Copied to
clipboard

G = C612C8order 488 = 23·61

The semidirect product of C61 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C612C8, C4.2D61, C2.Dic61, C122.2C4, C244.2C2, SmallGroup(488,1)

Series: Derived Chief Lower central Upper central

C1C61 — C612C8
C1C61C122C244 — C612C8
C61 — C612C8
C1C4

Generators and relations for C612C8
 G = < a,b | a61=b8=1, bab-1=a-1 >

61C8

Smallest permutation representation of C612C8
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)
(1 428 217 306 95 367 152 245)(2 488 218 366 96 427 153 305)(3 487 219 365 97 426 154 304)(4 486 220 364 98 425 155 303)(5 485 221 363 99 424 156 302)(6 484 222 362 100 423 157 301)(7 483 223 361 101 422 158 300)(8 482 224 360 102 421 159 299)(9 481 225 359 103 420 160 298)(10 480 226 358 104 419 161 297)(11 479 227 357 105 418 162 296)(12 478 228 356 106 417 163 295)(13 477 229 355 107 416 164 294)(14 476 230 354 108 415 165 293)(15 475 231 353 109 414 166 292)(16 474 232 352 110 413 167 291)(17 473 233 351 111 412 168 290)(18 472 234 350 112 411 169 289)(19 471 235 349 113 410 170 288)(20 470 236 348 114 409 171 287)(21 469 237 347 115 408 172 286)(22 468 238 346 116 407 173 285)(23 467 239 345 117 406 174 284)(24 466 240 344 118 405 175 283)(25 465 241 343 119 404 176 282)(26 464 242 342 120 403 177 281)(27 463 243 341 121 402 178 280)(28 462 244 340 122 401 179 279)(29 461 184 339 62 400 180 278)(30 460 185 338 63 399 181 277)(31 459 186 337 64 398 182 276)(32 458 187 336 65 397 183 275)(33 457 188 335 66 396 123 274)(34 456 189 334 67 395 124 273)(35 455 190 333 68 394 125 272)(36 454 191 332 69 393 126 271)(37 453 192 331 70 392 127 270)(38 452 193 330 71 391 128 269)(39 451 194 329 72 390 129 268)(40 450 195 328 73 389 130 267)(41 449 196 327 74 388 131 266)(42 448 197 326 75 387 132 265)(43 447 198 325 76 386 133 264)(44 446 199 324 77 385 134 263)(45 445 200 323 78 384 135 262)(46 444 201 322 79 383 136 261)(47 443 202 321 80 382 137 260)(48 442 203 320 81 381 138 259)(49 441 204 319 82 380 139 258)(50 440 205 318 83 379 140 257)(51 439 206 317 84 378 141 256)(52 438 207 316 85 377 142 255)(53 437 208 315 86 376 143 254)(54 436 209 314 87 375 144 253)(55 435 210 313 88 374 145 252)(56 434 211 312 89 373 146 251)(57 433 212 311 90 372 147 250)(58 432 213 310 91 371 148 249)(59 431 214 309 92 370 149 248)(60 430 215 308 93 369 150 247)(61 429 216 307 94 368 151 246)

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

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

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

128 conjugacy classes

class 1  2 4A4B8A8B8C8D61A···61AD122A···122AD244A···244BH
order1244888861···61122···122244···244
size1111616161612···22···22···2

128 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D61Dic61C612C8
kernelC612C8C244C122C61C4C2C1
# reps1124303060

Matrix representation of C612C8 in GL2(𝔽977) generated by

9761
252724
,
760165
403217
G:=sub<GL(2,GF(977))| [976,252,1,724],[760,403,165,217] >;

C612C8 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

Subgroup lattice of C612C8 in TeX

׿
×
𝔽