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G = C2×C220order 440 = 23·5·11

Abelian group of type [2,220]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C220, SmallGroup(440,39)

Series: Derived Chief Lower central Upper central

C1 — C2×C220
C1C2C22C110C220 — C2×C220
C1 — C2×C220
C1 — C2×C220

Generators and relations for C2×C220
 G = < a,b | a2=b220=1, ab=ba >


Smallest permutation representation of C2×C220
Regular action on 440 points
Generators in S440
(1 328)(2 329)(3 330)(4 331)(5 332)(6 333)(7 334)(8 335)(9 336)(10 337)(11 338)(12 339)(13 340)(14 341)(15 342)(16 343)(17 344)(18 345)(19 346)(20 347)(21 348)(22 349)(23 350)(24 351)(25 352)(26 353)(27 354)(28 355)(29 356)(30 357)(31 358)(32 359)(33 360)(34 361)(35 362)(36 363)(37 364)(38 365)(39 366)(40 367)(41 368)(42 369)(43 370)(44 371)(45 372)(46 373)(47 374)(48 375)(49 376)(50 377)(51 378)(52 379)(53 380)(54 381)(55 382)(56 383)(57 384)(58 385)(59 386)(60 387)(61 388)(62 389)(63 390)(64 391)(65 392)(66 393)(67 394)(68 395)(69 396)(70 397)(71 398)(72 399)(73 400)(74 401)(75 402)(76 403)(77 404)(78 405)(79 406)(80 407)(81 408)(82 409)(83 410)(84 411)(85 412)(86 413)(87 414)(88 415)(89 416)(90 417)(91 418)(92 419)(93 420)(94 421)(95 422)(96 423)(97 424)(98 425)(99 426)(100 427)(101 428)(102 429)(103 430)(104 431)(105 432)(106 433)(107 434)(108 435)(109 436)(110 437)(111 438)(112 439)(113 440)(114 221)(115 222)(116 223)(117 224)(118 225)(119 226)(120 227)(121 228)(122 229)(123 230)(124 231)(125 232)(126 233)(127 234)(128 235)(129 236)(130 237)(131 238)(132 239)(133 240)(134 241)(135 242)(136 243)(137 244)(138 245)(139 246)(140 247)(141 248)(142 249)(143 250)(144 251)(145 252)(146 253)(147 254)(148 255)(149 256)(150 257)(151 258)(152 259)(153 260)(154 261)(155 262)(156 263)(157 264)(158 265)(159 266)(160 267)(161 268)(162 269)(163 270)(164 271)(165 272)(166 273)(167 274)(168 275)(169 276)(170 277)(171 278)(172 279)(173 280)(174 281)(175 282)(176 283)(177 284)(178 285)(179 286)(180 287)(181 288)(182 289)(183 290)(184 291)(185 292)(186 293)(187 294)(188 295)(189 296)(190 297)(191 298)(192 299)(193 300)(194 301)(195 302)(196 303)(197 304)(198 305)(199 306)(200 307)(201 308)(202 309)(203 310)(204 311)(205 312)(206 313)(207 314)(208 315)(209 316)(210 317)(211 318)(212 319)(213 320)(214 321)(215 322)(216 323)(217 324)(218 325)(219 326)(220 327)
(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)

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

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

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

440 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D10A···10L11A···11J20A···20P22A···22AD44A···44AN55A···55AN110A···110DP220A···220FD
order12224444555510···1011···1120···2022···2244···4455···55110···110220···220
size1111111111111···11···11···11···11···11···11···11···1

440 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C4C5C10C10C11C20C22C22C44C55C110C110C220
kernelC2×C220C220C2×C110C110C2×C44C44C2×C22C2×C20C22C20C2×C10C10C2×C4C4C22C2
# reps12144841016201040408040160

Matrix representation of C2×C220 in GL2(𝔽661) generated by

10
0660
,
6030
0201
G:=sub<GL(2,GF(661))| [1,0,0,660],[603,0,0,201] >;

C2×C220 in GAP, Magma, Sage, TeX

C_2\times C_{220}
% in TeX

G:=Group("C2xC220");
// GroupNames label

G:=SmallGroup(440,39);
// by ID

G=gap.SmallGroup(440,39);
# by ID

G:=PCGroup([5,-2,-2,-5,-11,-2,1100]);
// Polycyclic

G:=Group<a,b|a^2=b^220=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C220 in TeX

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