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G = Dic122order 488 = 23·61

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic122, C61⋊Q8, C4.D61, C244.1C2, C2.3D122, C122.1C22, Dic61.1C2, SmallGroup(488,4)

Series: Derived Chief Lower central Upper central

C1C122 — Dic122
C1C61C122Dic61 — Dic122
C61C122 — Dic122
C1C2C4

Generators and relations for Dic122
 G = < a,b | a244=1, b2=a122, bab-1=a-1 >

61C4
61C4
61Q8

Smallest permutation representation of Dic122
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)
(1 439 123 317)(2 438 124 316)(3 437 125 315)(4 436 126 314)(5 435 127 313)(6 434 128 312)(7 433 129 311)(8 432 130 310)(9 431 131 309)(10 430 132 308)(11 429 133 307)(12 428 134 306)(13 427 135 305)(14 426 136 304)(15 425 137 303)(16 424 138 302)(17 423 139 301)(18 422 140 300)(19 421 141 299)(20 420 142 298)(21 419 143 297)(22 418 144 296)(23 417 145 295)(24 416 146 294)(25 415 147 293)(26 414 148 292)(27 413 149 291)(28 412 150 290)(29 411 151 289)(30 410 152 288)(31 409 153 287)(32 408 154 286)(33 407 155 285)(34 406 156 284)(35 405 157 283)(36 404 158 282)(37 403 159 281)(38 402 160 280)(39 401 161 279)(40 400 162 278)(41 399 163 277)(42 398 164 276)(43 397 165 275)(44 396 166 274)(45 395 167 273)(46 394 168 272)(47 393 169 271)(48 392 170 270)(49 391 171 269)(50 390 172 268)(51 389 173 267)(52 388 174 266)(53 387 175 265)(54 386 176 264)(55 385 177 263)(56 384 178 262)(57 383 179 261)(58 382 180 260)(59 381 181 259)(60 380 182 258)(61 379 183 257)(62 378 184 256)(63 377 185 255)(64 376 186 254)(65 375 187 253)(66 374 188 252)(67 373 189 251)(68 372 190 250)(69 371 191 249)(70 370 192 248)(71 369 193 247)(72 368 194 246)(73 367 195 245)(74 366 196 488)(75 365 197 487)(76 364 198 486)(77 363 199 485)(78 362 200 484)(79 361 201 483)(80 360 202 482)(81 359 203 481)(82 358 204 480)(83 357 205 479)(84 356 206 478)(85 355 207 477)(86 354 208 476)(87 353 209 475)(88 352 210 474)(89 351 211 473)(90 350 212 472)(91 349 213 471)(92 348 214 470)(93 347 215 469)(94 346 216 468)(95 345 217 467)(96 344 218 466)(97 343 219 465)(98 342 220 464)(99 341 221 463)(100 340 222 462)(101 339 223 461)(102 338 224 460)(103 337 225 459)(104 336 226 458)(105 335 227 457)(106 334 228 456)(107 333 229 455)(108 332 230 454)(109 331 231 453)(110 330 232 452)(111 329 233 451)(112 328 234 450)(113 327 235 449)(114 326 236 448)(115 325 237 447)(116 324 238 446)(117 323 239 445)(118 322 240 444)(119 321 241 443)(120 320 242 442)(121 319 243 441)(122 318 244 440)

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

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

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

125 conjugacy classes

class 1  2 4A4B4C61A···61AD122A···122AD244A···244BH
order1244461···61122···122244···244
size1121221222···22···22···2

125 irreducible representations

dim1112222
type+++-++-
imageC1C2C2Q8D61D122Dic122
kernelDic122Dic61C244C61C4C2C1
# reps1211303060

Matrix representation of Dic122 in GL2(𝔽733) generated by

610444
289536
,
107570
430626
G:=sub<GL(2,GF(733))| [610,289,444,536],[107,430,570,626] >;

Dic122 in GAP, Magma, Sage, TeX

{\rm Dic}_{122}
% in TeX

G:=Group("Dic122");
// GroupNames label

G:=SmallGroup(488,4);
// by ID

G=gap.SmallGroup(488,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-61,16,49,21,7683]);
// Polycyclic

G:=Group<a,b|a^244=1,b^2=a^122,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic122 in TeX

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