metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic113, C113⋊2C4, C226.C2, C2.D113, SmallGroup(452,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C113 — Dic113 |
Generators and relations for Dic113
G = < a,b | a226=1, b2=a113, bab-1=a-1 >
Smallest permutation representation of Dic113
►Regular action on 452 points
Generators in S452
(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)
(1 313 114 426)(2 312 115 425)(3 311 116 424)(4 310 117 423)(5 309 118 422)(6 308 119 421)(7 307 120 420)(8 306 121 419)(9 305 122 418)(10 304 123 417)(11 303 124 416)(12 302 125 415)(13 301 126 414)(14 300 127 413)(15 299 128 412)(16 298 129 411)(17 297 130 410)(18 296 131 409)(19 295 132 408)(20 294 133 407)(21 293 134 406)(22 292 135 405)(23 291 136 404)(24 290 137 403)(25 289 138 402)(26 288 139 401)(27 287 140 400)(28 286 141 399)(29 285 142 398)(30 284 143 397)(31 283 144 396)(32 282 145 395)(33 281 146 394)(34 280 147 393)(35 279 148 392)(36 278 149 391)(37 277 150 390)(38 276 151 389)(39 275 152 388)(40 274 153 387)(41 273 154 386)(42 272 155 385)(43 271 156 384)(44 270 157 383)(45 269 158 382)(46 268 159 381)(47 267 160 380)(48 266 161 379)(49 265 162 378)(50 264 163 377)(51 263 164 376)(52 262 165 375)(53 261 166 374)(54 260 167 373)(55 259 168 372)(56 258 169 371)(57 257 170 370)(58 256 171 369)(59 255 172 368)(60 254 173 367)(61 253 174 366)(62 252 175 365)(63 251 176 364)(64 250 177 363)(65 249 178 362)(66 248 179 361)(67 247 180 360)(68 246 181 359)(69 245 182 358)(70 244 183 357)(71 243 184 356)(72 242 185 355)(73 241 186 354)(74 240 187 353)(75 239 188 352)(76 238 189 351)(77 237 190 350)(78 236 191 349)(79 235 192 348)(80 234 193 347)(81 233 194 346)(82 232 195 345)(83 231 196 344)(84 230 197 343)(85 229 198 342)(86 228 199 341)(87 227 200 340)(88 452 201 339)(89 451 202 338)(90 450 203 337)(91 449 204 336)(92 448 205 335)(93 447 206 334)(94 446 207 333)(95 445 208 332)(96 444 209 331)(97 443 210 330)(98 442 211 329)(99 441 212 328)(100 440 213 327)(101 439 214 326)(102 438 215 325)(103 437 216 324)(104 436 217 323)(105 435 218 322)(106 434 219 321)(107 433 220 320)(108 432 221 319)(109 431 222 318)(110 430 223 317)(111 429 224 316)(112 428 225 315)(113 427 226 314)
G:=sub<Sym(452)| (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), (1,313,114,426)(2,312,115,425)(3,311,116,424)(4,310,117,423)(5,309,118,422)(6,308,119,421)(7,307,120,420)(8,306,121,419)(9,305,122,418)(10,304,123,417)(11,303,124,416)(12,302,125,415)(13,301,126,414)(14,300,127,413)(15,299,128,412)(16,298,129,411)(17,297,130,410)(18,296,131,409)(19,295,132,408)(20,294,133,407)(21,293,134,406)(22,292,135,405)(23,291,136,404)(24,290,137,403)(25,289,138,402)(26,288,139,401)(27,287,140,400)(28,286,141,399)(29,285,142,398)(30,284,143,397)(31,283,144,396)(32,282,145,395)(33,281,146,394)(34,280,147,393)(35,279,148,392)(36,278,149,391)(37,277,150,390)(38,276,151,389)(39,275,152,388)(40,274,153,387)(41,273,154,386)(42,272,155,385)(43,271,156,384)(44,270,157,383)(45,269,158,382)(46,268,159,381)(47,267,160,380)(48,266,161,379)(49,265,162,378)(50,264,163,377)(51,263,164,376)(52,262,165,375)(53,261,166,374)(54,260,167,373)(55,259,168,372)(56,258,169,371)(57,257,170,370)(58,256,171,369)(59,255,172,368)(60,254,173,367)(61,253,174,366)(62,252,175,365)(63,251,176,364)(64,250,177,363)(65,249,178,362)(66,248,179,361)(67,247,180,360)(68,246,181,359)(69,245,182,358)(70,244,183,357)(71,243,184,356)(72,242,185,355)(73,241,186,354)(74,240,187,353)(75,239,188,352)(76,238,189,351)(77,237,190,350)(78,236,191,349)(79,235,192,348)(80,234,193,347)(81,233,194,346)(82,232,195,345)(83,231,196,344)(84,230,197,343)(85,229,198,342)(86,228,199,341)(87,227,200,340)(88,452,201,339)(89,451,202,338)(90,450,203,337)(91,449,204,336)(92,448,205,335)(93,447,206,334)(94,446,207,333)(95,445,208,332)(96,444,209,331)(97,443,210,330)(98,442,211,329)(99,441,212,328)(100,440,213,327)(101,439,214,326)(102,438,215,325)(103,437,216,324)(104,436,217,323)(105,435,218,322)(106,434,219,321)(107,433,220,320)(108,432,221,319)(109,431,222,318)(110,430,223,317)(111,429,224,316)(112,428,225,315)(113,427,226,314)>;
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), (1,313,114,426)(2,312,115,425)(3,311,116,424)(4,310,117,423)(5,309,118,422)(6,308,119,421)(7,307,120,420)(8,306,121,419)(9,305,122,418)(10,304,123,417)(11,303,124,416)(12,302,125,415)(13,301,126,414)(14,300,127,413)(15,299,128,412)(16,298,129,411)(17,297,130,410)(18,296,131,409)(19,295,132,408)(20,294,133,407)(21,293,134,406)(22,292,135,405)(23,291,136,404)(24,290,137,403)(25,289,138,402)(26,288,139,401)(27,287,140,400)(28,286,141,399)(29,285,142,398)(30,284,143,397)(31,283,144,396)(32,282,145,395)(33,281,146,394)(34,280,147,393)(35,279,148,392)(36,278,149,391)(37,277,150,390)(38,276,151,389)(39,275,152,388)(40,274,153,387)(41,273,154,386)(42,272,155,385)(43,271,156,384)(44,270,157,383)(45,269,158,382)(46,268,159,381)(47,267,160,380)(48,266,161,379)(49,265,162,378)(50,264,163,377)(51,263,164,376)(52,262,165,375)(53,261,166,374)(54,260,167,373)(55,259,168,372)(56,258,169,371)(57,257,170,370)(58,256,171,369)(59,255,172,368)(60,254,173,367)(61,253,174,366)(62,252,175,365)(63,251,176,364)(64,250,177,363)(65,249,178,362)(66,248,179,361)(67,247,180,360)(68,246,181,359)(69,245,182,358)(70,244,183,357)(71,243,184,356)(72,242,185,355)(73,241,186,354)(74,240,187,353)(75,239,188,352)(76,238,189,351)(77,237,190,350)(78,236,191,349)(79,235,192,348)(80,234,193,347)(81,233,194,346)(82,232,195,345)(83,231,196,344)(84,230,197,343)(85,229,198,342)(86,228,199,341)(87,227,200,340)(88,452,201,339)(89,451,202,338)(90,450,203,337)(91,449,204,336)(92,448,205,335)(93,447,206,334)(94,446,207,333)(95,445,208,332)(96,444,209,331)(97,443,210,330)(98,442,211,329)(99,441,212,328)(100,440,213,327)(101,439,214,326)(102,438,215,325)(103,437,216,324)(104,436,217,323)(105,435,218,322)(106,434,219,321)(107,433,220,320)(108,432,221,319)(109,431,222,318)(110,430,223,317)(111,429,224,316)(112,428,225,315)(113,427,226,314) );
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)], [(1,313,114,426),(2,312,115,425),(3,311,116,424),(4,310,117,423),(5,309,118,422),(6,308,119,421),(7,307,120,420),(8,306,121,419),(9,305,122,418),(10,304,123,417),(11,303,124,416),(12,302,125,415),(13,301,126,414),(14,300,127,413),(15,299,128,412),(16,298,129,411),(17,297,130,410),(18,296,131,409),(19,295,132,408),(20,294,133,407),(21,293,134,406),(22,292,135,405),(23,291,136,404),(24,290,137,403),(25,289,138,402),(26,288,139,401),(27,287,140,400),(28,286,141,399),(29,285,142,398),(30,284,143,397),(31,283,144,396),(32,282,145,395),(33,281,146,394),(34,280,147,393),(35,279,148,392),(36,278,149,391),(37,277,150,390),(38,276,151,389),(39,275,152,388),(40,274,153,387),(41,273,154,386),(42,272,155,385),(43,271,156,384),(44,270,157,383),(45,269,158,382),(46,268,159,381),(47,267,160,380),(48,266,161,379),(49,265,162,378),(50,264,163,377),(51,263,164,376),(52,262,165,375),(53,261,166,374),(54,260,167,373),(55,259,168,372),(56,258,169,371),(57,257,170,370),(58,256,171,369),(59,255,172,368),(60,254,173,367),(61,253,174,366),(62,252,175,365),(63,251,176,364),(64,250,177,363),(65,249,178,362),(66,248,179,361),(67,247,180,360),(68,246,181,359),(69,245,182,358),(70,244,183,357),(71,243,184,356),(72,242,185,355),(73,241,186,354),(74,240,187,353),(75,239,188,352),(76,238,189,351),(77,237,190,350),(78,236,191,349),(79,235,192,348),(80,234,193,347),(81,233,194,346),(82,232,195,345),(83,231,196,344),(84,230,197,343),(85,229,198,342),(86,228,199,341),(87,227,200,340),(88,452,201,339),(89,451,202,338),(90,450,203,337),(91,449,204,336),(92,448,205,335),(93,447,206,334),(94,446,207,333),(95,445,208,332),(96,444,209,331),(97,443,210,330),(98,442,211,329),(99,441,212,328),(100,440,213,327),(101,439,214,326),(102,438,215,325),(103,437,216,324),(104,436,217,323),(105,435,218,322),(106,434,219,321),(107,433,220,320),(108,432,221,319),(109,431,222,318),(110,430,223,317),(111,429,224,316),(112,428,225,315),(113,427,226,314)]])
116 conjugacy classes
| class | 1 | 2 | 4A | 4B | 113A | ··· | 113BD | 226A | ··· | 226BD |
| order | 1 | 2 | 4 | 4 | 113 | ··· | 113 | 226 | ··· | 226 |
| size | 1 | 1 | 113 | 113 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D113 | Dic113 |
| kernel | Dic113 | C226 | C113 | C2 | C1 |
| # reps | 1 | 1 | 2 | 56 | 56 |
Matrix representation of Dic113 ►in GL2(𝔽2713) generated by
| 612 | 1 |
| 2712 | 0 |
| 2028 | 582 |
| 2000 | 685 |
G:=sub<GL(2,GF(2713))| [612,2712,1,0],[2028,2000,582,685] >;
Dic113 in GAP, Magma, Sage, TeX
{\rm Dic}_{113} % in TeX
G:=Group("Dic113"); // GroupNames label
G:=SmallGroup(452,1);
// by ID
G=gap.SmallGroup(452,1);
# by ID
G:=PCGroup([3,-2,-2,-113,6,4034]);
// Polycyclic
G:=Group<a,b|a^226=1,b^2=a^113,b*a*b^-1=a^-1>;
// generators/relations
Export