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G = C2×C232order 464 = 24·29

Abelian group of type [2,232]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C232, SmallGroup(464,23)

Series: Derived Chief Lower central Upper central

C1 — C2×C232
C1C2C4C116C232 — 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
(1 341)(2 342)(3 343)(4 344)(5 345)(6 346)(7 347)(8 348)(9 349)(10 350)(11 351)(12 352)(13 353)(14 354)(15 355)(16 356)(17 357)(18 358)(19 359)(20 360)(21 361)(22 362)(23 363)(24 364)(25 365)(26 366)(27 367)(28 368)(29 369)(30 370)(31 371)(32 372)(33 373)(34 374)(35 375)(36 376)(37 377)(38 378)(39 379)(40 380)(41 381)(42 382)(43 383)(44 384)(45 385)(46 386)(47 387)(48 388)(49 389)(50 390)(51 391)(52 392)(53 393)(54 394)(55 395)(56 396)(57 397)(58 398)(59 399)(60 400)(61 401)(62 402)(63 403)(64 404)(65 405)(66 406)(67 407)(68 408)(69 409)(70 410)(71 411)(72 412)(73 413)(74 414)(75 415)(76 416)(77 417)(78 418)(79 419)(80 420)(81 421)(82 422)(83 423)(84 424)(85 425)(86 426)(87 427)(88 428)(89 429)(90 430)(91 431)(92 432)(93 433)(94 434)(95 435)(96 436)(97 437)(98 438)(99 439)(100 440)(101 441)(102 442)(103 443)(104 444)(105 445)(106 446)(107 447)(108 448)(109 449)(110 450)(111 451)(112 452)(113 453)(114 454)(115 455)(116 456)(117 457)(118 458)(119 459)(120 460)(121 461)(122 462)(123 463)(124 464)(125 233)(126 234)(127 235)(128 236)(129 237)(130 238)(131 239)(132 240)(133 241)(134 242)(135 243)(136 244)(137 245)(138 246)(139 247)(140 248)(141 249)(142 250)(143 251)(144 252)(145 253)(146 254)(147 255)(148 256)(149 257)(150 258)(151 259)(152 260)(153 261)(154 262)(155 263)(156 264)(157 265)(158 266)(159 267)(160 268)(161 269)(162 270)(163 271)(164 272)(165 273)(166 274)(167 275)(168 276)(169 277)(170 278)(171 279)(172 280)(173 281)(174 282)(175 283)(176 284)(177 285)(178 286)(179 287)(180 288)(181 289)(182 290)(183 291)(184 292)(185 293)(186 294)(187 295)(188 296)(189 297)(190 298)(191 299)(192 300)(193 301)(194 302)(195 303)(196 304)(197 305)(198 306)(199 307)(200 308)(201 309)(202 310)(203 311)(204 312)(205 313)(206 314)(207 315)(208 316)(209 317)(210 318)(211 319)(212 320)(213 321)(214 322)(215 323)(216 324)(217 325)(218 326)(219 327)(220 328)(221 329)(222 330)(223 331)(224 332)(225 333)(226 334)(227 335)(228 336)(229 337)(230 338)(231 339)(232 340)
(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)| (1,341)(2,342)(3,343)(4,344)(5,345)(6,346)(7,347)(8,348)(9,349)(10,350)(11,351)(12,352)(13,353)(14,354)(15,355)(16,356)(17,357)(18,358)(19,359)(20,360)(21,361)(22,362)(23,363)(24,364)(25,365)(26,366)(27,367)(28,368)(29,369)(30,370)(31,371)(32,372)(33,373)(34,374)(35,375)(36,376)(37,377)(38,378)(39,379)(40,380)(41,381)(42,382)(43,383)(44,384)(45,385)(46,386)(47,387)(48,388)(49,389)(50,390)(51,391)(52,392)(53,393)(54,394)(55,395)(56,396)(57,397)(58,398)(59,399)(60,400)(61,401)(62,402)(63,403)(64,404)(65,405)(66,406)(67,407)(68,408)(69,409)(70,410)(71,411)(72,412)(73,413)(74,414)(75,415)(76,416)(77,417)(78,418)(79,419)(80,420)(81,421)(82,422)(83,423)(84,424)(85,425)(86,426)(87,427)(88,428)(89,429)(90,430)(91,431)(92,432)(93,433)(94,434)(95,435)(96,436)(97,437)(98,438)(99,439)(100,440)(101,441)(102,442)(103,443)(104,444)(105,445)(106,446)(107,447)(108,448)(109,449)(110,450)(111,451)(112,452)(113,453)(114,454)(115,455)(116,456)(117,457)(118,458)(119,459)(120,460)(121,461)(122,462)(123,463)(124,464)(125,233)(126,234)(127,235)(128,236)(129,237)(130,238)(131,239)(132,240)(133,241)(134,242)(135,243)(136,244)(137,245)(138,246)(139,247)(140,248)(141,249)(142,250)(143,251)(144,252)(145,253)(146,254)(147,255)(148,256)(149,257)(150,258)(151,259)(152,260)(153,261)(154,262)(155,263)(156,264)(157,265)(158,266)(159,267)(160,268)(161,269)(162,270)(163,271)(164,272)(165,273)(166,274)(167,275)(168,276)(169,277)(170,278)(171,279)(172,280)(173,281)(174,282)(175,283)(176,284)(177,285)(178,286)(179,287)(180,288)(181,289)(182,290)(183,291)(184,292)(185,293)(186,294)(187,295)(188,296)(189,297)(190,298)(191,299)(192,300)(193,301)(194,302)(195,303)(196,304)(197,305)(198,306)(199,307)(200,308)(201,309)(202,310)(203,311)(204,312)(205,313)(206,314)(207,315)(208,316)(209,317)(210,318)(211,319)(212,320)(213,321)(214,322)(215,323)(216,324)(217,325)(218,326)(219,327)(220,328)(221,329)(222,330)(223,331)(224,332)(225,333)(226,334)(227,335)(228,336)(229,337)(230,338)(231,339)(232,340), (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( (1,341)(2,342)(3,343)(4,344)(5,345)(6,346)(7,347)(8,348)(9,349)(10,350)(11,351)(12,352)(13,353)(14,354)(15,355)(16,356)(17,357)(18,358)(19,359)(20,360)(21,361)(22,362)(23,363)(24,364)(25,365)(26,366)(27,367)(28,368)(29,369)(30,370)(31,371)(32,372)(33,373)(34,374)(35,375)(36,376)(37,377)(38,378)(39,379)(40,380)(41,381)(42,382)(43,383)(44,384)(45,385)(46,386)(47,387)(48,388)(49,389)(50,390)(51,391)(52,392)(53,393)(54,394)(55,395)(56,396)(57,397)(58,398)(59,399)(60,400)(61,401)(62,402)(63,403)(64,404)(65,405)(66,406)(67,407)(68,408)(69,409)(70,410)(71,411)(72,412)(73,413)(74,414)(75,415)(76,416)(77,417)(78,418)(79,419)(80,420)(81,421)(82,422)(83,423)(84,424)(85,425)(86,426)(87,427)(88,428)(89,429)(90,430)(91,431)(92,432)(93,433)(94,434)(95,435)(96,436)(97,437)(98,438)(99,439)(100,440)(101,441)(102,442)(103,443)(104,444)(105,445)(106,446)(107,447)(108,448)(109,449)(110,450)(111,451)(112,452)(113,453)(114,454)(115,455)(116,456)(117,457)(118,458)(119,459)(120,460)(121,461)(122,462)(123,463)(124,464)(125,233)(126,234)(127,235)(128,236)(129,237)(130,238)(131,239)(132,240)(133,241)(134,242)(135,243)(136,244)(137,245)(138,246)(139,247)(140,248)(141,249)(142,250)(143,251)(144,252)(145,253)(146,254)(147,255)(148,256)(149,257)(150,258)(151,259)(152,260)(153,261)(154,262)(155,263)(156,264)(157,265)(158,266)(159,267)(160,268)(161,269)(162,270)(163,271)(164,272)(165,273)(166,274)(167,275)(168,276)(169,277)(170,278)(171,279)(172,280)(173,281)(174,282)(175,283)(176,284)(177,285)(178,286)(179,287)(180,288)(181,289)(182,290)(183,291)(184,292)(185,293)(186,294)(187,295)(188,296)(189,297)(190,298)(191,299)(192,300)(193,301)(194,302)(195,303)(196,304)(197,305)(198,306)(199,307)(200,308)(201,309)(202,310)(203,311)(204,312)(205,313)(206,314)(207,315)(208,316)(209,317)(210,318)(211,319)(212,320)(213,321)(214,322)(215,323)(216,324)(217,325)(218,326)(219,327)(220,328)(221,329)(222,330)(223,331)(224,332)(225,333)(226,334)(227,335)(228,336)(229,337)(230,338)(231,339)(232,340), (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([(1,341),(2,342),(3,343),(4,344),(5,345),(6,346),(7,347),(8,348),(9,349),(10,350),(11,351),(12,352),(13,353),(14,354),(15,355),(16,356),(17,357),(18,358),(19,359),(20,360),(21,361),(22,362),(23,363),(24,364),(25,365),(26,366),(27,367),(28,368),(29,369),(30,370),(31,371),(32,372),(33,373),(34,374),(35,375),(36,376),(37,377),(38,378),(39,379),(40,380),(41,381),(42,382),(43,383),(44,384),(45,385),(46,386),(47,387),(48,388),(49,389),(50,390),(51,391),(52,392),(53,393),(54,394),(55,395),(56,396),(57,397),(58,398),(59,399),(60,400),(61,401),(62,402),(63,403),(64,404),(65,405),(66,406),(67,407),(68,408),(69,409),(70,410),(71,411),(72,412),(73,413),(74,414),(75,415),(76,416),(77,417),(78,418),(79,419),(80,420),(81,421),(82,422),(83,423),(84,424),(85,425),(86,426),(87,427),(88,428),(89,429),(90,430),(91,431),(92,432),(93,433),(94,434),(95,435),(96,436),(97,437),(98,438),(99,439),(100,440),(101,441),(102,442),(103,443),(104,444),(105,445),(106,446),(107,447),(108,448),(109,449),(110,450),(111,451),(112,452),(113,453),(114,454),(115,455),(116,456),(117,457),(118,458),(119,459),(120,460),(121,461),(122,462),(123,463),(124,464),(125,233),(126,234),(127,235),(128,236),(129,237),(130,238),(131,239),(132,240),(133,241),(134,242),(135,243),(136,244),(137,245),(138,246),(139,247),(140,248),(141,249),(142,250),(143,251),(144,252),(145,253),(146,254),(147,255),(148,256),(149,257),(150,258),(151,259),(152,260),(153,261),(154,262),(155,263),(156,264),(157,265),(158,266),(159,267),(160,268),(161,269),(162,270),(163,271),(164,272),(165,273),(166,274),(167,275),(168,276),(169,277),(170,278),(171,279),(172,280),(173,281),(174,282),(175,283),(176,284),(177,285),(178,286),(179,287),(180,288),(181,289),(182,290),(183,291),(184,292),(185,293),(186,294),(187,295),(188,296),(189,297),(190,298),(191,299),(192,300),(193,301),(194,302),(195,303),(196,304),(197,305),(198,306),(199,307),(200,308),(201,309),(202,310),(203,311),(204,312),(205,313),(206,314),(207,315),(208,316),(209,317),(210,318),(211,319),(212,320),(213,321),(214,322),(215,323),(216,324),(217,325),(218,326),(219,327),(220,328),(221,329),(222,330),(223,331),(224,332),(225,333),(226,334),(227,335),(228,336),(229,337),(230,338),(231,339),(232,340)], [(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 2A2B2C4A4B4C4D8A···8H29A···29AB58A···58CF116A···116DH232A···232HP
order122244448···829···2958···58116···116232···232
size111111111···11···11···11···11···1

464 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C8C29C58C58C116C116C232
kernelC2×C232C232C2×C116C116C2×C58C58C2×C8C8C2×C4C4C22C2
# reps1212282856285656224

Matrix representation of C2×C232 in GL2(𝔽233) generated by

2320
01
,
500
039
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

Subgroup lattice of C2×C232 in TeX

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