metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C61⋊2C8, C4.2D61, C2.Dic61, C122.2C4, C244.2C2, SmallGroup(488,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C61 — C61⋊2C8 |
Generators and relations for C61⋊2C8
G = < a,b | a61=b8=1, bab-1=a-1 >
Smallest permutation representation of C61⋊2C8
►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)
(1 428 213 306 97 367 139 245)(2 488 214 366 98 427 140 305)(3 487 215 365 99 426 141 304)(4 486 216 364 100 425 142 303)(5 485 217 363 101 424 143 302)(6 484 218 362 102 423 144 301)(7 483 219 361 103 422 145 300)(8 482 220 360 104 421 146 299)(9 481 221 359 105 420 147 298)(10 480 222 358 106 419 148 297)(11 479 223 357 107 418 149 296)(12 478 224 356 108 417 150 295)(13 477 225 355 109 416 151 294)(14 476 226 354 110 415 152 293)(15 475 227 353 111 414 153 292)(16 474 228 352 112 413 154 291)(17 473 229 351 113 412 155 290)(18 472 230 350 114 411 156 289)(19 471 231 349 115 410 157 288)(20 470 232 348 116 409 158 287)(21 469 233 347 117 408 159 286)(22 468 234 346 118 407 160 285)(23 467 235 345 119 406 161 284)(24 466 236 344 120 405 162 283)(25 465 237 343 121 404 163 282)(26 464 238 342 122 403 164 281)(27 463 239 341 62 402 165 280)(28 462 240 340 63 401 166 279)(29 461 241 339 64 400 167 278)(30 460 242 338 65 399 168 277)(31 459 243 337 66 398 169 276)(32 458 244 336 67 397 170 275)(33 457 184 335 68 396 171 274)(34 456 185 334 69 395 172 273)(35 455 186 333 70 394 173 272)(36 454 187 332 71 393 174 271)(37 453 188 331 72 392 175 270)(38 452 189 330 73 391 176 269)(39 451 190 329 74 390 177 268)(40 450 191 328 75 389 178 267)(41 449 192 327 76 388 179 266)(42 448 193 326 77 387 180 265)(43 447 194 325 78 386 181 264)(44 446 195 324 79 385 182 263)(45 445 196 323 80 384 183 262)(46 444 197 322 81 383 123 261)(47 443 198 321 82 382 124 260)(48 442 199 320 83 381 125 259)(49 441 200 319 84 380 126 258)(50 440 201 318 85 379 127 257)(51 439 202 317 86 378 128 256)(52 438 203 316 87 377 129 255)(53 437 204 315 88 376 130 254)(54 436 205 314 89 375 131 253)(55 435 206 313 90 374 132 252)(56 434 207 312 91 373 133 251)(57 433 208 311 92 372 134 250)(58 432 209 310 93 371 135 249)(59 431 210 309 94 370 136 248)(60 430 211 308 95 369 137 247)(61 429 212 307 96 368 138 246)
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), (1,428,213,306,97,367,139,245)(2,488,214,366,98,427,140,305)(3,487,215,365,99,426,141,304)(4,486,216,364,100,425,142,303)(5,485,217,363,101,424,143,302)(6,484,218,362,102,423,144,301)(7,483,219,361,103,422,145,300)(8,482,220,360,104,421,146,299)(9,481,221,359,105,420,147,298)(10,480,222,358,106,419,148,297)(11,479,223,357,107,418,149,296)(12,478,224,356,108,417,150,295)(13,477,225,355,109,416,151,294)(14,476,226,354,110,415,152,293)(15,475,227,353,111,414,153,292)(16,474,228,352,112,413,154,291)(17,473,229,351,113,412,155,290)(18,472,230,350,114,411,156,289)(19,471,231,349,115,410,157,288)(20,470,232,348,116,409,158,287)(21,469,233,347,117,408,159,286)(22,468,234,346,118,407,160,285)(23,467,235,345,119,406,161,284)(24,466,236,344,120,405,162,283)(25,465,237,343,121,404,163,282)(26,464,238,342,122,403,164,281)(27,463,239,341,62,402,165,280)(28,462,240,340,63,401,166,279)(29,461,241,339,64,400,167,278)(30,460,242,338,65,399,168,277)(31,459,243,337,66,398,169,276)(32,458,244,336,67,397,170,275)(33,457,184,335,68,396,171,274)(34,456,185,334,69,395,172,273)(35,455,186,333,70,394,173,272)(36,454,187,332,71,393,174,271)(37,453,188,331,72,392,175,270)(38,452,189,330,73,391,176,269)(39,451,190,329,74,390,177,268)(40,450,191,328,75,389,178,267)(41,449,192,327,76,388,179,266)(42,448,193,326,77,387,180,265)(43,447,194,325,78,386,181,264)(44,446,195,324,79,385,182,263)(45,445,196,323,80,384,183,262)(46,444,197,322,81,383,123,261)(47,443,198,321,82,382,124,260)(48,442,199,320,83,381,125,259)(49,441,200,319,84,380,126,258)(50,440,201,318,85,379,127,257)(51,439,202,317,86,378,128,256)(52,438,203,316,87,377,129,255)(53,437,204,315,88,376,130,254)(54,436,205,314,89,375,131,253)(55,435,206,313,90,374,132,252)(56,434,207,312,91,373,133,251)(57,433,208,311,92,372,134,250)(58,432,209,310,93,371,135,249)(59,431,210,309,94,370,136,248)(60,430,211,308,95,369,137,247)(61,429,212,307,96,368,138,246)>;
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), (1,428,213,306,97,367,139,245)(2,488,214,366,98,427,140,305)(3,487,215,365,99,426,141,304)(4,486,216,364,100,425,142,303)(5,485,217,363,101,424,143,302)(6,484,218,362,102,423,144,301)(7,483,219,361,103,422,145,300)(8,482,220,360,104,421,146,299)(9,481,221,359,105,420,147,298)(10,480,222,358,106,419,148,297)(11,479,223,357,107,418,149,296)(12,478,224,356,108,417,150,295)(13,477,225,355,109,416,151,294)(14,476,226,354,110,415,152,293)(15,475,227,353,111,414,153,292)(16,474,228,352,112,413,154,291)(17,473,229,351,113,412,155,290)(18,472,230,350,114,411,156,289)(19,471,231,349,115,410,157,288)(20,470,232,348,116,409,158,287)(21,469,233,347,117,408,159,286)(22,468,234,346,118,407,160,285)(23,467,235,345,119,406,161,284)(24,466,236,344,120,405,162,283)(25,465,237,343,121,404,163,282)(26,464,238,342,122,403,164,281)(27,463,239,341,62,402,165,280)(28,462,240,340,63,401,166,279)(29,461,241,339,64,400,167,278)(30,460,242,338,65,399,168,277)(31,459,243,337,66,398,169,276)(32,458,244,336,67,397,170,275)(33,457,184,335,68,396,171,274)(34,456,185,334,69,395,172,273)(35,455,186,333,70,394,173,272)(36,454,187,332,71,393,174,271)(37,453,188,331,72,392,175,270)(38,452,189,330,73,391,176,269)(39,451,190,329,74,390,177,268)(40,450,191,328,75,389,178,267)(41,449,192,327,76,388,179,266)(42,448,193,326,77,387,180,265)(43,447,194,325,78,386,181,264)(44,446,195,324,79,385,182,263)(45,445,196,323,80,384,183,262)(46,444,197,322,81,383,123,261)(47,443,198,321,82,382,124,260)(48,442,199,320,83,381,125,259)(49,441,200,319,84,380,126,258)(50,440,201,318,85,379,127,257)(51,439,202,317,86,378,128,256)(52,438,203,316,87,377,129,255)(53,437,204,315,88,376,130,254)(54,436,205,314,89,375,131,253)(55,435,206,313,90,374,132,252)(56,434,207,312,91,373,133,251)(57,433,208,311,92,372,134,250)(58,432,209,310,93,371,135,249)(59,431,210,309,94,370,136,248)(60,430,211,308,95,369,137,247)(61,429,212,307,96,368,138,246) );
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)], [(1,428,213,306,97,367,139,245),(2,488,214,366,98,427,140,305),(3,487,215,365,99,426,141,304),(4,486,216,364,100,425,142,303),(5,485,217,363,101,424,143,302),(6,484,218,362,102,423,144,301),(7,483,219,361,103,422,145,300),(8,482,220,360,104,421,146,299),(9,481,221,359,105,420,147,298),(10,480,222,358,106,419,148,297),(11,479,223,357,107,418,149,296),(12,478,224,356,108,417,150,295),(13,477,225,355,109,416,151,294),(14,476,226,354,110,415,152,293),(15,475,227,353,111,414,153,292),(16,474,228,352,112,413,154,291),(17,473,229,351,113,412,155,290),(18,472,230,350,114,411,156,289),(19,471,231,349,115,410,157,288),(20,470,232,348,116,409,158,287),(21,469,233,347,117,408,159,286),(22,468,234,346,118,407,160,285),(23,467,235,345,119,406,161,284),(24,466,236,344,120,405,162,283),(25,465,237,343,121,404,163,282),(26,464,238,342,122,403,164,281),(27,463,239,341,62,402,165,280),(28,462,240,340,63,401,166,279),(29,461,241,339,64,400,167,278),(30,460,242,338,65,399,168,277),(31,459,243,337,66,398,169,276),(32,458,244,336,67,397,170,275),(33,457,184,335,68,396,171,274),(34,456,185,334,69,395,172,273),(35,455,186,333,70,394,173,272),(36,454,187,332,71,393,174,271),(37,453,188,331,72,392,175,270),(38,452,189,330,73,391,176,269),(39,451,190,329,74,390,177,268),(40,450,191,328,75,389,178,267),(41,449,192,327,76,388,179,266),(42,448,193,326,77,387,180,265),(43,447,194,325,78,386,181,264),(44,446,195,324,79,385,182,263),(45,445,196,323,80,384,183,262),(46,444,197,322,81,383,123,261),(47,443,198,321,82,382,124,260),(48,442,199,320,83,381,125,259),(49,441,200,319,84,380,126,258),(50,440,201,318,85,379,127,257),(51,439,202,317,86,378,128,256),(52,438,203,316,87,377,129,255),(53,437,204,315,88,376,130,254),(54,436,205,314,89,375,131,253),(55,435,206,313,90,374,132,252),(56,434,207,312,91,373,133,251),(57,433,208,311,92,372,134,250),(58,432,209,310,93,371,135,249),(59,431,210,309,94,370,136,248),(60,430,211,308,95,369,137,247),(61,429,212,307,96,368,138,246)]])
128 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 61A | ··· | 61AD | 122A | ··· | 122AD | 244A | ··· | 244BH |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 61 | ··· | 61 | 122 | ··· | 122 | 244 | ··· | 244 |
| size | 1 | 1 | 1 | 1 | 61 | 61 | 61 | 61 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D61 | Dic61 | C61⋊2C8 |
| kernel | C61⋊2C8 | C244 | C122 | C61 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 30 | 30 | 60 |
Matrix representation of C61⋊2C8 ►in GL2(𝔽977) generated by
| 976 | 1 |
| 252 | 724 |
| 760 | 165 |
| 403 | 217 |
G:=sub<GL(2,GF(977))| [976,252,1,724],[760,403,165,217] >;
C61⋊2C8 in GAP, Magma, Sage, TeX
C_{61}\rtimes_2C_8 % in TeX
G:=Group("C61:2C8"); // GroupNames label
G:=SmallGroup(488,1);
// by ID
G=gap.SmallGroup(488,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-61,8,21,7683]);
// Polycyclic
G:=Group<a,b|a^61=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export