metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic123, C82.S3, C6.D41, C3⋊Dic41, C123⋊3C4, C2.D123, C41⋊2Dic3, C246.1C2, SmallGroup(492,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C123 — Dic123 |
Generators and relations for Dic123
G = < a,b | a246=1, b2=a123, bab-1=a-1 >
Smallest permutation representation of Dic123
►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)
(1 406 124 283)(2 405 125 282)(3 404 126 281)(4 403 127 280)(5 402 128 279)(6 401 129 278)(7 400 130 277)(8 399 131 276)(9 398 132 275)(10 397 133 274)(11 396 134 273)(12 395 135 272)(13 394 136 271)(14 393 137 270)(15 392 138 269)(16 391 139 268)(17 390 140 267)(18 389 141 266)(19 388 142 265)(20 387 143 264)(21 386 144 263)(22 385 145 262)(23 384 146 261)(24 383 147 260)(25 382 148 259)(26 381 149 258)(27 380 150 257)(28 379 151 256)(29 378 152 255)(30 377 153 254)(31 376 154 253)(32 375 155 252)(33 374 156 251)(34 373 157 250)(35 372 158 249)(36 371 159 248)(37 370 160 247)(38 369 161 492)(39 368 162 491)(40 367 163 490)(41 366 164 489)(42 365 165 488)(43 364 166 487)(44 363 167 486)(45 362 168 485)(46 361 169 484)(47 360 170 483)(48 359 171 482)(49 358 172 481)(50 357 173 480)(51 356 174 479)(52 355 175 478)(53 354 176 477)(54 353 177 476)(55 352 178 475)(56 351 179 474)(57 350 180 473)(58 349 181 472)(59 348 182 471)(60 347 183 470)(61 346 184 469)(62 345 185 468)(63 344 186 467)(64 343 187 466)(65 342 188 465)(66 341 189 464)(67 340 190 463)(68 339 191 462)(69 338 192 461)(70 337 193 460)(71 336 194 459)(72 335 195 458)(73 334 196 457)(74 333 197 456)(75 332 198 455)(76 331 199 454)(77 330 200 453)(78 329 201 452)(79 328 202 451)(80 327 203 450)(81 326 204 449)(82 325 205 448)(83 324 206 447)(84 323 207 446)(85 322 208 445)(86 321 209 444)(87 320 210 443)(88 319 211 442)(89 318 212 441)(90 317 213 440)(91 316 214 439)(92 315 215 438)(93 314 216 437)(94 313 217 436)(95 312 218 435)(96 311 219 434)(97 310 220 433)(98 309 221 432)(99 308 222 431)(100 307 223 430)(101 306 224 429)(102 305 225 428)(103 304 226 427)(104 303 227 426)(105 302 228 425)(106 301 229 424)(107 300 230 423)(108 299 231 422)(109 298 232 421)(110 297 233 420)(111 296 234 419)(112 295 235 418)(113 294 236 417)(114 293 237 416)(115 292 238 415)(116 291 239 414)(117 290 240 413)(118 289 241 412)(119 288 242 411)(120 287 243 410)(121 286 244 409)(122 285 245 408)(123 284 246 407)
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), (1,406,124,283)(2,405,125,282)(3,404,126,281)(4,403,127,280)(5,402,128,279)(6,401,129,278)(7,400,130,277)(8,399,131,276)(9,398,132,275)(10,397,133,274)(11,396,134,273)(12,395,135,272)(13,394,136,271)(14,393,137,270)(15,392,138,269)(16,391,139,268)(17,390,140,267)(18,389,141,266)(19,388,142,265)(20,387,143,264)(21,386,144,263)(22,385,145,262)(23,384,146,261)(24,383,147,260)(25,382,148,259)(26,381,149,258)(27,380,150,257)(28,379,151,256)(29,378,152,255)(30,377,153,254)(31,376,154,253)(32,375,155,252)(33,374,156,251)(34,373,157,250)(35,372,158,249)(36,371,159,248)(37,370,160,247)(38,369,161,492)(39,368,162,491)(40,367,163,490)(41,366,164,489)(42,365,165,488)(43,364,166,487)(44,363,167,486)(45,362,168,485)(46,361,169,484)(47,360,170,483)(48,359,171,482)(49,358,172,481)(50,357,173,480)(51,356,174,479)(52,355,175,478)(53,354,176,477)(54,353,177,476)(55,352,178,475)(56,351,179,474)(57,350,180,473)(58,349,181,472)(59,348,182,471)(60,347,183,470)(61,346,184,469)(62,345,185,468)(63,344,186,467)(64,343,187,466)(65,342,188,465)(66,341,189,464)(67,340,190,463)(68,339,191,462)(69,338,192,461)(70,337,193,460)(71,336,194,459)(72,335,195,458)(73,334,196,457)(74,333,197,456)(75,332,198,455)(76,331,199,454)(77,330,200,453)(78,329,201,452)(79,328,202,451)(80,327,203,450)(81,326,204,449)(82,325,205,448)(83,324,206,447)(84,323,207,446)(85,322,208,445)(86,321,209,444)(87,320,210,443)(88,319,211,442)(89,318,212,441)(90,317,213,440)(91,316,214,439)(92,315,215,438)(93,314,216,437)(94,313,217,436)(95,312,218,435)(96,311,219,434)(97,310,220,433)(98,309,221,432)(99,308,222,431)(100,307,223,430)(101,306,224,429)(102,305,225,428)(103,304,226,427)(104,303,227,426)(105,302,228,425)(106,301,229,424)(107,300,230,423)(108,299,231,422)(109,298,232,421)(110,297,233,420)(111,296,234,419)(112,295,235,418)(113,294,236,417)(114,293,237,416)(115,292,238,415)(116,291,239,414)(117,290,240,413)(118,289,241,412)(119,288,242,411)(120,287,243,410)(121,286,244,409)(122,285,245,408)(123,284,246,407)>;
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), (1,406,124,283)(2,405,125,282)(3,404,126,281)(4,403,127,280)(5,402,128,279)(6,401,129,278)(7,400,130,277)(8,399,131,276)(9,398,132,275)(10,397,133,274)(11,396,134,273)(12,395,135,272)(13,394,136,271)(14,393,137,270)(15,392,138,269)(16,391,139,268)(17,390,140,267)(18,389,141,266)(19,388,142,265)(20,387,143,264)(21,386,144,263)(22,385,145,262)(23,384,146,261)(24,383,147,260)(25,382,148,259)(26,381,149,258)(27,380,150,257)(28,379,151,256)(29,378,152,255)(30,377,153,254)(31,376,154,253)(32,375,155,252)(33,374,156,251)(34,373,157,250)(35,372,158,249)(36,371,159,248)(37,370,160,247)(38,369,161,492)(39,368,162,491)(40,367,163,490)(41,366,164,489)(42,365,165,488)(43,364,166,487)(44,363,167,486)(45,362,168,485)(46,361,169,484)(47,360,170,483)(48,359,171,482)(49,358,172,481)(50,357,173,480)(51,356,174,479)(52,355,175,478)(53,354,176,477)(54,353,177,476)(55,352,178,475)(56,351,179,474)(57,350,180,473)(58,349,181,472)(59,348,182,471)(60,347,183,470)(61,346,184,469)(62,345,185,468)(63,344,186,467)(64,343,187,466)(65,342,188,465)(66,341,189,464)(67,340,190,463)(68,339,191,462)(69,338,192,461)(70,337,193,460)(71,336,194,459)(72,335,195,458)(73,334,196,457)(74,333,197,456)(75,332,198,455)(76,331,199,454)(77,330,200,453)(78,329,201,452)(79,328,202,451)(80,327,203,450)(81,326,204,449)(82,325,205,448)(83,324,206,447)(84,323,207,446)(85,322,208,445)(86,321,209,444)(87,320,210,443)(88,319,211,442)(89,318,212,441)(90,317,213,440)(91,316,214,439)(92,315,215,438)(93,314,216,437)(94,313,217,436)(95,312,218,435)(96,311,219,434)(97,310,220,433)(98,309,221,432)(99,308,222,431)(100,307,223,430)(101,306,224,429)(102,305,225,428)(103,304,226,427)(104,303,227,426)(105,302,228,425)(106,301,229,424)(107,300,230,423)(108,299,231,422)(109,298,232,421)(110,297,233,420)(111,296,234,419)(112,295,235,418)(113,294,236,417)(114,293,237,416)(115,292,238,415)(116,291,239,414)(117,290,240,413)(118,289,241,412)(119,288,242,411)(120,287,243,410)(121,286,244,409)(122,285,245,408)(123,284,246,407) );
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)], [(1,406,124,283),(2,405,125,282),(3,404,126,281),(4,403,127,280),(5,402,128,279),(6,401,129,278),(7,400,130,277),(8,399,131,276),(9,398,132,275),(10,397,133,274),(11,396,134,273),(12,395,135,272),(13,394,136,271),(14,393,137,270),(15,392,138,269),(16,391,139,268),(17,390,140,267),(18,389,141,266),(19,388,142,265),(20,387,143,264),(21,386,144,263),(22,385,145,262),(23,384,146,261),(24,383,147,260),(25,382,148,259),(26,381,149,258),(27,380,150,257),(28,379,151,256),(29,378,152,255),(30,377,153,254),(31,376,154,253),(32,375,155,252),(33,374,156,251),(34,373,157,250),(35,372,158,249),(36,371,159,248),(37,370,160,247),(38,369,161,492),(39,368,162,491),(40,367,163,490),(41,366,164,489),(42,365,165,488),(43,364,166,487),(44,363,167,486),(45,362,168,485),(46,361,169,484),(47,360,170,483),(48,359,171,482),(49,358,172,481),(50,357,173,480),(51,356,174,479),(52,355,175,478),(53,354,176,477),(54,353,177,476),(55,352,178,475),(56,351,179,474),(57,350,180,473),(58,349,181,472),(59,348,182,471),(60,347,183,470),(61,346,184,469),(62,345,185,468),(63,344,186,467),(64,343,187,466),(65,342,188,465),(66,341,189,464),(67,340,190,463),(68,339,191,462),(69,338,192,461),(70,337,193,460),(71,336,194,459),(72,335,195,458),(73,334,196,457),(74,333,197,456),(75,332,198,455),(76,331,199,454),(77,330,200,453),(78,329,201,452),(79,328,202,451),(80,327,203,450),(81,326,204,449),(82,325,205,448),(83,324,206,447),(84,323,207,446),(85,322,208,445),(86,321,209,444),(87,320,210,443),(88,319,211,442),(89,318,212,441),(90,317,213,440),(91,316,214,439),(92,315,215,438),(93,314,216,437),(94,313,217,436),(95,312,218,435),(96,311,219,434),(97,310,220,433),(98,309,221,432),(99,308,222,431),(100,307,223,430),(101,306,224,429),(102,305,225,428),(103,304,226,427),(104,303,227,426),(105,302,228,425),(106,301,229,424),(107,300,230,423),(108,299,231,422),(109,298,232,421),(110,297,233,420),(111,296,234,419),(112,295,235,418),(113,294,236,417),(114,293,237,416),(115,292,238,415),(116,291,239,414),(117,290,240,413),(118,289,241,412),(119,288,242,411),(120,287,243,410),(121,286,244,409),(122,285,245,408),(123,284,246,407)]])
126 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 41A | ··· | 41T | 82A | ··· | 82T | 123A | ··· | 123AN | 246A | ··· | 246AN |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 41 | ··· | 41 | 82 | ··· | 82 | 123 | ··· | 123 | 246 | ··· | 246 |
| size | 1 | 1 | 2 | 123 | 123 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D41 | Dic41 | D123 | Dic123 |
| kernel | Dic123 | C246 | C123 | C82 | C41 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 20 | 20 | 40 | 40 |
Matrix representation of Dic123 ►in GL3(𝔽2953) generated by
| 2952 | 0 | 0 |
| 0 | 2819 | 2283 |
| 0 | 670 | 992 |
| 1226 | 0 | 0 |
| 0 | 35 | 1802 |
| 0 | 2228 | 2918 |
G:=sub<GL(3,GF(2953))| [2952,0,0,0,2819,670,0,2283,992],[1226,0,0,0,35,2228,0,1802,2918] >;
Dic123 in GAP, Magma, Sage, TeX
{\rm Dic}_{123} % in TeX
G:=Group("Dic123"); // GroupNames label
G:=SmallGroup(492,3);
// by ID
G=gap.SmallGroup(492,3);
# by ID
G:=PCGroup([4,-2,-2,-3,-41,8,98,7683]);
// Polycyclic
G:=Group<a,b|a^246=1,b^2=a^123,b*a*b^-1=a^-1>;
// generators/relations
Export