metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic120, C16.D15, C15⋊4Q32, C80.1S3, C4.3D60, C48.1D5, C6.3D40, C3⋊1Dic40, C5⋊1Dic24, 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
Generators and relations for Dic120
G = < a,b | a240=1, b2=a120, bab-1=a-1 >
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)
(1 376 121 256)(2 375 122 255)(3 374 123 254)(4 373 124 253)(5 372 125 252)(6 371 126 251)(7 370 127 250)(8 369 128 249)(9 368 129 248)(10 367 130 247)(11 366 131 246)(12 365 132 245)(13 364 133 244)(14 363 134 243)(15 362 135 242)(16 361 136 241)(17 360 137 480)(18 359 138 479)(19 358 139 478)(20 357 140 477)(21 356 141 476)(22 355 142 475)(23 354 143 474)(24 353 144 473)(25 352 145 472)(26 351 146 471)(27 350 147 470)(28 349 148 469)(29 348 149 468)(30 347 150 467)(31 346 151 466)(32 345 152 465)(33 344 153 464)(34 343 154 463)(35 342 155 462)(36 341 156 461)(37 340 157 460)(38 339 158 459)(39 338 159 458)(40 337 160 457)(41 336 161 456)(42 335 162 455)(43 334 163 454)(44 333 164 453)(45 332 165 452)(46 331 166 451)(47 330 167 450)(48 329 168 449)(49 328 169 448)(50 327 170 447)(51 326 171 446)(52 325 172 445)(53 324 173 444)(54 323 174 443)(55 322 175 442)(56 321 176 441)(57 320 177 440)(58 319 178 439)(59 318 179 438)(60 317 180 437)(61 316 181 436)(62 315 182 435)(63 314 183 434)(64 313 184 433)(65 312 185 432)(66 311 186 431)(67 310 187 430)(68 309 188 429)(69 308 189 428)(70 307 190 427)(71 306 191 426)(72 305 192 425)(73 304 193 424)(74 303 194 423)(75 302 195 422)(76 301 196 421)(77 300 197 420)(78 299 198 419)(79 298 199 418)(80 297 200 417)(81 296 201 416)(82 295 202 415)(83 294 203 414)(84 293 204 413)(85 292 205 412)(86 291 206 411)(87 290 207 410)(88 289 208 409)(89 288 209 408)(90 287 210 407)(91 286 211 406)(92 285 212 405)(93 284 213 404)(94 283 214 403)(95 282 215 402)(96 281 216 401)(97 280 217 400)(98 279 218 399)(99 278 219 398)(100 277 220 397)(101 276 221 396)(102 275 222 395)(103 274 223 394)(104 273 224 393)(105 272 225 392)(106 271 226 391)(107 270 227 390)(108 269 228 389)(109 268 229 388)(110 267 230 387)(111 266 231 386)(112 265 232 385)(113 264 233 384)(114 263 234 383)(115 262 235 382)(116 261 236 381)(117 260 237 380)(118 259 238 379)(119 258 239 378)(120 257 240 377)
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), (1,376,121,256)(2,375,122,255)(3,374,123,254)(4,373,124,253)(5,372,125,252)(6,371,126,251)(7,370,127,250)(8,369,128,249)(9,368,129,248)(10,367,130,247)(11,366,131,246)(12,365,132,245)(13,364,133,244)(14,363,134,243)(15,362,135,242)(16,361,136,241)(17,360,137,480)(18,359,138,479)(19,358,139,478)(20,357,140,477)(21,356,141,476)(22,355,142,475)(23,354,143,474)(24,353,144,473)(25,352,145,472)(26,351,146,471)(27,350,147,470)(28,349,148,469)(29,348,149,468)(30,347,150,467)(31,346,151,466)(32,345,152,465)(33,344,153,464)(34,343,154,463)(35,342,155,462)(36,341,156,461)(37,340,157,460)(38,339,158,459)(39,338,159,458)(40,337,160,457)(41,336,161,456)(42,335,162,455)(43,334,163,454)(44,333,164,453)(45,332,165,452)(46,331,166,451)(47,330,167,450)(48,329,168,449)(49,328,169,448)(50,327,170,447)(51,326,171,446)(52,325,172,445)(53,324,173,444)(54,323,174,443)(55,322,175,442)(56,321,176,441)(57,320,177,440)(58,319,178,439)(59,318,179,438)(60,317,180,437)(61,316,181,436)(62,315,182,435)(63,314,183,434)(64,313,184,433)(65,312,185,432)(66,311,186,431)(67,310,187,430)(68,309,188,429)(69,308,189,428)(70,307,190,427)(71,306,191,426)(72,305,192,425)(73,304,193,424)(74,303,194,423)(75,302,195,422)(76,301,196,421)(77,300,197,420)(78,299,198,419)(79,298,199,418)(80,297,200,417)(81,296,201,416)(82,295,202,415)(83,294,203,414)(84,293,204,413)(85,292,205,412)(86,291,206,411)(87,290,207,410)(88,289,208,409)(89,288,209,408)(90,287,210,407)(91,286,211,406)(92,285,212,405)(93,284,213,404)(94,283,214,403)(95,282,215,402)(96,281,216,401)(97,280,217,400)(98,279,218,399)(99,278,219,398)(100,277,220,397)(101,276,221,396)(102,275,222,395)(103,274,223,394)(104,273,224,393)(105,272,225,392)(106,271,226,391)(107,270,227,390)(108,269,228,389)(109,268,229,388)(110,267,230,387)(111,266,231,386)(112,265,232,385)(113,264,233,384)(114,263,234,383)(115,262,235,382)(116,261,236,381)(117,260,237,380)(118,259,238,379)(119,258,239,378)(120,257,240,377)>;
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), (1,376,121,256)(2,375,122,255)(3,374,123,254)(4,373,124,253)(5,372,125,252)(6,371,126,251)(7,370,127,250)(8,369,128,249)(9,368,129,248)(10,367,130,247)(11,366,131,246)(12,365,132,245)(13,364,133,244)(14,363,134,243)(15,362,135,242)(16,361,136,241)(17,360,137,480)(18,359,138,479)(19,358,139,478)(20,357,140,477)(21,356,141,476)(22,355,142,475)(23,354,143,474)(24,353,144,473)(25,352,145,472)(26,351,146,471)(27,350,147,470)(28,349,148,469)(29,348,149,468)(30,347,150,467)(31,346,151,466)(32,345,152,465)(33,344,153,464)(34,343,154,463)(35,342,155,462)(36,341,156,461)(37,340,157,460)(38,339,158,459)(39,338,159,458)(40,337,160,457)(41,336,161,456)(42,335,162,455)(43,334,163,454)(44,333,164,453)(45,332,165,452)(46,331,166,451)(47,330,167,450)(48,329,168,449)(49,328,169,448)(50,327,170,447)(51,326,171,446)(52,325,172,445)(53,324,173,444)(54,323,174,443)(55,322,175,442)(56,321,176,441)(57,320,177,440)(58,319,178,439)(59,318,179,438)(60,317,180,437)(61,316,181,436)(62,315,182,435)(63,314,183,434)(64,313,184,433)(65,312,185,432)(66,311,186,431)(67,310,187,430)(68,309,188,429)(69,308,189,428)(70,307,190,427)(71,306,191,426)(72,305,192,425)(73,304,193,424)(74,303,194,423)(75,302,195,422)(76,301,196,421)(77,300,197,420)(78,299,198,419)(79,298,199,418)(80,297,200,417)(81,296,201,416)(82,295,202,415)(83,294,203,414)(84,293,204,413)(85,292,205,412)(86,291,206,411)(87,290,207,410)(88,289,208,409)(89,288,209,408)(90,287,210,407)(91,286,211,406)(92,285,212,405)(93,284,213,404)(94,283,214,403)(95,282,215,402)(96,281,216,401)(97,280,217,400)(98,279,218,399)(99,278,219,398)(100,277,220,397)(101,276,221,396)(102,275,222,395)(103,274,223,394)(104,273,224,393)(105,272,225,392)(106,271,226,391)(107,270,227,390)(108,269,228,389)(109,268,229,388)(110,267,230,387)(111,266,231,386)(112,265,232,385)(113,264,233,384)(114,263,234,383)(115,262,235,382)(116,261,236,381)(117,260,237,380)(118,259,238,379)(119,258,239,378)(120,257,240,377) );
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)], [(1,376,121,256),(2,375,122,255),(3,374,123,254),(4,373,124,253),(5,372,125,252),(6,371,126,251),(7,370,127,250),(8,369,128,249),(9,368,129,248),(10,367,130,247),(11,366,131,246),(12,365,132,245),(13,364,133,244),(14,363,134,243),(15,362,135,242),(16,361,136,241),(17,360,137,480),(18,359,138,479),(19,358,139,478),(20,357,140,477),(21,356,141,476),(22,355,142,475),(23,354,143,474),(24,353,144,473),(25,352,145,472),(26,351,146,471),(27,350,147,470),(28,349,148,469),(29,348,149,468),(30,347,150,467),(31,346,151,466),(32,345,152,465),(33,344,153,464),(34,343,154,463),(35,342,155,462),(36,341,156,461),(37,340,157,460),(38,339,158,459),(39,338,159,458),(40,337,160,457),(41,336,161,456),(42,335,162,455),(43,334,163,454),(44,333,164,453),(45,332,165,452),(46,331,166,451),(47,330,167,450),(48,329,168,449),(49,328,169,448),(50,327,170,447),(51,326,171,446),(52,325,172,445),(53,324,173,444),(54,323,174,443),(55,322,175,442),(56,321,176,441),(57,320,177,440),(58,319,178,439),(59,318,179,438),(60,317,180,437),(61,316,181,436),(62,315,182,435),(63,314,183,434),(64,313,184,433),(65,312,185,432),(66,311,186,431),(67,310,187,430),(68,309,188,429),(69,308,189,428),(70,307,190,427),(71,306,191,426),(72,305,192,425),(73,304,193,424),(74,303,194,423),(75,302,195,422),(76,301,196,421),(77,300,197,420),(78,299,198,419),(79,298,199,418),(80,297,200,417),(81,296,201,416),(82,295,202,415),(83,294,203,414),(84,293,204,413),(85,292,205,412),(86,291,206,411),(87,290,207,410),(88,289,208,409),(89,288,209,408),(90,287,210,407),(91,286,211,406),(92,285,212,405),(93,284,213,404),(94,283,214,403),(95,282,215,402),(96,281,216,401),(97,280,217,400),(98,279,218,399),(99,278,219,398),(100,277,220,397),(101,276,221,396),(102,275,222,395),(103,274,223,394),(104,273,224,393),(105,272,225,392),(106,271,226,391),(107,270,227,390),(108,269,228,389),(109,268,229,388),(110,267,230,387),(111,266,231,386),(112,265,232,385),(113,264,233,384),(114,263,234,383),(115,262,235,382),(116,261,236,381),(117,260,237,380),(118,259,238,379),(119,258,239,378),(120,257,240,377)]])
123 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 2 | 2 | 120 | 120 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
123 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | D12 | D15 | Q32 | D20 | D24 | D30 | D40 | Dic24 | D60 | Dic40 | D120 | Dic120 |
| kernel | Dic120 | C240 | Dic60 | C80 | C60 | C48 | C40 | C30 | C24 | C20 | C16 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of Dic120 ►in GL2(𝔽241) generated by
| 41 | 114 |
| 127 | 71 |
| 94 | 210 |
| 184 | 147 |
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