direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic3×C41, C3⋊C164, C6.C82, C123⋊5C4, C82.2S3, C246.3C2, C2.(S3×C41), SmallGroup(492,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C41 |
Generators and relations for Dic3×C41
G = < a,b,c | a41=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C41
►Regular action on 492 points
Generators in S492
(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)(452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492)
(1 358 451 193 282 76)(2 359 411 194 283 77)(3 360 412 195 284 78)(4 361 413 196 285 79)(5 362 414 197 286 80)(6 363 415 198 287 81)(7 364 416 199 247 82)(8 365 417 200 248 42)(9 366 418 201 249 43)(10 367 419 202 250 44)(11 368 420 203 251 45)(12 369 421 204 252 46)(13 329 422 205 253 47)(14 330 423 165 254 48)(15 331 424 166 255 49)(16 332 425 167 256 50)(17 333 426 168 257 51)(18 334 427 169 258 52)(19 335 428 170 259 53)(20 336 429 171 260 54)(21 337 430 172 261 55)(22 338 431 173 262 56)(23 339 432 174 263 57)(24 340 433 175 264 58)(25 341 434 176 265 59)(26 342 435 177 266 60)(27 343 436 178 267 61)(28 344 437 179 268 62)(29 345 438 180 269 63)(30 346 439 181 270 64)(31 347 440 182 271 65)(32 348 441 183 272 66)(33 349 442 184 273 67)(34 350 443 185 274 68)(35 351 444 186 275 69)(36 352 445 187 276 70)(37 353 446 188 277 71)(38 354 447 189 278 72)(39 355 448 190 279 73)(40 356 449 191 280 74)(41 357 450 192 281 75)(83 408 323 153 479 225)(84 409 324 154 480 226)(85 410 325 155 481 227)(86 370 326 156 482 228)(87 371 327 157 483 229)(88 372 328 158 484 230)(89 373 288 159 485 231)(90 374 289 160 486 232)(91 375 290 161 487 233)(92 376 291 162 488 234)(93 377 292 163 489 235)(94 378 293 164 490 236)(95 379 294 124 491 237)(96 380 295 125 492 238)(97 381 296 126 452 239)(98 382 297 127 453 240)(99 383 298 128 454 241)(100 384 299 129 455 242)(101 385 300 130 456 243)(102 386 301 131 457 244)(103 387 302 132 458 245)(104 388 303 133 459 246)(105 389 304 134 460 206)(106 390 305 135 461 207)(107 391 306 136 462 208)(108 392 307 137 463 209)(109 393 308 138 464 210)(110 394 309 139 465 211)(111 395 310 140 466 212)(112 396 311 141 467 213)(113 397 312 142 468 214)(114 398 313 143 469 215)(115 399 314 144 470 216)(116 400 315 145 471 217)(117 401 316 146 472 218)(118 402 317 147 473 219)(119 403 318 148 474 220)(120 404 319 149 475 221)(121 405 320 150 476 222)(122 406 321 151 477 223)(123 407 322 152 478 224)
(1 299 193 242)(2 300 194 243)(3 301 195 244)(4 302 196 245)(5 303 197 246)(6 304 198 206)(7 305 199 207)(8 306 200 208)(9 307 201 209)(10 308 202 210)(11 309 203 211)(12 310 204 212)(13 311 205 213)(14 312 165 214)(15 313 166 215)(16 314 167 216)(17 315 168 217)(18 316 169 218)(19 317 170 219)(20 318 171 220)(21 319 172 221)(22 320 173 222)(23 321 174 223)(24 322 175 224)(25 323 176 225)(26 324 177 226)(27 325 178 227)(28 326 179 228)(29 327 180 229)(30 328 181 230)(31 288 182 231)(32 289 183 232)(33 290 184 233)(34 291 185 234)(35 292 186 235)(36 293 187 236)(37 294 188 237)(38 295 189 238)(39 296 190 239)(40 297 191 240)(41 298 192 241)(42 136 417 107)(43 137 418 108)(44 138 419 109)(45 139 420 110)(46 140 421 111)(47 141 422 112)(48 142 423 113)(49 143 424 114)(50 144 425 115)(51 145 426 116)(52 146 427 117)(53 147 428 118)(54 148 429 119)(55 149 430 120)(56 150 431 121)(57 151 432 122)(58 152 433 123)(59 153 434 83)(60 154 435 84)(61 155 436 85)(62 156 437 86)(63 157 438 87)(64 158 439 88)(65 159 440 89)(66 160 441 90)(67 161 442 91)(68 162 443 92)(69 163 444 93)(70 164 445 94)(71 124 446 95)(72 125 447 96)(73 126 448 97)(74 127 449 98)(75 128 450 99)(76 129 451 100)(77 130 411 101)(78 131 412 102)(79 132 413 103)(80 133 414 104)(81 134 415 105)(82 135 416 106)(247 461 364 390)(248 462 365 391)(249 463 366 392)(250 464 367 393)(251 465 368 394)(252 466 369 395)(253 467 329 396)(254 468 330 397)(255 469 331 398)(256 470 332 399)(257 471 333 400)(258 472 334 401)(259 473 335 402)(260 474 336 403)(261 475 337 404)(262 476 338 405)(263 477 339 406)(264 478 340 407)(265 479 341 408)(266 480 342 409)(267 481 343 410)(268 482 344 370)(269 483 345 371)(270 484 346 372)(271 485 347 373)(272 486 348 374)(273 487 349 375)(274 488 350 376)(275 489 351 377)(276 490 352 378)(277 491 353 379)(278 492 354 380)(279 452 355 381)(280 453 356 382)(281 454 357 383)(282 455 358 384)(283 456 359 385)(284 457 360 386)(285 458 361 387)(286 459 362 388)(287 460 363 389)
G:=sub<Sym(492)| (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)(452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492), (1,358,451,193,282,76)(2,359,411,194,283,77)(3,360,412,195,284,78)(4,361,413,196,285,79)(5,362,414,197,286,80)(6,363,415,198,287,81)(7,364,416,199,247,82)(8,365,417,200,248,42)(9,366,418,201,249,43)(10,367,419,202,250,44)(11,368,420,203,251,45)(12,369,421,204,252,46)(13,329,422,205,253,47)(14,330,423,165,254,48)(15,331,424,166,255,49)(16,332,425,167,256,50)(17,333,426,168,257,51)(18,334,427,169,258,52)(19,335,428,170,259,53)(20,336,429,171,260,54)(21,337,430,172,261,55)(22,338,431,173,262,56)(23,339,432,174,263,57)(24,340,433,175,264,58)(25,341,434,176,265,59)(26,342,435,177,266,60)(27,343,436,178,267,61)(28,344,437,179,268,62)(29,345,438,180,269,63)(30,346,439,181,270,64)(31,347,440,182,271,65)(32,348,441,183,272,66)(33,349,442,184,273,67)(34,350,443,185,274,68)(35,351,444,186,275,69)(36,352,445,187,276,70)(37,353,446,188,277,71)(38,354,447,189,278,72)(39,355,448,190,279,73)(40,356,449,191,280,74)(41,357,450,192,281,75)(83,408,323,153,479,225)(84,409,324,154,480,226)(85,410,325,155,481,227)(86,370,326,156,482,228)(87,371,327,157,483,229)(88,372,328,158,484,230)(89,373,288,159,485,231)(90,374,289,160,486,232)(91,375,290,161,487,233)(92,376,291,162,488,234)(93,377,292,163,489,235)(94,378,293,164,490,236)(95,379,294,124,491,237)(96,380,295,125,492,238)(97,381,296,126,452,239)(98,382,297,127,453,240)(99,383,298,128,454,241)(100,384,299,129,455,242)(101,385,300,130,456,243)(102,386,301,131,457,244)(103,387,302,132,458,245)(104,388,303,133,459,246)(105,389,304,134,460,206)(106,390,305,135,461,207)(107,391,306,136,462,208)(108,392,307,137,463,209)(109,393,308,138,464,210)(110,394,309,139,465,211)(111,395,310,140,466,212)(112,396,311,141,467,213)(113,397,312,142,468,214)(114,398,313,143,469,215)(115,399,314,144,470,216)(116,400,315,145,471,217)(117,401,316,146,472,218)(118,402,317,147,473,219)(119,403,318,148,474,220)(120,404,319,149,475,221)(121,405,320,150,476,222)(122,406,321,151,477,223)(123,407,322,152,478,224), (1,299,193,242)(2,300,194,243)(3,301,195,244)(4,302,196,245)(5,303,197,246)(6,304,198,206)(7,305,199,207)(8,306,200,208)(9,307,201,209)(10,308,202,210)(11,309,203,211)(12,310,204,212)(13,311,205,213)(14,312,165,214)(15,313,166,215)(16,314,167,216)(17,315,168,217)(18,316,169,218)(19,317,170,219)(20,318,171,220)(21,319,172,221)(22,320,173,222)(23,321,174,223)(24,322,175,224)(25,323,176,225)(26,324,177,226)(27,325,178,227)(28,326,179,228)(29,327,180,229)(30,328,181,230)(31,288,182,231)(32,289,183,232)(33,290,184,233)(34,291,185,234)(35,292,186,235)(36,293,187,236)(37,294,188,237)(38,295,189,238)(39,296,190,239)(40,297,191,240)(41,298,192,241)(42,136,417,107)(43,137,418,108)(44,138,419,109)(45,139,420,110)(46,140,421,111)(47,141,422,112)(48,142,423,113)(49,143,424,114)(50,144,425,115)(51,145,426,116)(52,146,427,117)(53,147,428,118)(54,148,429,119)(55,149,430,120)(56,150,431,121)(57,151,432,122)(58,152,433,123)(59,153,434,83)(60,154,435,84)(61,155,436,85)(62,156,437,86)(63,157,438,87)(64,158,439,88)(65,159,440,89)(66,160,441,90)(67,161,442,91)(68,162,443,92)(69,163,444,93)(70,164,445,94)(71,124,446,95)(72,125,447,96)(73,126,448,97)(74,127,449,98)(75,128,450,99)(76,129,451,100)(77,130,411,101)(78,131,412,102)(79,132,413,103)(80,133,414,104)(81,134,415,105)(82,135,416,106)(247,461,364,390)(248,462,365,391)(249,463,366,392)(250,464,367,393)(251,465,368,394)(252,466,369,395)(253,467,329,396)(254,468,330,397)(255,469,331,398)(256,470,332,399)(257,471,333,400)(258,472,334,401)(259,473,335,402)(260,474,336,403)(261,475,337,404)(262,476,338,405)(263,477,339,406)(264,478,340,407)(265,479,341,408)(266,480,342,409)(267,481,343,410)(268,482,344,370)(269,483,345,371)(270,484,346,372)(271,485,347,373)(272,486,348,374)(273,487,349,375)(274,488,350,376)(275,489,351,377)(276,490,352,378)(277,491,353,379)(278,492,354,380)(279,452,355,381)(280,453,356,382)(281,454,357,383)(282,455,358,384)(283,456,359,385)(284,457,360,386)(285,458,361,387)(286,459,362,388)(287,460,363,389)>;
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)(452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492), (1,358,451,193,282,76)(2,359,411,194,283,77)(3,360,412,195,284,78)(4,361,413,196,285,79)(5,362,414,197,286,80)(6,363,415,198,287,81)(7,364,416,199,247,82)(8,365,417,200,248,42)(9,366,418,201,249,43)(10,367,419,202,250,44)(11,368,420,203,251,45)(12,369,421,204,252,46)(13,329,422,205,253,47)(14,330,423,165,254,48)(15,331,424,166,255,49)(16,332,425,167,256,50)(17,333,426,168,257,51)(18,334,427,169,258,52)(19,335,428,170,259,53)(20,336,429,171,260,54)(21,337,430,172,261,55)(22,338,431,173,262,56)(23,339,432,174,263,57)(24,340,433,175,264,58)(25,341,434,176,265,59)(26,342,435,177,266,60)(27,343,436,178,267,61)(28,344,437,179,268,62)(29,345,438,180,269,63)(30,346,439,181,270,64)(31,347,440,182,271,65)(32,348,441,183,272,66)(33,349,442,184,273,67)(34,350,443,185,274,68)(35,351,444,186,275,69)(36,352,445,187,276,70)(37,353,446,188,277,71)(38,354,447,189,278,72)(39,355,448,190,279,73)(40,356,449,191,280,74)(41,357,450,192,281,75)(83,408,323,153,479,225)(84,409,324,154,480,226)(85,410,325,155,481,227)(86,370,326,156,482,228)(87,371,327,157,483,229)(88,372,328,158,484,230)(89,373,288,159,485,231)(90,374,289,160,486,232)(91,375,290,161,487,233)(92,376,291,162,488,234)(93,377,292,163,489,235)(94,378,293,164,490,236)(95,379,294,124,491,237)(96,380,295,125,492,238)(97,381,296,126,452,239)(98,382,297,127,453,240)(99,383,298,128,454,241)(100,384,299,129,455,242)(101,385,300,130,456,243)(102,386,301,131,457,244)(103,387,302,132,458,245)(104,388,303,133,459,246)(105,389,304,134,460,206)(106,390,305,135,461,207)(107,391,306,136,462,208)(108,392,307,137,463,209)(109,393,308,138,464,210)(110,394,309,139,465,211)(111,395,310,140,466,212)(112,396,311,141,467,213)(113,397,312,142,468,214)(114,398,313,143,469,215)(115,399,314,144,470,216)(116,400,315,145,471,217)(117,401,316,146,472,218)(118,402,317,147,473,219)(119,403,318,148,474,220)(120,404,319,149,475,221)(121,405,320,150,476,222)(122,406,321,151,477,223)(123,407,322,152,478,224), (1,299,193,242)(2,300,194,243)(3,301,195,244)(4,302,196,245)(5,303,197,246)(6,304,198,206)(7,305,199,207)(8,306,200,208)(9,307,201,209)(10,308,202,210)(11,309,203,211)(12,310,204,212)(13,311,205,213)(14,312,165,214)(15,313,166,215)(16,314,167,216)(17,315,168,217)(18,316,169,218)(19,317,170,219)(20,318,171,220)(21,319,172,221)(22,320,173,222)(23,321,174,223)(24,322,175,224)(25,323,176,225)(26,324,177,226)(27,325,178,227)(28,326,179,228)(29,327,180,229)(30,328,181,230)(31,288,182,231)(32,289,183,232)(33,290,184,233)(34,291,185,234)(35,292,186,235)(36,293,187,236)(37,294,188,237)(38,295,189,238)(39,296,190,239)(40,297,191,240)(41,298,192,241)(42,136,417,107)(43,137,418,108)(44,138,419,109)(45,139,420,110)(46,140,421,111)(47,141,422,112)(48,142,423,113)(49,143,424,114)(50,144,425,115)(51,145,426,116)(52,146,427,117)(53,147,428,118)(54,148,429,119)(55,149,430,120)(56,150,431,121)(57,151,432,122)(58,152,433,123)(59,153,434,83)(60,154,435,84)(61,155,436,85)(62,156,437,86)(63,157,438,87)(64,158,439,88)(65,159,440,89)(66,160,441,90)(67,161,442,91)(68,162,443,92)(69,163,444,93)(70,164,445,94)(71,124,446,95)(72,125,447,96)(73,126,448,97)(74,127,449,98)(75,128,450,99)(76,129,451,100)(77,130,411,101)(78,131,412,102)(79,132,413,103)(80,133,414,104)(81,134,415,105)(82,135,416,106)(247,461,364,390)(248,462,365,391)(249,463,366,392)(250,464,367,393)(251,465,368,394)(252,466,369,395)(253,467,329,396)(254,468,330,397)(255,469,331,398)(256,470,332,399)(257,471,333,400)(258,472,334,401)(259,473,335,402)(260,474,336,403)(261,475,337,404)(262,476,338,405)(263,477,339,406)(264,478,340,407)(265,479,341,408)(266,480,342,409)(267,481,343,410)(268,482,344,370)(269,483,345,371)(270,484,346,372)(271,485,347,373)(272,486,348,374)(273,487,349,375)(274,488,350,376)(275,489,351,377)(276,490,352,378)(277,491,353,379)(278,492,354,380)(279,452,355,381)(280,453,356,382)(281,454,357,383)(282,455,358,384)(283,456,359,385)(284,457,360,386)(285,458,361,387)(286,459,362,388)(287,460,363,389) );
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),(452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492)], [(1,358,451,193,282,76),(2,359,411,194,283,77),(3,360,412,195,284,78),(4,361,413,196,285,79),(5,362,414,197,286,80),(6,363,415,198,287,81),(7,364,416,199,247,82),(8,365,417,200,248,42),(9,366,418,201,249,43),(10,367,419,202,250,44),(11,368,420,203,251,45),(12,369,421,204,252,46),(13,329,422,205,253,47),(14,330,423,165,254,48),(15,331,424,166,255,49),(16,332,425,167,256,50),(17,333,426,168,257,51),(18,334,427,169,258,52),(19,335,428,170,259,53),(20,336,429,171,260,54),(21,337,430,172,261,55),(22,338,431,173,262,56),(23,339,432,174,263,57),(24,340,433,175,264,58),(25,341,434,176,265,59),(26,342,435,177,266,60),(27,343,436,178,267,61),(28,344,437,179,268,62),(29,345,438,180,269,63),(30,346,439,181,270,64),(31,347,440,182,271,65),(32,348,441,183,272,66),(33,349,442,184,273,67),(34,350,443,185,274,68),(35,351,444,186,275,69),(36,352,445,187,276,70),(37,353,446,188,277,71),(38,354,447,189,278,72),(39,355,448,190,279,73),(40,356,449,191,280,74),(41,357,450,192,281,75),(83,408,323,153,479,225),(84,409,324,154,480,226),(85,410,325,155,481,227),(86,370,326,156,482,228),(87,371,327,157,483,229),(88,372,328,158,484,230),(89,373,288,159,485,231),(90,374,289,160,486,232),(91,375,290,161,487,233),(92,376,291,162,488,234),(93,377,292,163,489,235),(94,378,293,164,490,236),(95,379,294,124,491,237),(96,380,295,125,492,238),(97,381,296,126,452,239),(98,382,297,127,453,240),(99,383,298,128,454,241),(100,384,299,129,455,242),(101,385,300,130,456,243),(102,386,301,131,457,244),(103,387,302,132,458,245),(104,388,303,133,459,246),(105,389,304,134,460,206),(106,390,305,135,461,207),(107,391,306,136,462,208),(108,392,307,137,463,209),(109,393,308,138,464,210),(110,394,309,139,465,211),(111,395,310,140,466,212),(112,396,311,141,467,213),(113,397,312,142,468,214),(114,398,313,143,469,215),(115,399,314,144,470,216),(116,400,315,145,471,217),(117,401,316,146,472,218),(118,402,317,147,473,219),(119,403,318,148,474,220),(120,404,319,149,475,221),(121,405,320,150,476,222),(122,406,321,151,477,223),(123,407,322,152,478,224)], [(1,299,193,242),(2,300,194,243),(3,301,195,244),(4,302,196,245),(5,303,197,246),(6,304,198,206),(7,305,199,207),(8,306,200,208),(9,307,201,209),(10,308,202,210),(11,309,203,211),(12,310,204,212),(13,311,205,213),(14,312,165,214),(15,313,166,215),(16,314,167,216),(17,315,168,217),(18,316,169,218),(19,317,170,219),(20,318,171,220),(21,319,172,221),(22,320,173,222),(23,321,174,223),(24,322,175,224),(25,323,176,225),(26,324,177,226),(27,325,178,227),(28,326,179,228),(29,327,180,229),(30,328,181,230),(31,288,182,231),(32,289,183,232),(33,290,184,233),(34,291,185,234),(35,292,186,235),(36,293,187,236),(37,294,188,237),(38,295,189,238),(39,296,190,239),(40,297,191,240),(41,298,192,241),(42,136,417,107),(43,137,418,108),(44,138,419,109),(45,139,420,110),(46,140,421,111),(47,141,422,112),(48,142,423,113),(49,143,424,114),(50,144,425,115),(51,145,426,116),(52,146,427,117),(53,147,428,118),(54,148,429,119),(55,149,430,120),(56,150,431,121),(57,151,432,122),(58,152,433,123),(59,153,434,83),(60,154,435,84),(61,155,436,85),(62,156,437,86),(63,157,438,87),(64,158,439,88),(65,159,440,89),(66,160,441,90),(67,161,442,91),(68,162,443,92),(69,163,444,93),(70,164,445,94),(71,124,446,95),(72,125,447,96),(73,126,448,97),(74,127,449,98),(75,128,450,99),(76,129,451,100),(77,130,411,101),(78,131,412,102),(79,132,413,103),(80,133,414,104),(81,134,415,105),(82,135,416,106),(247,461,364,390),(248,462,365,391),(249,463,366,392),(250,464,367,393),(251,465,368,394),(252,466,369,395),(253,467,329,396),(254,468,330,397),(255,469,331,398),(256,470,332,399),(257,471,333,400),(258,472,334,401),(259,473,335,402),(260,474,336,403),(261,475,337,404),(262,476,338,405),(263,477,339,406),(264,478,340,407),(265,479,341,408),(266,480,342,409),(267,481,343,410),(268,482,344,370),(269,483,345,371),(270,484,346,372),(271,485,347,373),(272,486,348,374),(273,487,349,375),(274,488,350,376),(275,489,351,377),(276,490,352,378),(277,491,353,379),(278,492,354,380),(279,452,355,381),(280,453,356,382),(281,454,357,383),(282,455,358,384),(283,456,359,385),(284,457,360,386),(285,458,361,387),(286,459,362,388),(287,460,363,389)]])
246 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 41A | ··· | 41AN | 82A | ··· | 82AN | 123A | ··· | 123AN | 164A | ··· | 164CB | 246A | ··· | 246AN |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 41 | ··· | 41 | 82 | ··· | 82 | 123 | ··· | 123 | 164 | ··· | 164 | 246 | ··· | 246 |
| size | 1 | 1 | 2 | 3 | 3 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
246 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C41 | C82 | C164 | S3 | Dic3 | S3×C41 | Dic3×C41 |
| kernel | Dic3×C41 | C246 | C123 | Dic3 | C6 | C3 | C82 | C41 | C2 | C1 |
| # reps | 1 | 1 | 2 | 40 | 40 | 80 | 1 | 1 | 40 | 40 |
Matrix representation of Dic3×C41 ►in GL2(𝔽2953) generated by
| 1277 | 0 |
| 0 | 1277 |
| 0 | 2952 |
| 1 | 1 |
| 401 | 2629 |
| 2228 | 2552 |
G:=sub<GL(2,GF(2953))| [1277,0,0,1277],[0,1,2952,1],[401,2228,2629,2552] >;
Dic3×C41 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{41} % in TeX
G:=Group("Dic3xC41"); // GroupNames label
G:=SmallGroup(492,1);
// by ID
G=gap.SmallGroup(492,1);
# by ID
G:=PCGroup([4,-2,-41,-2,-3,328,5251]);
// Polycyclic
G:=Group<a,b,c|a^41=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export