metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic104, C16.D13, C13⋊1Q32, C4.3D52, C26.3D8, C208.1C2, C8.15D26, C52.26D4, C2.5D104, Dic52.1C2, C104.16C22, SmallGroup(416,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic104
G = < a,b | a208=1, b2=a104, bab-1=a-1 >
Smallest permutation representation of Dic104
►Regular action on 416 points
Generators in S416
(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)
(1 405 105 301)(2 404 106 300)(3 403 107 299)(4 402 108 298)(5 401 109 297)(6 400 110 296)(7 399 111 295)(8 398 112 294)(9 397 113 293)(10 396 114 292)(11 395 115 291)(12 394 116 290)(13 393 117 289)(14 392 118 288)(15 391 119 287)(16 390 120 286)(17 389 121 285)(18 388 122 284)(19 387 123 283)(20 386 124 282)(21 385 125 281)(22 384 126 280)(23 383 127 279)(24 382 128 278)(25 381 129 277)(26 380 130 276)(27 379 131 275)(28 378 132 274)(29 377 133 273)(30 376 134 272)(31 375 135 271)(32 374 136 270)(33 373 137 269)(34 372 138 268)(35 371 139 267)(36 370 140 266)(37 369 141 265)(38 368 142 264)(39 367 143 263)(40 366 144 262)(41 365 145 261)(42 364 146 260)(43 363 147 259)(44 362 148 258)(45 361 149 257)(46 360 150 256)(47 359 151 255)(48 358 152 254)(49 357 153 253)(50 356 154 252)(51 355 155 251)(52 354 156 250)(53 353 157 249)(54 352 158 248)(55 351 159 247)(56 350 160 246)(57 349 161 245)(58 348 162 244)(59 347 163 243)(60 346 164 242)(61 345 165 241)(62 344 166 240)(63 343 167 239)(64 342 168 238)(65 341 169 237)(66 340 170 236)(67 339 171 235)(68 338 172 234)(69 337 173 233)(70 336 174 232)(71 335 175 231)(72 334 176 230)(73 333 177 229)(74 332 178 228)(75 331 179 227)(76 330 180 226)(77 329 181 225)(78 328 182 224)(79 327 183 223)(80 326 184 222)(81 325 185 221)(82 324 186 220)(83 323 187 219)(84 322 188 218)(85 321 189 217)(86 320 190 216)(87 319 191 215)(88 318 192 214)(89 317 193 213)(90 316 194 212)(91 315 195 211)(92 314 196 210)(93 313 197 209)(94 312 198 416)(95 311 199 415)(96 310 200 414)(97 309 201 413)(98 308 202 412)(99 307 203 411)(100 306 204 410)(101 305 205 409)(102 304 206 408)(103 303 207 407)(104 302 208 406)
G:=sub<Sym(416)| (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), (1,405,105,301)(2,404,106,300)(3,403,107,299)(4,402,108,298)(5,401,109,297)(6,400,110,296)(7,399,111,295)(8,398,112,294)(9,397,113,293)(10,396,114,292)(11,395,115,291)(12,394,116,290)(13,393,117,289)(14,392,118,288)(15,391,119,287)(16,390,120,286)(17,389,121,285)(18,388,122,284)(19,387,123,283)(20,386,124,282)(21,385,125,281)(22,384,126,280)(23,383,127,279)(24,382,128,278)(25,381,129,277)(26,380,130,276)(27,379,131,275)(28,378,132,274)(29,377,133,273)(30,376,134,272)(31,375,135,271)(32,374,136,270)(33,373,137,269)(34,372,138,268)(35,371,139,267)(36,370,140,266)(37,369,141,265)(38,368,142,264)(39,367,143,263)(40,366,144,262)(41,365,145,261)(42,364,146,260)(43,363,147,259)(44,362,148,258)(45,361,149,257)(46,360,150,256)(47,359,151,255)(48,358,152,254)(49,357,153,253)(50,356,154,252)(51,355,155,251)(52,354,156,250)(53,353,157,249)(54,352,158,248)(55,351,159,247)(56,350,160,246)(57,349,161,245)(58,348,162,244)(59,347,163,243)(60,346,164,242)(61,345,165,241)(62,344,166,240)(63,343,167,239)(64,342,168,238)(65,341,169,237)(66,340,170,236)(67,339,171,235)(68,338,172,234)(69,337,173,233)(70,336,174,232)(71,335,175,231)(72,334,176,230)(73,333,177,229)(74,332,178,228)(75,331,179,227)(76,330,180,226)(77,329,181,225)(78,328,182,224)(79,327,183,223)(80,326,184,222)(81,325,185,221)(82,324,186,220)(83,323,187,219)(84,322,188,218)(85,321,189,217)(86,320,190,216)(87,319,191,215)(88,318,192,214)(89,317,193,213)(90,316,194,212)(91,315,195,211)(92,314,196,210)(93,313,197,209)(94,312,198,416)(95,311,199,415)(96,310,200,414)(97,309,201,413)(98,308,202,412)(99,307,203,411)(100,306,204,410)(101,305,205,409)(102,304,206,408)(103,303,207,407)(104,302,208,406)>;
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), (1,405,105,301)(2,404,106,300)(3,403,107,299)(4,402,108,298)(5,401,109,297)(6,400,110,296)(7,399,111,295)(8,398,112,294)(9,397,113,293)(10,396,114,292)(11,395,115,291)(12,394,116,290)(13,393,117,289)(14,392,118,288)(15,391,119,287)(16,390,120,286)(17,389,121,285)(18,388,122,284)(19,387,123,283)(20,386,124,282)(21,385,125,281)(22,384,126,280)(23,383,127,279)(24,382,128,278)(25,381,129,277)(26,380,130,276)(27,379,131,275)(28,378,132,274)(29,377,133,273)(30,376,134,272)(31,375,135,271)(32,374,136,270)(33,373,137,269)(34,372,138,268)(35,371,139,267)(36,370,140,266)(37,369,141,265)(38,368,142,264)(39,367,143,263)(40,366,144,262)(41,365,145,261)(42,364,146,260)(43,363,147,259)(44,362,148,258)(45,361,149,257)(46,360,150,256)(47,359,151,255)(48,358,152,254)(49,357,153,253)(50,356,154,252)(51,355,155,251)(52,354,156,250)(53,353,157,249)(54,352,158,248)(55,351,159,247)(56,350,160,246)(57,349,161,245)(58,348,162,244)(59,347,163,243)(60,346,164,242)(61,345,165,241)(62,344,166,240)(63,343,167,239)(64,342,168,238)(65,341,169,237)(66,340,170,236)(67,339,171,235)(68,338,172,234)(69,337,173,233)(70,336,174,232)(71,335,175,231)(72,334,176,230)(73,333,177,229)(74,332,178,228)(75,331,179,227)(76,330,180,226)(77,329,181,225)(78,328,182,224)(79,327,183,223)(80,326,184,222)(81,325,185,221)(82,324,186,220)(83,323,187,219)(84,322,188,218)(85,321,189,217)(86,320,190,216)(87,319,191,215)(88,318,192,214)(89,317,193,213)(90,316,194,212)(91,315,195,211)(92,314,196,210)(93,313,197,209)(94,312,198,416)(95,311,199,415)(96,310,200,414)(97,309,201,413)(98,308,202,412)(99,307,203,411)(100,306,204,410)(101,305,205,409)(102,304,206,408)(103,303,207,407)(104,302,208,406) );
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)], [(1,405,105,301),(2,404,106,300),(3,403,107,299),(4,402,108,298),(5,401,109,297),(6,400,110,296),(7,399,111,295),(8,398,112,294),(9,397,113,293),(10,396,114,292),(11,395,115,291),(12,394,116,290),(13,393,117,289),(14,392,118,288),(15,391,119,287),(16,390,120,286),(17,389,121,285),(18,388,122,284),(19,387,123,283),(20,386,124,282),(21,385,125,281),(22,384,126,280),(23,383,127,279),(24,382,128,278),(25,381,129,277),(26,380,130,276),(27,379,131,275),(28,378,132,274),(29,377,133,273),(30,376,134,272),(31,375,135,271),(32,374,136,270),(33,373,137,269),(34,372,138,268),(35,371,139,267),(36,370,140,266),(37,369,141,265),(38,368,142,264),(39,367,143,263),(40,366,144,262),(41,365,145,261),(42,364,146,260),(43,363,147,259),(44,362,148,258),(45,361,149,257),(46,360,150,256),(47,359,151,255),(48,358,152,254),(49,357,153,253),(50,356,154,252),(51,355,155,251),(52,354,156,250),(53,353,157,249),(54,352,158,248),(55,351,159,247),(56,350,160,246),(57,349,161,245),(58,348,162,244),(59,347,163,243),(60,346,164,242),(61,345,165,241),(62,344,166,240),(63,343,167,239),(64,342,168,238),(65,341,169,237),(66,340,170,236),(67,339,171,235),(68,338,172,234),(69,337,173,233),(70,336,174,232),(71,335,175,231),(72,334,176,230),(73,333,177,229),(74,332,178,228),(75,331,179,227),(76,330,180,226),(77,329,181,225),(78,328,182,224),(79,327,183,223),(80,326,184,222),(81,325,185,221),(82,324,186,220),(83,323,187,219),(84,322,188,218),(85,321,189,217),(86,320,190,216),(87,319,191,215),(88,318,192,214),(89,317,193,213),(90,316,194,212),(91,315,195,211),(92,314,196,210),(93,313,197,209),(94,312,198,416),(95,311,199,415),(96,310,200,414),(97,309,201,413),(98,308,202,412),(99,307,203,411),(100,306,204,410),(101,305,205,409),(102,304,206,408),(103,303,207,407),(104,302,208,406)]])
107 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 13A | ··· | 13F | 16A | 16B | 16C | 16D | 26A | ··· | 26F | 52A | ··· | 52L | 104A | ··· | 104X | 208A | ··· | 208AV |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 16 | 16 | 16 | 16 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 | 208 | ··· | 208 |
| size | 1 | 1 | 2 | 104 | 104 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
107 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | D4 | D8 | D13 | Q32 | D26 | D52 | D104 | Dic104 |
| kernel | Dic104 | C208 | Dic52 | C52 | C26 | C16 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 6 | 4 | 6 | 12 | 24 | 48 |
Matrix representation of Dic104 ►in GL2(𝔽1249) generated by
| 801 | 304 |
| 945 | 1012 |
| 1052 | 478 |
| 1126 | 197 |
G:=sub<GL(2,GF(1249))| [801,945,304,1012],[1052,1126,478,197] >;
Dic104 in GAP, Magma, Sage, TeX
{\rm Dic}_{104} % in TeX
G:=Group("Dic104"); // GroupNames label
G:=SmallGroup(416,8);
// by ID
G=gap.SmallGroup(416,8);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,73,79,218,122,579,69,13829]);
// Polycyclic
G:=Group<a,b|a^208=1,b^2=a^104,b*a*b^-1=a^-1>;
// generators/relations
Export