direct product, abelian, monomial, 2-elementary
Aliases: C2×C232, SmallGroup(464,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C232 |
| C1 — C2×C232 |
| C1 — C2×C232 |
Generators and relations for C2×C232
G = < a,b | a2=b232=1, ab=ba >
Smallest permutation representation of C2×C232
►Regular action on 464 points
Generators in S464
(1 235)(2 236)(3 237)(4 238)(5 239)(6 240)(7 241)(8 242)(9 243)(10 244)(11 245)(12 246)(13 247)(14 248)(15 249)(16 250)(17 251)(18 252)(19 253)(20 254)(21 255)(22 256)(23 257)(24 258)(25 259)(26 260)(27 261)(28 262)(29 263)(30 264)(31 265)(32 266)(33 267)(34 268)(35 269)(36 270)(37 271)(38 272)(39 273)(40 274)(41 275)(42 276)(43 277)(44 278)(45 279)(46 280)(47 281)(48 282)(49 283)(50 284)(51 285)(52 286)(53 287)(54 288)(55 289)(56 290)(57 291)(58 292)(59 293)(60 294)(61 295)(62 296)(63 297)(64 298)(65 299)(66 300)(67 301)(68 302)(69 303)(70 304)(71 305)(72 306)(73 307)(74 308)(75 309)(76 310)(77 311)(78 312)(79 313)(80 314)(81 315)(82 316)(83 317)(84 318)(85 319)(86 320)(87 321)(88 322)(89 323)(90 324)(91 325)(92 326)(93 327)(94 328)(95 329)(96 330)(97 331)(98 332)(99 333)(100 334)(101 335)(102 336)(103 337)(104 338)(105 339)(106 340)(107 341)(108 342)(109 343)(110 344)(111 345)(112 346)(113 347)(114 348)(115 349)(116 350)(117 351)(118 352)(119 353)(120 354)(121 355)(122 356)(123 357)(124 358)(125 359)(126 360)(127 361)(128 362)(129 363)(130 364)(131 365)(132 366)(133 367)(134 368)(135 369)(136 370)(137 371)(138 372)(139 373)(140 374)(141 375)(142 376)(143 377)(144 378)(145 379)(146 380)(147 381)(148 382)(149 383)(150 384)(151 385)(152 386)(153 387)(154 388)(155 389)(156 390)(157 391)(158 392)(159 393)(160 394)(161 395)(162 396)(163 397)(164 398)(165 399)(166 400)(167 401)(168 402)(169 403)(170 404)(171 405)(172 406)(173 407)(174 408)(175 409)(176 410)(177 411)(178 412)(179 413)(180 414)(181 415)(182 416)(183 417)(184 418)(185 419)(186 420)(187 421)(188 422)(189 423)(190 424)(191 425)(192 426)(193 427)(194 428)(195 429)(196 430)(197 431)(198 432)(199 433)(200 434)(201 435)(202 436)(203 437)(204 438)(205 439)(206 440)(207 441)(208 442)(209 443)(210 444)(211 445)(212 446)(213 447)(214 448)(215 449)(216 450)(217 451)(218 452)(219 453)(220 454)(221 455)(222 456)(223 457)(224 458)(225 459)(226 460)(227 461)(228 462)(229 463)(230 464)(231 233)(232 234)
(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)(233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464)
G:=sub<Sym(464)| (1,235)(2,236)(3,237)(4,238)(5,239)(6,240)(7,241)(8,242)(9,243)(10,244)(11,245)(12,246)(13,247)(14,248)(15,249)(16,250)(17,251)(18,252)(19,253)(20,254)(21,255)(22,256)(23,257)(24,258)(25,259)(26,260)(27,261)(28,262)(29,263)(30,264)(31,265)(32,266)(33,267)(34,268)(35,269)(36,270)(37,271)(38,272)(39,273)(40,274)(41,275)(42,276)(43,277)(44,278)(45,279)(46,280)(47,281)(48,282)(49,283)(50,284)(51,285)(52,286)(53,287)(54,288)(55,289)(56,290)(57,291)(58,292)(59,293)(60,294)(61,295)(62,296)(63,297)(64,298)(65,299)(66,300)(67,301)(68,302)(69,303)(70,304)(71,305)(72,306)(73,307)(74,308)(75,309)(76,310)(77,311)(78,312)(79,313)(80,314)(81,315)(82,316)(83,317)(84,318)(85,319)(86,320)(87,321)(88,322)(89,323)(90,324)(91,325)(92,326)(93,327)(94,328)(95,329)(96,330)(97,331)(98,332)(99,333)(100,334)(101,335)(102,336)(103,337)(104,338)(105,339)(106,340)(107,341)(108,342)(109,343)(110,344)(111,345)(112,346)(113,347)(114,348)(115,349)(116,350)(117,351)(118,352)(119,353)(120,354)(121,355)(122,356)(123,357)(124,358)(125,359)(126,360)(127,361)(128,362)(129,363)(130,364)(131,365)(132,366)(133,367)(134,368)(135,369)(136,370)(137,371)(138,372)(139,373)(140,374)(141,375)(142,376)(143,377)(144,378)(145,379)(146,380)(147,381)(148,382)(149,383)(150,384)(151,385)(152,386)(153,387)(154,388)(155,389)(156,390)(157,391)(158,392)(159,393)(160,394)(161,395)(162,396)(163,397)(164,398)(165,399)(166,400)(167,401)(168,402)(169,403)(170,404)(171,405)(172,406)(173,407)(174,408)(175,409)(176,410)(177,411)(178,412)(179,413)(180,414)(181,415)(182,416)(183,417)(184,418)(185,419)(186,420)(187,421)(188,422)(189,423)(190,424)(191,425)(192,426)(193,427)(194,428)(195,429)(196,430)(197,431)(198,432)(199,433)(200,434)(201,435)(202,436)(203,437)(204,438)(205,439)(206,440)(207,441)(208,442)(209,443)(210,444)(211,445)(212,446)(213,447)(214,448)(215,449)(216,450)(217,451)(218,452)(219,453)(220,454)(221,455)(222,456)(223,457)(224,458)(225,459)(226,460)(227,461)(228,462)(229,463)(230,464)(231,233)(232,234), (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)(233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464)>;
G:=Group( (1,235)(2,236)(3,237)(4,238)(5,239)(6,240)(7,241)(8,242)(9,243)(10,244)(11,245)(12,246)(13,247)(14,248)(15,249)(16,250)(17,251)(18,252)(19,253)(20,254)(21,255)(22,256)(23,257)(24,258)(25,259)(26,260)(27,261)(28,262)(29,263)(30,264)(31,265)(32,266)(33,267)(34,268)(35,269)(36,270)(37,271)(38,272)(39,273)(40,274)(41,275)(42,276)(43,277)(44,278)(45,279)(46,280)(47,281)(48,282)(49,283)(50,284)(51,285)(52,286)(53,287)(54,288)(55,289)(56,290)(57,291)(58,292)(59,293)(60,294)(61,295)(62,296)(63,297)(64,298)(65,299)(66,300)(67,301)(68,302)(69,303)(70,304)(71,305)(72,306)(73,307)(74,308)(75,309)(76,310)(77,311)(78,312)(79,313)(80,314)(81,315)(82,316)(83,317)(84,318)(85,319)(86,320)(87,321)(88,322)(89,323)(90,324)(91,325)(92,326)(93,327)(94,328)(95,329)(96,330)(97,331)(98,332)(99,333)(100,334)(101,335)(102,336)(103,337)(104,338)(105,339)(106,340)(107,341)(108,342)(109,343)(110,344)(111,345)(112,346)(113,347)(114,348)(115,349)(116,350)(117,351)(118,352)(119,353)(120,354)(121,355)(122,356)(123,357)(124,358)(125,359)(126,360)(127,361)(128,362)(129,363)(130,364)(131,365)(132,366)(133,367)(134,368)(135,369)(136,370)(137,371)(138,372)(139,373)(140,374)(141,375)(142,376)(143,377)(144,378)(145,379)(146,380)(147,381)(148,382)(149,383)(150,384)(151,385)(152,386)(153,387)(154,388)(155,389)(156,390)(157,391)(158,392)(159,393)(160,394)(161,395)(162,396)(163,397)(164,398)(165,399)(166,400)(167,401)(168,402)(169,403)(170,404)(171,405)(172,406)(173,407)(174,408)(175,409)(176,410)(177,411)(178,412)(179,413)(180,414)(181,415)(182,416)(183,417)(184,418)(185,419)(186,420)(187,421)(188,422)(189,423)(190,424)(191,425)(192,426)(193,427)(194,428)(195,429)(196,430)(197,431)(198,432)(199,433)(200,434)(201,435)(202,436)(203,437)(204,438)(205,439)(206,440)(207,441)(208,442)(209,443)(210,444)(211,445)(212,446)(213,447)(214,448)(215,449)(216,450)(217,451)(218,452)(219,453)(220,454)(221,455)(222,456)(223,457)(224,458)(225,459)(226,460)(227,461)(228,462)(229,463)(230,464)(231,233)(232,234), (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)(233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464) );
G=PermutationGroup([[(1,235),(2,236),(3,237),(4,238),(5,239),(6,240),(7,241),(8,242),(9,243),(10,244),(11,245),(12,246),(13,247),(14,248),(15,249),(16,250),(17,251),(18,252),(19,253),(20,254),(21,255),(22,256),(23,257),(24,258),(25,259),(26,260),(27,261),(28,262),(29,263),(30,264),(31,265),(32,266),(33,267),(34,268),(35,269),(36,270),(37,271),(38,272),(39,273),(40,274),(41,275),(42,276),(43,277),(44,278),(45,279),(46,280),(47,281),(48,282),(49,283),(50,284),(51,285),(52,286),(53,287),(54,288),(55,289),(56,290),(57,291),(58,292),(59,293),(60,294),(61,295),(62,296),(63,297),(64,298),(65,299),(66,300),(67,301),(68,302),(69,303),(70,304),(71,305),(72,306),(73,307),(74,308),(75,309),(76,310),(77,311),(78,312),(79,313),(80,314),(81,315),(82,316),(83,317),(84,318),(85,319),(86,320),(87,321),(88,322),(89,323),(90,324),(91,325),(92,326),(93,327),(94,328),(95,329),(96,330),(97,331),(98,332),(99,333),(100,334),(101,335),(102,336),(103,337),(104,338),(105,339),(106,340),(107,341),(108,342),(109,343),(110,344),(111,345),(112,346),(113,347),(114,348),(115,349),(116,350),(117,351),(118,352),(119,353),(120,354),(121,355),(122,356),(123,357),(124,358),(125,359),(126,360),(127,361),(128,362),(129,363),(130,364),(131,365),(132,366),(133,367),(134,368),(135,369),(136,370),(137,371),(138,372),(139,373),(140,374),(141,375),(142,376),(143,377),(144,378),(145,379),(146,380),(147,381),(148,382),(149,383),(150,384),(151,385),(152,386),(153,387),(154,388),(155,389),(156,390),(157,391),(158,392),(159,393),(160,394),(161,395),(162,396),(163,397),(164,398),(165,399),(166,400),(167,401),(168,402),(169,403),(170,404),(171,405),(172,406),(173,407),(174,408),(175,409),(176,410),(177,411),(178,412),(179,413),(180,414),(181,415),(182,416),(183,417),(184,418),(185,419),(186,420),(187,421),(188,422),(189,423),(190,424),(191,425),(192,426),(193,427),(194,428),(195,429),(196,430),(197,431),(198,432),(199,433),(200,434),(201,435),(202,436),(203,437),(204,438),(205,439),(206,440),(207,441),(208,442),(209,443),(210,444),(211,445),(212,446),(213,447),(214,448),(215,449),(216,450),(217,451),(218,452),(219,453),(220,454),(221,455),(222,456),(223,457),(224,458),(225,459),(226,460),(227,461),(228,462),(229,463),(230,464),(231,233),(232,234)], [(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),(233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464)]])
464 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 29A | ··· | 29AB | 58A | ··· | 58CF | 116A | ··· | 116DH | 232A | ··· | 232HP |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 | 232 | ··· | 232 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
464 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C29 | C58 | C58 | C116 | C116 | C232 |
| kernel | C2×C232 | C232 | C2×C116 | C116 | C2×C58 | C58 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 28 | 56 | 28 | 56 | 56 | 224 |
Matrix representation of C2×C232 ►in GL2(𝔽233) generated by
| 232 | 0 |
| 0 | 1 |
| 50 | 0 |
| 0 | 39 |
G:=sub<GL(2,GF(233))| [232,0,0,1],[50,0,0,39] >;
C2×C232 in GAP, Magma, Sage, TeX
C_2\times C_{232} % in TeX
G:=Group("C2xC232"); // GroupNames label
G:=SmallGroup(464,23);
// by ID
G=gap.SmallGroup(464,23);
# by ID
G:=PCGroup([5,-2,-2,-29,-2,-2,580,58]);
// Polycyclic
G:=Group<a,b|a^2=b^232=1,a*b=b*a>;
// generators/relations
Export