metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C61⋊C8, C122.C4, Dic61.2C2, C2.(C61⋊C4), SmallGroup(488,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C61 — C61⋊C8 |
Generators and relations for C61⋊C8
G = < a,b | a61=b8=1, bab-1=a50 >
Smallest permutation representation of C61⋊C8
►Regular action on 488 points
Generators in S488
(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)(428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488)
(1 479 184 306 72 426 123 279)(2 429 244 356 73 376 183 268)(3 440 243 345 74 387 182 257)(4 451 242 334 75 398 181 246)(5 462 241 323 76 409 180 296)(6 473 240 312 77 420 179 285)(7 484 239 362 78 370 178 274)(8 434 238 351 79 381 177 263)(9 445 237 340 80 392 176 252)(10 456 236 329 81 403 175 302)(11 467 235 318 82 414 174 291)(12 478 234 307 83 425 173 280)(13 428 233 357 84 375 172 269)(14 439 232 346 85 386 171 258)(15 450 231 335 86 397 170 247)(16 461 230 324 87 408 169 297)(17 472 229 313 88 419 168 286)(18 483 228 363 89 369 167 275)(19 433 227 352 90 380 166 264)(20 444 226 341 91 391 165 253)(21 455 225 330 92 402 164 303)(22 466 224 319 93 413 163 292)(23 477 223 308 94 424 162 281)(24 488 222 358 95 374 161 270)(25 438 221 347 96 385 160 259)(26 449 220 336 97 396 159 248)(27 460 219 325 98 407 158 298)(28 471 218 314 99 418 157 287)(29 482 217 364 100 368 156 276)(30 432 216 353 101 379 155 265)(31 443 215 342 102 390 154 254)(32 454 214 331 103 401 153 304)(33 465 213 320 104 412 152 293)(34 476 212 309 105 423 151 282)(35 487 211 359 106 373 150 271)(36 437 210 348 107 384 149 260)(37 448 209 337 108 395 148 249)(38 459 208 326 109 406 147 299)(39 470 207 315 110 417 146 288)(40 481 206 365 111 367 145 277)(41 431 205 354 112 378 144 266)(42 442 204 343 113 389 143 255)(43 453 203 332 114 400 142 305)(44 464 202 321 115 411 141 294)(45 475 201 310 116 422 140 283)(46 486 200 360 117 372 139 272)(47 436 199 349 118 383 138 261)(48 447 198 338 119 394 137 250)(49 458 197 327 120 405 136 300)(50 469 196 316 121 416 135 289)(51 480 195 366 122 427 134 278)(52 430 194 355 62 377 133 267)(53 441 193 344 63 388 132 256)(54 452 192 333 64 399 131 245)(55 463 191 322 65 410 130 295)(56 474 190 311 66 421 129 284)(57 485 189 361 67 371 128 273)(58 435 188 350 68 382 127 262)(59 446 187 339 69 393 126 251)(60 457 186 328 70 404 125 301)(61 468 185 317 71 415 124 290)
G:=sub<Sym(488)| (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)(428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488), (1,479,184,306,72,426,123,279)(2,429,244,356,73,376,183,268)(3,440,243,345,74,387,182,257)(4,451,242,334,75,398,181,246)(5,462,241,323,76,409,180,296)(6,473,240,312,77,420,179,285)(7,484,239,362,78,370,178,274)(8,434,238,351,79,381,177,263)(9,445,237,340,80,392,176,252)(10,456,236,329,81,403,175,302)(11,467,235,318,82,414,174,291)(12,478,234,307,83,425,173,280)(13,428,233,357,84,375,172,269)(14,439,232,346,85,386,171,258)(15,450,231,335,86,397,170,247)(16,461,230,324,87,408,169,297)(17,472,229,313,88,419,168,286)(18,483,228,363,89,369,167,275)(19,433,227,352,90,380,166,264)(20,444,226,341,91,391,165,253)(21,455,225,330,92,402,164,303)(22,466,224,319,93,413,163,292)(23,477,223,308,94,424,162,281)(24,488,222,358,95,374,161,270)(25,438,221,347,96,385,160,259)(26,449,220,336,97,396,159,248)(27,460,219,325,98,407,158,298)(28,471,218,314,99,418,157,287)(29,482,217,364,100,368,156,276)(30,432,216,353,101,379,155,265)(31,443,215,342,102,390,154,254)(32,454,214,331,103,401,153,304)(33,465,213,320,104,412,152,293)(34,476,212,309,105,423,151,282)(35,487,211,359,106,373,150,271)(36,437,210,348,107,384,149,260)(37,448,209,337,108,395,148,249)(38,459,208,326,109,406,147,299)(39,470,207,315,110,417,146,288)(40,481,206,365,111,367,145,277)(41,431,205,354,112,378,144,266)(42,442,204,343,113,389,143,255)(43,453,203,332,114,400,142,305)(44,464,202,321,115,411,141,294)(45,475,201,310,116,422,140,283)(46,486,200,360,117,372,139,272)(47,436,199,349,118,383,138,261)(48,447,198,338,119,394,137,250)(49,458,197,327,120,405,136,300)(50,469,196,316,121,416,135,289)(51,480,195,366,122,427,134,278)(52,430,194,355,62,377,133,267)(53,441,193,344,63,388,132,256)(54,452,192,333,64,399,131,245)(55,463,191,322,65,410,130,295)(56,474,190,311,66,421,129,284)(57,485,189,361,67,371,128,273)(58,435,188,350,68,382,127,262)(59,446,187,339,69,393,126,251)(60,457,186,328,70,404,125,301)(61,468,185,317,71,415,124,290)>;
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)(428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488), (1,479,184,306,72,426,123,279)(2,429,244,356,73,376,183,268)(3,440,243,345,74,387,182,257)(4,451,242,334,75,398,181,246)(5,462,241,323,76,409,180,296)(6,473,240,312,77,420,179,285)(7,484,239,362,78,370,178,274)(8,434,238,351,79,381,177,263)(9,445,237,340,80,392,176,252)(10,456,236,329,81,403,175,302)(11,467,235,318,82,414,174,291)(12,478,234,307,83,425,173,280)(13,428,233,357,84,375,172,269)(14,439,232,346,85,386,171,258)(15,450,231,335,86,397,170,247)(16,461,230,324,87,408,169,297)(17,472,229,313,88,419,168,286)(18,483,228,363,89,369,167,275)(19,433,227,352,90,380,166,264)(20,444,226,341,91,391,165,253)(21,455,225,330,92,402,164,303)(22,466,224,319,93,413,163,292)(23,477,223,308,94,424,162,281)(24,488,222,358,95,374,161,270)(25,438,221,347,96,385,160,259)(26,449,220,336,97,396,159,248)(27,460,219,325,98,407,158,298)(28,471,218,314,99,418,157,287)(29,482,217,364,100,368,156,276)(30,432,216,353,101,379,155,265)(31,443,215,342,102,390,154,254)(32,454,214,331,103,401,153,304)(33,465,213,320,104,412,152,293)(34,476,212,309,105,423,151,282)(35,487,211,359,106,373,150,271)(36,437,210,348,107,384,149,260)(37,448,209,337,108,395,148,249)(38,459,208,326,109,406,147,299)(39,470,207,315,110,417,146,288)(40,481,206,365,111,367,145,277)(41,431,205,354,112,378,144,266)(42,442,204,343,113,389,143,255)(43,453,203,332,114,400,142,305)(44,464,202,321,115,411,141,294)(45,475,201,310,116,422,140,283)(46,486,200,360,117,372,139,272)(47,436,199,349,118,383,138,261)(48,447,198,338,119,394,137,250)(49,458,197,327,120,405,136,300)(50,469,196,316,121,416,135,289)(51,480,195,366,122,427,134,278)(52,430,194,355,62,377,133,267)(53,441,193,344,63,388,132,256)(54,452,192,333,64,399,131,245)(55,463,191,322,65,410,130,295)(56,474,190,311,66,421,129,284)(57,485,189,361,67,371,128,273)(58,435,188,350,68,382,127,262)(59,446,187,339,69,393,126,251)(60,457,186,328,70,404,125,301)(61,468,185,317,71,415,124,290) );
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),(428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488)], [(1,479,184,306,72,426,123,279),(2,429,244,356,73,376,183,268),(3,440,243,345,74,387,182,257),(4,451,242,334,75,398,181,246),(5,462,241,323,76,409,180,296),(6,473,240,312,77,420,179,285),(7,484,239,362,78,370,178,274),(8,434,238,351,79,381,177,263),(9,445,237,340,80,392,176,252),(10,456,236,329,81,403,175,302),(11,467,235,318,82,414,174,291),(12,478,234,307,83,425,173,280),(13,428,233,357,84,375,172,269),(14,439,232,346,85,386,171,258),(15,450,231,335,86,397,170,247),(16,461,230,324,87,408,169,297),(17,472,229,313,88,419,168,286),(18,483,228,363,89,369,167,275),(19,433,227,352,90,380,166,264),(20,444,226,341,91,391,165,253),(21,455,225,330,92,402,164,303),(22,466,224,319,93,413,163,292),(23,477,223,308,94,424,162,281),(24,488,222,358,95,374,161,270),(25,438,221,347,96,385,160,259),(26,449,220,336,97,396,159,248),(27,460,219,325,98,407,158,298),(28,471,218,314,99,418,157,287),(29,482,217,364,100,368,156,276),(30,432,216,353,101,379,155,265),(31,443,215,342,102,390,154,254),(32,454,214,331,103,401,153,304),(33,465,213,320,104,412,152,293),(34,476,212,309,105,423,151,282),(35,487,211,359,106,373,150,271),(36,437,210,348,107,384,149,260),(37,448,209,337,108,395,148,249),(38,459,208,326,109,406,147,299),(39,470,207,315,110,417,146,288),(40,481,206,365,111,367,145,277),(41,431,205,354,112,378,144,266),(42,442,204,343,113,389,143,255),(43,453,203,332,114,400,142,305),(44,464,202,321,115,411,141,294),(45,475,201,310,116,422,140,283),(46,486,200,360,117,372,139,272),(47,436,199,349,118,383,138,261),(48,447,198,338,119,394,137,250),(49,458,197,327,120,405,136,300),(50,469,196,316,121,416,135,289),(51,480,195,366,122,427,134,278),(52,430,194,355,62,377,133,267),(53,441,193,344,63,388,132,256),(54,452,192,333,64,399,131,245),(55,463,191,322,65,410,130,295),(56,474,190,311,66,421,129,284),(57,485,189,361,67,371,128,273),(58,435,188,350,68,382,127,262),(59,446,187,339,69,393,126,251),(60,457,186,328,70,404,125,301),(61,468,185,317,71,415,124,290)]])
38 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 61A | ··· | 61O | 122A | ··· | 122O |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 61 | ··· | 61 | 122 | ··· | 122 |
| size | 1 | 1 | 61 | 61 | 61 | 61 | 61 | 61 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | - | ||
| image | C1 | C2 | C4 | C8 | C61⋊C4 | C61⋊C8 |
| kernel | C61⋊C8 | Dic61 | C122 | C61 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 15 | 15 |
Matrix representation of C61⋊C8 ►in GL4(𝔽977) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 976 | 583 | 192 | 583 |
| 168 | 849 | 356 | 214 |
| 879 | 741 | 918 | 793 |
| 861 | 509 | 486 | 724 |
| 313 | 355 | 729 | 559 |
G:=sub<GL(4,GF(977))| [0,0,0,976,1,0,0,583,0,1,0,192,0,0,1,583],[168,879,861,313,849,741,509,355,356,918,486,729,214,793,724,559] >;
C61⋊C8 in GAP, Magma, Sage, TeX
C_{61}\rtimes C_8 % in TeX
G:=Group("C61:C8"); // GroupNames label
G:=SmallGroup(488,3);
// by ID
G=gap.SmallGroup(488,3);
# by ID
G:=PCGroup([4,-2,-2,-2,-61,8,21,1411,3847]);
// Polycyclic
G:=Group<a,b|a^61=b^8=1,b*a*b^-1=a^50>;
// generators/relations
Export