direct product, abelian, monomial, 2-elementary
Aliases: C2×C206, SmallGroup(412,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C206 |
| C1 — C2×C206 |
| C1 — C2×C206 |
Generators and relations for C2×C206
G = < a,b | a2=b206=1, ab=ba >
Smallest permutation representation of C2×C206
►Regular action on 412 points
Generators in S412
(1 384)(2 385)(3 386)(4 387)(5 388)(6 389)(7 390)(8 391)(9 392)(10 393)(11 394)(12 395)(13 396)(14 397)(15 398)(16 399)(17 400)(18 401)(19 402)(20 403)(21 404)(22 405)(23 406)(24 407)(25 408)(26 409)(27 410)(28 411)(29 412)(30 207)(31 208)(32 209)(33 210)(34 211)(35 212)(36 213)(37 214)(38 215)(39 216)(40 217)(41 218)(42 219)(43 220)(44 221)(45 222)(46 223)(47 224)(48 225)(49 226)(50 227)(51 228)(52 229)(53 230)(54 231)(55 232)(56 233)(57 234)(58 235)(59 236)(60 237)(61 238)(62 239)(63 240)(64 241)(65 242)(66 243)(67 244)(68 245)(69 246)(70 247)(71 248)(72 249)(73 250)(74 251)(75 252)(76 253)(77 254)(78 255)(79 256)(80 257)(81 258)(82 259)(83 260)(84 261)(85 262)(86 263)(87 264)(88 265)(89 266)(90 267)(91 268)(92 269)(93 270)(94 271)(95 272)(96 273)(97 274)(98 275)(99 276)(100 277)(101 278)(102 279)(103 280)(104 281)(105 282)(106 283)(107 284)(108 285)(109 286)(110 287)(111 288)(112 289)(113 290)(114 291)(115 292)(116 293)(117 294)(118 295)(119 296)(120 297)(121 298)(122 299)(123 300)(124 301)(125 302)(126 303)(127 304)(128 305)(129 306)(130 307)(131 308)(132 309)(133 310)(134 311)(135 312)(136 313)(137 314)(138 315)(139 316)(140 317)(141 318)(142 319)(143 320)(144 321)(145 322)(146 323)(147 324)(148 325)(149 326)(150 327)(151 328)(152 329)(153 330)(154 331)(155 332)(156 333)(157 334)(158 335)(159 336)(160 337)(161 338)(162 339)(163 340)(164 341)(165 342)(166 343)(167 344)(168 345)(169 346)(170 347)(171 348)(172 349)(173 350)(174 351)(175 352)(176 353)(177 354)(178 355)(179 356)(180 357)(181 358)(182 359)(183 360)(184 361)(185 362)(186 363)(187 364)(188 365)(189 366)(190 367)(191 368)(192 369)(193 370)(194 371)(195 372)(196 373)(197 374)(198 375)(199 376)(200 377)(201 378)(202 379)(203 380)(204 381)(205 382)(206 383)
(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)
G:=sub<Sym(412)| (1,384)(2,385)(3,386)(4,387)(5,388)(6,389)(7,390)(8,391)(9,392)(10,393)(11,394)(12,395)(13,396)(14,397)(15,398)(16,399)(17,400)(18,401)(19,402)(20,403)(21,404)(22,405)(23,406)(24,407)(25,408)(26,409)(27,410)(28,411)(29,412)(30,207)(31,208)(32,209)(33,210)(34,211)(35,212)(36,213)(37,214)(38,215)(39,216)(40,217)(41,218)(42,219)(43,220)(44,221)(45,222)(46,223)(47,224)(48,225)(49,226)(50,227)(51,228)(52,229)(53,230)(54,231)(55,232)(56,233)(57,234)(58,235)(59,236)(60,237)(61,238)(62,239)(63,240)(64,241)(65,242)(66,243)(67,244)(68,245)(69,246)(70,247)(71,248)(72,249)(73,250)(74,251)(75,252)(76,253)(77,254)(78,255)(79,256)(80,257)(81,258)(82,259)(83,260)(84,261)(85,262)(86,263)(87,264)(88,265)(89,266)(90,267)(91,268)(92,269)(93,270)(94,271)(95,272)(96,273)(97,274)(98,275)(99,276)(100,277)(101,278)(102,279)(103,280)(104,281)(105,282)(106,283)(107,284)(108,285)(109,286)(110,287)(111,288)(112,289)(113,290)(114,291)(115,292)(116,293)(117,294)(118,295)(119,296)(120,297)(121,298)(122,299)(123,300)(124,301)(125,302)(126,303)(127,304)(128,305)(129,306)(130,307)(131,308)(132,309)(133,310)(134,311)(135,312)(136,313)(137,314)(138,315)(139,316)(140,317)(141,318)(142,319)(143,320)(144,321)(145,322)(146,323)(147,324)(148,325)(149,326)(150,327)(151,328)(152,329)(153,330)(154,331)(155,332)(156,333)(157,334)(158,335)(159,336)(160,337)(161,338)(162,339)(163,340)(164,341)(165,342)(166,343)(167,344)(168,345)(169,346)(170,347)(171,348)(172,349)(173,350)(174,351)(175,352)(176,353)(177,354)(178,355)(179,356)(180,357)(181,358)(182,359)(183,360)(184,361)(185,362)(186,363)(187,364)(188,365)(189,366)(190,367)(191,368)(192,369)(193,370)(194,371)(195,372)(196,373)(197,374)(198,375)(199,376)(200,377)(201,378)(202,379)(203,380)(204,381)(205,382)(206,383), (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)>;
G:=Group( (1,384)(2,385)(3,386)(4,387)(5,388)(6,389)(7,390)(8,391)(9,392)(10,393)(11,394)(12,395)(13,396)(14,397)(15,398)(16,399)(17,400)(18,401)(19,402)(20,403)(21,404)(22,405)(23,406)(24,407)(25,408)(26,409)(27,410)(28,411)(29,412)(30,207)(31,208)(32,209)(33,210)(34,211)(35,212)(36,213)(37,214)(38,215)(39,216)(40,217)(41,218)(42,219)(43,220)(44,221)(45,222)(46,223)(47,224)(48,225)(49,226)(50,227)(51,228)(52,229)(53,230)(54,231)(55,232)(56,233)(57,234)(58,235)(59,236)(60,237)(61,238)(62,239)(63,240)(64,241)(65,242)(66,243)(67,244)(68,245)(69,246)(70,247)(71,248)(72,249)(73,250)(74,251)(75,252)(76,253)(77,254)(78,255)(79,256)(80,257)(81,258)(82,259)(83,260)(84,261)(85,262)(86,263)(87,264)(88,265)(89,266)(90,267)(91,268)(92,269)(93,270)(94,271)(95,272)(96,273)(97,274)(98,275)(99,276)(100,277)(101,278)(102,279)(103,280)(104,281)(105,282)(106,283)(107,284)(108,285)(109,286)(110,287)(111,288)(112,289)(113,290)(114,291)(115,292)(116,293)(117,294)(118,295)(119,296)(120,297)(121,298)(122,299)(123,300)(124,301)(125,302)(126,303)(127,304)(128,305)(129,306)(130,307)(131,308)(132,309)(133,310)(134,311)(135,312)(136,313)(137,314)(138,315)(139,316)(140,317)(141,318)(142,319)(143,320)(144,321)(145,322)(146,323)(147,324)(148,325)(149,326)(150,327)(151,328)(152,329)(153,330)(154,331)(155,332)(156,333)(157,334)(158,335)(159,336)(160,337)(161,338)(162,339)(163,340)(164,341)(165,342)(166,343)(167,344)(168,345)(169,346)(170,347)(171,348)(172,349)(173,350)(174,351)(175,352)(176,353)(177,354)(178,355)(179,356)(180,357)(181,358)(182,359)(183,360)(184,361)(185,362)(186,363)(187,364)(188,365)(189,366)(190,367)(191,368)(192,369)(193,370)(194,371)(195,372)(196,373)(197,374)(198,375)(199,376)(200,377)(201,378)(202,379)(203,380)(204,381)(205,382)(206,383), (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) );
G=PermutationGroup([[(1,384),(2,385),(3,386),(4,387),(5,388),(6,389),(7,390),(8,391),(9,392),(10,393),(11,394),(12,395),(13,396),(14,397),(15,398),(16,399),(17,400),(18,401),(19,402),(20,403),(21,404),(22,405),(23,406),(24,407),(25,408),(26,409),(27,410),(28,411),(29,412),(30,207),(31,208),(32,209),(33,210),(34,211),(35,212),(36,213),(37,214),(38,215),(39,216),(40,217),(41,218),(42,219),(43,220),(44,221),(45,222),(46,223),(47,224),(48,225),(49,226),(50,227),(51,228),(52,229),(53,230),(54,231),(55,232),(56,233),(57,234),(58,235),(59,236),(60,237),(61,238),(62,239),(63,240),(64,241),(65,242),(66,243),(67,244),(68,245),(69,246),(70,247),(71,248),(72,249),(73,250),(74,251),(75,252),(76,253),(77,254),(78,255),(79,256),(80,257),(81,258),(82,259),(83,260),(84,261),(85,262),(86,263),(87,264),(88,265),(89,266),(90,267),(91,268),(92,269),(93,270),(94,271),(95,272),(96,273),(97,274),(98,275),(99,276),(100,277),(101,278),(102,279),(103,280),(104,281),(105,282),(106,283),(107,284),(108,285),(109,286),(110,287),(111,288),(112,289),(113,290),(114,291),(115,292),(116,293),(117,294),(118,295),(119,296),(120,297),(121,298),(122,299),(123,300),(124,301),(125,302),(126,303),(127,304),(128,305),(129,306),(130,307),(131,308),(132,309),(133,310),(134,311),(135,312),(136,313),(137,314),(138,315),(139,316),(140,317),(141,318),(142,319),(143,320),(144,321),(145,322),(146,323),(147,324),(148,325),(149,326),(150,327),(151,328),(152,329),(153,330),(154,331),(155,332),(156,333),(157,334),(158,335),(159,336),(160,337),(161,338),(162,339),(163,340),(164,341),(165,342),(166,343),(167,344),(168,345),(169,346),(170,347),(171,348),(172,349),(173,350),(174,351),(175,352),(176,353),(177,354),(178,355),(179,356),(180,357),(181,358),(182,359),(183,360),(184,361),(185,362),(186,363),(187,364),(188,365),(189,366),(190,367),(191,368),(192,369),(193,370),(194,371),(195,372),(196,373),(197,374),(198,375),(199,376),(200,377),(201,378),(202,379),(203,380),(204,381),(205,382),(206,383)], [(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)]])
412 conjugacy classes
| class | 1 | 2A | 2B | 2C | 103A | ··· | 103CX | 206A | ··· | 206KT |
| order | 1 | 2 | 2 | 2 | 103 | ··· | 103 | 206 | ··· | 206 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
412 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C103 | C206 |
| kernel | C2×C206 | C206 | C22 | C2 |
| # reps | 1 | 3 | 102 | 306 |
Matrix representation of C2×C206 ►in GL2(𝔽619) generated by
| 618 | 0 |
| 0 | 1 |
| 439 | 0 |
| 0 | 588 |
G:=sub<GL(2,GF(619))| [618,0,0,1],[439,0,0,588] >;
C2×C206 in GAP, Magma, Sage, TeX
C_2\times C_{206} % in TeX
G:=Group("C2xC206"); // GroupNames label
G:=SmallGroup(412,4);
// by ID
G=gap.SmallGroup(412,4);
# by ID
G:=PCGroup([3,-2,-2,-103]);
// Polycyclic
G:=Group<a,b|a^2=b^206=1,a*b=b*a>;
// generators/relations
Export