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G = C2×C208order 416 = 25·13

Abelian group of type [2,208]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C208, SmallGroup(416,59)

Series: Derived Chief Lower central Upper central

C1 — C2×C208
C1C2C4C8C104C208 — C2×C208
C1 — C2×C208
C1 — C2×C208

Generators and relations for C2×C208
 G = < a,b | a2=b208=1, ab=ba >


Smallest permutation representation of C2×C208
Regular action on 416 points
Generators in S416
(1 256)(2 257)(3 258)(4 259)(5 260)(6 261)(7 262)(8 263)(9 264)(10 265)(11 266)(12 267)(13 268)(14 269)(15 270)(16 271)(17 272)(18 273)(19 274)(20 275)(21 276)(22 277)(23 278)(24 279)(25 280)(26 281)(27 282)(28 283)(29 284)(30 285)(31 286)(32 287)(33 288)(34 289)(35 290)(36 291)(37 292)(38 293)(39 294)(40 295)(41 296)(42 297)(43 298)(44 299)(45 300)(46 301)(47 302)(48 303)(49 304)(50 305)(51 306)(52 307)(53 308)(54 309)(55 310)(56 311)(57 312)(58 313)(59 314)(60 315)(61 316)(62 317)(63 318)(64 319)(65 320)(66 321)(67 322)(68 323)(69 324)(70 325)(71 326)(72 327)(73 328)(74 329)(75 330)(76 331)(77 332)(78 333)(79 334)(80 335)(81 336)(82 337)(83 338)(84 339)(85 340)(86 341)(87 342)(88 343)(89 344)(90 345)(91 346)(92 347)(93 348)(94 349)(95 350)(96 351)(97 352)(98 353)(99 354)(100 355)(101 356)(102 357)(103 358)(104 359)(105 360)(106 361)(107 362)(108 363)(109 364)(110 365)(111 366)(112 367)(113 368)(114 369)(115 370)(116 371)(117 372)(118 373)(119 374)(120 375)(121 376)(122 377)(123 378)(124 379)(125 380)(126 381)(127 382)(128 383)(129 384)(130 385)(131 386)(132 387)(133 388)(134 389)(135 390)(136 391)(137 392)(138 393)(139 394)(140 395)(141 396)(142 397)(143 398)(144 399)(145 400)(146 401)(147 402)(148 403)(149 404)(150 405)(151 406)(152 407)(153 408)(154 409)(155 410)(156 411)(157 412)(158 413)(159 414)(160 415)(161 416)(162 209)(163 210)(164 211)(165 212)(166 213)(167 214)(168 215)(169 216)(170 217)(171 218)(172 219)(173 220)(174 221)(175 222)(176 223)(177 224)(178 225)(179 226)(180 227)(181 228)(182 229)(183 230)(184 231)(185 232)(186 233)(187 234)(188 235)(189 236)(190 237)(191 238)(192 239)(193 240)(194 241)(195 242)(196 243)(197 244)(198 245)(199 246)(200 247)(201 248)(202 249)(203 250)(204 251)(205 252)(206 253)(207 254)(208 255)
(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)

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

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

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

416 conjugacy classes

class 1 2A2B2C4A4B4C4D8A···8H13A···13L16A···16P26A···26AJ52A···52AV104A···104CR208A···208GJ
order122244448···813···1316···1626···2652···52104···104208···208
size111111111···11···11···11···11···11···11···1

416 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C4C4C8C8C13C16C26C26C52C52C104C104C208
kernelC2×C208C208C2×C104C104C2×C52C52C2×C26C2×C16C26C16C2×C8C8C2×C4C4C22C2
# reps12122441216241224244848192

Matrix representation of C2×C208 in GL2(𝔽1249) generated by

12480
01248
,
7710
0992
G:=sub<GL(2,GF(1249))| [1248,0,0,1248],[771,0,0,992] >;

C2×C208 in GAP, Magma, Sage, TeX

C_2\times C_{208}
% in TeX

G:=Group("C2xC208");
// GroupNames label

G:=SmallGroup(416,59);
// by ID

G=gap.SmallGroup(416,59);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,312,69,88]);
// Polycyclic

G:=Group<a,b|a^2=b^208=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C208 in TeX

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