metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic116, C8.D29, C29⋊1Q16, C58.3D4, C232.1C2, C2.5D116, C4.10D58, Dic58.1C2, C116.10C22, SmallGroup(464,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic116
G = < a,b | a232=1, b2=a116, bab-1=a-1 >
Smallest permutation representation of Dic116
►Regular action on 464 points
Generators in S464
(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)
(1 268 117 384)(2 267 118 383)(3 266 119 382)(4 265 120 381)(5 264 121 380)(6 263 122 379)(7 262 123 378)(8 261 124 377)(9 260 125 376)(10 259 126 375)(11 258 127 374)(12 257 128 373)(13 256 129 372)(14 255 130 371)(15 254 131 370)(16 253 132 369)(17 252 133 368)(18 251 134 367)(19 250 135 366)(20 249 136 365)(21 248 137 364)(22 247 138 363)(23 246 139 362)(24 245 140 361)(25 244 141 360)(26 243 142 359)(27 242 143 358)(28 241 144 357)(29 240 145 356)(30 239 146 355)(31 238 147 354)(32 237 148 353)(33 236 149 352)(34 235 150 351)(35 234 151 350)(36 233 152 349)(37 464 153 348)(38 463 154 347)(39 462 155 346)(40 461 156 345)(41 460 157 344)(42 459 158 343)(43 458 159 342)(44 457 160 341)(45 456 161 340)(46 455 162 339)(47 454 163 338)(48 453 164 337)(49 452 165 336)(50 451 166 335)(51 450 167 334)(52 449 168 333)(53 448 169 332)(54 447 170 331)(55 446 171 330)(56 445 172 329)(57 444 173 328)(58 443 174 327)(59 442 175 326)(60 441 176 325)(61 440 177 324)(62 439 178 323)(63 438 179 322)(64 437 180 321)(65 436 181 320)(66 435 182 319)(67 434 183 318)(68 433 184 317)(69 432 185 316)(70 431 186 315)(71 430 187 314)(72 429 188 313)(73 428 189 312)(74 427 190 311)(75 426 191 310)(76 425 192 309)(77 424 193 308)(78 423 194 307)(79 422 195 306)(80 421 196 305)(81 420 197 304)(82 419 198 303)(83 418 199 302)(84 417 200 301)(85 416 201 300)(86 415 202 299)(87 414 203 298)(88 413 204 297)(89 412 205 296)(90 411 206 295)(91 410 207 294)(92 409 208 293)(93 408 209 292)(94 407 210 291)(95 406 211 290)(96 405 212 289)(97 404 213 288)(98 403 214 287)(99 402 215 286)(100 401 216 285)(101 400 217 284)(102 399 218 283)(103 398 219 282)(104 397 220 281)(105 396 221 280)(106 395 222 279)(107 394 223 278)(108 393 224 277)(109 392 225 276)(110 391 226 275)(111 390 227 274)(112 389 228 273)(113 388 229 272)(114 387 230 271)(115 386 231 270)(116 385 232 269)
G:=sub<Sym(464)| (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), (1,268,117,384)(2,267,118,383)(3,266,119,382)(4,265,120,381)(5,264,121,380)(6,263,122,379)(7,262,123,378)(8,261,124,377)(9,260,125,376)(10,259,126,375)(11,258,127,374)(12,257,128,373)(13,256,129,372)(14,255,130,371)(15,254,131,370)(16,253,132,369)(17,252,133,368)(18,251,134,367)(19,250,135,366)(20,249,136,365)(21,248,137,364)(22,247,138,363)(23,246,139,362)(24,245,140,361)(25,244,141,360)(26,243,142,359)(27,242,143,358)(28,241,144,357)(29,240,145,356)(30,239,146,355)(31,238,147,354)(32,237,148,353)(33,236,149,352)(34,235,150,351)(35,234,151,350)(36,233,152,349)(37,464,153,348)(38,463,154,347)(39,462,155,346)(40,461,156,345)(41,460,157,344)(42,459,158,343)(43,458,159,342)(44,457,160,341)(45,456,161,340)(46,455,162,339)(47,454,163,338)(48,453,164,337)(49,452,165,336)(50,451,166,335)(51,450,167,334)(52,449,168,333)(53,448,169,332)(54,447,170,331)(55,446,171,330)(56,445,172,329)(57,444,173,328)(58,443,174,327)(59,442,175,326)(60,441,176,325)(61,440,177,324)(62,439,178,323)(63,438,179,322)(64,437,180,321)(65,436,181,320)(66,435,182,319)(67,434,183,318)(68,433,184,317)(69,432,185,316)(70,431,186,315)(71,430,187,314)(72,429,188,313)(73,428,189,312)(74,427,190,311)(75,426,191,310)(76,425,192,309)(77,424,193,308)(78,423,194,307)(79,422,195,306)(80,421,196,305)(81,420,197,304)(82,419,198,303)(83,418,199,302)(84,417,200,301)(85,416,201,300)(86,415,202,299)(87,414,203,298)(88,413,204,297)(89,412,205,296)(90,411,206,295)(91,410,207,294)(92,409,208,293)(93,408,209,292)(94,407,210,291)(95,406,211,290)(96,405,212,289)(97,404,213,288)(98,403,214,287)(99,402,215,286)(100,401,216,285)(101,400,217,284)(102,399,218,283)(103,398,219,282)(104,397,220,281)(105,396,221,280)(106,395,222,279)(107,394,223,278)(108,393,224,277)(109,392,225,276)(110,391,226,275)(111,390,227,274)(112,389,228,273)(113,388,229,272)(114,387,230,271)(115,386,231,270)(116,385,232,269)>;
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), (1,268,117,384)(2,267,118,383)(3,266,119,382)(4,265,120,381)(5,264,121,380)(6,263,122,379)(7,262,123,378)(8,261,124,377)(9,260,125,376)(10,259,126,375)(11,258,127,374)(12,257,128,373)(13,256,129,372)(14,255,130,371)(15,254,131,370)(16,253,132,369)(17,252,133,368)(18,251,134,367)(19,250,135,366)(20,249,136,365)(21,248,137,364)(22,247,138,363)(23,246,139,362)(24,245,140,361)(25,244,141,360)(26,243,142,359)(27,242,143,358)(28,241,144,357)(29,240,145,356)(30,239,146,355)(31,238,147,354)(32,237,148,353)(33,236,149,352)(34,235,150,351)(35,234,151,350)(36,233,152,349)(37,464,153,348)(38,463,154,347)(39,462,155,346)(40,461,156,345)(41,460,157,344)(42,459,158,343)(43,458,159,342)(44,457,160,341)(45,456,161,340)(46,455,162,339)(47,454,163,338)(48,453,164,337)(49,452,165,336)(50,451,166,335)(51,450,167,334)(52,449,168,333)(53,448,169,332)(54,447,170,331)(55,446,171,330)(56,445,172,329)(57,444,173,328)(58,443,174,327)(59,442,175,326)(60,441,176,325)(61,440,177,324)(62,439,178,323)(63,438,179,322)(64,437,180,321)(65,436,181,320)(66,435,182,319)(67,434,183,318)(68,433,184,317)(69,432,185,316)(70,431,186,315)(71,430,187,314)(72,429,188,313)(73,428,189,312)(74,427,190,311)(75,426,191,310)(76,425,192,309)(77,424,193,308)(78,423,194,307)(79,422,195,306)(80,421,196,305)(81,420,197,304)(82,419,198,303)(83,418,199,302)(84,417,200,301)(85,416,201,300)(86,415,202,299)(87,414,203,298)(88,413,204,297)(89,412,205,296)(90,411,206,295)(91,410,207,294)(92,409,208,293)(93,408,209,292)(94,407,210,291)(95,406,211,290)(96,405,212,289)(97,404,213,288)(98,403,214,287)(99,402,215,286)(100,401,216,285)(101,400,217,284)(102,399,218,283)(103,398,219,282)(104,397,220,281)(105,396,221,280)(106,395,222,279)(107,394,223,278)(108,393,224,277)(109,392,225,276)(110,391,226,275)(111,390,227,274)(112,389,228,273)(113,388,229,272)(114,387,230,271)(115,386,231,270)(116,385,232,269) );
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)], [(1,268,117,384),(2,267,118,383),(3,266,119,382),(4,265,120,381),(5,264,121,380),(6,263,122,379),(7,262,123,378),(8,261,124,377),(9,260,125,376),(10,259,126,375),(11,258,127,374),(12,257,128,373),(13,256,129,372),(14,255,130,371),(15,254,131,370),(16,253,132,369),(17,252,133,368),(18,251,134,367),(19,250,135,366),(20,249,136,365),(21,248,137,364),(22,247,138,363),(23,246,139,362),(24,245,140,361),(25,244,141,360),(26,243,142,359),(27,242,143,358),(28,241,144,357),(29,240,145,356),(30,239,146,355),(31,238,147,354),(32,237,148,353),(33,236,149,352),(34,235,150,351),(35,234,151,350),(36,233,152,349),(37,464,153,348),(38,463,154,347),(39,462,155,346),(40,461,156,345),(41,460,157,344),(42,459,158,343),(43,458,159,342),(44,457,160,341),(45,456,161,340),(46,455,162,339),(47,454,163,338),(48,453,164,337),(49,452,165,336),(50,451,166,335),(51,450,167,334),(52,449,168,333),(53,448,169,332),(54,447,170,331),(55,446,171,330),(56,445,172,329),(57,444,173,328),(58,443,174,327),(59,442,175,326),(60,441,176,325),(61,440,177,324),(62,439,178,323),(63,438,179,322),(64,437,180,321),(65,436,181,320),(66,435,182,319),(67,434,183,318),(68,433,184,317),(69,432,185,316),(70,431,186,315),(71,430,187,314),(72,429,188,313),(73,428,189,312),(74,427,190,311),(75,426,191,310),(76,425,192,309),(77,424,193,308),(78,423,194,307),(79,422,195,306),(80,421,196,305),(81,420,197,304),(82,419,198,303),(83,418,199,302),(84,417,200,301),(85,416,201,300),(86,415,202,299),(87,414,203,298),(88,413,204,297),(89,412,205,296),(90,411,206,295),(91,410,207,294),(92,409,208,293),(93,408,209,292),(94,407,210,291),(95,406,211,290),(96,405,212,289),(97,404,213,288),(98,403,214,287),(99,402,215,286),(100,401,216,285),(101,400,217,284),(102,399,218,283),(103,398,219,282),(104,397,220,281),(105,396,221,280),(106,395,222,279),(107,394,223,278),(108,393,224,277),(109,392,225,276),(110,391,226,275),(111,390,227,274),(112,389,228,273),(113,388,229,272),(114,387,230,271),(115,386,231,270),(116,385,232,269)]])
119 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 29A | ··· | 29N | 58A | ··· | 58N | 116A | ··· | 116AB | 232A | ··· | 232BD |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 | 232 | ··· | 232 |
| size | 1 | 1 | 2 | 116 | 116 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
119 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | D4 | Q16 | D29 | D58 | D116 | Dic116 |
| kernel | Dic116 | C232 | Dic58 | C58 | C29 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 14 | 14 | 28 | 56 |
Matrix representation of Dic116 ►in GL2(𝔽233) generated by
| 2 | 26 |
| 73 | 134 |
| 222 | 46 |
| 48 | 11 |
G:=sub<GL(2,GF(233))| [2,73,26,134],[222,48,46,11] >;
Dic116 in GAP, Magma, Sage, TeX
{\rm Dic}_{116} % in TeX
G:=Group("Dic116"); // GroupNames label
G:=SmallGroup(464,8);
// by ID
G=gap.SmallGroup(464,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,40,61,66,182,42,11204]);
// Polycyclic
G:=Group<a,b|a^232=1,b^2=a^116,b*a*b^-1=a^-1>;
// generators/relations
Export