metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic110, C4.D55, C55⋊2Q8, C44.1D5, C5⋊2Dic22, C220.1C2, C20.1D11, C10.8D22, C2.3D110, C22.8D10, C11⋊2Dic10, C110.8C22, Dic55.1C2, SmallGroup(440,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic110
G = < a,b | a220=1, b2=a110, bab-1=a-1 >
Smallest permutation representation of Dic110
►Regular action on 440 points
Generators in S440
(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)
(1 221 111 331)(2 440 112 330)(3 439 113 329)(4 438 114 328)(5 437 115 327)(6 436 116 326)(7 435 117 325)(8 434 118 324)(9 433 119 323)(10 432 120 322)(11 431 121 321)(12 430 122 320)(13 429 123 319)(14 428 124 318)(15 427 125 317)(16 426 126 316)(17 425 127 315)(18 424 128 314)(19 423 129 313)(20 422 130 312)(21 421 131 311)(22 420 132 310)(23 419 133 309)(24 418 134 308)(25 417 135 307)(26 416 136 306)(27 415 137 305)(28 414 138 304)(29 413 139 303)(30 412 140 302)(31 411 141 301)(32 410 142 300)(33 409 143 299)(34 408 144 298)(35 407 145 297)(36 406 146 296)(37 405 147 295)(38 404 148 294)(39 403 149 293)(40 402 150 292)(41 401 151 291)(42 400 152 290)(43 399 153 289)(44 398 154 288)(45 397 155 287)(46 396 156 286)(47 395 157 285)(48 394 158 284)(49 393 159 283)(50 392 160 282)(51 391 161 281)(52 390 162 280)(53 389 163 279)(54 388 164 278)(55 387 165 277)(56 386 166 276)(57 385 167 275)(58 384 168 274)(59 383 169 273)(60 382 170 272)(61 381 171 271)(62 380 172 270)(63 379 173 269)(64 378 174 268)(65 377 175 267)(66 376 176 266)(67 375 177 265)(68 374 178 264)(69 373 179 263)(70 372 180 262)(71 371 181 261)(72 370 182 260)(73 369 183 259)(74 368 184 258)(75 367 185 257)(76 366 186 256)(77 365 187 255)(78 364 188 254)(79 363 189 253)(80 362 190 252)(81 361 191 251)(82 360 192 250)(83 359 193 249)(84 358 194 248)(85 357 195 247)(86 356 196 246)(87 355 197 245)(88 354 198 244)(89 353 199 243)(90 352 200 242)(91 351 201 241)(92 350 202 240)(93 349 203 239)(94 348 204 238)(95 347 205 237)(96 346 206 236)(97 345 207 235)(98 344 208 234)(99 343 209 233)(100 342 210 232)(101 341 211 231)(102 340 212 230)(103 339 213 229)(104 338 214 228)(105 337 215 227)(106 336 216 226)(107 335 217 225)(108 334 218 224)(109 333 219 223)(110 332 220 222)
G:=sub<Sym(440)| (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), (1,221,111,331)(2,440,112,330)(3,439,113,329)(4,438,114,328)(5,437,115,327)(6,436,116,326)(7,435,117,325)(8,434,118,324)(9,433,119,323)(10,432,120,322)(11,431,121,321)(12,430,122,320)(13,429,123,319)(14,428,124,318)(15,427,125,317)(16,426,126,316)(17,425,127,315)(18,424,128,314)(19,423,129,313)(20,422,130,312)(21,421,131,311)(22,420,132,310)(23,419,133,309)(24,418,134,308)(25,417,135,307)(26,416,136,306)(27,415,137,305)(28,414,138,304)(29,413,139,303)(30,412,140,302)(31,411,141,301)(32,410,142,300)(33,409,143,299)(34,408,144,298)(35,407,145,297)(36,406,146,296)(37,405,147,295)(38,404,148,294)(39,403,149,293)(40,402,150,292)(41,401,151,291)(42,400,152,290)(43,399,153,289)(44,398,154,288)(45,397,155,287)(46,396,156,286)(47,395,157,285)(48,394,158,284)(49,393,159,283)(50,392,160,282)(51,391,161,281)(52,390,162,280)(53,389,163,279)(54,388,164,278)(55,387,165,277)(56,386,166,276)(57,385,167,275)(58,384,168,274)(59,383,169,273)(60,382,170,272)(61,381,171,271)(62,380,172,270)(63,379,173,269)(64,378,174,268)(65,377,175,267)(66,376,176,266)(67,375,177,265)(68,374,178,264)(69,373,179,263)(70,372,180,262)(71,371,181,261)(72,370,182,260)(73,369,183,259)(74,368,184,258)(75,367,185,257)(76,366,186,256)(77,365,187,255)(78,364,188,254)(79,363,189,253)(80,362,190,252)(81,361,191,251)(82,360,192,250)(83,359,193,249)(84,358,194,248)(85,357,195,247)(86,356,196,246)(87,355,197,245)(88,354,198,244)(89,353,199,243)(90,352,200,242)(91,351,201,241)(92,350,202,240)(93,349,203,239)(94,348,204,238)(95,347,205,237)(96,346,206,236)(97,345,207,235)(98,344,208,234)(99,343,209,233)(100,342,210,232)(101,341,211,231)(102,340,212,230)(103,339,213,229)(104,338,214,228)(105,337,215,227)(106,336,216,226)(107,335,217,225)(108,334,218,224)(109,333,219,223)(110,332,220,222)>;
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), (1,221,111,331)(2,440,112,330)(3,439,113,329)(4,438,114,328)(5,437,115,327)(6,436,116,326)(7,435,117,325)(8,434,118,324)(9,433,119,323)(10,432,120,322)(11,431,121,321)(12,430,122,320)(13,429,123,319)(14,428,124,318)(15,427,125,317)(16,426,126,316)(17,425,127,315)(18,424,128,314)(19,423,129,313)(20,422,130,312)(21,421,131,311)(22,420,132,310)(23,419,133,309)(24,418,134,308)(25,417,135,307)(26,416,136,306)(27,415,137,305)(28,414,138,304)(29,413,139,303)(30,412,140,302)(31,411,141,301)(32,410,142,300)(33,409,143,299)(34,408,144,298)(35,407,145,297)(36,406,146,296)(37,405,147,295)(38,404,148,294)(39,403,149,293)(40,402,150,292)(41,401,151,291)(42,400,152,290)(43,399,153,289)(44,398,154,288)(45,397,155,287)(46,396,156,286)(47,395,157,285)(48,394,158,284)(49,393,159,283)(50,392,160,282)(51,391,161,281)(52,390,162,280)(53,389,163,279)(54,388,164,278)(55,387,165,277)(56,386,166,276)(57,385,167,275)(58,384,168,274)(59,383,169,273)(60,382,170,272)(61,381,171,271)(62,380,172,270)(63,379,173,269)(64,378,174,268)(65,377,175,267)(66,376,176,266)(67,375,177,265)(68,374,178,264)(69,373,179,263)(70,372,180,262)(71,371,181,261)(72,370,182,260)(73,369,183,259)(74,368,184,258)(75,367,185,257)(76,366,186,256)(77,365,187,255)(78,364,188,254)(79,363,189,253)(80,362,190,252)(81,361,191,251)(82,360,192,250)(83,359,193,249)(84,358,194,248)(85,357,195,247)(86,356,196,246)(87,355,197,245)(88,354,198,244)(89,353,199,243)(90,352,200,242)(91,351,201,241)(92,350,202,240)(93,349,203,239)(94,348,204,238)(95,347,205,237)(96,346,206,236)(97,345,207,235)(98,344,208,234)(99,343,209,233)(100,342,210,232)(101,341,211,231)(102,340,212,230)(103,339,213,229)(104,338,214,228)(105,337,215,227)(106,336,216,226)(107,335,217,225)(108,334,218,224)(109,333,219,223)(110,332,220,222) );
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)], [(1,221,111,331),(2,440,112,330),(3,439,113,329),(4,438,114,328),(5,437,115,327),(6,436,116,326),(7,435,117,325),(8,434,118,324),(9,433,119,323),(10,432,120,322),(11,431,121,321),(12,430,122,320),(13,429,123,319),(14,428,124,318),(15,427,125,317),(16,426,126,316),(17,425,127,315),(18,424,128,314),(19,423,129,313),(20,422,130,312),(21,421,131,311),(22,420,132,310),(23,419,133,309),(24,418,134,308),(25,417,135,307),(26,416,136,306),(27,415,137,305),(28,414,138,304),(29,413,139,303),(30,412,140,302),(31,411,141,301),(32,410,142,300),(33,409,143,299),(34,408,144,298),(35,407,145,297),(36,406,146,296),(37,405,147,295),(38,404,148,294),(39,403,149,293),(40,402,150,292),(41,401,151,291),(42,400,152,290),(43,399,153,289),(44,398,154,288),(45,397,155,287),(46,396,156,286),(47,395,157,285),(48,394,158,284),(49,393,159,283),(50,392,160,282),(51,391,161,281),(52,390,162,280),(53,389,163,279),(54,388,164,278),(55,387,165,277),(56,386,166,276),(57,385,167,275),(58,384,168,274),(59,383,169,273),(60,382,170,272),(61,381,171,271),(62,380,172,270),(63,379,173,269),(64,378,174,268),(65,377,175,267),(66,376,176,266),(67,375,177,265),(68,374,178,264),(69,373,179,263),(70,372,180,262),(71,371,181,261),(72,370,182,260),(73,369,183,259),(74,368,184,258),(75,367,185,257),(76,366,186,256),(77,365,187,255),(78,364,188,254),(79,363,189,253),(80,362,190,252),(81,361,191,251),(82,360,192,250),(83,359,193,249),(84,358,194,248),(85,357,195,247),(86,356,196,246),(87,355,197,245),(88,354,198,244),(89,353,199,243),(90,352,200,242),(91,351,201,241),(92,350,202,240),(93,349,203,239),(94,348,204,238),(95,347,205,237),(96,346,206,236),(97,345,207,235),(98,344,208,234),(99,343,209,233),(100,342,210,232),(101,341,211,231),(102,340,212,230),(103,339,213,229),(104,338,214,228),(105,337,215,227),(106,336,216,226),(107,335,217,225),(108,334,218,224),(109,333,219,223),(110,332,220,222)]])
113 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 10A | 10B | 11A | ··· | 11E | 20A | 20B | 20C | 20D | 22A | ··· | 22E | 44A | ··· | 44J | 55A | ··· | 55T | 110A | ··· | 110T | 220A | ··· | 220AN |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 10 | 10 | 11 | ··· | 11 | 20 | 20 | 20 | 20 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 220 | ··· | 220 |
| size | 1 | 1 | 2 | 110 | 110 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
113 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | + | - | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D5 | D10 | D11 | Dic10 | D22 | Dic22 | D55 | D110 | Dic110 |
| kernel | Dic110 | Dic55 | C220 | C55 | C44 | C22 | C20 | C11 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 5 | 4 | 5 | 10 | 20 | 20 | 40 |
Matrix representation of Dic110 ►in GL2(𝔽661) generated by
| 15 | 381 |
| 508 | 168 |
| 350 | 431 |
| 501 | 311 |
G:=sub<GL(2,GF(661))| [15,508,381,168],[350,501,431,311] >;
Dic110 in GAP, Magma, Sage, TeX
{\rm Dic}_{110} % in TeX
G:=Group("Dic110"); // GroupNames label
G:=SmallGroup(440,34);
// by ID
G=gap.SmallGroup(440,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,20,61,26,643,10004]);
// Polycyclic
G:=Group<a,b|a^220=1,b^2=a^110,b*a*b^-1=a^-1>;
// generators/relations
Export