metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic118, C59⋊Q8, C4.D59, C236.1C2, C2.3D118, Dic59.C2, C118.1C22, SmallGroup(472,3)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic118
G = < a,b | a236=1, b2=a118, bab-1=a-1 >
Smallest permutation representation of Dic118
►Regular action on 472 points
Generators in S472
(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)
(1 439 119 321)(2 438 120 320)(3 437 121 319)(4 436 122 318)(5 435 123 317)(6 434 124 316)(7 433 125 315)(8 432 126 314)(9 431 127 313)(10 430 128 312)(11 429 129 311)(12 428 130 310)(13 427 131 309)(14 426 132 308)(15 425 133 307)(16 424 134 306)(17 423 135 305)(18 422 136 304)(19 421 137 303)(20 420 138 302)(21 419 139 301)(22 418 140 300)(23 417 141 299)(24 416 142 298)(25 415 143 297)(26 414 144 296)(27 413 145 295)(28 412 146 294)(29 411 147 293)(30 410 148 292)(31 409 149 291)(32 408 150 290)(33 407 151 289)(34 406 152 288)(35 405 153 287)(36 404 154 286)(37 403 155 285)(38 402 156 284)(39 401 157 283)(40 400 158 282)(41 399 159 281)(42 398 160 280)(43 397 161 279)(44 396 162 278)(45 395 163 277)(46 394 164 276)(47 393 165 275)(48 392 166 274)(49 391 167 273)(50 390 168 272)(51 389 169 271)(52 388 170 270)(53 387 171 269)(54 386 172 268)(55 385 173 267)(56 384 174 266)(57 383 175 265)(58 382 176 264)(59 381 177 263)(60 380 178 262)(61 379 179 261)(62 378 180 260)(63 377 181 259)(64 376 182 258)(65 375 183 257)(66 374 184 256)(67 373 185 255)(68 372 186 254)(69 371 187 253)(70 370 188 252)(71 369 189 251)(72 368 190 250)(73 367 191 249)(74 366 192 248)(75 365 193 247)(76 364 194 246)(77 363 195 245)(78 362 196 244)(79 361 197 243)(80 360 198 242)(81 359 199 241)(82 358 200 240)(83 357 201 239)(84 356 202 238)(85 355 203 237)(86 354 204 472)(87 353 205 471)(88 352 206 470)(89 351 207 469)(90 350 208 468)(91 349 209 467)(92 348 210 466)(93 347 211 465)(94 346 212 464)(95 345 213 463)(96 344 214 462)(97 343 215 461)(98 342 216 460)(99 341 217 459)(100 340 218 458)(101 339 219 457)(102 338 220 456)(103 337 221 455)(104 336 222 454)(105 335 223 453)(106 334 224 452)(107 333 225 451)(108 332 226 450)(109 331 227 449)(110 330 228 448)(111 329 229 447)(112 328 230 446)(113 327 231 445)(114 326 232 444)(115 325 233 443)(116 324 234 442)(117 323 235 441)(118 322 236 440)
G:=sub<Sym(472)| (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), (1,439,119,321)(2,438,120,320)(3,437,121,319)(4,436,122,318)(5,435,123,317)(6,434,124,316)(7,433,125,315)(8,432,126,314)(9,431,127,313)(10,430,128,312)(11,429,129,311)(12,428,130,310)(13,427,131,309)(14,426,132,308)(15,425,133,307)(16,424,134,306)(17,423,135,305)(18,422,136,304)(19,421,137,303)(20,420,138,302)(21,419,139,301)(22,418,140,300)(23,417,141,299)(24,416,142,298)(25,415,143,297)(26,414,144,296)(27,413,145,295)(28,412,146,294)(29,411,147,293)(30,410,148,292)(31,409,149,291)(32,408,150,290)(33,407,151,289)(34,406,152,288)(35,405,153,287)(36,404,154,286)(37,403,155,285)(38,402,156,284)(39,401,157,283)(40,400,158,282)(41,399,159,281)(42,398,160,280)(43,397,161,279)(44,396,162,278)(45,395,163,277)(46,394,164,276)(47,393,165,275)(48,392,166,274)(49,391,167,273)(50,390,168,272)(51,389,169,271)(52,388,170,270)(53,387,171,269)(54,386,172,268)(55,385,173,267)(56,384,174,266)(57,383,175,265)(58,382,176,264)(59,381,177,263)(60,380,178,262)(61,379,179,261)(62,378,180,260)(63,377,181,259)(64,376,182,258)(65,375,183,257)(66,374,184,256)(67,373,185,255)(68,372,186,254)(69,371,187,253)(70,370,188,252)(71,369,189,251)(72,368,190,250)(73,367,191,249)(74,366,192,248)(75,365,193,247)(76,364,194,246)(77,363,195,245)(78,362,196,244)(79,361,197,243)(80,360,198,242)(81,359,199,241)(82,358,200,240)(83,357,201,239)(84,356,202,238)(85,355,203,237)(86,354,204,472)(87,353,205,471)(88,352,206,470)(89,351,207,469)(90,350,208,468)(91,349,209,467)(92,348,210,466)(93,347,211,465)(94,346,212,464)(95,345,213,463)(96,344,214,462)(97,343,215,461)(98,342,216,460)(99,341,217,459)(100,340,218,458)(101,339,219,457)(102,338,220,456)(103,337,221,455)(104,336,222,454)(105,335,223,453)(106,334,224,452)(107,333,225,451)(108,332,226,450)(109,331,227,449)(110,330,228,448)(111,329,229,447)(112,328,230,446)(113,327,231,445)(114,326,232,444)(115,325,233,443)(116,324,234,442)(117,323,235,441)(118,322,236,440)>;
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), (1,439,119,321)(2,438,120,320)(3,437,121,319)(4,436,122,318)(5,435,123,317)(6,434,124,316)(7,433,125,315)(8,432,126,314)(9,431,127,313)(10,430,128,312)(11,429,129,311)(12,428,130,310)(13,427,131,309)(14,426,132,308)(15,425,133,307)(16,424,134,306)(17,423,135,305)(18,422,136,304)(19,421,137,303)(20,420,138,302)(21,419,139,301)(22,418,140,300)(23,417,141,299)(24,416,142,298)(25,415,143,297)(26,414,144,296)(27,413,145,295)(28,412,146,294)(29,411,147,293)(30,410,148,292)(31,409,149,291)(32,408,150,290)(33,407,151,289)(34,406,152,288)(35,405,153,287)(36,404,154,286)(37,403,155,285)(38,402,156,284)(39,401,157,283)(40,400,158,282)(41,399,159,281)(42,398,160,280)(43,397,161,279)(44,396,162,278)(45,395,163,277)(46,394,164,276)(47,393,165,275)(48,392,166,274)(49,391,167,273)(50,390,168,272)(51,389,169,271)(52,388,170,270)(53,387,171,269)(54,386,172,268)(55,385,173,267)(56,384,174,266)(57,383,175,265)(58,382,176,264)(59,381,177,263)(60,380,178,262)(61,379,179,261)(62,378,180,260)(63,377,181,259)(64,376,182,258)(65,375,183,257)(66,374,184,256)(67,373,185,255)(68,372,186,254)(69,371,187,253)(70,370,188,252)(71,369,189,251)(72,368,190,250)(73,367,191,249)(74,366,192,248)(75,365,193,247)(76,364,194,246)(77,363,195,245)(78,362,196,244)(79,361,197,243)(80,360,198,242)(81,359,199,241)(82,358,200,240)(83,357,201,239)(84,356,202,238)(85,355,203,237)(86,354,204,472)(87,353,205,471)(88,352,206,470)(89,351,207,469)(90,350,208,468)(91,349,209,467)(92,348,210,466)(93,347,211,465)(94,346,212,464)(95,345,213,463)(96,344,214,462)(97,343,215,461)(98,342,216,460)(99,341,217,459)(100,340,218,458)(101,339,219,457)(102,338,220,456)(103,337,221,455)(104,336,222,454)(105,335,223,453)(106,334,224,452)(107,333,225,451)(108,332,226,450)(109,331,227,449)(110,330,228,448)(111,329,229,447)(112,328,230,446)(113,327,231,445)(114,326,232,444)(115,325,233,443)(116,324,234,442)(117,323,235,441)(118,322,236,440) );
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)], [(1,439,119,321),(2,438,120,320),(3,437,121,319),(4,436,122,318),(5,435,123,317),(6,434,124,316),(7,433,125,315),(8,432,126,314),(9,431,127,313),(10,430,128,312),(11,429,129,311),(12,428,130,310),(13,427,131,309),(14,426,132,308),(15,425,133,307),(16,424,134,306),(17,423,135,305),(18,422,136,304),(19,421,137,303),(20,420,138,302),(21,419,139,301),(22,418,140,300),(23,417,141,299),(24,416,142,298),(25,415,143,297),(26,414,144,296),(27,413,145,295),(28,412,146,294),(29,411,147,293),(30,410,148,292),(31,409,149,291),(32,408,150,290),(33,407,151,289),(34,406,152,288),(35,405,153,287),(36,404,154,286),(37,403,155,285),(38,402,156,284),(39,401,157,283),(40,400,158,282),(41,399,159,281),(42,398,160,280),(43,397,161,279),(44,396,162,278),(45,395,163,277),(46,394,164,276),(47,393,165,275),(48,392,166,274),(49,391,167,273),(50,390,168,272),(51,389,169,271),(52,388,170,270),(53,387,171,269),(54,386,172,268),(55,385,173,267),(56,384,174,266),(57,383,175,265),(58,382,176,264),(59,381,177,263),(60,380,178,262),(61,379,179,261),(62,378,180,260),(63,377,181,259),(64,376,182,258),(65,375,183,257),(66,374,184,256),(67,373,185,255),(68,372,186,254),(69,371,187,253),(70,370,188,252),(71,369,189,251),(72,368,190,250),(73,367,191,249),(74,366,192,248),(75,365,193,247),(76,364,194,246),(77,363,195,245),(78,362,196,244),(79,361,197,243),(80,360,198,242),(81,359,199,241),(82,358,200,240),(83,357,201,239),(84,356,202,238),(85,355,203,237),(86,354,204,472),(87,353,205,471),(88,352,206,470),(89,351,207,469),(90,350,208,468),(91,349,209,467),(92,348,210,466),(93,347,211,465),(94,346,212,464),(95,345,213,463),(96,344,214,462),(97,343,215,461),(98,342,216,460),(99,341,217,459),(100,340,218,458),(101,339,219,457),(102,338,220,456),(103,337,221,455),(104,336,222,454),(105,335,223,453),(106,334,224,452),(107,333,225,451),(108,332,226,450),(109,331,227,449),(110,330,228,448),(111,329,229,447),(112,328,230,446),(113,327,231,445),(114,326,232,444),(115,325,233,443),(116,324,234,442),(117,323,235,441),(118,322,236,440)]])
121 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 59A | ··· | 59AC | 118A | ··· | 118AC | 236A | ··· | 236BF |
| order | 1 | 2 | 4 | 4 | 4 | 59 | ··· | 59 | 118 | ··· | 118 | 236 | ··· | 236 |
| size | 1 | 1 | 2 | 118 | 118 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
121 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D59 | D118 | Dic118 |
| kernel | Dic118 | Dic59 | C236 | C59 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 29 | 29 | 58 |
Matrix representation of Dic118 ►in GL2(𝔽709) generated by
| 351 | 501 |
| 208 | 4 |
| 287 | 140 |
| 70 | 422 |
G:=sub<GL(2,GF(709))| [351,208,501,4],[287,70,140,422] >;
Dic118 in GAP, Magma, Sage, TeX
{\rm Dic}_{118} % in TeX
G:=Group("Dic118"); // GroupNames label
G:=SmallGroup(472,3);
// by ID
G=gap.SmallGroup(472,3);
# by ID
G:=PCGroup([4,-2,-2,-2,-59,16,49,21,7427]);
// Polycyclic
G:=Group<a,b|a^236=1,b^2=a^118,b*a*b^-1=a^-1>;
// generators/relations
Export