metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic102, C4.D51, C51⋊2Q8, C68.1S3, C34.8D6, C6.8D34, C17⋊2Dic6, C3⋊2Dic34, C204.1C2, C12.1D17, C2.3D102, C102.8C22, Dic51.1C2, SmallGroup(408,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic102
G = < a,b | a204=1, b2=a102, bab-1=a-1 >
Smallest permutation representation of Dic102
►Regular action on 408 points
Generators in S408
(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)
(1 217 103 319)(2 216 104 318)(3 215 105 317)(4 214 106 316)(5 213 107 315)(6 212 108 314)(7 211 109 313)(8 210 110 312)(9 209 111 311)(10 208 112 310)(11 207 113 309)(12 206 114 308)(13 205 115 307)(14 408 116 306)(15 407 117 305)(16 406 118 304)(17 405 119 303)(18 404 120 302)(19 403 121 301)(20 402 122 300)(21 401 123 299)(22 400 124 298)(23 399 125 297)(24 398 126 296)(25 397 127 295)(26 396 128 294)(27 395 129 293)(28 394 130 292)(29 393 131 291)(30 392 132 290)(31 391 133 289)(32 390 134 288)(33 389 135 287)(34 388 136 286)(35 387 137 285)(36 386 138 284)(37 385 139 283)(38 384 140 282)(39 383 141 281)(40 382 142 280)(41 381 143 279)(42 380 144 278)(43 379 145 277)(44 378 146 276)(45 377 147 275)(46 376 148 274)(47 375 149 273)(48 374 150 272)(49 373 151 271)(50 372 152 270)(51 371 153 269)(52 370 154 268)(53 369 155 267)(54 368 156 266)(55 367 157 265)(56 366 158 264)(57 365 159 263)(58 364 160 262)(59 363 161 261)(60 362 162 260)(61 361 163 259)(62 360 164 258)(63 359 165 257)(64 358 166 256)(65 357 167 255)(66 356 168 254)(67 355 169 253)(68 354 170 252)(69 353 171 251)(70 352 172 250)(71 351 173 249)(72 350 174 248)(73 349 175 247)(74 348 176 246)(75 347 177 245)(76 346 178 244)(77 345 179 243)(78 344 180 242)(79 343 181 241)(80 342 182 240)(81 341 183 239)(82 340 184 238)(83 339 185 237)(84 338 186 236)(85 337 187 235)(86 336 188 234)(87 335 189 233)(88 334 190 232)(89 333 191 231)(90 332 192 230)(91 331 193 229)(92 330 194 228)(93 329 195 227)(94 328 196 226)(95 327 197 225)(96 326 198 224)(97 325 199 223)(98 324 200 222)(99 323 201 221)(100 322 202 220)(101 321 203 219)(102 320 204 218)
G:=sub<Sym(408)| (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), (1,217,103,319)(2,216,104,318)(3,215,105,317)(4,214,106,316)(5,213,107,315)(6,212,108,314)(7,211,109,313)(8,210,110,312)(9,209,111,311)(10,208,112,310)(11,207,113,309)(12,206,114,308)(13,205,115,307)(14,408,116,306)(15,407,117,305)(16,406,118,304)(17,405,119,303)(18,404,120,302)(19,403,121,301)(20,402,122,300)(21,401,123,299)(22,400,124,298)(23,399,125,297)(24,398,126,296)(25,397,127,295)(26,396,128,294)(27,395,129,293)(28,394,130,292)(29,393,131,291)(30,392,132,290)(31,391,133,289)(32,390,134,288)(33,389,135,287)(34,388,136,286)(35,387,137,285)(36,386,138,284)(37,385,139,283)(38,384,140,282)(39,383,141,281)(40,382,142,280)(41,381,143,279)(42,380,144,278)(43,379,145,277)(44,378,146,276)(45,377,147,275)(46,376,148,274)(47,375,149,273)(48,374,150,272)(49,373,151,271)(50,372,152,270)(51,371,153,269)(52,370,154,268)(53,369,155,267)(54,368,156,266)(55,367,157,265)(56,366,158,264)(57,365,159,263)(58,364,160,262)(59,363,161,261)(60,362,162,260)(61,361,163,259)(62,360,164,258)(63,359,165,257)(64,358,166,256)(65,357,167,255)(66,356,168,254)(67,355,169,253)(68,354,170,252)(69,353,171,251)(70,352,172,250)(71,351,173,249)(72,350,174,248)(73,349,175,247)(74,348,176,246)(75,347,177,245)(76,346,178,244)(77,345,179,243)(78,344,180,242)(79,343,181,241)(80,342,182,240)(81,341,183,239)(82,340,184,238)(83,339,185,237)(84,338,186,236)(85,337,187,235)(86,336,188,234)(87,335,189,233)(88,334,190,232)(89,333,191,231)(90,332,192,230)(91,331,193,229)(92,330,194,228)(93,329,195,227)(94,328,196,226)(95,327,197,225)(96,326,198,224)(97,325,199,223)(98,324,200,222)(99,323,201,221)(100,322,202,220)(101,321,203,219)(102,320,204,218)>;
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), (1,217,103,319)(2,216,104,318)(3,215,105,317)(4,214,106,316)(5,213,107,315)(6,212,108,314)(7,211,109,313)(8,210,110,312)(9,209,111,311)(10,208,112,310)(11,207,113,309)(12,206,114,308)(13,205,115,307)(14,408,116,306)(15,407,117,305)(16,406,118,304)(17,405,119,303)(18,404,120,302)(19,403,121,301)(20,402,122,300)(21,401,123,299)(22,400,124,298)(23,399,125,297)(24,398,126,296)(25,397,127,295)(26,396,128,294)(27,395,129,293)(28,394,130,292)(29,393,131,291)(30,392,132,290)(31,391,133,289)(32,390,134,288)(33,389,135,287)(34,388,136,286)(35,387,137,285)(36,386,138,284)(37,385,139,283)(38,384,140,282)(39,383,141,281)(40,382,142,280)(41,381,143,279)(42,380,144,278)(43,379,145,277)(44,378,146,276)(45,377,147,275)(46,376,148,274)(47,375,149,273)(48,374,150,272)(49,373,151,271)(50,372,152,270)(51,371,153,269)(52,370,154,268)(53,369,155,267)(54,368,156,266)(55,367,157,265)(56,366,158,264)(57,365,159,263)(58,364,160,262)(59,363,161,261)(60,362,162,260)(61,361,163,259)(62,360,164,258)(63,359,165,257)(64,358,166,256)(65,357,167,255)(66,356,168,254)(67,355,169,253)(68,354,170,252)(69,353,171,251)(70,352,172,250)(71,351,173,249)(72,350,174,248)(73,349,175,247)(74,348,176,246)(75,347,177,245)(76,346,178,244)(77,345,179,243)(78,344,180,242)(79,343,181,241)(80,342,182,240)(81,341,183,239)(82,340,184,238)(83,339,185,237)(84,338,186,236)(85,337,187,235)(86,336,188,234)(87,335,189,233)(88,334,190,232)(89,333,191,231)(90,332,192,230)(91,331,193,229)(92,330,194,228)(93,329,195,227)(94,328,196,226)(95,327,197,225)(96,326,198,224)(97,325,199,223)(98,324,200,222)(99,323,201,221)(100,322,202,220)(101,321,203,219)(102,320,204,218) );
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)], [(1,217,103,319),(2,216,104,318),(3,215,105,317),(4,214,106,316),(5,213,107,315),(6,212,108,314),(7,211,109,313),(8,210,110,312),(9,209,111,311),(10,208,112,310),(11,207,113,309),(12,206,114,308),(13,205,115,307),(14,408,116,306),(15,407,117,305),(16,406,118,304),(17,405,119,303),(18,404,120,302),(19,403,121,301),(20,402,122,300),(21,401,123,299),(22,400,124,298),(23,399,125,297),(24,398,126,296),(25,397,127,295),(26,396,128,294),(27,395,129,293),(28,394,130,292),(29,393,131,291),(30,392,132,290),(31,391,133,289),(32,390,134,288),(33,389,135,287),(34,388,136,286),(35,387,137,285),(36,386,138,284),(37,385,139,283),(38,384,140,282),(39,383,141,281),(40,382,142,280),(41,381,143,279),(42,380,144,278),(43,379,145,277),(44,378,146,276),(45,377,147,275),(46,376,148,274),(47,375,149,273),(48,374,150,272),(49,373,151,271),(50,372,152,270),(51,371,153,269),(52,370,154,268),(53,369,155,267),(54,368,156,266),(55,367,157,265),(56,366,158,264),(57,365,159,263),(58,364,160,262),(59,363,161,261),(60,362,162,260),(61,361,163,259),(62,360,164,258),(63,359,165,257),(64,358,166,256),(65,357,167,255),(66,356,168,254),(67,355,169,253),(68,354,170,252),(69,353,171,251),(70,352,172,250),(71,351,173,249),(72,350,174,248),(73,349,175,247),(74,348,176,246),(75,347,177,245),(76,346,178,244),(77,345,179,243),(78,344,180,242),(79,343,181,241),(80,342,182,240),(81,341,183,239),(82,340,184,238),(83,339,185,237),(84,338,186,236),(85,337,187,235),(86,336,188,234),(87,335,189,233),(88,334,190,232),(89,333,191,231),(90,332,192,230),(91,331,193,229),(92,330,194,228),(93,329,195,227),(94,328,196,226),(95,327,197,225),(96,326,198,224),(97,325,199,223),(98,324,200,222),(99,323,201,221),(100,322,202,220),(101,321,203,219),(102,320,204,218)]])
105 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 12A | 12B | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51P | 68A | ··· | 68P | 102A | ··· | 102P | 204A | ··· | 204AF |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 | 204 | ··· | 204 |
| size | 1 | 1 | 2 | 2 | 102 | 102 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | Q8 | D6 | Dic6 | D17 | D34 | D51 | Dic34 | D102 | Dic102 |
| kernel | Dic102 | Dic51 | C204 | C68 | C51 | C34 | C17 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 16 | 16 | 16 | 32 |
Matrix representation of Dic102 ►in GL2(𝔽409) generated by
| 281 | 257 |
| 342 | 77 |
| 109 | 238 |
| 36 | 300 |
G:=sub<GL(2,GF(409))| [281,342,257,77],[109,36,238,300] >;
Dic102 in GAP, Magma, Sage, TeX
{\rm Dic}_{102} % in TeX
G:=Group("Dic102"); // GroupNames label
G:=SmallGroup(408,25);
// by ID
G=gap.SmallGroup(408,25);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,20,61,26,323,9604]);
// Polycyclic
G:=Group<a,b|a^204=1,b^2=a^102,b*a*b^-1=a^-1>;
// generators/relations
Export