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G = C2×C214order 428 = 22·107

Abelian group of type [2,214]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C214, SmallGroup(428,4)

Series: Derived Chief Lower central Upper central

C1 — C2×C214
C1C107C214 — C2×C214
C1 — C2×C214
C1 — C2×C214

Generators and relations for C2×C214
 G = < a,b | a2=b214=1, ab=ba >


Smallest permutation representation of C2×C214
Regular action on 428 points
Generators in S428
(1 370)(2 371)(3 372)(4 373)(5 374)(6 375)(7 376)(8 377)(9 378)(10 379)(11 380)(12 381)(13 382)(14 383)(15 384)(16 385)(17 386)(18 387)(19 388)(20 389)(21 390)(22 391)(23 392)(24 393)(25 394)(26 395)(27 396)(28 397)(29 398)(30 399)(31 400)(32 401)(33 402)(34 403)(35 404)(36 405)(37 406)(38 407)(39 408)(40 409)(41 410)(42 411)(43 412)(44 413)(45 414)(46 415)(47 416)(48 417)(49 418)(50 419)(51 420)(52 421)(53 422)(54 423)(55 424)(56 425)(57 426)(58 427)(59 428)(60 215)(61 216)(62 217)(63 218)(64 219)(65 220)(66 221)(67 222)(68 223)(69 224)(70 225)(71 226)(72 227)(73 228)(74 229)(75 230)(76 231)(77 232)(78 233)(79 234)(80 235)(81 236)(82 237)(83 238)(84 239)(85 240)(86 241)(87 242)(88 243)(89 244)(90 245)(91 246)(92 247)(93 248)(94 249)(95 250)(96 251)(97 252)(98 253)(99 254)(100 255)(101 256)(102 257)(103 258)(104 259)(105 260)(106 261)(107 262)(108 263)(109 264)(110 265)(111 266)(112 267)(113 268)(114 269)(115 270)(116 271)(117 272)(118 273)(119 274)(120 275)(121 276)(122 277)(123 278)(124 279)(125 280)(126 281)(127 282)(128 283)(129 284)(130 285)(131 286)(132 287)(133 288)(134 289)(135 290)(136 291)(137 292)(138 293)(139 294)(140 295)(141 296)(142 297)(143 298)(144 299)(145 300)(146 301)(147 302)(148 303)(149 304)(150 305)(151 306)(152 307)(153 308)(154 309)(155 310)(156 311)(157 312)(158 313)(159 314)(160 315)(161 316)(162 317)(163 318)(164 319)(165 320)(166 321)(167 322)(168 323)(169 324)(170 325)(171 326)(172 327)(173 328)(174 329)(175 330)(176 331)(177 332)(178 333)(179 334)(180 335)(181 336)(182 337)(183 338)(184 339)(185 340)(186 341)(187 342)(188 343)(189 344)(190 345)(191 346)(192 347)(193 348)(194 349)(195 350)(196 351)(197 352)(198 353)(199 354)(200 355)(201 356)(202 357)(203 358)(204 359)(205 360)(206 361)(207 362)(208 363)(209 364)(210 365)(211 366)(212 367)(213 368)(214 369)
(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)

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

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

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

428 conjugacy classes

class 1 2A2B2C107A···107DB214A···214LF
order1222107···107214···214
size11111···11···1

428 irreducible representations

dim1111
type++
imageC1C2C107C214
kernelC2×C214C214C22C2
# reps13106318

Matrix representation of C2×C214 in GL2(𝔽643) generated by

6420
0642
,
4760
0536
G:=sub<GL(2,GF(643))| [642,0,0,642],[476,0,0,536] >;

C2×C214 in GAP, Magma, Sage, TeX

C_2\times C_{214}
% in TeX

G:=Group("C2xC214");
// GroupNames label

G:=SmallGroup(428,4);
// by ID

G=gap.SmallGroup(428,4);
# by ID

G:=PCGroup([3,-2,-2,-107]);
// Polycyclic

G:=Group<a,b|a^2=b^214=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C214 in TeX

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