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G = Dic121order 484 = 22·112

Dicyclic group

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

Aliases: Dic121, C121⋊C4, C242.C2, C2.D121, C22.1D11, C11.Dic11, SmallGroup(484,1)

Series: Derived Chief Lower central Upper central

C1C121 — Dic121
C1C11C121C242 — Dic121
C121 — Dic121
C1C2

Generators and relations for Dic121
 G = < a,b | a242=1, b2=a121, bab-1=a-1 >

121C4
11Dic11

Smallest permutation representation of Dic121
Regular action on 484 points
Generators in S484
(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)
(1 415 122 294)(2 414 123 293)(3 413 124 292)(4 412 125 291)(5 411 126 290)(6 410 127 289)(7 409 128 288)(8 408 129 287)(9 407 130 286)(10 406 131 285)(11 405 132 284)(12 404 133 283)(13 403 134 282)(14 402 135 281)(15 401 136 280)(16 400 137 279)(17 399 138 278)(18 398 139 277)(19 397 140 276)(20 396 141 275)(21 395 142 274)(22 394 143 273)(23 393 144 272)(24 392 145 271)(25 391 146 270)(26 390 147 269)(27 389 148 268)(28 388 149 267)(29 387 150 266)(30 386 151 265)(31 385 152 264)(32 384 153 263)(33 383 154 262)(34 382 155 261)(35 381 156 260)(36 380 157 259)(37 379 158 258)(38 378 159 257)(39 377 160 256)(40 376 161 255)(41 375 162 254)(42 374 163 253)(43 373 164 252)(44 372 165 251)(45 371 166 250)(46 370 167 249)(47 369 168 248)(48 368 169 247)(49 367 170 246)(50 366 171 245)(51 365 172 244)(52 364 173 243)(53 363 174 484)(54 362 175 483)(55 361 176 482)(56 360 177 481)(57 359 178 480)(58 358 179 479)(59 357 180 478)(60 356 181 477)(61 355 182 476)(62 354 183 475)(63 353 184 474)(64 352 185 473)(65 351 186 472)(66 350 187 471)(67 349 188 470)(68 348 189 469)(69 347 190 468)(70 346 191 467)(71 345 192 466)(72 344 193 465)(73 343 194 464)(74 342 195 463)(75 341 196 462)(76 340 197 461)(77 339 198 460)(78 338 199 459)(79 337 200 458)(80 336 201 457)(81 335 202 456)(82 334 203 455)(83 333 204 454)(84 332 205 453)(85 331 206 452)(86 330 207 451)(87 329 208 450)(88 328 209 449)(89 327 210 448)(90 326 211 447)(91 325 212 446)(92 324 213 445)(93 323 214 444)(94 322 215 443)(95 321 216 442)(96 320 217 441)(97 319 218 440)(98 318 219 439)(99 317 220 438)(100 316 221 437)(101 315 222 436)(102 314 223 435)(103 313 224 434)(104 312 225 433)(105 311 226 432)(106 310 227 431)(107 309 228 430)(108 308 229 429)(109 307 230 428)(110 306 231 427)(111 305 232 426)(112 304 233 425)(113 303 234 424)(114 302 235 423)(115 301 236 422)(116 300 237 421)(117 299 238 420)(118 298 239 419)(119 297 240 418)(120 296 241 417)(121 295 242 416)

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

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

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

124 conjugacy classes

class 1  2 4A4B11A···11E22A···22E121A···121BC242A···242BC
order124411···1122···22121···121242···242
size111211212···22···22···22···2

124 irreducible representations

dim1112222
type+++-+-
imageC1C2C4D11Dic11D121Dic121
kernelDic121C242C121C22C11C2C1
# reps112555555

Matrix representation of Dic121 in GL2(𝔽1453) generated by

197450
1003875
,
43998
361410
G:=sub<GL(2,GF(1453))| [197,1003,450,875],[43,36,998,1410] >;

Dic121 in GAP, Magma, Sage, TeX

{\rm Dic}_{121}
% in TeX

G:=Group("Dic121");
// GroupNames label

G:=SmallGroup(484,1);
// by ID

G=gap.SmallGroup(484,1);
# by ID

G:=PCGroup([4,-2,-2,-11,-11,8,1010,1330,7043]);
// Polycyclic

G:=Group<a,b|a^242=1,b^2=a^121,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic121 in TeX

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