metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic109, C109⋊2C4, C218.C2, C2.D109, SmallGroup(436,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C109 — Dic109 |
Generators and relations for Dic109
G = < a,b | a218=1, b2=a109, bab-1=a-1 >
Smallest permutation representation of Dic109
►Regular action on 436 points
Generators in S436
(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)
(1 222 110 331)(2 221 111 330)(3 220 112 329)(4 219 113 328)(5 436 114 327)(6 435 115 326)(7 434 116 325)(8 433 117 324)(9 432 118 323)(10 431 119 322)(11 430 120 321)(12 429 121 320)(13 428 122 319)(14 427 123 318)(15 426 124 317)(16 425 125 316)(17 424 126 315)(18 423 127 314)(19 422 128 313)(20 421 129 312)(21 420 130 311)(22 419 131 310)(23 418 132 309)(24 417 133 308)(25 416 134 307)(26 415 135 306)(27 414 136 305)(28 413 137 304)(29 412 138 303)(30 411 139 302)(31 410 140 301)(32 409 141 300)(33 408 142 299)(34 407 143 298)(35 406 144 297)(36 405 145 296)(37 404 146 295)(38 403 147 294)(39 402 148 293)(40 401 149 292)(41 400 150 291)(42 399 151 290)(43 398 152 289)(44 397 153 288)(45 396 154 287)(46 395 155 286)(47 394 156 285)(48 393 157 284)(49 392 158 283)(50 391 159 282)(51 390 160 281)(52 389 161 280)(53 388 162 279)(54 387 163 278)(55 386 164 277)(56 385 165 276)(57 384 166 275)(58 383 167 274)(59 382 168 273)(60 381 169 272)(61 380 170 271)(62 379 171 270)(63 378 172 269)(64 377 173 268)(65 376 174 267)(66 375 175 266)(67 374 176 265)(68 373 177 264)(69 372 178 263)(70 371 179 262)(71 370 180 261)(72 369 181 260)(73 368 182 259)(74 367 183 258)(75 366 184 257)(76 365 185 256)(77 364 186 255)(78 363 187 254)(79 362 188 253)(80 361 189 252)(81 360 190 251)(82 359 191 250)(83 358 192 249)(84 357 193 248)(85 356 194 247)(86 355 195 246)(87 354 196 245)(88 353 197 244)(89 352 198 243)(90 351 199 242)(91 350 200 241)(92 349 201 240)(93 348 202 239)(94 347 203 238)(95 346 204 237)(96 345 205 236)(97 344 206 235)(98 343 207 234)(99 342 208 233)(100 341 209 232)(101 340 210 231)(102 339 211 230)(103 338 212 229)(104 337 213 228)(105 336 214 227)(106 335 215 226)(107 334 216 225)(108 333 217 224)(109 332 218 223)
G:=sub<Sym(436)| (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), (1,222,110,331)(2,221,111,330)(3,220,112,329)(4,219,113,328)(5,436,114,327)(6,435,115,326)(7,434,116,325)(8,433,117,324)(9,432,118,323)(10,431,119,322)(11,430,120,321)(12,429,121,320)(13,428,122,319)(14,427,123,318)(15,426,124,317)(16,425,125,316)(17,424,126,315)(18,423,127,314)(19,422,128,313)(20,421,129,312)(21,420,130,311)(22,419,131,310)(23,418,132,309)(24,417,133,308)(25,416,134,307)(26,415,135,306)(27,414,136,305)(28,413,137,304)(29,412,138,303)(30,411,139,302)(31,410,140,301)(32,409,141,300)(33,408,142,299)(34,407,143,298)(35,406,144,297)(36,405,145,296)(37,404,146,295)(38,403,147,294)(39,402,148,293)(40,401,149,292)(41,400,150,291)(42,399,151,290)(43,398,152,289)(44,397,153,288)(45,396,154,287)(46,395,155,286)(47,394,156,285)(48,393,157,284)(49,392,158,283)(50,391,159,282)(51,390,160,281)(52,389,161,280)(53,388,162,279)(54,387,163,278)(55,386,164,277)(56,385,165,276)(57,384,166,275)(58,383,167,274)(59,382,168,273)(60,381,169,272)(61,380,170,271)(62,379,171,270)(63,378,172,269)(64,377,173,268)(65,376,174,267)(66,375,175,266)(67,374,176,265)(68,373,177,264)(69,372,178,263)(70,371,179,262)(71,370,180,261)(72,369,181,260)(73,368,182,259)(74,367,183,258)(75,366,184,257)(76,365,185,256)(77,364,186,255)(78,363,187,254)(79,362,188,253)(80,361,189,252)(81,360,190,251)(82,359,191,250)(83,358,192,249)(84,357,193,248)(85,356,194,247)(86,355,195,246)(87,354,196,245)(88,353,197,244)(89,352,198,243)(90,351,199,242)(91,350,200,241)(92,349,201,240)(93,348,202,239)(94,347,203,238)(95,346,204,237)(96,345,205,236)(97,344,206,235)(98,343,207,234)(99,342,208,233)(100,341,209,232)(101,340,210,231)(102,339,211,230)(103,338,212,229)(104,337,213,228)(105,336,214,227)(106,335,215,226)(107,334,216,225)(108,333,217,224)(109,332,218,223)>;
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), (1,222,110,331)(2,221,111,330)(3,220,112,329)(4,219,113,328)(5,436,114,327)(6,435,115,326)(7,434,116,325)(8,433,117,324)(9,432,118,323)(10,431,119,322)(11,430,120,321)(12,429,121,320)(13,428,122,319)(14,427,123,318)(15,426,124,317)(16,425,125,316)(17,424,126,315)(18,423,127,314)(19,422,128,313)(20,421,129,312)(21,420,130,311)(22,419,131,310)(23,418,132,309)(24,417,133,308)(25,416,134,307)(26,415,135,306)(27,414,136,305)(28,413,137,304)(29,412,138,303)(30,411,139,302)(31,410,140,301)(32,409,141,300)(33,408,142,299)(34,407,143,298)(35,406,144,297)(36,405,145,296)(37,404,146,295)(38,403,147,294)(39,402,148,293)(40,401,149,292)(41,400,150,291)(42,399,151,290)(43,398,152,289)(44,397,153,288)(45,396,154,287)(46,395,155,286)(47,394,156,285)(48,393,157,284)(49,392,158,283)(50,391,159,282)(51,390,160,281)(52,389,161,280)(53,388,162,279)(54,387,163,278)(55,386,164,277)(56,385,165,276)(57,384,166,275)(58,383,167,274)(59,382,168,273)(60,381,169,272)(61,380,170,271)(62,379,171,270)(63,378,172,269)(64,377,173,268)(65,376,174,267)(66,375,175,266)(67,374,176,265)(68,373,177,264)(69,372,178,263)(70,371,179,262)(71,370,180,261)(72,369,181,260)(73,368,182,259)(74,367,183,258)(75,366,184,257)(76,365,185,256)(77,364,186,255)(78,363,187,254)(79,362,188,253)(80,361,189,252)(81,360,190,251)(82,359,191,250)(83,358,192,249)(84,357,193,248)(85,356,194,247)(86,355,195,246)(87,354,196,245)(88,353,197,244)(89,352,198,243)(90,351,199,242)(91,350,200,241)(92,349,201,240)(93,348,202,239)(94,347,203,238)(95,346,204,237)(96,345,205,236)(97,344,206,235)(98,343,207,234)(99,342,208,233)(100,341,209,232)(101,340,210,231)(102,339,211,230)(103,338,212,229)(104,337,213,228)(105,336,214,227)(106,335,215,226)(107,334,216,225)(108,333,217,224)(109,332,218,223) );
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)], [(1,222,110,331),(2,221,111,330),(3,220,112,329),(4,219,113,328),(5,436,114,327),(6,435,115,326),(7,434,116,325),(8,433,117,324),(9,432,118,323),(10,431,119,322),(11,430,120,321),(12,429,121,320),(13,428,122,319),(14,427,123,318),(15,426,124,317),(16,425,125,316),(17,424,126,315),(18,423,127,314),(19,422,128,313),(20,421,129,312),(21,420,130,311),(22,419,131,310),(23,418,132,309),(24,417,133,308),(25,416,134,307),(26,415,135,306),(27,414,136,305),(28,413,137,304),(29,412,138,303),(30,411,139,302),(31,410,140,301),(32,409,141,300),(33,408,142,299),(34,407,143,298),(35,406,144,297),(36,405,145,296),(37,404,146,295),(38,403,147,294),(39,402,148,293),(40,401,149,292),(41,400,150,291),(42,399,151,290),(43,398,152,289),(44,397,153,288),(45,396,154,287),(46,395,155,286),(47,394,156,285),(48,393,157,284),(49,392,158,283),(50,391,159,282),(51,390,160,281),(52,389,161,280),(53,388,162,279),(54,387,163,278),(55,386,164,277),(56,385,165,276),(57,384,166,275),(58,383,167,274),(59,382,168,273),(60,381,169,272),(61,380,170,271),(62,379,171,270),(63,378,172,269),(64,377,173,268),(65,376,174,267),(66,375,175,266),(67,374,176,265),(68,373,177,264),(69,372,178,263),(70,371,179,262),(71,370,180,261),(72,369,181,260),(73,368,182,259),(74,367,183,258),(75,366,184,257),(76,365,185,256),(77,364,186,255),(78,363,187,254),(79,362,188,253),(80,361,189,252),(81,360,190,251),(82,359,191,250),(83,358,192,249),(84,357,193,248),(85,356,194,247),(86,355,195,246),(87,354,196,245),(88,353,197,244),(89,352,198,243),(90,351,199,242),(91,350,200,241),(92,349,201,240),(93,348,202,239),(94,347,203,238),(95,346,204,237),(96,345,205,236),(97,344,206,235),(98,343,207,234),(99,342,208,233),(100,341,209,232),(101,340,210,231),(102,339,211,230),(103,338,212,229),(104,337,213,228),(105,336,214,227),(106,335,215,226),(107,334,216,225),(108,333,217,224),(109,332,218,223)]])
112 conjugacy classes
| class | 1 | 2 | 4A | 4B | 109A | ··· | 109BB | 218A | ··· | 218BB |
| order | 1 | 2 | 4 | 4 | 109 | ··· | 109 | 218 | ··· | 218 |
| size | 1 | 1 | 109 | 109 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D109 | Dic109 |
| kernel | Dic109 | C218 | C109 | C2 | C1 |
| # reps | 1 | 1 | 2 | 54 | 54 |
Matrix representation of Dic109 ►in GL3(𝔽2617) generated by
| 2616 | 0 | 0 |
| 0 | 1731 | 2616 |
| 0 | 1 | 0 |
| 667 | 0 | 0 |
| 0 | 1768 | 1445 |
| 0 | 2580 | 849 |
G:=sub<GL(3,GF(2617))| [2616,0,0,0,1731,1,0,2616,0],[667,0,0,0,1768,2580,0,1445,849] >;
Dic109 in GAP, Magma, Sage, TeX
{\rm Dic}_{109} % in TeX
G:=Group("Dic109"); // GroupNames label
G:=SmallGroup(436,1);
// by ID
G=gap.SmallGroup(436,1);
# by ID
G:=PCGroup([3,-2,-2,-109,6,3890]);
// Polycyclic
G:=Group<a,b|a^218=1,b^2=a^109,b*a*b^-1=a^-1>;
// generators/relations
Export