metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic103, C103⋊C4, C206.C2, C2.D103, SmallGroup(412,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C103 — Dic103 |
Generators and relations for Dic103
G = < a,b | a206=1, b2=a103, bab-1=a-1 >
Smallest permutation representation of Dic103
►Regular action on 412 points
Generators in S412
(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)
(1 248 104 351)(2 247 105 350)(3 246 106 349)(4 245 107 348)(5 244 108 347)(6 243 109 346)(7 242 110 345)(8 241 111 344)(9 240 112 343)(10 239 113 342)(11 238 114 341)(12 237 115 340)(13 236 116 339)(14 235 117 338)(15 234 118 337)(16 233 119 336)(17 232 120 335)(18 231 121 334)(19 230 122 333)(20 229 123 332)(21 228 124 331)(22 227 125 330)(23 226 126 329)(24 225 127 328)(25 224 128 327)(26 223 129 326)(27 222 130 325)(28 221 131 324)(29 220 132 323)(30 219 133 322)(31 218 134 321)(32 217 135 320)(33 216 136 319)(34 215 137 318)(35 214 138 317)(36 213 139 316)(37 212 140 315)(38 211 141 314)(39 210 142 313)(40 209 143 312)(41 208 144 311)(42 207 145 310)(43 412 146 309)(44 411 147 308)(45 410 148 307)(46 409 149 306)(47 408 150 305)(48 407 151 304)(49 406 152 303)(50 405 153 302)(51 404 154 301)(52 403 155 300)(53 402 156 299)(54 401 157 298)(55 400 158 297)(56 399 159 296)(57 398 160 295)(58 397 161 294)(59 396 162 293)(60 395 163 292)(61 394 164 291)(62 393 165 290)(63 392 166 289)(64 391 167 288)(65 390 168 287)(66 389 169 286)(67 388 170 285)(68 387 171 284)(69 386 172 283)(70 385 173 282)(71 384 174 281)(72 383 175 280)(73 382 176 279)(74 381 177 278)(75 380 178 277)(76 379 179 276)(77 378 180 275)(78 377 181 274)(79 376 182 273)(80 375 183 272)(81 374 184 271)(82 373 185 270)(83 372 186 269)(84 371 187 268)(85 370 188 267)(86 369 189 266)(87 368 190 265)(88 367 191 264)(89 366 192 263)(90 365 193 262)(91 364 194 261)(92 363 195 260)(93 362 196 259)(94 361 197 258)(95 360 198 257)(96 359 199 256)(97 358 200 255)(98 357 201 254)(99 356 202 253)(100 355 203 252)(101 354 204 251)(102 353 205 250)(103 352 206 249)
G:=sub<Sym(412)| (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), (1,248,104,351)(2,247,105,350)(3,246,106,349)(4,245,107,348)(5,244,108,347)(6,243,109,346)(7,242,110,345)(8,241,111,344)(9,240,112,343)(10,239,113,342)(11,238,114,341)(12,237,115,340)(13,236,116,339)(14,235,117,338)(15,234,118,337)(16,233,119,336)(17,232,120,335)(18,231,121,334)(19,230,122,333)(20,229,123,332)(21,228,124,331)(22,227,125,330)(23,226,126,329)(24,225,127,328)(25,224,128,327)(26,223,129,326)(27,222,130,325)(28,221,131,324)(29,220,132,323)(30,219,133,322)(31,218,134,321)(32,217,135,320)(33,216,136,319)(34,215,137,318)(35,214,138,317)(36,213,139,316)(37,212,140,315)(38,211,141,314)(39,210,142,313)(40,209,143,312)(41,208,144,311)(42,207,145,310)(43,412,146,309)(44,411,147,308)(45,410,148,307)(46,409,149,306)(47,408,150,305)(48,407,151,304)(49,406,152,303)(50,405,153,302)(51,404,154,301)(52,403,155,300)(53,402,156,299)(54,401,157,298)(55,400,158,297)(56,399,159,296)(57,398,160,295)(58,397,161,294)(59,396,162,293)(60,395,163,292)(61,394,164,291)(62,393,165,290)(63,392,166,289)(64,391,167,288)(65,390,168,287)(66,389,169,286)(67,388,170,285)(68,387,171,284)(69,386,172,283)(70,385,173,282)(71,384,174,281)(72,383,175,280)(73,382,176,279)(74,381,177,278)(75,380,178,277)(76,379,179,276)(77,378,180,275)(78,377,181,274)(79,376,182,273)(80,375,183,272)(81,374,184,271)(82,373,185,270)(83,372,186,269)(84,371,187,268)(85,370,188,267)(86,369,189,266)(87,368,190,265)(88,367,191,264)(89,366,192,263)(90,365,193,262)(91,364,194,261)(92,363,195,260)(93,362,196,259)(94,361,197,258)(95,360,198,257)(96,359,199,256)(97,358,200,255)(98,357,201,254)(99,356,202,253)(100,355,203,252)(101,354,204,251)(102,353,205,250)(103,352,206,249)>;
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), (1,248,104,351)(2,247,105,350)(3,246,106,349)(4,245,107,348)(5,244,108,347)(6,243,109,346)(7,242,110,345)(8,241,111,344)(9,240,112,343)(10,239,113,342)(11,238,114,341)(12,237,115,340)(13,236,116,339)(14,235,117,338)(15,234,118,337)(16,233,119,336)(17,232,120,335)(18,231,121,334)(19,230,122,333)(20,229,123,332)(21,228,124,331)(22,227,125,330)(23,226,126,329)(24,225,127,328)(25,224,128,327)(26,223,129,326)(27,222,130,325)(28,221,131,324)(29,220,132,323)(30,219,133,322)(31,218,134,321)(32,217,135,320)(33,216,136,319)(34,215,137,318)(35,214,138,317)(36,213,139,316)(37,212,140,315)(38,211,141,314)(39,210,142,313)(40,209,143,312)(41,208,144,311)(42,207,145,310)(43,412,146,309)(44,411,147,308)(45,410,148,307)(46,409,149,306)(47,408,150,305)(48,407,151,304)(49,406,152,303)(50,405,153,302)(51,404,154,301)(52,403,155,300)(53,402,156,299)(54,401,157,298)(55,400,158,297)(56,399,159,296)(57,398,160,295)(58,397,161,294)(59,396,162,293)(60,395,163,292)(61,394,164,291)(62,393,165,290)(63,392,166,289)(64,391,167,288)(65,390,168,287)(66,389,169,286)(67,388,170,285)(68,387,171,284)(69,386,172,283)(70,385,173,282)(71,384,174,281)(72,383,175,280)(73,382,176,279)(74,381,177,278)(75,380,178,277)(76,379,179,276)(77,378,180,275)(78,377,181,274)(79,376,182,273)(80,375,183,272)(81,374,184,271)(82,373,185,270)(83,372,186,269)(84,371,187,268)(85,370,188,267)(86,369,189,266)(87,368,190,265)(88,367,191,264)(89,366,192,263)(90,365,193,262)(91,364,194,261)(92,363,195,260)(93,362,196,259)(94,361,197,258)(95,360,198,257)(96,359,199,256)(97,358,200,255)(98,357,201,254)(99,356,202,253)(100,355,203,252)(101,354,204,251)(102,353,205,250)(103,352,206,249) );
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)], [(1,248,104,351),(2,247,105,350),(3,246,106,349),(4,245,107,348),(5,244,108,347),(6,243,109,346),(7,242,110,345),(8,241,111,344),(9,240,112,343),(10,239,113,342),(11,238,114,341),(12,237,115,340),(13,236,116,339),(14,235,117,338),(15,234,118,337),(16,233,119,336),(17,232,120,335),(18,231,121,334),(19,230,122,333),(20,229,123,332),(21,228,124,331),(22,227,125,330),(23,226,126,329),(24,225,127,328),(25,224,128,327),(26,223,129,326),(27,222,130,325),(28,221,131,324),(29,220,132,323),(30,219,133,322),(31,218,134,321),(32,217,135,320),(33,216,136,319),(34,215,137,318),(35,214,138,317),(36,213,139,316),(37,212,140,315),(38,211,141,314),(39,210,142,313),(40,209,143,312),(41,208,144,311),(42,207,145,310),(43,412,146,309),(44,411,147,308),(45,410,148,307),(46,409,149,306),(47,408,150,305),(48,407,151,304),(49,406,152,303),(50,405,153,302),(51,404,154,301),(52,403,155,300),(53,402,156,299),(54,401,157,298),(55,400,158,297),(56,399,159,296),(57,398,160,295),(58,397,161,294),(59,396,162,293),(60,395,163,292),(61,394,164,291),(62,393,165,290),(63,392,166,289),(64,391,167,288),(65,390,168,287),(66,389,169,286),(67,388,170,285),(68,387,171,284),(69,386,172,283),(70,385,173,282),(71,384,174,281),(72,383,175,280),(73,382,176,279),(74,381,177,278),(75,380,178,277),(76,379,179,276),(77,378,180,275),(78,377,181,274),(79,376,182,273),(80,375,183,272),(81,374,184,271),(82,373,185,270),(83,372,186,269),(84,371,187,268),(85,370,188,267),(86,369,189,266),(87,368,190,265),(88,367,191,264),(89,366,192,263),(90,365,193,262),(91,364,194,261),(92,363,195,260),(93,362,196,259),(94,361,197,258),(95,360,198,257),(96,359,199,256),(97,358,200,255),(98,357,201,254),(99,356,202,253),(100,355,203,252),(101,354,204,251),(102,353,205,250),(103,352,206,249)]])
106 conjugacy classes
| class | 1 | 2 | 4A | 4B | 103A | ··· | 103AY | 206A | ··· | 206AY |
| order | 1 | 2 | 4 | 4 | 103 | ··· | 103 | 206 | ··· | 206 |
| size | 1 | 1 | 103 | 103 | 2 | ··· | 2 | 2 | ··· | 2 |
106 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D103 | Dic103 |
| kernel | Dic103 | C206 | C103 | C2 | C1 |
| # reps | 1 | 1 | 2 | 51 | 51 |
Matrix representation of Dic103 ►in GL3(𝔽1237) generated by
| 1236 | 0 | 0 |
| 0 | 531 | 1236 |
| 0 | 1 | 0 |
| 546 | 0 | 0 |
| 0 | 680 | 204 |
| 0 | 80 | 557 |
G:=sub<GL(3,GF(1237))| [1236,0,0,0,531,1,0,1236,0],[546,0,0,0,680,80,0,204,557] >;
Dic103 in GAP, Magma, Sage, TeX
{\rm Dic}_{103} % in TeX
G:=Group("Dic103"); // GroupNames label
G:=SmallGroup(412,1);
// by ID
G=gap.SmallGroup(412,1);
# by ID
G:=PCGroup([3,-2,-2,-103,6,3674]);
// Polycyclic
G:=Group<a,b|a^206=1,b^2=a^103,b*a*b^-1=a^-1>;
// generators/relations
Export