direct product, abelian, monomial, 2-elementary
Aliases: C2×C208, SmallGroup(416,59)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C208 |
| C1 — C2×C208 |
| C1 — C2×C208 |
Generators and relations for C2×C208
G = < a,b | a2=b208=1, ab=ba >
Smallest permutation representation of C2×C208
►Regular action on 416 points
Generators in S416
(1 289)(2 290)(3 291)(4 292)(5 293)(6 294)(7 295)(8 296)(9 297)(10 298)(11 299)(12 300)(13 301)(14 302)(15 303)(16 304)(17 305)(18 306)(19 307)(20 308)(21 309)(22 310)(23 311)(24 312)(25 313)(26 314)(27 315)(28 316)(29 317)(30 318)(31 319)(32 320)(33 321)(34 322)(35 323)(36 324)(37 325)(38 326)(39 327)(40 328)(41 329)(42 330)(43 331)(44 332)(45 333)(46 334)(47 335)(48 336)(49 337)(50 338)(51 339)(52 340)(53 341)(54 342)(55 343)(56 344)(57 345)(58 346)(59 347)(60 348)(61 349)(62 350)(63 351)(64 352)(65 353)(66 354)(67 355)(68 356)(69 357)(70 358)(71 359)(72 360)(73 361)(74 362)(75 363)(76 364)(77 365)(78 366)(79 367)(80 368)(81 369)(82 370)(83 371)(84 372)(85 373)(86 374)(87 375)(88 376)(89 377)(90 378)(91 379)(92 380)(93 381)(94 382)(95 383)(96 384)(97 385)(98 386)(99 387)(100 388)(101 389)(102 390)(103 391)(104 392)(105 393)(106 394)(107 395)(108 396)(109 397)(110 398)(111 399)(112 400)(113 401)(114 402)(115 403)(116 404)(117 405)(118 406)(119 407)(120 408)(121 409)(122 410)(123 411)(124 412)(125 413)(126 414)(127 415)(128 416)(129 209)(130 210)(131 211)(132 212)(133 213)(134 214)(135 215)(136 216)(137 217)(138 218)(139 219)(140 220)(141 221)(142 222)(143 223)(144 224)(145 225)(146 226)(147 227)(148 228)(149 229)(150 230)(151 231)(152 232)(153 233)(154 234)(155 235)(156 236)(157 237)(158 238)(159 239)(160 240)(161 241)(162 242)(163 243)(164 244)(165 245)(166 246)(167 247)(168 248)(169 249)(170 250)(171 251)(172 252)(173 253)(174 254)(175 255)(176 256)(177 257)(178 258)(179 259)(180 260)(181 261)(182 262)(183 263)(184 264)(185 265)(186 266)(187 267)(188 268)(189 269)(190 270)(191 271)(192 272)(193 273)(194 274)(195 275)(196 276)(197 277)(198 278)(199 279)(200 280)(201 281)(202 282)(203 283)(204 284)(205 285)(206 286)(207 287)(208 288)
(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)
G:=sub<Sym(416)| (1,289)(2,290)(3,291)(4,292)(5,293)(6,294)(7,295)(8,296)(9,297)(10,298)(11,299)(12,300)(13,301)(14,302)(15,303)(16,304)(17,305)(18,306)(19,307)(20,308)(21,309)(22,310)(23,311)(24,312)(25,313)(26,314)(27,315)(28,316)(29,317)(30,318)(31,319)(32,320)(33,321)(34,322)(35,323)(36,324)(37,325)(38,326)(39,327)(40,328)(41,329)(42,330)(43,331)(44,332)(45,333)(46,334)(47,335)(48,336)(49,337)(50,338)(51,339)(52,340)(53,341)(54,342)(55,343)(56,344)(57,345)(58,346)(59,347)(60,348)(61,349)(62,350)(63,351)(64,352)(65,353)(66,354)(67,355)(68,356)(69,357)(70,358)(71,359)(72,360)(73,361)(74,362)(75,363)(76,364)(77,365)(78,366)(79,367)(80,368)(81,369)(82,370)(83,371)(84,372)(85,373)(86,374)(87,375)(88,376)(89,377)(90,378)(91,379)(92,380)(93,381)(94,382)(95,383)(96,384)(97,385)(98,386)(99,387)(100,388)(101,389)(102,390)(103,391)(104,392)(105,393)(106,394)(107,395)(108,396)(109,397)(110,398)(111,399)(112,400)(113,401)(114,402)(115,403)(116,404)(117,405)(118,406)(119,407)(120,408)(121,409)(122,410)(123,411)(124,412)(125,413)(126,414)(127,415)(128,416)(129,209)(130,210)(131,211)(132,212)(133,213)(134,214)(135,215)(136,216)(137,217)(138,218)(139,219)(140,220)(141,221)(142,222)(143,223)(144,224)(145,225)(146,226)(147,227)(148,228)(149,229)(150,230)(151,231)(152,232)(153,233)(154,234)(155,235)(156,236)(157,237)(158,238)(159,239)(160,240)(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)(168,248)(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)(176,256)(177,257)(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)(184,264)(185,265)(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)(192,272)(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)(201,281)(202,282)(203,283)(204,284)(205,285)(206,286)(207,287)(208,288), (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)>;
G:=Group( (1,289)(2,290)(3,291)(4,292)(5,293)(6,294)(7,295)(8,296)(9,297)(10,298)(11,299)(12,300)(13,301)(14,302)(15,303)(16,304)(17,305)(18,306)(19,307)(20,308)(21,309)(22,310)(23,311)(24,312)(25,313)(26,314)(27,315)(28,316)(29,317)(30,318)(31,319)(32,320)(33,321)(34,322)(35,323)(36,324)(37,325)(38,326)(39,327)(40,328)(41,329)(42,330)(43,331)(44,332)(45,333)(46,334)(47,335)(48,336)(49,337)(50,338)(51,339)(52,340)(53,341)(54,342)(55,343)(56,344)(57,345)(58,346)(59,347)(60,348)(61,349)(62,350)(63,351)(64,352)(65,353)(66,354)(67,355)(68,356)(69,357)(70,358)(71,359)(72,360)(73,361)(74,362)(75,363)(76,364)(77,365)(78,366)(79,367)(80,368)(81,369)(82,370)(83,371)(84,372)(85,373)(86,374)(87,375)(88,376)(89,377)(90,378)(91,379)(92,380)(93,381)(94,382)(95,383)(96,384)(97,385)(98,386)(99,387)(100,388)(101,389)(102,390)(103,391)(104,392)(105,393)(106,394)(107,395)(108,396)(109,397)(110,398)(111,399)(112,400)(113,401)(114,402)(115,403)(116,404)(117,405)(118,406)(119,407)(120,408)(121,409)(122,410)(123,411)(124,412)(125,413)(126,414)(127,415)(128,416)(129,209)(130,210)(131,211)(132,212)(133,213)(134,214)(135,215)(136,216)(137,217)(138,218)(139,219)(140,220)(141,221)(142,222)(143,223)(144,224)(145,225)(146,226)(147,227)(148,228)(149,229)(150,230)(151,231)(152,232)(153,233)(154,234)(155,235)(156,236)(157,237)(158,238)(159,239)(160,240)(161,241)(162,242)(163,243)(164,244)(165,245)(166,246)(167,247)(168,248)(169,249)(170,250)(171,251)(172,252)(173,253)(174,254)(175,255)(176,256)(177,257)(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)(184,264)(185,265)(186,266)(187,267)(188,268)(189,269)(190,270)(191,271)(192,272)(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)(201,281)(202,282)(203,283)(204,284)(205,285)(206,286)(207,287)(208,288), (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) );
G=PermutationGroup([[(1,289),(2,290),(3,291),(4,292),(5,293),(6,294),(7,295),(8,296),(9,297),(10,298),(11,299),(12,300),(13,301),(14,302),(15,303),(16,304),(17,305),(18,306),(19,307),(20,308),(21,309),(22,310),(23,311),(24,312),(25,313),(26,314),(27,315),(28,316),(29,317),(30,318),(31,319),(32,320),(33,321),(34,322),(35,323),(36,324),(37,325),(38,326),(39,327),(40,328),(41,329),(42,330),(43,331),(44,332),(45,333),(46,334),(47,335),(48,336),(49,337),(50,338),(51,339),(52,340),(53,341),(54,342),(55,343),(56,344),(57,345),(58,346),(59,347),(60,348),(61,349),(62,350),(63,351),(64,352),(65,353),(66,354),(67,355),(68,356),(69,357),(70,358),(71,359),(72,360),(73,361),(74,362),(75,363),(76,364),(77,365),(78,366),(79,367),(80,368),(81,369),(82,370),(83,371),(84,372),(85,373),(86,374),(87,375),(88,376),(89,377),(90,378),(91,379),(92,380),(93,381),(94,382),(95,383),(96,384),(97,385),(98,386),(99,387),(100,388),(101,389),(102,390),(103,391),(104,392),(105,393),(106,394),(107,395),(108,396),(109,397),(110,398),(111,399),(112,400),(113,401),(114,402),(115,403),(116,404),(117,405),(118,406),(119,407),(120,408),(121,409),(122,410),(123,411),(124,412),(125,413),(126,414),(127,415),(128,416),(129,209),(130,210),(131,211),(132,212),(133,213),(134,214),(135,215),(136,216),(137,217),(138,218),(139,219),(140,220),(141,221),(142,222),(143,223),(144,224),(145,225),(146,226),(147,227),(148,228),(149,229),(150,230),(151,231),(152,232),(153,233),(154,234),(155,235),(156,236),(157,237),(158,238),(159,239),(160,240),(161,241),(162,242),(163,243),(164,244),(165,245),(166,246),(167,247),(168,248),(169,249),(170,250),(171,251),(172,252),(173,253),(174,254),(175,255),(176,256),(177,257),(178,258),(179,259),(180,260),(181,261),(182,262),(183,263),(184,264),(185,265),(186,266),(187,267),(188,268),(189,269),(190,270),(191,271),(192,272),(193,273),(194,274),(195,275),(196,276),(197,277),(198,278),(199,279),(200,280),(201,281),(202,282),(203,283),(204,284),(205,285),(206,286),(207,287),(208,288)], [(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)]])
416 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 13A | ··· | 13L | 16A | ··· | 16P | 26A | ··· | 26AJ | 52A | ··· | 52AV | 104A | ··· | 104CR | 208A | ··· | 208GJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | ··· | 13 | 16 | ··· | 16 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 | 208 | ··· | 208 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
416 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C13 | C16 | C26 | C26 | C52 | C52 | C104 | C104 | C208 |
| kernel | C2×C208 | C208 | C2×C104 | C104 | C2×C52 | C52 | C2×C26 | C2×C16 | C26 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 12 | 16 | 24 | 12 | 24 | 24 | 48 | 48 | 192 |
Matrix representation of C2×C208 ►in GL2(𝔽1249) generated by
| 1248 | 0 |
| 0 | 1248 |
| 771 | 0 |
| 0 | 992 |
G:=sub<GL(2,GF(1249))| [1248,0,0,1248],[771,0,0,992] >;
C2×C208 in GAP, Magma, Sage, TeX
C_2\times C_{208} % in TeX
G:=Group("C2xC208"); // GroupNames label
G:=SmallGroup(416,59);
// by ID
G=gap.SmallGroup(416,59);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,312,69,88]);
// Polycyclic
G:=Group<a,b|a^2=b^208=1,a*b=b*a>;
// generators/relations
Export