metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic112, C32.D7, C7⋊1Q64, C8.7D28, C4.3D56, C224.1C2, C28.28D8, C2.5D112, C14.3D16, C56.57D4, C16.15D14, Dic56.1C2, C112.16C22, SmallGroup(448,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic112
G = < a,b | a224=1, b2=a112, bab-1=a-1 >
Smallest permutation representation of Dic112
►Regular action on 448 points
Generators in S448
(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)
(1 413 113 301)(2 412 114 300)(3 411 115 299)(4 410 116 298)(5 409 117 297)(6 408 118 296)(7 407 119 295)(8 406 120 294)(9 405 121 293)(10 404 122 292)(11 403 123 291)(12 402 124 290)(13 401 125 289)(14 400 126 288)(15 399 127 287)(16 398 128 286)(17 397 129 285)(18 396 130 284)(19 395 131 283)(20 394 132 282)(21 393 133 281)(22 392 134 280)(23 391 135 279)(24 390 136 278)(25 389 137 277)(26 388 138 276)(27 387 139 275)(28 386 140 274)(29 385 141 273)(30 384 142 272)(31 383 143 271)(32 382 144 270)(33 381 145 269)(34 380 146 268)(35 379 147 267)(36 378 148 266)(37 377 149 265)(38 376 150 264)(39 375 151 263)(40 374 152 262)(41 373 153 261)(42 372 154 260)(43 371 155 259)(44 370 156 258)(45 369 157 257)(46 368 158 256)(47 367 159 255)(48 366 160 254)(49 365 161 253)(50 364 162 252)(51 363 163 251)(52 362 164 250)(53 361 165 249)(54 360 166 248)(55 359 167 247)(56 358 168 246)(57 357 169 245)(58 356 170 244)(59 355 171 243)(60 354 172 242)(61 353 173 241)(62 352 174 240)(63 351 175 239)(64 350 176 238)(65 349 177 237)(66 348 178 236)(67 347 179 235)(68 346 180 234)(69 345 181 233)(70 344 182 232)(71 343 183 231)(72 342 184 230)(73 341 185 229)(74 340 186 228)(75 339 187 227)(76 338 188 226)(77 337 189 225)(78 336 190 448)(79 335 191 447)(80 334 192 446)(81 333 193 445)(82 332 194 444)(83 331 195 443)(84 330 196 442)(85 329 197 441)(86 328 198 440)(87 327 199 439)(88 326 200 438)(89 325 201 437)(90 324 202 436)(91 323 203 435)(92 322 204 434)(93 321 205 433)(94 320 206 432)(95 319 207 431)(96 318 208 430)(97 317 209 429)(98 316 210 428)(99 315 211 427)(100 314 212 426)(101 313 213 425)(102 312 214 424)(103 311 215 423)(104 310 216 422)(105 309 217 421)(106 308 218 420)(107 307 219 419)(108 306 220 418)(109 305 221 417)(110 304 222 416)(111 303 223 415)(112 302 224 414)
G:=sub<Sym(448)| (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), (1,413,113,301)(2,412,114,300)(3,411,115,299)(4,410,116,298)(5,409,117,297)(6,408,118,296)(7,407,119,295)(8,406,120,294)(9,405,121,293)(10,404,122,292)(11,403,123,291)(12,402,124,290)(13,401,125,289)(14,400,126,288)(15,399,127,287)(16,398,128,286)(17,397,129,285)(18,396,130,284)(19,395,131,283)(20,394,132,282)(21,393,133,281)(22,392,134,280)(23,391,135,279)(24,390,136,278)(25,389,137,277)(26,388,138,276)(27,387,139,275)(28,386,140,274)(29,385,141,273)(30,384,142,272)(31,383,143,271)(32,382,144,270)(33,381,145,269)(34,380,146,268)(35,379,147,267)(36,378,148,266)(37,377,149,265)(38,376,150,264)(39,375,151,263)(40,374,152,262)(41,373,153,261)(42,372,154,260)(43,371,155,259)(44,370,156,258)(45,369,157,257)(46,368,158,256)(47,367,159,255)(48,366,160,254)(49,365,161,253)(50,364,162,252)(51,363,163,251)(52,362,164,250)(53,361,165,249)(54,360,166,248)(55,359,167,247)(56,358,168,246)(57,357,169,245)(58,356,170,244)(59,355,171,243)(60,354,172,242)(61,353,173,241)(62,352,174,240)(63,351,175,239)(64,350,176,238)(65,349,177,237)(66,348,178,236)(67,347,179,235)(68,346,180,234)(69,345,181,233)(70,344,182,232)(71,343,183,231)(72,342,184,230)(73,341,185,229)(74,340,186,228)(75,339,187,227)(76,338,188,226)(77,337,189,225)(78,336,190,448)(79,335,191,447)(80,334,192,446)(81,333,193,445)(82,332,194,444)(83,331,195,443)(84,330,196,442)(85,329,197,441)(86,328,198,440)(87,327,199,439)(88,326,200,438)(89,325,201,437)(90,324,202,436)(91,323,203,435)(92,322,204,434)(93,321,205,433)(94,320,206,432)(95,319,207,431)(96,318,208,430)(97,317,209,429)(98,316,210,428)(99,315,211,427)(100,314,212,426)(101,313,213,425)(102,312,214,424)(103,311,215,423)(104,310,216,422)(105,309,217,421)(106,308,218,420)(107,307,219,419)(108,306,220,418)(109,305,221,417)(110,304,222,416)(111,303,223,415)(112,302,224,414)>;
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), (1,413,113,301)(2,412,114,300)(3,411,115,299)(4,410,116,298)(5,409,117,297)(6,408,118,296)(7,407,119,295)(8,406,120,294)(9,405,121,293)(10,404,122,292)(11,403,123,291)(12,402,124,290)(13,401,125,289)(14,400,126,288)(15,399,127,287)(16,398,128,286)(17,397,129,285)(18,396,130,284)(19,395,131,283)(20,394,132,282)(21,393,133,281)(22,392,134,280)(23,391,135,279)(24,390,136,278)(25,389,137,277)(26,388,138,276)(27,387,139,275)(28,386,140,274)(29,385,141,273)(30,384,142,272)(31,383,143,271)(32,382,144,270)(33,381,145,269)(34,380,146,268)(35,379,147,267)(36,378,148,266)(37,377,149,265)(38,376,150,264)(39,375,151,263)(40,374,152,262)(41,373,153,261)(42,372,154,260)(43,371,155,259)(44,370,156,258)(45,369,157,257)(46,368,158,256)(47,367,159,255)(48,366,160,254)(49,365,161,253)(50,364,162,252)(51,363,163,251)(52,362,164,250)(53,361,165,249)(54,360,166,248)(55,359,167,247)(56,358,168,246)(57,357,169,245)(58,356,170,244)(59,355,171,243)(60,354,172,242)(61,353,173,241)(62,352,174,240)(63,351,175,239)(64,350,176,238)(65,349,177,237)(66,348,178,236)(67,347,179,235)(68,346,180,234)(69,345,181,233)(70,344,182,232)(71,343,183,231)(72,342,184,230)(73,341,185,229)(74,340,186,228)(75,339,187,227)(76,338,188,226)(77,337,189,225)(78,336,190,448)(79,335,191,447)(80,334,192,446)(81,333,193,445)(82,332,194,444)(83,331,195,443)(84,330,196,442)(85,329,197,441)(86,328,198,440)(87,327,199,439)(88,326,200,438)(89,325,201,437)(90,324,202,436)(91,323,203,435)(92,322,204,434)(93,321,205,433)(94,320,206,432)(95,319,207,431)(96,318,208,430)(97,317,209,429)(98,316,210,428)(99,315,211,427)(100,314,212,426)(101,313,213,425)(102,312,214,424)(103,311,215,423)(104,310,216,422)(105,309,217,421)(106,308,218,420)(107,307,219,419)(108,306,220,418)(109,305,221,417)(110,304,222,416)(111,303,223,415)(112,302,224,414) );
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)], [(1,413,113,301),(2,412,114,300),(3,411,115,299),(4,410,116,298),(5,409,117,297),(6,408,118,296),(7,407,119,295),(8,406,120,294),(9,405,121,293),(10,404,122,292),(11,403,123,291),(12,402,124,290),(13,401,125,289),(14,400,126,288),(15,399,127,287),(16,398,128,286),(17,397,129,285),(18,396,130,284),(19,395,131,283),(20,394,132,282),(21,393,133,281),(22,392,134,280),(23,391,135,279),(24,390,136,278),(25,389,137,277),(26,388,138,276),(27,387,139,275),(28,386,140,274),(29,385,141,273),(30,384,142,272),(31,383,143,271),(32,382,144,270),(33,381,145,269),(34,380,146,268),(35,379,147,267),(36,378,148,266),(37,377,149,265),(38,376,150,264),(39,375,151,263),(40,374,152,262),(41,373,153,261),(42,372,154,260),(43,371,155,259),(44,370,156,258),(45,369,157,257),(46,368,158,256),(47,367,159,255),(48,366,160,254),(49,365,161,253),(50,364,162,252),(51,363,163,251),(52,362,164,250),(53,361,165,249),(54,360,166,248),(55,359,167,247),(56,358,168,246),(57,357,169,245),(58,356,170,244),(59,355,171,243),(60,354,172,242),(61,353,173,241),(62,352,174,240),(63,351,175,239),(64,350,176,238),(65,349,177,237),(66,348,178,236),(67,347,179,235),(68,346,180,234),(69,345,181,233),(70,344,182,232),(71,343,183,231),(72,342,184,230),(73,341,185,229),(74,340,186,228),(75,339,187,227),(76,338,188,226),(77,337,189,225),(78,336,190,448),(79,335,191,447),(80,334,192,446),(81,333,193,445),(82,332,194,444),(83,331,195,443),(84,330,196,442),(85,329,197,441),(86,328,198,440),(87,327,199,439),(88,326,200,438),(89,325,201,437),(90,324,202,436),(91,323,203,435),(92,322,204,434),(93,321,205,433),(94,320,206,432),(95,319,207,431),(96,318,208,430),(97,317,209,429),(98,316,210,428),(99,315,211,427),(100,314,212,426),(101,313,213,425),(102,312,214,424),(103,311,215,423),(104,310,216,422),(105,309,217,421),(106,308,218,420),(107,307,219,419),(108,306,220,418),(109,305,221,417),(110,304,222,416),(111,303,223,415),(112,302,224,414)]])
115 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 32A | ··· | 32H | 56A | ··· | 56L | 112A | ··· | 112X | 224A | ··· | 224AV |
| order | 1 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 | 224 | ··· | 224 |
| size | 1 | 1 | 2 | 112 | 112 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
115 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | D4 | D7 | D8 | D14 | D16 | D28 | Q64 | D56 | D112 | Dic112 |
| kernel | Dic112 | C224 | Dic56 | C56 | C32 | C28 | C16 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 3 | 2 | 3 | 4 | 6 | 8 | 12 | 24 | 48 |
Matrix representation of Dic112 ►in GL2(𝔽449) generated by
| 163 | 44 |
| 265 | 347 |
| 260 | 356 |
| 331 | 189 |
G:=sub<GL(2,GF(449))| [163,265,44,347],[260,331,356,189] >;
Dic112 in GAP, Magma, Sage, TeX
{\rm Dic}_{112} % in TeX
G:=Group("Dic112"); // GroupNames label
G:=SmallGroup(448,7);
// by ID
G=gap.SmallGroup(448,7);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,85,92,254,142,675,192,1684,102,18822]);
// Polycyclic
G:=Group<a,b|a^224=1,b^2=a^112,b*a*b^-1=a^-1>;
// generators/relations
Export