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G = Dic123order 492 = 22·3·41

Dicyclic group

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

Aliases: Dic123, C82.S3, C6.D41, C3⋊Dic41, C1233C4, C2.D123, C412Dic3, C246.1C2, SmallGroup(492,3)

Series: Derived Chief Lower central Upper central

C1C123 — Dic123
C1C41C123C246 — Dic123
C123 — Dic123
C1C2

Generators and relations for Dic123
 G = < a,b | a246=1, b2=a123, bab-1=a-1 >

123C4
41Dic3
3Dic41

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

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

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

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

126 conjugacy classes

class 1  2  3 4A4B 6 41A···41T82A···82T123A···123AN246A···246AN
order12344641···4182···82123···123246···246
size11212312322···22···22···22···2

126 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4S3Dic3D41Dic41D123Dic123
kernelDic123C246C123C82C41C6C3C2C1
# reps1121120204040

Matrix representation of Dic123 in GL3(𝔽2953) generated by

295200
028192283
0670992
,
122600
0351802
022282918
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

Subgroup lattice of Dic123 in TeX

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