Copied to
clipboard

G = Dic120order 480 = 25·3·5

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic120, C16.D15, C154Q32, C80.1S3, C4.3D60, C48.1D5, C6.3D40, C31Dic40, C51Dic24, C240.1C2, C10.3D24, C30.24D8, C40.67D6, C2.5D120, C8.15D30, C60.159D4, C20.28D12, C12.28D20, C24.67D10, Dic60.1C2, C120.80C22, SmallGroup(480,161)

Series: Derived Chief Lower central Upper central

C1C120 — Dic120
C1C5C15C30C60C120Dic60 — Dic120
C15C30C60C120 — Dic120
C1C2C4C8C16

Generators and relations for Dic120
 G = < a,b | a240=1, b2=a120, bab-1=a-1 >

60C4
60C4
30Q8
30Q8
20Dic3
20Dic3
12Dic5
12Dic5
15Q16
15Q16
10Dic6
10Dic6
6Dic10
6Dic10
4Dic15
4Dic15
15Q32
5Dic12
5Dic12
3Dic20
3Dic20
2Dic30
2Dic30
5Dic24
3Dic40

Smallest permutation representation of Dic120
Regular action on 480 points
Generators in S480
(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)
(1 421 121 301)(2 420 122 300)(3 419 123 299)(4 418 124 298)(5 417 125 297)(6 416 126 296)(7 415 127 295)(8 414 128 294)(9 413 129 293)(10 412 130 292)(11 411 131 291)(12 410 132 290)(13 409 133 289)(14 408 134 288)(15 407 135 287)(16 406 136 286)(17 405 137 285)(18 404 138 284)(19 403 139 283)(20 402 140 282)(21 401 141 281)(22 400 142 280)(23 399 143 279)(24 398 144 278)(25 397 145 277)(26 396 146 276)(27 395 147 275)(28 394 148 274)(29 393 149 273)(30 392 150 272)(31 391 151 271)(32 390 152 270)(33 389 153 269)(34 388 154 268)(35 387 155 267)(36 386 156 266)(37 385 157 265)(38 384 158 264)(39 383 159 263)(40 382 160 262)(41 381 161 261)(42 380 162 260)(43 379 163 259)(44 378 164 258)(45 377 165 257)(46 376 166 256)(47 375 167 255)(48 374 168 254)(49 373 169 253)(50 372 170 252)(51 371 171 251)(52 370 172 250)(53 369 173 249)(54 368 174 248)(55 367 175 247)(56 366 176 246)(57 365 177 245)(58 364 178 244)(59 363 179 243)(60 362 180 242)(61 361 181 241)(62 360 182 480)(63 359 183 479)(64 358 184 478)(65 357 185 477)(66 356 186 476)(67 355 187 475)(68 354 188 474)(69 353 189 473)(70 352 190 472)(71 351 191 471)(72 350 192 470)(73 349 193 469)(74 348 194 468)(75 347 195 467)(76 346 196 466)(77 345 197 465)(78 344 198 464)(79 343 199 463)(80 342 200 462)(81 341 201 461)(82 340 202 460)(83 339 203 459)(84 338 204 458)(85 337 205 457)(86 336 206 456)(87 335 207 455)(88 334 208 454)(89 333 209 453)(90 332 210 452)(91 331 211 451)(92 330 212 450)(93 329 213 449)(94 328 214 448)(95 327 215 447)(96 326 216 446)(97 325 217 445)(98 324 218 444)(99 323 219 443)(100 322 220 442)(101 321 221 441)(102 320 222 440)(103 319 223 439)(104 318 224 438)(105 317 225 437)(106 316 226 436)(107 315 227 435)(108 314 228 434)(109 313 229 433)(110 312 230 432)(111 311 231 431)(112 310 232 430)(113 309 233 429)(114 308 234 428)(115 307 235 427)(116 306 236 426)(117 305 237 425)(118 304 238 424)(119 303 239 423)(120 302 240 422)

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

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

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

123 conjugacy classes

class 1  2  3 4A4B4C5A5B 6 8A8B10A10B12A12B15A15B15C15D16A16B16C16D20A20B20C20D24A24B24C24D30A30B30C30D40A···40H48A···48H60A···60H80A···80P120A···120P240A···240AF
order1234445568810101212151515151616161620202020242424243030303040···4048···4860···6080···80120···120240···240
size1122120120222222222222222222222222222222···22···22···22···22···22···2

123 irreducible representations

dim111222222222222222222
type+++++++++++-++++-+-+-
imageC1C2C2S3D4D5D6D8D10D12D15Q32D20D24D30D40Dic24D60Dic40D120Dic120
kernelDic120C240Dic60C80C60C48C40C30C24C20C16C15C12C10C8C6C5C4C3C2C1
# reps112112122244444888161632

Matrix representation of Dic120 in GL2(𝔽241) generated by

41114
12771
,
94210
184147
G:=sub<GL(2,GF(241))| [41,127,114,71],[94,184,210,147] >;

Dic120 in GAP, Magma, Sage, TeX

{\rm Dic}_{120}
% in TeX

G:=Group("Dic120");
// GroupNames label

G:=SmallGroup(480,161);
// by ID

G=gap.SmallGroup(480,161);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,85,92,254,142,675,80,2693,18822]);
// Polycyclic

G:=Group<a,b|a^240=1,b^2=a^120,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic120 in TeX

׿
×
𝔽