Copied to
clipboard

G = C2×C210order 420 = 22·3·5·7

Abelian group of type [2,210]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C210, SmallGroup(420,41)

Series: Derived Chief Lower central Upper central

C1 — C2×C210
C1C7C35C105C210 — C2×C210
C1 — C2×C210
C1 — C2×C210

Generators and relations for C2×C210
 G = < a,b | a2=b210=1, ab=ba >


Smallest permutation representation of C2×C210
Regular action on 420 points
Generators in S420
(1 274)(2 275)(3 276)(4 277)(5 278)(6 279)(7 280)(8 281)(9 282)(10 283)(11 284)(12 285)(13 286)(14 287)(15 288)(16 289)(17 290)(18 291)(19 292)(20 293)(21 294)(22 295)(23 296)(24 297)(25 298)(26 299)(27 300)(28 301)(29 302)(30 303)(31 304)(32 305)(33 306)(34 307)(35 308)(36 309)(37 310)(38 311)(39 312)(40 313)(41 314)(42 315)(43 316)(44 317)(45 318)(46 319)(47 320)(48 321)(49 322)(50 323)(51 324)(52 325)(53 326)(54 327)(55 328)(56 329)(57 330)(58 331)(59 332)(60 333)(61 334)(62 335)(63 336)(64 337)(65 338)(66 339)(67 340)(68 341)(69 342)(70 343)(71 344)(72 345)(73 346)(74 347)(75 348)(76 349)(77 350)(78 351)(79 352)(80 353)(81 354)(82 355)(83 356)(84 357)(85 358)(86 359)(87 360)(88 361)(89 362)(90 363)(91 364)(92 365)(93 366)(94 367)(95 368)(96 369)(97 370)(98 371)(99 372)(100 373)(101 374)(102 375)(103 376)(104 377)(105 378)(106 379)(107 380)(108 381)(109 382)(110 383)(111 384)(112 385)(113 386)(114 387)(115 388)(116 389)(117 390)(118 391)(119 392)(120 393)(121 394)(122 395)(123 396)(124 397)(125 398)(126 399)(127 400)(128 401)(129 402)(130 403)(131 404)(132 405)(133 406)(134 407)(135 408)(136 409)(137 410)(138 411)(139 412)(140 413)(141 414)(142 415)(143 416)(144 417)(145 418)(146 419)(147 420)(148 211)(149 212)(150 213)(151 214)(152 215)(153 216)(154 217)(155 218)(156 219)(157 220)(158 221)(159 222)(160 223)(161 224)(162 225)(163 226)(164 227)(165 228)(166 229)(167 230)(168 231)(169 232)(170 233)(171 234)(172 235)(173 236)(174 237)(175 238)(176 239)(177 240)(178 241)(179 242)(180 243)(181 244)(182 245)(183 246)(184 247)(185 248)(186 249)(187 250)(188 251)(189 252)(190 253)(191 254)(192 255)(193 256)(194 257)(195 258)(196 259)(197 260)(198 261)(199 262)(200 263)(201 264)(202 265)(203 266)(204 267)(205 268)(206 269)(207 270)(208 271)(209 272)(210 273)
(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)

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

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

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

420 conjugacy classes

class 1 2A2B2C3A3B5A5B5C5D6A···6F7A···7F10A···10L14A···14R15A···15H21A···21L30A···30X35A···35X42A···42AJ70A···70BT105A···105AV210A···210EN
order12223355556···67···710···1014···1415···1521···2130···3035···3542···4270···70105···105210···210
size11111111111···11···11···11···11···11···11···11···11···11···11···11···1

420 irreducible representations

dim1111111111111111
type++
imageC1C2C3C5C6C7C10C14C15C21C30C35C42C70C105C210
kernelC2×C210C210C2×C70C2×C42C70C2×C30C42C30C2×C14C2×C10C14C2×C6C10C6C22C2
# reps13246612188122424367248144

Matrix representation of C2×C210 in GL3(𝔽211) generated by

100
02100
001
,
4200
080
00201
G:=sub<GL(3,GF(211))| [1,0,0,0,210,0,0,0,1],[42,0,0,0,8,0,0,0,201] >;

C2×C210 in GAP, Magma, Sage, TeX

C_2\times C_{210}
% in TeX

G:=Group("C2xC210");
// GroupNames label

G:=SmallGroup(420,41);
// by ID

G=gap.SmallGroup(420,41);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-7]);
// Polycyclic

G:=Group<a,b|a^2=b^210=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C210 in TeX

׿
×
𝔽