metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic114, C4.D57, C57⋊2Q8, C76.1S3, C38.8D6, C6.8D38, C19⋊2Dic6, C3⋊2Dic38, C228.1C2, C12.1D19, C2.3D114, C114.8C22, Dic57.1C2, SmallGroup(456,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic114
G = < a,b | a228=1, b2=a114, bab-1=a-1 >
Smallest permutation representation of Dic114
►Regular action on 456 points
Generators in S456
(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)
(1 328 115 442)(2 327 116 441)(3 326 117 440)(4 325 118 439)(5 324 119 438)(6 323 120 437)(7 322 121 436)(8 321 122 435)(9 320 123 434)(10 319 124 433)(11 318 125 432)(12 317 126 431)(13 316 127 430)(14 315 128 429)(15 314 129 428)(16 313 130 427)(17 312 131 426)(18 311 132 425)(19 310 133 424)(20 309 134 423)(21 308 135 422)(22 307 136 421)(23 306 137 420)(24 305 138 419)(25 304 139 418)(26 303 140 417)(27 302 141 416)(28 301 142 415)(29 300 143 414)(30 299 144 413)(31 298 145 412)(32 297 146 411)(33 296 147 410)(34 295 148 409)(35 294 149 408)(36 293 150 407)(37 292 151 406)(38 291 152 405)(39 290 153 404)(40 289 154 403)(41 288 155 402)(42 287 156 401)(43 286 157 400)(44 285 158 399)(45 284 159 398)(46 283 160 397)(47 282 161 396)(48 281 162 395)(49 280 163 394)(50 279 164 393)(51 278 165 392)(52 277 166 391)(53 276 167 390)(54 275 168 389)(55 274 169 388)(56 273 170 387)(57 272 171 386)(58 271 172 385)(59 270 173 384)(60 269 174 383)(61 268 175 382)(62 267 176 381)(63 266 177 380)(64 265 178 379)(65 264 179 378)(66 263 180 377)(67 262 181 376)(68 261 182 375)(69 260 183 374)(70 259 184 373)(71 258 185 372)(72 257 186 371)(73 256 187 370)(74 255 188 369)(75 254 189 368)(76 253 190 367)(77 252 191 366)(78 251 192 365)(79 250 193 364)(80 249 194 363)(81 248 195 362)(82 247 196 361)(83 246 197 360)(84 245 198 359)(85 244 199 358)(86 243 200 357)(87 242 201 356)(88 241 202 355)(89 240 203 354)(90 239 204 353)(91 238 205 352)(92 237 206 351)(93 236 207 350)(94 235 208 349)(95 234 209 348)(96 233 210 347)(97 232 211 346)(98 231 212 345)(99 230 213 344)(100 229 214 343)(101 456 215 342)(102 455 216 341)(103 454 217 340)(104 453 218 339)(105 452 219 338)(106 451 220 337)(107 450 221 336)(108 449 222 335)(109 448 223 334)(110 447 224 333)(111 446 225 332)(112 445 226 331)(113 444 227 330)(114 443 228 329)
G:=sub<Sym(456)| (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), (1,328,115,442)(2,327,116,441)(3,326,117,440)(4,325,118,439)(5,324,119,438)(6,323,120,437)(7,322,121,436)(8,321,122,435)(9,320,123,434)(10,319,124,433)(11,318,125,432)(12,317,126,431)(13,316,127,430)(14,315,128,429)(15,314,129,428)(16,313,130,427)(17,312,131,426)(18,311,132,425)(19,310,133,424)(20,309,134,423)(21,308,135,422)(22,307,136,421)(23,306,137,420)(24,305,138,419)(25,304,139,418)(26,303,140,417)(27,302,141,416)(28,301,142,415)(29,300,143,414)(30,299,144,413)(31,298,145,412)(32,297,146,411)(33,296,147,410)(34,295,148,409)(35,294,149,408)(36,293,150,407)(37,292,151,406)(38,291,152,405)(39,290,153,404)(40,289,154,403)(41,288,155,402)(42,287,156,401)(43,286,157,400)(44,285,158,399)(45,284,159,398)(46,283,160,397)(47,282,161,396)(48,281,162,395)(49,280,163,394)(50,279,164,393)(51,278,165,392)(52,277,166,391)(53,276,167,390)(54,275,168,389)(55,274,169,388)(56,273,170,387)(57,272,171,386)(58,271,172,385)(59,270,173,384)(60,269,174,383)(61,268,175,382)(62,267,176,381)(63,266,177,380)(64,265,178,379)(65,264,179,378)(66,263,180,377)(67,262,181,376)(68,261,182,375)(69,260,183,374)(70,259,184,373)(71,258,185,372)(72,257,186,371)(73,256,187,370)(74,255,188,369)(75,254,189,368)(76,253,190,367)(77,252,191,366)(78,251,192,365)(79,250,193,364)(80,249,194,363)(81,248,195,362)(82,247,196,361)(83,246,197,360)(84,245,198,359)(85,244,199,358)(86,243,200,357)(87,242,201,356)(88,241,202,355)(89,240,203,354)(90,239,204,353)(91,238,205,352)(92,237,206,351)(93,236,207,350)(94,235,208,349)(95,234,209,348)(96,233,210,347)(97,232,211,346)(98,231,212,345)(99,230,213,344)(100,229,214,343)(101,456,215,342)(102,455,216,341)(103,454,217,340)(104,453,218,339)(105,452,219,338)(106,451,220,337)(107,450,221,336)(108,449,222,335)(109,448,223,334)(110,447,224,333)(111,446,225,332)(112,445,226,331)(113,444,227,330)(114,443,228,329)>;
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), (1,328,115,442)(2,327,116,441)(3,326,117,440)(4,325,118,439)(5,324,119,438)(6,323,120,437)(7,322,121,436)(8,321,122,435)(9,320,123,434)(10,319,124,433)(11,318,125,432)(12,317,126,431)(13,316,127,430)(14,315,128,429)(15,314,129,428)(16,313,130,427)(17,312,131,426)(18,311,132,425)(19,310,133,424)(20,309,134,423)(21,308,135,422)(22,307,136,421)(23,306,137,420)(24,305,138,419)(25,304,139,418)(26,303,140,417)(27,302,141,416)(28,301,142,415)(29,300,143,414)(30,299,144,413)(31,298,145,412)(32,297,146,411)(33,296,147,410)(34,295,148,409)(35,294,149,408)(36,293,150,407)(37,292,151,406)(38,291,152,405)(39,290,153,404)(40,289,154,403)(41,288,155,402)(42,287,156,401)(43,286,157,400)(44,285,158,399)(45,284,159,398)(46,283,160,397)(47,282,161,396)(48,281,162,395)(49,280,163,394)(50,279,164,393)(51,278,165,392)(52,277,166,391)(53,276,167,390)(54,275,168,389)(55,274,169,388)(56,273,170,387)(57,272,171,386)(58,271,172,385)(59,270,173,384)(60,269,174,383)(61,268,175,382)(62,267,176,381)(63,266,177,380)(64,265,178,379)(65,264,179,378)(66,263,180,377)(67,262,181,376)(68,261,182,375)(69,260,183,374)(70,259,184,373)(71,258,185,372)(72,257,186,371)(73,256,187,370)(74,255,188,369)(75,254,189,368)(76,253,190,367)(77,252,191,366)(78,251,192,365)(79,250,193,364)(80,249,194,363)(81,248,195,362)(82,247,196,361)(83,246,197,360)(84,245,198,359)(85,244,199,358)(86,243,200,357)(87,242,201,356)(88,241,202,355)(89,240,203,354)(90,239,204,353)(91,238,205,352)(92,237,206,351)(93,236,207,350)(94,235,208,349)(95,234,209,348)(96,233,210,347)(97,232,211,346)(98,231,212,345)(99,230,213,344)(100,229,214,343)(101,456,215,342)(102,455,216,341)(103,454,217,340)(104,453,218,339)(105,452,219,338)(106,451,220,337)(107,450,221,336)(108,449,222,335)(109,448,223,334)(110,447,224,333)(111,446,225,332)(112,445,226,331)(113,444,227,330)(114,443,228,329) );
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)], [(1,328,115,442),(2,327,116,441),(3,326,117,440),(4,325,118,439),(5,324,119,438),(6,323,120,437),(7,322,121,436),(8,321,122,435),(9,320,123,434),(10,319,124,433),(11,318,125,432),(12,317,126,431),(13,316,127,430),(14,315,128,429),(15,314,129,428),(16,313,130,427),(17,312,131,426),(18,311,132,425),(19,310,133,424),(20,309,134,423),(21,308,135,422),(22,307,136,421),(23,306,137,420),(24,305,138,419),(25,304,139,418),(26,303,140,417),(27,302,141,416),(28,301,142,415),(29,300,143,414),(30,299,144,413),(31,298,145,412),(32,297,146,411),(33,296,147,410),(34,295,148,409),(35,294,149,408),(36,293,150,407),(37,292,151,406),(38,291,152,405),(39,290,153,404),(40,289,154,403),(41,288,155,402),(42,287,156,401),(43,286,157,400),(44,285,158,399),(45,284,159,398),(46,283,160,397),(47,282,161,396),(48,281,162,395),(49,280,163,394),(50,279,164,393),(51,278,165,392),(52,277,166,391),(53,276,167,390),(54,275,168,389),(55,274,169,388),(56,273,170,387),(57,272,171,386),(58,271,172,385),(59,270,173,384),(60,269,174,383),(61,268,175,382),(62,267,176,381),(63,266,177,380),(64,265,178,379),(65,264,179,378),(66,263,180,377),(67,262,181,376),(68,261,182,375),(69,260,183,374),(70,259,184,373),(71,258,185,372),(72,257,186,371),(73,256,187,370),(74,255,188,369),(75,254,189,368),(76,253,190,367),(77,252,191,366),(78,251,192,365),(79,250,193,364),(80,249,194,363),(81,248,195,362),(82,247,196,361),(83,246,197,360),(84,245,198,359),(85,244,199,358),(86,243,200,357),(87,242,201,356),(88,241,202,355),(89,240,203,354),(90,239,204,353),(91,238,205,352),(92,237,206,351),(93,236,207,350),(94,235,208,349),(95,234,209,348),(96,233,210,347),(97,232,211,346),(98,231,212,345),(99,230,213,344),(100,229,214,343),(101,456,215,342),(102,455,216,341),(103,454,217,340),(104,453,218,339),(105,452,219,338),(106,451,220,337),(107,450,221,336),(108,449,222,335),(109,448,223,334),(110,447,224,333),(111,446,225,332),(112,445,226,331),(113,444,227,330),(114,443,228,329)]])
117 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 12A | 12B | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57R | 76A | ··· | 76R | 114A | ··· | 114R | 228A | ··· | 228AJ |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 | 228 | ··· | 228 |
| size | 1 | 1 | 2 | 2 | 114 | 114 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | Q8 | D6 | Dic6 | D19 | D38 | D57 | Dic38 | D114 | Dic114 |
| kernel | Dic114 | Dic57 | C228 | C76 | C57 | C38 | C19 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 18 | 18 | 18 | 36 |
Matrix representation of Dic114 ►in GL4(𝔽229) generated by
| 111 | 106 | 0 | 0 |
| 123 | 4 | 0 | 0 |
| 0 | 0 | 165 | 23 |
| 0 | 0 | 162 | 67 |
| 45 | 100 | 0 | 0 |
| 195 | 184 | 0 | 0 |
| 0 | 0 | 48 | 161 |
| 0 | 0 | 145 | 181 |
G:=sub<GL(4,GF(229))| [111,123,0,0,106,4,0,0,0,0,165,162,0,0,23,67],[45,195,0,0,100,184,0,0,0,0,48,145,0,0,161,181] >;
Dic114 in GAP, Magma, Sage, TeX
{\rm Dic}_{114} % in TeX
G:=Group("Dic114"); // GroupNames label
G:=SmallGroup(456,34);
// by ID
G=gap.SmallGroup(456,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,20,61,26,323,10804]);
// Polycyclic
G:=Group<a,b|a^228=1,b^2=a^114,b*a*b^-1=a^-1>;
// generators/relations
Export