metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic111, C74.S3, C6.D37, C3⋊Dic37, C111⋊3C4, C2.D111, C37⋊2Dic3, C222.1C2, SmallGroup(444,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C111 — Dic111 |
Generators and relations for Dic111
G = < a,b | a222=1, b2=a111, bab-1=a-1 >
Smallest permutation representation of Dic111
►Regular action on 444 points
Generators in S444
(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)
(1 391 112 280)(2 390 113 279)(3 389 114 278)(4 388 115 277)(5 387 116 276)(6 386 117 275)(7 385 118 274)(8 384 119 273)(9 383 120 272)(10 382 121 271)(11 381 122 270)(12 380 123 269)(13 379 124 268)(14 378 125 267)(15 377 126 266)(16 376 127 265)(17 375 128 264)(18 374 129 263)(19 373 130 262)(20 372 131 261)(21 371 132 260)(22 370 133 259)(23 369 134 258)(24 368 135 257)(25 367 136 256)(26 366 137 255)(27 365 138 254)(28 364 139 253)(29 363 140 252)(30 362 141 251)(31 361 142 250)(32 360 143 249)(33 359 144 248)(34 358 145 247)(35 357 146 246)(36 356 147 245)(37 355 148 244)(38 354 149 243)(39 353 150 242)(40 352 151 241)(41 351 152 240)(42 350 153 239)(43 349 154 238)(44 348 155 237)(45 347 156 236)(46 346 157 235)(47 345 158 234)(48 344 159 233)(49 343 160 232)(50 342 161 231)(51 341 162 230)(52 340 163 229)(53 339 164 228)(54 338 165 227)(55 337 166 226)(56 336 167 225)(57 335 168 224)(58 334 169 223)(59 333 170 444)(60 332 171 443)(61 331 172 442)(62 330 173 441)(63 329 174 440)(64 328 175 439)(65 327 176 438)(66 326 177 437)(67 325 178 436)(68 324 179 435)(69 323 180 434)(70 322 181 433)(71 321 182 432)(72 320 183 431)(73 319 184 430)(74 318 185 429)(75 317 186 428)(76 316 187 427)(77 315 188 426)(78 314 189 425)(79 313 190 424)(80 312 191 423)(81 311 192 422)(82 310 193 421)(83 309 194 420)(84 308 195 419)(85 307 196 418)(86 306 197 417)(87 305 198 416)(88 304 199 415)(89 303 200 414)(90 302 201 413)(91 301 202 412)(92 300 203 411)(93 299 204 410)(94 298 205 409)(95 297 206 408)(96 296 207 407)(97 295 208 406)(98 294 209 405)(99 293 210 404)(100 292 211 403)(101 291 212 402)(102 290 213 401)(103 289 214 400)(104 288 215 399)(105 287 216 398)(106 286 217 397)(107 285 218 396)(108 284 219 395)(109 283 220 394)(110 282 221 393)(111 281 222 392)
G:=sub<Sym(444)| (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), (1,391,112,280)(2,390,113,279)(3,389,114,278)(4,388,115,277)(5,387,116,276)(6,386,117,275)(7,385,118,274)(8,384,119,273)(9,383,120,272)(10,382,121,271)(11,381,122,270)(12,380,123,269)(13,379,124,268)(14,378,125,267)(15,377,126,266)(16,376,127,265)(17,375,128,264)(18,374,129,263)(19,373,130,262)(20,372,131,261)(21,371,132,260)(22,370,133,259)(23,369,134,258)(24,368,135,257)(25,367,136,256)(26,366,137,255)(27,365,138,254)(28,364,139,253)(29,363,140,252)(30,362,141,251)(31,361,142,250)(32,360,143,249)(33,359,144,248)(34,358,145,247)(35,357,146,246)(36,356,147,245)(37,355,148,244)(38,354,149,243)(39,353,150,242)(40,352,151,241)(41,351,152,240)(42,350,153,239)(43,349,154,238)(44,348,155,237)(45,347,156,236)(46,346,157,235)(47,345,158,234)(48,344,159,233)(49,343,160,232)(50,342,161,231)(51,341,162,230)(52,340,163,229)(53,339,164,228)(54,338,165,227)(55,337,166,226)(56,336,167,225)(57,335,168,224)(58,334,169,223)(59,333,170,444)(60,332,171,443)(61,331,172,442)(62,330,173,441)(63,329,174,440)(64,328,175,439)(65,327,176,438)(66,326,177,437)(67,325,178,436)(68,324,179,435)(69,323,180,434)(70,322,181,433)(71,321,182,432)(72,320,183,431)(73,319,184,430)(74,318,185,429)(75,317,186,428)(76,316,187,427)(77,315,188,426)(78,314,189,425)(79,313,190,424)(80,312,191,423)(81,311,192,422)(82,310,193,421)(83,309,194,420)(84,308,195,419)(85,307,196,418)(86,306,197,417)(87,305,198,416)(88,304,199,415)(89,303,200,414)(90,302,201,413)(91,301,202,412)(92,300,203,411)(93,299,204,410)(94,298,205,409)(95,297,206,408)(96,296,207,407)(97,295,208,406)(98,294,209,405)(99,293,210,404)(100,292,211,403)(101,291,212,402)(102,290,213,401)(103,289,214,400)(104,288,215,399)(105,287,216,398)(106,286,217,397)(107,285,218,396)(108,284,219,395)(109,283,220,394)(110,282,221,393)(111,281,222,392)>;
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), (1,391,112,280)(2,390,113,279)(3,389,114,278)(4,388,115,277)(5,387,116,276)(6,386,117,275)(7,385,118,274)(8,384,119,273)(9,383,120,272)(10,382,121,271)(11,381,122,270)(12,380,123,269)(13,379,124,268)(14,378,125,267)(15,377,126,266)(16,376,127,265)(17,375,128,264)(18,374,129,263)(19,373,130,262)(20,372,131,261)(21,371,132,260)(22,370,133,259)(23,369,134,258)(24,368,135,257)(25,367,136,256)(26,366,137,255)(27,365,138,254)(28,364,139,253)(29,363,140,252)(30,362,141,251)(31,361,142,250)(32,360,143,249)(33,359,144,248)(34,358,145,247)(35,357,146,246)(36,356,147,245)(37,355,148,244)(38,354,149,243)(39,353,150,242)(40,352,151,241)(41,351,152,240)(42,350,153,239)(43,349,154,238)(44,348,155,237)(45,347,156,236)(46,346,157,235)(47,345,158,234)(48,344,159,233)(49,343,160,232)(50,342,161,231)(51,341,162,230)(52,340,163,229)(53,339,164,228)(54,338,165,227)(55,337,166,226)(56,336,167,225)(57,335,168,224)(58,334,169,223)(59,333,170,444)(60,332,171,443)(61,331,172,442)(62,330,173,441)(63,329,174,440)(64,328,175,439)(65,327,176,438)(66,326,177,437)(67,325,178,436)(68,324,179,435)(69,323,180,434)(70,322,181,433)(71,321,182,432)(72,320,183,431)(73,319,184,430)(74,318,185,429)(75,317,186,428)(76,316,187,427)(77,315,188,426)(78,314,189,425)(79,313,190,424)(80,312,191,423)(81,311,192,422)(82,310,193,421)(83,309,194,420)(84,308,195,419)(85,307,196,418)(86,306,197,417)(87,305,198,416)(88,304,199,415)(89,303,200,414)(90,302,201,413)(91,301,202,412)(92,300,203,411)(93,299,204,410)(94,298,205,409)(95,297,206,408)(96,296,207,407)(97,295,208,406)(98,294,209,405)(99,293,210,404)(100,292,211,403)(101,291,212,402)(102,290,213,401)(103,289,214,400)(104,288,215,399)(105,287,216,398)(106,286,217,397)(107,285,218,396)(108,284,219,395)(109,283,220,394)(110,282,221,393)(111,281,222,392) );
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)], [(1,391,112,280),(2,390,113,279),(3,389,114,278),(4,388,115,277),(5,387,116,276),(6,386,117,275),(7,385,118,274),(8,384,119,273),(9,383,120,272),(10,382,121,271),(11,381,122,270),(12,380,123,269),(13,379,124,268),(14,378,125,267),(15,377,126,266),(16,376,127,265),(17,375,128,264),(18,374,129,263),(19,373,130,262),(20,372,131,261),(21,371,132,260),(22,370,133,259),(23,369,134,258),(24,368,135,257),(25,367,136,256),(26,366,137,255),(27,365,138,254),(28,364,139,253),(29,363,140,252),(30,362,141,251),(31,361,142,250),(32,360,143,249),(33,359,144,248),(34,358,145,247),(35,357,146,246),(36,356,147,245),(37,355,148,244),(38,354,149,243),(39,353,150,242),(40,352,151,241),(41,351,152,240),(42,350,153,239),(43,349,154,238),(44,348,155,237),(45,347,156,236),(46,346,157,235),(47,345,158,234),(48,344,159,233),(49,343,160,232),(50,342,161,231),(51,341,162,230),(52,340,163,229),(53,339,164,228),(54,338,165,227),(55,337,166,226),(56,336,167,225),(57,335,168,224),(58,334,169,223),(59,333,170,444),(60,332,171,443),(61,331,172,442),(62,330,173,441),(63,329,174,440),(64,328,175,439),(65,327,176,438),(66,326,177,437),(67,325,178,436),(68,324,179,435),(69,323,180,434),(70,322,181,433),(71,321,182,432),(72,320,183,431),(73,319,184,430),(74,318,185,429),(75,317,186,428),(76,316,187,427),(77,315,188,426),(78,314,189,425),(79,313,190,424),(80,312,191,423),(81,311,192,422),(82,310,193,421),(83,309,194,420),(84,308,195,419),(85,307,196,418),(86,306,197,417),(87,305,198,416),(88,304,199,415),(89,303,200,414),(90,302,201,413),(91,301,202,412),(92,300,203,411),(93,299,204,410),(94,298,205,409),(95,297,206,408),(96,296,207,407),(97,295,208,406),(98,294,209,405),(99,293,210,404),(100,292,211,403),(101,291,212,402),(102,290,213,401),(103,289,214,400),(104,288,215,399),(105,287,216,398),(106,286,217,397),(107,285,218,396),(108,284,219,395),(109,283,220,394),(110,282,221,393),(111,281,222,392)]])
114 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 37A | ··· | 37R | 74A | ··· | 74R | 111A | ··· | 111AJ | 222A | ··· | 222AJ |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 37 | ··· | 37 | 74 | ··· | 74 | 111 | ··· | 111 | 222 | ··· | 222 |
| size | 1 | 1 | 2 | 111 | 111 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D37 | Dic37 | D111 | Dic111 |
| kernel | Dic111 | C222 | C111 | C74 | C37 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 18 | 18 | 36 | 36 |
Matrix representation of Dic111 ►in GL2(𝔽1777) generated by
| 983 | 1316 |
| 461 | 1700 |
| 102 | 959 |
| 1579 | 1675 |
G:=sub<GL(2,GF(1777))| [983,461,1316,1700],[102,1579,959,1675] >;
Dic111 in GAP, Magma, Sage, TeX
{\rm Dic}_{111} % in TeX
G:=Group("Dic111"); // GroupNames label
G:=SmallGroup(444,5);
// by ID
G=gap.SmallGroup(444,5);
# by ID
G:=PCGroup([4,-2,-2,-3,-37,8,98,6915]);
// Polycyclic
G:=Group<a,b|a^222=1,b^2=a^111,b*a*b^-1=a^-1>;
// generators/relations
Export