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G = C2×C226order 452 = 22·113

Abelian group of type [2,226]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C226, SmallGroup(452,5)

Series: Derived Chief Lower central Upper central

C1 — C2×C226
C1C113C226 — C2×C226
C1 — C2×C226
C1 — C2×C226

Generators and relations for C2×C226
 G = < a,b | a2=b226=1, ab=ba >


Smallest permutation representation of C2×C226
Regular action on 452 points
Generators in S452
(1 365)(2 366)(3 367)(4 368)(5 369)(6 370)(7 371)(8 372)(9 373)(10 374)(11 375)(12 376)(13 377)(14 378)(15 379)(16 380)(17 381)(18 382)(19 383)(20 384)(21 385)(22 386)(23 387)(24 388)(25 389)(26 390)(27 391)(28 392)(29 393)(30 394)(31 395)(32 396)(33 397)(34 398)(35 399)(36 400)(37 401)(38 402)(39 403)(40 404)(41 405)(42 406)(43 407)(44 408)(45 409)(46 410)(47 411)(48 412)(49 413)(50 414)(51 415)(52 416)(53 417)(54 418)(55 419)(56 420)(57 421)(58 422)(59 423)(60 424)(61 425)(62 426)(63 427)(64 428)(65 429)(66 430)(67 431)(68 432)(69 433)(70 434)(71 435)(72 436)(73 437)(74 438)(75 439)(76 440)(77 441)(78 442)(79 443)(80 444)(81 445)(82 446)(83 447)(84 448)(85 449)(86 450)(87 451)(88 452)(89 227)(90 228)(91 229)(92 230)(93 231)(94 232)(95 233)(96 234)(97 235)(98 236)(99 237)(100 238)(101 239)(102 240)(103 241)(104 242)(105 243)(106 244)(107 245)(108 246)(109 247)(110 248)(111 249)(112 250)(113 251)(114 252)(115 253)(116 254)(117 255)(118 256)(119 257)(120 258)(121 259)(122 260)(123 261)(124 262)(125 263)(126 264)(127 265)(128 266)(129 267)(130 268)(131 269)(132 270)(133 271)(134 272)(135 273)(136 274)(137 275)(138 276)(139 277)(140 278)(141 279)(142 280)(143 281)(144 282)(145 283)(146 284)(147 285)(148 286)(149 287)(150 288)(151 289)(152 290)(153 291)(154 292)(155 293)(156 294)(157 295)(158 296)(159 297)(160 298)(161 299)(162 300)(163 301)(164 302)(165 303)(166 304)(167 305)(168 306)(169 307)(170 308)(171 309)(172 310)(173 311)(174 312)(175 313)(176 314)(177 315)(178 316)(179 317)(180 318)(181 319)(182 320)(183 321)(184 322)(185 323)(186 324)(187 325)(188 326)(189 327)(190 328)(191 329)(192 330)(193 331)(194 332)(195 333)(196 334)(197 335)(198 336)(199 337)(200 338)(201 339)(202 340)(203 341)(204 342)(205 343)(206 344)(207 345)(208 346)(209 347)(210 348)(211 349)(212 350)(213 351)(214 352)(215 353)(216 354)(217 355)(218 356)(219 357)(220 358)(221 359)(222 360)(223 361)(224 362)(225 363)(226 364)
(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)

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

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

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

452 conjugacy classes

class 1 2A2B2C113A···113DH226A···226LX
order1222113···113226···226
size11111···11···1

452 irreducible representations

dim1111
type++
imageC1C2C113C226
kernelC2×C226C226C22C2
# reps13112336

Matrix representation of C2×C226 in GL2(𝔽227) generated by

10
0226
,
550
015
G:=sub<GL(2,GF(227))| [1,0,0,226],[55,0,0,15] >;

C2×C226 in GAP, Magma, Sage, TeX

C_2\times C_{226}
% in TeX

G:=Group("C2xC226");
// GroupNames label

G:=SmallGroup(452,5);
// by ID

G=gap.SmallGroup(452,5);
# by ID

G:=PCGroup([3,-2,-2,-113]);
// Polycyclic

G:=Group<a,b|a^2=b^226=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C226 in TeX

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