metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C49⋊C9, C147.C3, C7.(C7⋊C9), C3.(C49⋊C3), C21.1(C7⋊C3), SmallGroup(441,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C49 — C49⋊C9 |
Generators and relations for C49⋊C9
G = < a,b | a49=b9=1, bab-1=a18 >
Smallest permutation representation of C49⋊C9
►Regular action on 441 points
Generators in S441
(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)
(1 436 282 126 355 245 88 335 173)(2 417 251 127 385 214 89 316 191)(3 398 269 128 366 232 90 297 160)(4 428 287 129 347 201 91 327 178)(5 409 256 130 377 219 92 308 196)(6 439 274 131 358 237 93 338 165)(7 420 292 132 388 206 94 319 183)(8 401 261 133 369 224 95 300 152)(9 431 279 134 350 242 96 330 170)(10 412 248 135 380 211 97 311 188)(11 393 266 136 361 229 98 341 157)(12 423 284 137 391 198 50 322 175)(13 404 253 138 372 216 51 303 193)(14 434 271 139 353 234 52 333 162)(15 415 289 140 383 203 53 314 180)(16 396 258 141 364 221 54 295 149)(17 426 276 142 345 239 55 325 167)(18 407 294 143 375 208 56 306 185)(19 437 263 144 356 226 57 336 154)(20 418 281 145 386 244 58 317 172)(21 399 250 146 367 213 59 298 190)(22 429 268 147 348 231 60 328 159)(23 410 286 99 378 200 61 309 177)(24 440 255 100 359 218 62 339 195)(25 421 273 101 389 236 63 320 164)(26 402 291 102 370 205 64 301 182)(27 432 260 103 351 223 65 331 151)(28 413 278 104 381 241 66 312 169)(29 394 247 105 362 210 67 342 187)(30 424 265 106 392 228 68 323 156)(31 405 283 107 373 197 69 304 174)(32 435 252 108 354 215 70 334 192)(33 416 270 109 384 233 71 315 161)(34 397 288 110 365 202 72 296 179)(35 427 257 111 346 220 73 326 148)(36 408 275 112 376 238 74 307 166)(37 438 293 113 357 207 75 337 184)(38 419 262 114 387 225 76 318 153)(39 400 280 115 368 243 77 299 171)(40 430 249 116 349 212 78 329 189)(41 411 267 117 379 230 79 310 158)(42 441 285 118 360 199 80 340 176)(43 422 254 119 390 217 81 321 194)(44 403 272 120 371 235 82 302 163)(45 433 290 121 352 204 83 332 181)(46 414 259 122 382 222 84 313 150)(47 395 277 123 363 240 85 343 168)(48 425 246 124 344 209 86 324 186)(49 406 264 125 374 227 87 305 155)
G:=sub<Sym(441)| (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), (1,436,282,126,355,245,88,335,173)(2,417,251,127,385,214,89,316,191)(3,398,269,128,366,232,90,297,160)(4,428,287,129,347,201,91,327,178)(5,409,256,130,377,219,92,308,196)(6,439,274,131,358,237,93,338,165)(7,420,292,132,388,206,94,319,183)(8,401,261,133,369,224,95,300,152)(9,431,279,134,350,242,96,330,170)(10,412,248,135,380,211,97,311,188)(11,393,266,136,361,229,98,341,157)(12,423,284,137,391,198,50,322,175)(13,404,253,138,372,216,51,303,193)(14,434,271,139,353,234,52,333,162)(15,415,289,140,383,203,53,314,180)(16,396,258,141,364,221,54,295,149)(17,426,276,142,345,239,55,325,167)(18,407,294,143,375,208,56,306,185)(19,437,263,144,356,226,57,336,154)(20,418,281,145,386,244,58,317,172)(21,399,250,146,367,213,59,298,190)(22,429,268,147,348,231,60,328,159)(23,410,286,99,378,200,61,309,177)(24,440,255,100,359,218,62,339,195)(25,421,273,101,389,236,63,320,164)(26,402,291,102,370,205,64,301,182)(27,432,260,103,351,223,65,331,151)(28,413,278,104,381,241,66,312,169)(29,394,247,105,362,210,67,342,187)(30,424,265,106,392,228,68,323,156)(31,405,283,107,373,197,69,304,174)(32,435,252,108,354,215,70,334,192)(33,416,270,109,384,233,71,315,161)(34,397,288,110,365,202,72,296,179)(35,427,257,111,346,220,73,326,148)(36,408,275,112,376,238,74,307,166)(37,438,293,113,357,207,75,337,184)(38,419,262,114,387,225,76,318,153)(39,400,280,115,368,243,77,299,171)(40,430,249,116,349,212,78,329,189)(41,411,267,117,379,230,79,310,158)(42,441,285,118,360,199,80,340,176)(43,422,254,119,390,217,81,321,194)(44,403,272,120,371,235,82,302,163)(45,433,290,121,352,204,83,332,181)(46,414,259,122,382,222,84,313,150)(47,395,277,123,363,240,85,343,168)(48,425,246,124,344,209,86,324,186)(49,406,264,125,374,227,87,305,155)>;
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), (1,436,282,126,355,245,88,335,173)(2,417,251,127,385,214,89,316,191)(3,398,269,128,366,232,90,297,160)(4,428,287,129,347,201,91,327,178)(5,409,256,130,377,219,92,308,196)(6,439,274,131,358,237,93,338,165)(7,420,292,132,388,206,94,319,183)(8,401,261,133,369,224,95,300,152)(9,431,279,134,350,242,96,330,170)(10,412,248,135,380,211,97,311,188)(11,393,266,136,361,229,98,341,157)(12,423,284,137,391,198,50,322,175)(13,404,253,138,372,216,51,303,193)(14,434,271,139,353,234,52,333,162)(15,415,289,140,383,203,53,314,180)(16,396,258,141,364,221,54,295,149)(17,426,276,142,345,239,55,325,167)(18,407,294,143,375,208,56,306,185)(19,437,263,144,356,226,57,336,154)(20,418,281,145,386,244,58,317,172)(21,399,250,146,367,213,59,298,190)(22,429,268,147,348,231,60,328,159)(23,410,286,99,378,200,61,309,177)(24,440,255,100,359,218,62,339,195)(25,421,273,101,389,236,63,320,164)(26,402,291,102,370,205,64,301,182)(27,432,260,103,351,223,65,331,151)(28,413,278,104,381,241,66,312,169)(29,394,247,105,362,210,67,342,187)(30,424,265,106,392,228,68,323,156)(31,405,283,107,373,197,69,304,174)(32,435,252,108,354,215,70,334,192)(33,416,270,109,384,233,71,315,161)(34,397,288,110,365,202,72,296,179)(35,427,257,111,346,220,73,326,148)(36,408,275,112,376,238,74,307,166)(37,438,293,113,357,207,75,337,184)(38,419,262,114,387,225,76,318,153)(39,400,280,115,368,243,77,299,171)(40,430,249,116,349,212,78,329,189)(41,411,267,117,379,230,79,310,158)(42,441,285,118,360,199,80,340,176)(43,422,254,119,390,217,81,321,194)(44,403,272,120,371,235,82,302,163)(45,433,290,121,352,204,83,332,181)(46,414,259,122,382,222,84,313,150)(47,395,277,123,363,240,85,343,168)(48,425,246,124,344,209,86,324,186)(49,406,264,125,374,227,87,305,155) );
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)], [(1,436,282,126,355,245,88,335,173),(2,417,251,127,385,214,89,316,191),(3,398,269,128,366,232,90,297,160),(4,428,287,129,347,201,91,327,178),(5,409,256,130,377,219,92,308,196),(6,439,274,131,358,237,93,338,165),(7,420,292,132,388,206,94,319,183),(8,401,261,133,369,224,95,300,152),(9,431,279,134,350,242,96,330,170),(10,412,248,135,380,211,97,311,188),(11,393,266,136,361,229,98,341,157),(12,423,284,137,391,198,50,322,175),(13,404,253,138,372,216,51,303,193),(14,434,271,139,353,234,52,333,162),(15,415,289,140,383,203,53,314,180),(16,396,258,141,364,221,54,295,149),(17,426,276,142,345,239,55,325,167),(18,407,294,143,375,208,56,306,185),(19,437,263,144,356,226,57,336,154),(20,418,281,145,386,244,58,317,172),(21,399,250,146,367,213,59,298,190),(22,429,268,147,348,231,60,328,159),(23,410,286,99,378,200,61,309,177),(24,440,255,100,359,218,62,339,195),(25,421,273,101,389,236,63,320,164),(26,402,291,102,370,205,64,301,182),(27,432,260,103,351,223,65,331,151),(28,413,278,104,381,241,66,312,169),(29,394,247,105,362,210,67,342,187),(30,424,265,106,392,228,68,323,156),(31,405,283,107,373,197,69,304,174),(32,435,252,108,354,215,70,334,192),(33,416,270,109,384,233,71,315,161),(34,397,288,110,365,202,72,296,179),(35,427,257,111,346,220,73,326,148),(36,408,275,112,376,238,74,307,166),(37,438,293,113,357,207,75,337,184),(38,419,262,114,387,225,76,318,153),(39,400,280,115,368,243,77,299,171),(40,430,249,116,349,212,78,329,189),(41,411,267,117,379,230,79,310,158),(42,441,285,118,360,199,80,340,176),(43,422,254,119,390,217,81,321,194),(44,403,272,120,371,235,82,302,163),(45,433,290,121,352,204,83,332,181),(46,414,259,122,382,222,84,313,150),(47,395,277,123,363,240,85,343,168),(48,425,246,124,344,209,86,324,186),(49,406,264,125,374,227,87,305,155)]])
57 conjugacy classes
| class | 1 | 3A | 3B | 7A | 7B | 9A | ··· | 9F | 21A | 21B | 21C | 21D | 49A | ··· | 49N | 147A | ··· | 147AB |
| order | 1 | 3 | 3 | 7 | 7 | 9 | ··· | 9 | 21 | 21 | 21 | 21 | 49 | ··· | 49 | 147 | ··· | 147 |
| size | 1 | 1 | 1 | 3 | 3 | 49 | ··· | 49 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
57 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C9 | C7⋊C3 | C7⋊C9 | C49⋊C3 | C49⋊C9 |
| kernel | C49⋊C9 | C147 | C49 | C21 | C7 | C3 | C1 |
| # reps | 1 | 2 | 6 | 2 | 4 | 14 | 28 |
Matrix representation of C49⋊C9 ►in GL4(𝔽883) generated by
| 1 | 0 | 0 | 0 |
| 0 | 23 | 67 | 652 |
| 0 | 652 | 286 | 561 |
| 0 | 561 | 644 | 600 |
| 286 | 0 | 0 | 0 |
| 0 | 813 | 837 | 129 |
| 0 | 484 | 346 | 74 |
| 0 | 129 | 712 | 607 |
G:=sub<GL(4,GF(883))| [1,0,0,0,0,23,652,561,0,67,286,644,0,652,561,600],[286,0,0,0,0,813,484,129,0,837,346,712,0,129,74,607] >;
C49⋊C9 in GAP, Magma, Sage, TeX
C_{49}\rtimes C_9 % in TeX
G:=Group("C49:C9"); // GroupNames label
G:=SmallGroup(441,1);
// by ID
G=gap.SmallGroup(441,1);
# by ID
G:=PCGroup([4,-3,-3,-7,-7,12,974,178,2019]);
// Polycyclic
G:=Group<a,b|a^49=b^9=1,b*a*b^-1=a^18>;
// generators/relations
Export