(31,381,257,67,236,342,182),(32,382,258,68,237,343,183),(33,383,259,69,238,344,184),(34,384,260,70,239,345,185),(35,385,261,71,240,346,186),(36,386,262,72,241,347,187),(37,387,263,73,242,348,188),(38,388,264,74,243,349,189),(39,389,265,75,244,350,127),(40,390,266,76,245,351,128),(41,391,267,77,246,352,129),(42,392,268,78,247,353,130),(43,393,269,79,248,354,131),(44,394,270,80,249,355,132),(45,395,271,81,250,356,133),(46,396,272,82,251,357,134),(47,397,273,83,252,358,135),(48,398,274,84,190,359,136),(49,399,275,85,191,360,137),(50,400,276,86,192,361,138),(51,401,277,87,193,362,139),(52,402,278,88,194,363,140),(53,403,279,89,195,364,141),(54,404,280,90,196,365,142),(55,405,281,91,197,366,143),(56,406,282,92,198,367,144),(57,407,283,93,199,368,145),(58,408,284,94,200,369,146),(59,409,285,95,201,370,147),(60,410,286,96,202,371,148),(61,411,287,97,203,372,149),(62,412,288,98,204,373,150),(63,413,289,99,205,374,151)], [(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)]])
441 conjugacy classes
| class | 1 | 3A | 3B | 7A | ··· | 7AV | 9A | ··· | 9F | 21A | ··· | 21CR | 63A | ··· | 63KB |
| order | 1 | 3 | 3 | 7 | ··· | 7 | 9 | ··· | 9 | 21 | ··· | 21 | 63 | ··· | 63 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
441 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||
| image | C1 | C3 | C7 | C9 | C21 | C63 |
| kernel | C7×C63 | C7×C21 | C63 | C72 | C21 | C7 |
| # reps | 1 | 2 | 48 | 6 | 96 | 288 |
Matrix representation of C7×C63 ►in GL2(𝔽127) generated by
| 64 | 0 |
| 0 | 8 |
| 74 | 0 |
| 0 | 88 |
G:=sub<GL(2,GF(127))| [64,0,0,8],[74,0,0,88] >;
C7×C63 in GAP, Magma, Sage, TeX
C_7\times C_{63} % in TeX
G:=Group("C7xC63"); // GroupNames label
G:=SmallGroup(441,8);
// by ID
G=gap.SmallGroup(441,8);
# by ID
G:=PCGroup([4,-3,-7,-7,-3,588]);
// Polycyclic
G:=Group<a,b|a^7=b^63=1,a*b=b*a>;
// generators/relations
Export