metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic117, C26.D9, C9⋊Dic13, C117⋊3C4, C2.D117, C18.D13, C78.1S3, C6.1D39, C13⋊2Dic9, C3.Dic39, C234.1C2, C39.2Dic3, SmallGroup(468,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C117 — Dic117 |
Generators and relations for Dic117
G = < a,b | a234=1, b2=a117, bab-1=a-1 >
Smallest permutation representation of Dic117
►Regular action on 468 points
Generators in S468
(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)
(1 326 118 443)(2 325 119 442)(3 324 120 441)(4 323 121 440)(5 322 122 439)(6 321 123 438)(7 320 124 437)(8 319 125 436)(9 318 126 435)(10 317 127 434)(11 316 128 433)(12 315 129 432)(13 314 130 431)(14 313 131 430)(15 312 132 429)(16 311 133 428)(17 310 134 427)(18 309 135 426)(19 308 136 425)(20 307 137 424)(21 306 138 423)(22 305 139 422)(23 304 140 421)(24 303 141 420)(25 302 142 419)(26 301 143 418)(27 300 144 417)(28 299 145 416)(29 298 146 415)(30 297 147 414)(31 296 148 413)(32 295 149 412)(33 294 150 411)(34 293 151 410)(35 292 152 409)(36 291 153 408)(37 290 154 407)(38 289 155 406)(39 288 156 405)(40 287 157 404)(41 286 158 403)(42 285 159 402)(43 284 160 401)(44 283 161 400)(45 282 162 399)(46 281 163 398)(47 280 164 397)(48 279 165 396)(49 278 166 395)(50 277 167 394)(51 276 168 393)(52 275 169 392)(53 274 170 391)(54 273 171 390)(55 272 172 389)(56 271 173 388)(57 270 174 387)(58 269 175 386)(59 268 176 385)(60 267 177 384)(61 266 178 383)(62 265 179 382)(63 264 180 381)(64 263 181 380)(65 262 182 379)(66 261 183 378)(67 260 184 377)(68 259 185 376)(69 258 186 375)(70 257 187 374)(71 256 188 373)(72 255 189 372)(73 254 190 371)(74 253 191 370)(75 252 192 369)(76 251 193 368)(77 250 194 367)(78 249 195 366)(79 248 196 365)(80 247 197 364)(81 246 198 363)(82 245 199 362)(83 244 200 361)(84 243 201 360)(85 242 202 359)(86 241 203 358)(87 240 204 357)(88 239 205 356)(89 238 206 355)(90 237 207 354)(91 236 208 353)(92 235 209 352)(93 468 210 351)(94 467 211 350)(95 466 212 349)(96 465 213 348)(97 464 214 347)(98 463 215 346)(99 462 216 345)(100 461 217 344)(101 460 218 343)(102 459 219 342)(103 458 220 341)(104 457 221 340)(105 456 222 339)(106 455 223 338)(107 454 224 337)(108 453 225 336)(109 452 226 335)(110 451 227 334)(111 450 228 333)(112 449 229 332)(113 448 230 331)(114 447 231 330)(115 446 232 329)(116 445 233 328)(117 444 234 327)
G:=sub<Sym(468)| (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), (1,326,118,443)(2,325,119,442)(3,324,120,441)(4,323,121,440)(5,322,122,439)(6,321,123,438)(7,320,124,437)(8,319,125,436)(9,318,126,435)(10,317,127,434)(11,316,128,433)(12,315,129,432)(13,314,130,431)(14,313,131,430)(15,312,132,429)(16,311,133,428)(17,310,134,427)(18,309,135,426)(19,308,136,425)(20,307,137,424)(21,306,138,423)(22,305,139,422)(23,304,140,421)(24,303,141,420)(25,302,142,419)(26,301,143,418)(27,300,144,417)(28,299,145,416)(29,298,146,415)(30,297,147,414)(31,296,148,413)(32,295,149,412)(33,294,150,411)(34,293,151,410)(35,292,152,409)(36,291,153,408)(37,290,154,407)(38,289,155,406)(39,288,156,405)(40,287,157,404)(41,286,158,403)(42,285,159,402)(43,284,160,401)(44,283,161,400)(45,282,162,399)(46,281,163,398)(47,280,164,397)(48,279,165,396)(49,278,166,395)(50,277,167,394)(51,276,168,393)(52,275,169,392)(53,274,170,391)(54,273,171,390)(55,272,172,389)(56,271,173,388)(57,270,174,387)(58,269,175,386)(59,268,176,385)(60,267,177,384)(61,266,178,383)(62,265,179,382)(63,264,180,381)(64,263,181,380)(65,262,182,379)(66,261,183,378)(67,260,184,377)(68,259,185,376)(69,258,186,375)(70,257,187,374)(71,256,188,373)(72,255,189,372)(73,254,190,371)(74,253,191,370)(75,252,192,369)(76,251,193,368)(77,250,194,367)(78,249,195,366)(79,248,196,365)(80,247,197,364)(81,246,198,363)(82,245,199,362)(83,244,200,361)(84,243,201,360)(85,242,202,359)(86,241,203,358)(87,240,204,357)(88,239,205,356)(89,238,206,355)(90,237,207,354)(91,236,208,353)(92,235,209,352)(93,468,210,351)(94,467,211,350)(95,466,212,349)(96,465,213,348)(97,464,214,347)(98,463,215,346)(99,462,216,345)(100,461,217,344)(101,460,218,343)(102,459,219,342)(103,458,220,341)(104,457,221,340)(105,456,222,339)(106,455,223,338)(107,454,224,337)(108,453,225,336)(109,452,226,335)(110,451,227,334)(111,450,228,333)(112,449,229,332)(113,448,230,331)(114,447,231,330)(115,446,232,329)(116,445,233,328)(117,444,234,327)>;
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), (1,326,118,443)(2,325,119,442)(3,324,120,441)(4,323,121,440)(5,322,122,439)(6,321,123,438)(7,320,124,437)(8,319,125,436)(9,318,126,435)(10,317,127,434)(11,316,128,433)(12,315,129,432)(13,314,130,431)(14,313,131,430)(15,312,132,429)(16,311,133,428)(17,310,134,427)(18,309,135,426)(19,308,136,425)(20,307,137,424)(21,306,138,423)(22,305,139,422)(23,304,140,421)(24,303,141,420)(25,302,142,419)(26,301,143,418)(27,300,144,417)(28,299,145,416)(29,298,146,415)(30,297,147,414)(31,296,148,413)(32,295,149,412)(33,294,150,411)(34,293,151,410)(35,292,152,409)(36,291,153,408)(37,290,154,407)(38,289,155,406)(39,288,156,405)(40,287,157,404)(41,286,158,403)(42,285,159,402)(43,284,160,401)(44,283,161,400)(45,282,162,399)(46,281,163,398)(47,280,164,397)(48,279,165,396)(49,278,166,395)(50,277,167,394)(51,276,168,393)(52,275,169,392)(53,274,170,391)(54,273,171,390)(55,272,172,389)(56,271,173,388)(57,270,174,387)(58,269,175,386)(59,268,176,385)(60,267,177,384)(61,266,178,383)(62,265,179,382)(63,264,180,381)(64,263,181,380)(65,262,182,379)(66,261,183,378)(67,260,184,377)(68,259,185,376)(69,258,186,375)(70,257,187,374)(71,256,188,373)(72,255,189,372)(73,254,190,371)(74,253,191,370)(75,252,192,369)(76,251,193,368)(77,250,194,367)(78,249,195,366)(79,248,196,365)(80,247,197,364)(81,246,198,363)(82,245,199,362)(83,244,200,361)(84,243,201,360)(85,242,202,359)(86,241,203,358)(87,240,204,357)(88,239,205,356)(89,238,206,355)(90,237,207,354)(91,236,208,353)(92,235,209,352)(93,468,210,351)(94,467,211,350)(95,466,212,349)(96,465,213,348)(97,464,214,347)(98,463,215,346)(99,462,216,345)(100,461,217,344)(101,460,218,343)(102,459,219,342)(103,458,220,341)(104,457,221,340)(105,456,222,339)(106,455,223,338)(107,454,224,337)(108,453,225,336)(109,452,226,335)(110,451,227,334)(111,450,228,333)(112,449,229,332)(113,448,230,331)(114,447,231,330)(115,446,232,329)(116,445,233,328)(117,444,234,327) );
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)], [(1,326,118,443),(2,325,119,442),(3,324,120,441),(4,323,121,440),(5,322,122,439),(6,321,123,438),(7,320,124,437),(8,319,125,436),(9,318,126,435),(10,317,127,434),(11,316,128,433),(12,315,129,432),(13,314,130,431),(14,313,131,430),(15,312,132,429),(16,311,133,428),(17,310,134,427),(18,309,135,426),(19,308,136,425),(20,307,137,424),(21,306,138,423),(22,305,139,422),(23,304,140,421),(24,303,141,420),(25,302,142,419),(26,301,143,418),(27,300,144,417),(28,299,145,416),(29,298,146,415),(30,297,147,414),(31,296,148,413),(32,295,149,412),(33,294,150,411),(34,293,151,410),(35,292,152,409),(36,291,153,408),(37,290,154,407),(38,289,155,406),(39,288,156,405),(40,287,157,404),(41,286,158,403),(42,285,159,402),(43,284,160,401),(44,283,161,400),(45,282,162,399),(46,281,163,398),(47,280,164,397),(48,279,165,396),(49,278,166,395),(50,277,167,394),(51,276,168,393),(52,275,169,392),(53,274,170,391),(54,273,171,390),(55,272,172,389),(56,271,173,388),(57,270,174,387),(58,269,175,386),(59,268,176,385),(60,267,177,384),(61,266,178,383),(62,265,179,382),(63,264,180,381),(64,263,181,380),(65,262,182,379),(66,261,183,378),(67,260,184,377),(68,259,185,376),(69,258,186,375),(70,257,187,374),(71,256,188,373),(72,255,189,372),(73,254,190,371),(74,253,191,370),(75,252,192,369),(76,251,193,368),(77,250,194,367),(78,249,195,366),(79,248,196,365),(80,247,197,364),(81,246,198,363),(82,245,199,362),(83,244,200,361),(84,243,201,360),(85,242,202,359),(86,241,203,358),(87,240,204,357),(88,239,205,356),(89,238,206,355),(90,237,207,354),(91,236,208,353),(92,235,209,352),(93,468,210,351),(94,467,211,350),(95,466,212,349),(96,465,213,348),(97,464,214,347),(98,463,215,346),(99,462,216,345),(100,461,217,344),(101,460,218,343),(102,459,219,342),(103,458,220,341),(104,457,221,340),(105,456,222,339),(106,455,223,338),(107,454,224,337),(108,453,225,336),(109,452,226,335),(110,451,227,334),(111,450,228,333),(112,449,229,332),(113,448,230,331),(114,447,231,330),(115,446,232,329),(116,445,233,328),(117,444,234,327)]])
120 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 9A | 9B | 9C | 13A | ··· | 13F | 18A | 18B | 18C | 26A | ··· | 26F | 39A | ··· | 39L | 78A | ··· | 78L | 117A | ··· | 117AJ | 234A | ··· | 234AJ |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 9 | 9 | 9 | 13 | ··· | 13 | 18 | 18 | 18 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 | 117 | ··· | 117 | 234 | ··· | 234 |
| size | 1 | 1 | 2 | 117 | 117 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D9 | D13 | Dic9 | Dic13 | D39 | Dic39 | D117 | Dic117 |
| kernel | Dic117 | C234 | C117 | C78 | C39 | C26 | C18 | C13 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 3 | 6 | 3 | 6 | 12 | 12 | 36 | 36 |
Matrix representation of Dic117 ►in GL2(𝔽937) generated by
| 739 | 194 |
| 743 | 545 |
| 328 | 216 |
| 825 | 609 |
G:=sub<GL(2,GF(937))| [739,743,194,545],[328,825,216,609] >;
Dic117 in GAP, Magma, Sage, TeX
{\rm Dic}_{117} % in TeX
G:=Group("Dic117"); // GroupNames label
G:=SmallGroup(468,5);
// by ID
G=gap.SmallGroup(468,5);
# by ID
G:=PCGroup([5,-2,-2,-3,-13,-3,10,2462,1182,2883,7804]);
// Polycyclic
G:=Group<a,b|a^234=1,b^2=a^117,b*a*b^-1=a^-1>;
// generators/relations
Export