metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic119, C34.D7, C7⋊Dic17, C119⋊3C4, C14.D17, C2.D119, C17⋊2Dic7, C238.1C2, SmallGroup(476,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C119 — Dic119 |
Generators and relations for Dic119
G = < a,b | a238=1, b2=a119, bab-1=a-1 >
Smallest permutation representation of Dic119
►Regular action on 476 points
Generators in S476
(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)
(1 268 120 387)(2 267 121 386)(3 266 122 385)(4 265 123 384)(5 264 124 383)(6 263 125 382)(7 262 126 381)(8 261 127 380)(9 260 128 379)(10 259 129 378)(11 258 130 377)(12 257 131 376)(13 256 132 375)(14 255 133 374)(15 254 134 373)(16 253 135 372)(17 252 136 371)(18 251 137 370)(19 250 138 369)(20 249 139 368)(21 248 140 367)(22 247 141 366)(23 246 142 365)(24 245 143 364)(25 244 144 363)(26 243 145 362)(27 242 146 361)(28 241 147 360)(29 240 148 359)(30 239 149 358)(31 476 150 357)(32 475 151 356)(33 474 152 355)(34 473 153 354)(35 472 154 353)(36 471 155 352)(37 470 156 351)(38 469 157 350)(39 468 158 349)(40 467 159 348)(41 466 160 347)(42 465 161 346)(43 464 162 345)(44 463 163 344)(45 462 164 343)(46 461 165 342)(47 460 166 341)(48 459 167 340)(49 458 168 339)(50 457 169 338)(51 456 170 337)(52 455 171 336)(53 454 172 335)(54 453 173 334)(55 452 174 333)(56 451 175 332)(57 450 176 331)(58 449 177 330)(59 448 178 329)(60 447 179 328)(61 446 180 327)(62 445 181 326)(63 444 182 325)(64 443 183 324)(65 442 184 323)(66 441 185 322)(67 440 186 321)(68 439 187 320)(69 438 188 319)(70 437 189 318)(71 436 190 317)(72 435 191 316)(73 434 192 315)(74 433 193 314)(75 432 194 313)(76 431 195 312)(77 430 196 311)(78 429 197 310)(79 428 198 309)(80 427 199 308)(81 426 200 307)(82 425 201 306)(83 424 202 305)(84 423 203 304)(85 422 204 303)(86 421 205 302)(87 420 206 301)(88 419 207 300)(89 418 208 299)(90 417 209 298)(91 416 210 297)(92 415 211 296)(93 414 212 295)(94 413 213 294)(95 412 214 293)(96 411 215 292)(97 410 216 291)(98 409 217 290)(99 408 218 289)(100 407 219 288)(101 406 220 287)(102 405 221 286)(103 404 222 285)(104 403 223 284)(105 402 224 283)(106 401 225 282)(107 400 226 281)(108 399 227 280)(109 398 228 279)(110 397 229 278)(111 396 230 277)(112 395 231 276)(113 394 232 275)(114 393 233 274)(115 392 234 273)(116 391 235 272)(117 390 236 271)(118 389 237 270)(119 388 238 269)
G:=sub<Sym(476)| (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), (1,268,120,387)(2,267,121,386)(3,266,122,385)(4,265,123,384)(5,264,124,383)(6,263,125,382)(7,262,126,381)(8,261,127,380)(9,260,128,379)(10,259,129,378)(11,258,130,377)(12,257,131,376)(13,256,132,375)(14,255,133,374)(15,254,134,373)(16,253,135,372)(17,252,136,371)(18,251,137,370)(19,250,138,369)(20,249,139,368)(21,248,140,367)(22,247,141,366)(23,246,142,365)(24,245,143,364)(25,244,144,363)(26,243,145,362)(27,242,146,361)(28,241,147,360)(29,240,148,359)(30,239,149,358)(31,476,150,357)(32,475,151,356)(33,474,152,355)(34,473,153,354)(35,472,154,353)(36,471,155,352)(37,470,156,351)(38,469,157,350)(39,468,158,349)(40,467,159,348)(41,466,160,347)(42,465,161,346)(43,464,162,345)(44,463,163,344)(45,462,164,343)(46,461,165,342)(47,460,166,341)(48,459,167,340)(49,458,168,339)(50,457,169,338)(51,456,170,337)(52,455,171,336)(53,454,172,335)(54,453,173,334)(55,452,174,333)(56,451,175,332)(57,450,176,331)(58,449,177,330)(59,448,178,329)(60,447,179,328)(61,446,180,327)(62,445,181,326)(63,444,182,325)(64,443,183,324)(65,442,184,323)(66,441,185,322)(67,440,186,321)(68,439,187,320)(69,438,188,319)(70,437,189,318)(71,436,190,317)(72,435,191,316)(73,434,192,315)(74,433,193,314)(75,432,194,313)(76,431,195,312)(77,430,196,311)(78,429,197,310)(79,428,198,309)(80,427,199,308)(81,426,200,307)(82,425,201,306)(83,424,202,305)(84,423,203,304)(85,422,204,303)(86,421,205,302)(87,420,206,301)(88,419,207,300)(89,418,208,299)(90,417,209,298)(91,416,210,297)(92,415,211,296)(93,414,212,295)(94,413,213,294)(95,412,214,293)(96,411,215,292)(97,410,216,291)(98,409,217,290)(99,408,218,289)(100,407,219,288)(101,406,220,287)(102,405,221,286)(103,404,222,285)(104,403,223,284)(105,402,224,283)(106,401,225,282)(107,400,226,281)(108,399,227,280)(109,398,228,279)(110,397,229,278)(111,396,230,277)(112,395,231,276)(113,394,232,275)(114,393,233,274)(115,392,234,273)(116,391,235,272)(117,390,236,271)(118,389,237,270)(119,388,238,269)>;
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), (1,268,120,387)(2,267,121,386)(3,266,122,385)(4,265,123,384)(5,264,124,383)(6,263,125,382)(7,262,126,381)(8,261,127,380)(9,260,128,379)(10,259,129,378)(11,258,130,377)(12,257,131,376)(13,256,132,375)(14,255,133,374)(15,254,134,373)(16,253,135,372)(17,252,136,371)(18,251,137,370)(19,250,138,369)(20,249,139,368)(21,248,140,367)(22,247,141,366)(23,246,142,365)(24,245,143,364)(25,244,144,363)(26,243,145,362)(27,242,146,361)(28,241,147,360)(29,240,148,359)(30,239,149,358)(31,476,150,357)(32,475,151,356)(33,474,152,355)(34,473,153,354)(35,472,154,353)(36,471,155,352)(37,470,156,351)(38,469,157,350)(39,468,158,349)(40,467,159,348)(41,466,160,347)(42,465,161,346)(43,464,162,345)(44,463,163,344)(45,462,164,343)(46,461,165,342)(47,460,166,341)(48,459,167,340)(49,458,168,339)(50,457,169,338)(51,456,170,337)(52,455,171,336)(53,454,172,335)(54,453,173,334)(55,452,174,333)(56,451,175,332)(57,450,176,331)(58,449,177,330)(59,448,178,329)(60,447,179,328)(61,446,180,327)(62,445,181,326)(63,444,182,325)(64,443,183,324)(65,442,184,323)(66,441,185,322)(67,440,186,321)(68,439,187,320)(69,438,188,319)(70,437,189,318)(71,436,190,317)(72,435,191,316)(73,434,192,315)(74,433,193,314)(75,432,194,313)(76,431,195,312)(77,430,196,311)(78,429,197,310)(79,428,198,309)(80,427,199,308)(81,426,200,307)(82,425,201,306)(83,424,202,305)(84,423,203,304)(85,422,204,303)(86,421,205,302)(87,420,206,301)(88,419,207,300)(89,418,208,299)(90,417,209,298)(91,416,210,297)(92,415,211,296)(93,414,212,295)(94,413,213,294)(95,412,214,293)(96,411,215,292)(97,410,216,291)(98,409,217,290)(99,408,218,289)(100,407,219,288)(101,406,220,287)(102,405,221,286)(103,404,222,285)(104,403,223,284)(105,402,224,283)(106,401,225,282)(107,400,226,281)(108,399,227,280)(109,398,228,279)(110,397,229,278)(111,396,230,277)(112,395,231,276)(113,394,232,275)(114,393,233,274)(115,392,234,273)(116,391,235,272)(117,390,236,271)(118,389,237,270)(119,388,238,269) );
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)], [(1,268,120,387),(2,267,121,386),(3,266,122,385),(4,265,123,384),(5,264,124,383),(6,263,125,382),(7,262,126,381),(8,261,127,380),(9,260,128,379),(10,259,129,378),(11,258,130,377),(12,257,131,376),(13,256,132,375),(14,255,133,374),(15,254,134,373),(16,253,135,372),(17,252,136,371),(18,251,137,370),(19,250,138,369),(20,249,139,368),(21,248,140,367),(22,247,141,366),(23,246,142,365),(24,245,143,364),(25,244,144,363),(26,243,145,362),(27,242,146,361),(28,241,147,360),(29,240,148,359),(30,239,149,358),(31,476,150,357),(32,475,151,356),(33,474,152,355),(34,473,153,354),(35,472,154,353),(36,471,155,352),(37,470,156,351),(38,469,157,350),(39,468,158,349),(40,467,159,348),(41,466,160,347),(42,465,161,346),(43,464,162,345),(44,463,163,344),(45,462,164,343),(46,461,165,342),(47,460,166,341),(48,459,167,340),(49,458,168,339),(50,457,169,338),(51,456,170,337),(52,455,171,336),(53,454,172,335),(54,453,173,334),(55,452,174,333),(56,451,175,332),(57,450,176,331),(58,449,177,330),(59,448,178,329),(60,447,179,328),(61,446,180,327),(62,445,181,326),(63,444,182,325),(64,443,183,324),(65,442,184,323),(66,441,185,322),(67,440,186,321),(68,439,187,320),(69,438,188,319),(70,437,189,318),(71,436,190,317),(72,435,191,316),(73,434,192,315),(74,433,193,314),(75,432,194,313),(76,431,195,312),(77,430,196,311),(78,429,197,310),(79,428,198,309),(80,427,199,308),(81,426,200,307),(82,425,201,306),(83,424,202,305),(84,423,203,304),(85,422,204,303),(86,421,205,302),(87,420,206,301),(88,419,207,300),(89,418,208,299),(90,417,209,298),(91,416,210,297),(92,415,211,296),(93,414,212,295),(94,413,213,294),(95,412,214,293),(96,411,215,292),(97,410,216,291),(98,409,217,290),(99,408,218,289),(100,407,219,288),(101,406,220,287),(102,405,221,286),(103,404,222,285),(104,403,223,284),(105,402,224,283),(106,401,225,282),(107,400,226,281),(108,399,227,280),(109,398,228,279),(110,397,229,278),(111,396,230,277),(112,395,231,276),(113,394,232,275),(114,393,233,274),(115,392,234,273),(116,391,235,272),(117,390,236,271),(118,389,237,270),(119,388,238,269)]])
122 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 14A | 14B | 14C | 17A | ··· | 17H | 34A | ··· | 34H | 119A | ··· | 119AV | 238A | ··· | 238AV |
| order | 1 | 2 | 4 | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 17 | ··· | 17 | 34 | ··· | 34 | 119 | ··· | 119 | 238 | ··· | 238 |
| size | 1 | 1 | 119 | 119 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
122 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | D7 | Dic7 | D17 | Dic17 | D119 | Dic119 |
| kernel | Dic119 | C238 | C119 | C34 | C17 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 3 | 3 | 8 | 8 | 48 | 48 |
Matrix representation of Dic119 ►in GL2(𝔽953) generated by
| 300 | 123 |
| 830 | 191 |
| 118 | 648 |
| 280 | 835 |
G:=sub<GL(2,GF(953))| [300,830,123,191],[118,280,648,835] >;
Dic119 in GAP, Magma, Sage, TeX
{\rm Dic}_{119} % in TeX
G:=Group("Dic119"); // GroupNames label
G:=SmallGroup(476,3);
// by ID
G=gap.SmallGroup(476,3);
# by ID
G:=PCGroup([4,-2,-2,-7,-17,8,290,7171]);
// Polycyclic
G:=Group<a,b|a^238=1,b^2=a^119,b*a*b^-1=a^-1>;
// generators/relations
Export