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G = C2×C222order 444 = 22·3·37

Abelian group of type [2,222]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C222, SmallGroup(444,18)

Series: Derived Chief Lower central Upper central

C1 — C2×C222
C1C37C111C222 — C2×C222
C1 — C2×C222
C1 — C2×C222

Generators and relations for C2×C222
 G = < a,b | a2=b222=1, ab=ba >


Smallest permutation representation of C2×C222
Regular action on 444 points
Generators in S444
(1 335)(2 336)(3 337)(4 338)(5 339)(6 340)(7 341)(8 342)(9 343)(10 344)(11 345)(12 346)(13 347)(14 348)(15 349)(16 350)(17 351)(18 352)(19 353)(20 354)(21 355)(22 356)(23 357)(24 358)(25 359)(26 360)(27 361)(28 362)(29 363)(30 364)(31 365)(32 366)(33 367)(34 368)(35 369)(36 370)(37 371)(38 372)(39 373)(40 374)(41 375)(42 376)(43 377)(44 378)(45 379)(46 380)(47 381)(48 382)(49 383)(50 384)(51 385)(52 386)(53 387)(54 388)(55 389)(56 390)(57 391)(58 392)(59 393)(60 394)(61 395)(62 396)(63 397)(64 398)(65 399)(66 400)(67 401)(68 402)(69 403)(70 404)(71 405)(72 406)(73 407)(74 408)(75 409)(76 410)(77 411)(78 412)(79 413)(80 414)(81 415)(82 416)(83 417)(84 418)(85 419)(86 420)(87 421)(88 422)(89 423)(90 424)(91 425)(92 426)(93 427)(94 428)(95 429)(96 430)(97 431)(98 432)(99 433)(100 434)(101 435)(102 436)(103 437)(104 438)(105 439)(106 440)(107 441)(108 442)(109 443)(110 444)(111 223)(112 224)(113 225)(114 226)(115 227)(116 228)(117 229)(118 230)(119 231)(120 232)(121 233)(122 234)(123 235)(124 236)(125 237)(126 238)(127 239)(128 240)(129 241)(130 242)(131 243)(132 244)(133 245)(134 246)(135 247)(136 248)(137 249)(138 250)(139 251)(140 252)(141 253)(142 254)(143 255)(144 256)(145 257)(146 258)(147 259)(148 260)(149 261)(150 262)(151 263)(152 264)(153 265)(154 266)(155 267)(156 268)(157 269)(158 270)(159 271)(160 272)(161 273)(162 274)(163 275)(164 276)(165 277)(166 278)(167 279)(168 280)(169 281)(170 282)(171 283)(172 284)(173 285)(174 286)(175 287)(176 288)(177 289)(178 290)(179 291)(180 292)(181 293)(182 294)(183 295)(184 296)(185 297)(186 298)(187 299)(188 300)(189 301)(190 302)(191 303)(192 304)(193 305)(194 306)(195 307)(196 308)(197 309)(198 310)(199 311)(200 312)(201 313)(202 314)(203 315)(204 316)(205 317)(206 318)(207 319)(208 320)(209 321)(210 322)(211 323)(212 324)(213 325)(214 326)(215 327)(216 328)(217 329)(218 330)(219 331)(220 332)(221 333)(222 334)
(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)

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

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

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

444 conjugacy classes

class 1 2A2B2C3A3B6A···6F37A···37AJ74A···74DD111A···111BT222A···222HH
order1222336···637···3774···74111···111222···222
size1111111···11···11···11···11···1

444 irreducible representations

dim11111111
type++
imageC1C2C3C6C37C74C111C222
kernelC2×C222C222C2×C74C74C2×C6C6C22C2
# reps13263610872216

Matrix representation of C2×C222 in GL2(𝔽223) generated by

10
0222
,
1510
0198
G:=sub<GL(2,GF(223))| [1,0,0,222],[151,0,0,198] >;

C2×C222 in GAP, Magma, Sage, TeX

C_2\times C_{222}
% in TeX

G:=Group("C2xC222");
// GroupNames label

G:=SmallGroup(444,18);
// by ID

G=gap.SmallGroup(444,18);
# by ID

G:=PCGroup([4,-2,-2,-3,-37]);
// Polycyclic

G:=Group<a,b|a^2=b^222=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C222 in TeX

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