direct product, abelian, monomial, 2-elementary
Aliases: C2×C204, SmallGroup(408,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C204 |
| C1 — C2×C204 |
| C1 — C2×C204 |
Generators and relations for C2×C204
G = < a,b | a2=b204=1, ab=ba >
Smallest permutation representation of C2×C204
►Regular action on 408 points
Generators in S408
(1 256)(2 257)(3 258)(4 259)(5 260)(6 261)(7 262)(8 263)(9 264)(10 265)(11 266)(12 267)(13 268)(14 269)(15 270)(16 271)(17 272)(18 273)(19 274)(20 275)(21 276)(22 277)(23 278)(24 279)(25 280)(26 281)(27 282)(28 283)(29 284)(30 285)(31 286)(32 287)(33 288)(34 289)(35 290)(36 291)(37 292)(38 293)(39 294)(40 295)(41 296)(42 297)(43 298)(44 299)(45 300)(46 301)(47 302)(48 303)(49 304)(50 305)(51 306)(52 307)(53 308)(54 309)(55 310)(56 311)(57 312)(58 313)(59 314)(60 315)(61 316)(62 317)(63 318)(64 319)(65 320)(66 321)(67 322)(68 323)(69 324)(70 325)(71 326)(72 327)(73 328)(74 329)(75 330)(76 331)(77 332)(78 333)(79 334)(80 335)(81 336)(82 337)(83 338)(84 339)(85 340)(86 341)(87 342)(88 343)(89 344)(90 345)(91 346)(92 347)(93 348)(94 349)(95 350)(96 351)(97 352)(98 353)(99 354)(100 355)(101 356)(102 357)(103 358)(104 359)(105 360)(106 361)(107 362)(108 363)(109 364)(110 365)(111 366)(112 367)(113 368)(114 369)(115 370)(116 371)(117 372)(118 373)(119 374)(120 375)(121 376)(122 377)(123 378)(124 379)(125 380)(126 381)(127 382)(128 383)(129 384)(130 385)(131 386)(132 387)(133 388)(134 389)(135 390)(136 391)(137 392)(138 393)(139 394)(140 395)(141 396)(142 397)(143 398)(144 399)(145 400)(146 401)(147 402)(148 403)(149 404)(150 405)(151 406)(152 407)(153 408)(154 205)(155 206)(156 207)(157 208)(158 209)(159 210)(160 211)(161 212)(162 213)(163 214)(164 215)(165 216)(166 217)(167 218)(168 219)(169 220)(170 221)(171 222)(172 223)(173 224)(174 225)(175 226)(176 227)(177 228)(178 229)(179 230)(180 231)(181 232)(182 233)(183 234)(184 235)(185 236)(186 237)(187 238)(188 239)(189 240)(190 241)(191 242)(192 243)(193 244)(194 245)(195 246)(196 247)(197 248)(198 249)(199 250)(200 251)(201 252)(202 253)(203 254)(204 255)
(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)
G:=sub<Sym(408)| (1,256)(2,257)(3,258)(4,259)(5,260)(6,261)(7,262)(8,263)(9,264)(10,265)(11,266)(12,267)(13,268)(14,269)(15,270)(16,271)(17,272)(18,273)(19,274)(20,275)(21,276)(22,277)(23,278)(24,279)(25,280)(26,281)(27,282)(28,283)(29,284)(30,285)(31,286)(32,287)(33,288)(34,289)(35,290)(36,291)(37,292)(38,293)(39,294)(40,295)(41,296)(42,297)(43,298)(44,299)(45,300)(46,301)(47,302)(48,303)(49,304)(50,305)(51,306)(52,307)(53,308)(54,309)(55,310)(56,311)(57,312)(58,313)(59,314)(60,315)(61,316)(62,317)(63,318)(64,319)(65,320)(66,321)(67,322)(68,323)(69,324)(70,325)(71,326)(72,327)(73,328)(74,329)(75,330)(76,331)(77,332)(78,333)(79,334)(80,335)(81,336)(82,337)(83,338)(84,339)(85,340)(86,341)(87,342)(88,343)(89,344)(90,345)(91,346)(92,347)(93,348)(94,349)(95,350)(96,351)(97,352)(98,353)(99,354)(100,355)(101,356)(102,357)(103,358)(104,359)(105,360)(106,361)(107,362)(108,363)(109,364)(110,365)(111,366)(112,367)(113,368)(114,369)(115,370)(116,371)(117,372)(118,373)(119,374)(120,375)(121,376)(122,377)(123,378)(124,379)(125,380)(126,381)(127,382)(128,383)(129,384)(130,385)(131,386)(132,387)(133,388)(134,389)(135,390)(136,391)(137,392)(138,393)(139,394)(140,395)(141,396)(142,397)(143,398)(144,399)(145,400)(146,401)(147,402)(148,403)(149,404)(150,405)(151,406)(152,407)(153,408)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213)(163,214)(164,215)(165,216)(166,217)(167,218)(168,219)(169,220)(170,221)(171,222)(172,223)(173,224)(174,225)(175,226)(176,227)(177,228)(178,229)(179,230)(180,231)(181,232)(182,233)(183,234)(184,235)(185,236)(186,237)(187,238)(188,239)(189,240)(190,241)(191,242)(192,243)(193,244)(194,245)(195,246)(196,247)(197,248)(198,249)(199,250)(200,251)(201,252)(202,253)(203,254)(204,255), (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)>;
G:=Group( (1,256)(2,257)(3,258)(4,259)(5,260)(6,261)(7,262)(8,263)(9,264)(10,265)(11,266)(12,267)(13,268)(14,269)(15,270)(16,271)(17,272)(18,273)(19,274)(20,275)(21,276)(22,277)(23,278)(24,279)(25,280)(26,281)(27,282)(28,283)(29,284)(30,285)(31,286)(32,287)(33,288)(34,289)(35,290)(36,291)(37,292)(38,293)(39,294)(40,295)(41,296)(42,297)(43,298)(44,299)(45,300)(46,301)(47,302)(48,303)(49,304)(50,305)(51,306)(52,307)(53,308)(54,309)(55,310)(56,311)(57,312)(58,313)(59,314)(60,315)(61,316)(62,317)(63,318)(64,319)(65,320)(66,321)(67,322)(68,323)(69,324)(70,325)(71,326)(72,327)(73,328)(74,329)(75,330)(76,331)(77,332)(78,333)(79,334)(80,335)(81,336)(82,337)(83,338)(84,339)(85,340)(86,341)(87,342)(88,343)(89,344)(90,345)(91,346)(92,347)(93,348)(94,349)(95,350)(96,351)(97,352)(98,353)(99,354)(100,355)(101,356)(102,357)(103,358)(104,359)(105,360)(106,361)(107,362)(108,363)(109,364)(110,365)(111,366)(112,367)(113,368)(114,369)(115,370)(116,371)(117,372)(118,373)(119,374)(120,375)(121,376)(122,377)(123,378)(124,379)(125,380)(126,381)(127,382)(128,383)(129,384)(130,385)(131,386)(132,387)(133,388)(134,389)(135,390)(136,391)(137,392)(138,393)(139,394)(140,395)(141,396)(142,397)(143,398)(144,399)(145,400)(146,401)(147,402)(148,403)(149,404)(150,405)(151,406)(152,407)(153,408)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213)(163,214)(164,215)(165,216)(166,217)(167,218)(168,219)(169,220)(170,221)(171,222)(172,223)(173,224)(174,225)(175,226)(176,227)(177,228)(178,229)(179,230)(180,231)(181,232)(182,233)(183,234)(184,235)(185,236)(186,237)(187,238)(188,239)(189,240)(190,241)(191,242)(192,243)(193,244)(194,245)(195,246)(196,247)(197,248)(198,249)(199,250)(200,251)(201,252)(202,253)(203,254)(204,255), (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) );
G=PermutationGroup([[(1,256),(2,257),(3,258),(4,259),(5,260),(6,261),(7,262),(8,263),(9,264),(10,265),(11,266),(12,267),(13,268),(14,269),(15,270),(16,271),(17,272),(18,273),(19,274),(20,275),(21,276),(22,277),(23,278),(24,279),(25,280),(26,281),(27,282),(28,283),(29,284),(30,285),(31,286),(32,287),(33,288),(34,289),(35,290),(36,291),(37,292),(38,293),(39,294),(40,295),(41,296),(42,297),(43,298),(44,299),(45,300),(46,301),(47,302),(48,303),(49,304),(50,305),(51,306),(52,307),(53,308),(54,309),(55,310),(56,311),(57,312),(58,313),(59,314),(60,315),(61,316),(62,317),(63,318),(64,319),(65,320),(66,321),(67,322),(68,323),(69,324),(70,325),(71,326),(72,327),(73,328),(74,329),(75,330),(76,331),(77,332),(78,333),(79,334),(80,335),(81,336),(82,337),(83,338),(84,339),(85,340),(86,341),(87,342),(88,343),(89,344),(90,345),(91,346),(92,347),(93,348),(94,349),(95,350),(96,351),(97,352),(98,353),(99,354),(100,355),(101,356),(102,357),(103,358),(104,359),(105,360),(106,361),(107,362),(108,363),(109,364),(110,365),(111,366),(112,367),(113,368),(114,369),(115,370),(116,371),(117,372),(118,373),(119,374),(120,375),(121,376),(122,377),(123,378),(124,379),(125,380),(126,381),(127,382),(128,383),(129,384),(130,385),(131,386),(132,387),(133,388),(134,389),(135,390),(136,391),(137,392),(138,393),(139,394),(140,395),(141,396),(142,397),(143,398),(144,399),(145,400),(146,401),(147,402),(148,403),(149,404),(150,405),(151,406),(152,407),(153,408),(154,205),(155,206),(156,207),(157,208),(158,209),(159,210),(160,211),(161,212),(162,213),(163,214),(164,215),(165,216),(166,217),(167,218),(168,219),(169,220),(170,221),(171,222),(172,223),(173,224),(174,225),(175,226),(176,227),(177,228),(178,229),(179,230),(180,231),(181,232),(182,233),(183,234),(184,235),(185,236),(186,237),(187,238),(188,239),(189,240),(190,241),(191,242),(192,243),(193,244),(194,245),(195,246),(196,247),(197,248),(198,249),(199,250),(200,251),(201,252),(202,253),(203,254),(204,255)], [(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)]])
408 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 12A | ··· | 12H | 17A | ··· | 17P | 34A | ··· | 34AV | 51A | ··· | 51AF | 68A | ··· | 68BL | 102A | ··· | 102CR | 204A | ··· | 204DX |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 | 204 | ··· | 204 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
408 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | C17 | C34 | C34 | C51 | C68 | C102 | C102 | C204 |
| kernel | C2×C204 | C204 | C2×C102 | C2×C68 | C102 | C68 | C2×C34 | C34 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 16 | 32 | 16 | 32 | 64 | 64 | 32 | 128 |
Matrix representation of C2×C204 ►in GL2(𝔽409) generated by
| 1 | 0 |
| 0 | 408 |
| 281 | 0 |
| 0 | 96 |
G:=sub<GL(2,GF(409))| [1,0,0,408],[281,0,0,96] >;
C2×C204 in GAP, Magma, Sage, TeX
C_2\times C_{204} % in TeX
G:=Group("C2xC204"); // GroupNames label
G:=SmallGroup(408,30);
// by ID
G=gap.SmallGroup(408,30);
# by ID
G:=PCGroup([5,-2,-2,-3,-17,-2,1020]);
// Polycyclic
G:=Group<a,b|a^2=b^204=1,a*b=b*a>;
// generators/relations
Export