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G = C2×C228order 456 = 23·3·19

Abelian group of type [2,228]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C228, SmallGroup(456,39)

Series: Derived Chief Lower central Upper central

C1 — C2×C228
C1C2C38C114C228 — C2×C228
C1 — C2×C228
C1 — C2×C228

Generators and relations for C2×C228
 G = < a,b | a2=b228=1, ab=ba >


Smallest permutation representation of C2×C228
Regular action on 456 points
Generators in S456
(1 242)(2 243)(3 244)(4 245)(5 246)(6 247)(7 248)(8 249)(9 250)(10 251)(11 252)(12 253)(13 254)(14 255)(15 256)(16 257)(17 258)(18 259)(19 260)(20 261)(21 262)(22 263)(23 264)(24 265)(25 266)(26 267)(27 268)(28 269)(29 270)(30 271)(31 272)(32 273)(33 274)(34 275)(35 276)(36 277)(37 278)(38 279)(39 280)(40 281)(41 282)(42 283)(43 284)(44 285)(45 286)(46 287)(47 288)(48 289)(49 290)(50 291)(51 292)(52 293)(53 294)(54 295)(55 296)(56 297)(57 298)(58 299)(59 300)(60 301)(61 302)(62 303)(63 304)(64 305)(65 306)(66 307)(67 308)(68 309)(69 310)(70 311)(71 312)(72 313)(73 314)(74 315)(75 316)(76 317)(77 318)(78 319)(79 320)(80 321)(81 322)(82 323)(83 324)(84 325)(85 326)(86 327)(87 328)(88 329)(89 330)(90 331)(91 332)(92 333)(93 334)(94 335)(95 336)(96 337)(97 338)(98 339)(99 340)(100 341)(101 342)(102 343)(103 344)(104 345)(105 346)(106 347)(107 348)(108 349)(109 350)(110 351)(111 352)(112 353)(113 354)(114 355)(115 356)(116 357)(117 358)(118 359)(119 360)(120 361)(121 362)(122 363)(123 364)(124 365)(125 366)(126 367)(127 368)(128 369)(129 370)(130 371)(131 372)(132 373)(133 374)(134 375)(135 376)(136 377)(137 378)(138 379)(139 380)(140 381)(141 382)(142 383)(143 384)(144 385)(145 386)(146 387)(147 388)(148 389)(149 390)(150 391)(151 392)(152 393)(153 394)(154 395)(155 396)(156 397)(157 398)(158 399)(159 400)(160 401)(161 402)(162 403)(163 404)(164 405)(165 406)(166 407)(167 408)(168 409)(169 410)(170 411)(171 412)(172 413)(173 414)(174 415)(175 416)(176 417)(177 418)(178 419)(179 420)(180 421)(181 422)(182 423)(183 424)(184 425)(185 426)(186 427)(187 428)(188 429)(189 430)(190 431)(191 432)(192 433)(193 434)(194 435)(195 436)(196 437)(197 438)(198 439)(199 440)(200 441)(201 442)(202 443)(203 444)(204 445)(205 446)(206 447)(207 448)(208 449)(209 450)(210 451)(211 452)(212 453)(213 454)(214 455)(215 456)(216 229)(217 230)(218 231)(219 232)(220 233)(221 234)(222 235)(223 236)(224 237)(225 238)(226 239)(227 240)(228 241)
(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)

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

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

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

456 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F12A···12H19A···19R38A···38BB57A···57AJ76A···76BT114A···114DD228A···228EN
order12223344446···612···1219···1938···3857···5776···76114···114228···228
size11111111111···11···11···11···11···11···11···11···1

456 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C4C6C6C12C19C38C38C57C76C114C114C228
kernelC2×C228C228C2×C114C2×C76C114C76C2×C38C38C2×C12C12C2×C6C2×C4C6C4C22C2
# reps1212442818361836727236144

Matrix representation of C2×C228 in GL2(𝔽229) generated by

2280
01
,
30
0211
G:=sub<GL(2,GF(229))| [228,0,0,1],[3,0,0,211] >;

C2×C228 in GAP, Magma, Sage, TeX

C_2\times C_{228}
% in TeX

G:=Group("C2xC228");
// GroupNames label

G:=SmallGroup(456,39);
// by ID

G=gap.SmallGroup(456,39);
# by ID

G:=PCGroup([5,-2,-2,-3,-19,-2,1140]);
// Polycyclic

G:=Group<a,b|a^2=b^228=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C228 in TeX

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