metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic115, C46.D5, C23⋊Dic5, C115⋊3C4, C2.D115, C10.D23, C5⋊2Dic23, C230.1C2, SmallGroup(460,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C115 — Dic115 |
Generators and relations for Dic115
G = < a,b | a230=1, b2=a115, bab-1=a-1 >
Smallest permutation representation of Dic115
►Regular action on 460 points
Generators in S460
(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)
(1 392 116 277)(2 391 117 276)(3 390 118 275)(4 389 119 274)(5 388 120 273)(6 387 121 272)(7 386 122 271)(8 385 123 270)(9 384 124 269)(10 383 125 268)(11 382 126 267)(12 381 127 266)(13 380 128 265)(14 379 129 264)(15 378 130 263)(16 377 131 262)(17 376 132 261)(18 375 133 260)(19 374 134 259)(20 373 135 258)(21 372 136 257)(22 371 137 256)(23 370 138 255)(24 369 139 254)(25 368 140 253)(26 367 141 252)(27 366 142 251)(28 365 143 250)(29 364 144 249)(30 363 145 248)(31 362 146 247)(32 361 147 246)(33 360 148 245)(34 359 149 244)(35 358 150 243)(36 357 151 242)(37 356 152 241)(38 355 153 240)(39 354 154 239)(40 353 155 238)(41 352 156 237)(42 351 157 236)(43 350 158 235)(44 349 159 234)(45 348 160 233)(46 347 161 232)(47 346 162 231)(48 345 163 460)(49 344 164 459)(50 343 165 458)(51 342 166 457)(52 341 167 456)(53 340 168 455)(54 339 169 454)(55 338 170 453)(56 337 171 452)(57 336 172 451)(58 335 173 450)(59 334 174 449)(60 333 175 448)(61 332 176 447)(62 331 177 446)(63 330 178 445)(64 329 179 444)(65 328 180 443)(66 327 181 442)(67 326 182 441)(68 325 183 440)(69 324 184 439)(70 323 185 438)(71 322 186 437)(72 321 187 436)(73 320 188 435)(74 319 189 434)(75 318 190 433)(76 317 191 432)(77 316 192 431)(78 315 193 430)(79 314 194 429)(80 313 195 428)(81 312 196 427)(82 311 197 426)(83 310 198 425)(84 309 199 424)(85 308 200 423)(86 307 201 422)(87 306 202 421)(88 305 203 420)(89 304 204 419)(90 303 205 418)(91 302 206 417)(92 301 207 416)(93 300 208 415)(94 299 209 414)(95 298 210 413)(96 297 211 412)(97 296 212 411)(98 295 213 410)(99 294 214 409)(100 293 215 408)(101 292 216 407)(102 291 217 406)(103 290 218 405)(104 289 219 404)(105 288 220 403)(106 287 221 402)(107 286 222 401)(108 285 223 400)(109 284 224 399)(110 283 225 398)(111 282 226 397)(112 281 227 396)(113 280 228 395)(114 279 229 394)(115 278 230 393)
G:=sub<Sym(460)| (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), (1,392,116,277)(2,391,117,276)(3,390,118,275)(4,389,119,274)(5,388,120,273)(6,387,121,272)(7,386,122,271)(8,385,123,270)(9,384,124,269)(10,383,125,268)(11,382,126,267)(12,381,127,266)(13,380,128,265)(14,379,129,264)(15,378,130,263)(16,377,131,262)(17,376,132,261)(18,375,133,260)(19,374,134,259)(20,373,135,258)(21,372,136,257)(22,371,137,256)(23,370,138,255)(24,369,139,254)(25,368,140,253)(26,367,141,252)(27,366,142,251)(28,365,143,250)(29,364,144,249)(30,363,145,248)(31,362,146,247)(32,361,147,246)(33,360,148,245)(34,359,149,244)(35,358,150,243)(36,357,151,242)(37,356,152,241)(38,355,153,240)(39,354,154,239)(40,353,155,238)(41,352,156,237)(42,351,157,236)(43,350,158,235)(44,349,159,234)(45,348,160,233)(46,347,161,232)(47,346,162,231)(48,345,163,460)(49,344,164,459)(50,343,165,458)(51,342,166,457)(52,341,167,456)(53,340,168,455)(54,339,169,454)(55,338,170,453)(56,337,171,452)(57,336,172,451)(58,335,173,450)(59,334,174,449)(60,333,175,448)(61,332,176,447)(62,331,177,446)(63,330,178,445)(64,329,179,444)(65,328,180,443)(66,327,181,442)(67,326,182,441)(68,325,183,440)(69,324,184,439)(70,323,185,438)(71,322,186,437)(72,321,187,436)(73,320,188,435)(74,319,189,434)(75,318,190,433)(76,317,191,432)(77,316,192,431)(78,315,193,430)(79,314,194,429)(80,313,195,428)(81,312,196,427)(82,311,197,426)(83,310,198,425)(84,309,199,424)(85,308,200,423)(86,307,201,422)(87,306,202,421)(88,305,203,420)(89,304,204,419)(90,303,205,418)(91,302,206,417)(92,301,207,416)(93,300,208,415)(94,299,209,414)(95,298,210,413)(96,297,211,412)(97,296,212,411)(98,295,213,410)(99,294,214,409)(100,293,215,408)(101,292,216,407)(102,291,217,406)(103,290,218,405)(104,289,219,404)(105,288,220,403)(106,287,221,402)(107,286,222,401)(108,285,223,400)(109,284,224,399)(110,283,225,398)(111,282,226,397)(112,281,227,396)(113,280,228,395)(114,279,229,394)(115,278,230,393)>;
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), (1,392,116,277)(2,391,117,276)(3,390,118,275)(4,389,119,274)(5,388,120,273)(6,387,121,272)(7,386,122,271)(8,385,123,270)(9,384,124,269)(10,383,125,268)(11,382,126,267)(12,381,127,266)(13,380,128,265)(14,379,129,264)(15,378,130,263)(16,377,131,262)(17,376,132,261)(18,375,133,260)(19,374,134,259)(20,373,135,258)(21,372,136,257)(22,371,137,256)(23,370,138,255)(24,369,139,254)(25,368,140,253)(26,367,141,252)(27,366,142,251)(28,365,143,250)(29,364,144,249)(30,363,145,248)(31,362,146,247)(32,361,147,246)(33,360,148,245)(34,359,149,244)(35,358,150,243)(36,357,151,242)(37,356,152,241)(38,355,153,240)(39,354,154,239)(40,353,155,238)(41,352,156,237)(42,351,157,236)(43,350,158,235)(44,349,159,234)(45,348,160,233)(46,347,161,232)(47,346,162,231)(48,345,163,460)(49,344,164,459)(50,343,165,458)(51,342,166,457)(52,341,167,456)(53,340,168,455)(54,339,169,454)(55,338,170,453)(56,337,171,452)(57,336,172,451)(58,335,173,450)(59,334,174,449)(60,333,175,448)(61,332,176,447)(62,331,177,446)(63,330,178,445)(64,329,179,444)(65,328,180,443)(66,327,181,442)(67,326,182,441)(68,325,183,440)(69,324,184,439)(70,323,185,438)(71,322,186,437)(72,321,187,436)(73,320,188,435)(74,319,189,434)(75,318,190,433)(76,317,191,432)(77,316,192,431)(78,315,193,430)(79,314,194,429)(80,313,195,428)(81,312,196,427)(82,311,197,426)(83,310,198,425)(84,309,199,424)(85,308,200,423)(86,307,201,422)(87,306,202,421)(88,305,203,420)(89,304,204,419)(90,303,205,418)(91,302,206,417)(92,301,207,416)(93,300,208,415)(94,299,209,414)(95,298,210,413)(96,297,211,412)(97,296,212,411)(98,295,213,410)(99,294,214,409)(100,293,215,408)(101,292,216,407)(102,291,217,406)(103,290,218,405)(104,289,219,404)(105,288,220,403)(106,287,221,402)(107,286,222,401)(108,285,223,400)(109,284,224,399)(110,283,225,398)(111,282,226,397)(112,281,227,396)(113,280,228,395)(114,279,229,394)(115,278,230,393) );
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)], [(1,392,116,277),(2,391,117,276),(3,390,118,275),(4,389,119,274),(5,388,120,273),(6,387,121,272),(7,386,122,271),(8,385,123,270),(9,384,124,269),(10,383,125,268),(11,382,126,267),(12,381,127,266),(13,380,128,265),(14,379,129,264),(15,378,130,263),(16,377,131,262),(17,376,132,261),(18,375,133,260),(19,374,134,259),(20,373,135,258),(21,372,136,257),(22,371,137,256),(23,370,138,255),(24,369,139,254),(25,368,140,253),(26,367,141,252),(27,366,142,251),(28,365,143,250),(29,364,144,249),(30,363,145,248),(31,362,146,247),(32,361,147,246),(33,360,148,245),(34,359,149,244),(35,358,150,243),(36,357,151,242),(37,356,152,241),(38,355,153,240),(39,354,154,239),(40,353,155,238),(41,352,156,237),(42,351,157,236),(43,350,158,235),(44,349,159,234),(45,348,160,233),(46,347,161,232),(47,346,162,231),(48,345,163,460),(49,344,164,459),(50,343,165,458),(51,342,166,457),(52,341,167,456),(53,340,168,455),(54,339,169,454),(55,338,170,453),(56,337,171,452),(57,336,172,451),(58,335,173,450),(59,334,174,449),(60,333,175,448),(61,332,176,447),(62,331,177,446),(63,330,178,445),(64,329,179,444),(65,328,180,443),(66,327,181,442),(67,326,182,441),(68,325,183,440),(69,324,184,439),(70,323,185,438),(71,322,186,437),(72,321,187,436),(73,320,188,435),(74,319,189,434),(75,318,190,433),(76,317,191,432),(77,316,192,431),(78,315,193,430),(79,314,194,429),(80,313,195,428),(81,312,196,427),(82,311,197,426),(83,310,198,425),(84,309,199,424),(85,308,200,423),(86,307,201,422),(87,306,202,421),(88,305,203,420),(89,304,204,419),(90,303,205,418),(91,302,206,417),(92,301,207,416),(93,300,208,415),(94,299,209,414),(95,298,210,413),(96,297,211,412),(97,296,212,411),(98,295,213,410),(99,294,214,409),(100,293,215,408),(101,292,216,407),(102,291,217,406),(103,290,218,405),(104,289,219,404),(105,288,220,403),(106,287,221,402),(107,286,222,401),(108,285,223,400),(109,284,224,399),(110,283,225,398),(111,282,226,397),(112,281,227,396),(113,280,228,395),(114,279,229,394),(115,278,230,393)]])
118 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 10A | 10B | 23A | ··· | 23K | 46A | ··· | 46K | 115A | ··· | 115AR | 230A | ··· | 230AR |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 10 | 10 | 23 | ··· | 23 | 46 | ··· | 46 | 115 | ··· | 115 | 230 | ··· | 230 |
| size | 1 | 1 | 115 | 115 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | D5 | Dic5 | D23 | Dic23 | D115 | Dic115 |
| kernel | Dic115 | C230 | C115 | C46 | C23 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 11 | 11 | 44 | 44 |
Matrix representation of Dic115 ►in GL3(𝔽461) generated by
| 460 | 0 | 0 |
| 0 | 332 | 275 |
| 0 | 186 | 211 |
| 413 | 0 | 0 |
| 0 | 432 | 21 |
| 0 | 421 | 29 |
G:=sub<GL(3,GF(461))| [460,0,0,0,332,186,0,275,211],[413,0,0,0,432,421,0,21,29] >;
Dic115 in GAP, Magma, Sage, TeX
{\rm Dic}_{115} % in TeX
G:=Group("Dic115"); // GroupNames label
G:=SmallGroup(460,3);
// by ID
G=gap.SmallGroup(460,3);
# by ID
G:=PCGroup([4,-2,-2,-5,-23,8,194,7043]);
// Polycyclic
G:=Group<a,b|a^230=1,b^2=a^115,b*a*b^-1=a^-1>;
// generators/relations
Export