Copied to
clipboard

G = C2×C236order 472 = 23·59

Abelian group of type [2,236]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C236, SmallGroup(472,8)

Series: Derived Chief Lower central Upper central

C1 — C2×C236
C1C2C118C236 — C2×C236
C1 — C2×C236
C1 — C2×C236

Generators and relations for C2×C236
 G = < a,b | a2=b236=1, ab=ba >


Smallest permutation representation of C2×C236
Regular action on 472 points
Generators in S472
(1 287)(2 288)(3 289)(4 290)(5 291)(6 292)(7 293)(8 294)(9 295)(10 296)(11 297)(12 298)(13 299)(14 300)(15 301)(16 302)(17 303)(18 304)(19 305)(20 306)(21 307)(22 308)(23 309)(24 310)(25 311)(26 312)(27 313)(28 314)(29 315)(30 316)(31 317)(32 318)(33 319)(34 320)(35 321)(36 322)(37 323)(38 324)(39 325)(40 326)(41 327)(42 328)(43 329)(44 330)(45 331)(46 332)(47 333)(48 334)(49 335)(50 336)(51 337)(52 338)(53 339)(54 340)(55 341)(56 342)(57 343)(58 344)(59 345)(60 346)(61 347)(62 348)(63 349)(64 350)(65 351)(66 352)(67 353)(68 354)(69 355)(70 356)(71 357)(72 358)(73 359)(74 360)(75 361)(76 362)(77 363)(78 364)(79 365)(80 366)(81 367)(82 368)(83 369)(84 370)(85 371)(86 372)(87 373)(88 374)(89 375)(90 376)(91 377)(92 378)(93 379)(94 380)(95 381)(96 382)(97 383)(98 384)(99 385)(100 386)(101 387)(102 388)(103 389)(104 390)(105 391)(106 392)(107 393)(108 394)(109 395)(110 396)(111 397)(112 398)(113 399)(114 400)(115 401)(116 402)(117 403)(118 404)(119 405)(120 406)(121 407)(122 408)(123 409)(124 410)(125 411)(126 412)(127 413)(128 414)(129 415)(130 416)(131 417)(132 418)(133 419)(134 420)(135 421)(136 422)(137 423)(138 424)(139 425)(140 426)(141 427)(142 428)(143 429)(144 430)(145 431)(146 432)(147 433)(148 434)(149 435)(150 436)(151 437)(152 438)(153 439)(154 440)(155 441)(156 442)(157 443)(158 444)(159 445)(160 446)(161 447)(162 448)(163 449)(164 450)(165 451)(166 452)(167 453)(168 454)(169 455)(170 456)(171 457)(172 458)(173 459)(174 460)(175 461)(176 462)(177 463)(178 464)(179 465)(180 466)(181 467)(182 468)(183 469)(184 470)(185 471)(186 472)(187 237)(188 238)(189 239)(190 240)(191 241)(192 242)(193 243)(194 244)(195 245)(196 246)(197 247)(198 248)(199 249)(200 250)(201 251)(202 252)(203 253)(204 254)(205 255)(206 256)(207 257)(208 258)(209 259)(210 260)(211 261)(212 262)(213 263)(214 264)(215 265)(216 266)(217 267)(218 268)(219 269)(220 270)(221 271)(222 272)(223 273)(224 274)(225 275)(226 276)(227 277)(228 278)(229 279)(230 280)(231 281)(232 282)(233 283)(234 284)(235 285)(236 286)
(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)(237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472)

G:=sub<Sym(472)| (1,287)(2,288)(3,289)(4,290)(5,291)(6,292)(7,293)(8,294)(9,295)(10,296)(11,297)(12,298)(13,299)(14,300)(15,301)(16,302)(17,303)(18,304)(19,305)(20,306)(21,307)(22,308)(23,309)(24,310)(25,311)(26,312)(27,313)(28,314)(29,315)(30,316)(31,317)(32,318)(33,319)(34,320)(35,321)(36,322)(37,323)(38,324)(39,325)(40,326)(41,327)(42,328)(43,329)(44,330)(45,331)(46,332)(47,333)(48,334)(49,335)(50,336)(51,337)(52,338)(53,339)(54,340)(55,341)(56,342)(57,343)(58,344)(59,345)(60,346)(61,347)(62,348)(63,349)(64,350)(65,351)(66,352)(67,353)(68,354)(69,355)(70,356)(71,357)(72,358)(73,359)(74,360)(75,361)(76,362)(77,363)(78,364)(79,365)(80,366)(81,367)(82,368)(83,369)(84,370)(85,371)(86,372)(87,373)(88,374)(89,375)(90,376)(91,377)(92,378)(93,379)(94,380)(95,381)(96,382)(97,383)(98,384)(99,385)(100,386)(101,387)(102,388)(103,389)(104,390)(105,391)(106,392)(107,393)(108,394)(109,395)(110,396)(111,397)(112,398)(113,399)(114,400)(115,401)(116,402)(117,403)(118,404)(119,405)(120,406)(121,407)(122,408)(123,409)(124,410)(125,411)(126,412)(127,413)(128,414)(129,415)(130,416)(131,417)(132,418)(133,419)(134,420)(135,421)(136,422)(137,423)(138,424)(139,425)(140,426)(141,427)(142,428)(143,429)(144,430)(145,431)(146,432)(147,433)(148,434)(149,435)(150,436)(151,437)(152,438)(153,439)(154,440)(155,441)(156,442)(157,443)(158,444)(159,445)(160,446)(161,447)(162,448)(163,449)(164,450)(165,451)(166,452)(167,453)(168,454)(169,455)(170,456)(171,457)(172,458)(173,459)(174,460)(175,461)(176,462)(177,463)(178,464)(179,465)(180,466)(181,467)(182,468)(183,469)(184,470)(185,471)(186,472)(187,237)(188,238)(189,239)(190,240)(191,241)(192,242)(193,243)(194,244)(195,245)(196,246)(197,247)(198,248)(199,249)(200,250)(201,251)(202,252)(203,253)(204,254)(205,255)(206,256)(207,257)(208,258)(209,259)(210,260)(211,261)(212,262)(213,263)(214,264)(215,265)(216,266)(217,267)(218,268)(219,269)(220,270)(221,271)(222,272)(223,273)(224,274)(225,275)(226,276)(227,277)(228,278)(229,279)(230,280)(231,281)(232,282)(233,283)(234,284)(235,285)(236,286), (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)(237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472)>;

G:=Group( (1,287)(2,288)(3,289)(4,290)(5,291)(6,292)(7,293)(8,294)(9,295)(10,296)(11,297)(12,298)(13,299)(14,300)(15,301)(16,302)(17,303)(18,304)(19,305)(20,306)(21,307)(22,308)(23,309)(24,310)(25,311)(26,312)(27,313)(28,314)(29,315)(30,316)(31,317)(32,318)(33,319)(34,320)(35,321)(36,322)(37,323)(38,324)(39,325)(40,326)(41,327)(42,328)(43,329)(44,330)(45,331)(46,332)(47,333)(48,334)(49,335)(50,336)(51,337)(52,338)(53,339)(54,340)(55,341)(56,342)(57,343)(58,344)(59,345)(60,346)(61,347)(62,348)(63,349)(64,350)(65,351)(66,352)(67,353)(68,354)(69,355)(70,356)(71,357)(72,358)(73,359)(74,360)(75,361)(76,362)(77,363)(78,364)(79,365)(80,366)(81,367)(82,368)(83,369)(84,370)(85,371)(86,372)(87,373)(88,374)(89,375)(90,376)(91,377)(92,378)(93,379)(94,380)(95,381)(96,382)(97,383)(98,384)(99,385)(100,386)(101,387)(102,388)(103,389)(104,390)(105,391)(106,392)(107,393)(108,394)(109,395)(110,396)(111,397)(112,398)(113,399)(114,400)(115,401)(116,402)(117,403)(118,404)(119,405)(120,406)(121,407)(122,408)(123,409)(124,410)(125,411)(126,412)(127,413)(128,414)(129,415)(130,416)(131,417)(132,418)(133,419)(134,420)(135,421)(136,422)(137,423)(138,424)(139,425)(140,426)(141,427)(142,428)(143,429)(144,430)(145,431)(146,432)(147,433)(148,434)(149,435)(150,436)(151,437)(152,438)(153,439)(154,440)(155,441)(156,442)(157,443)(158,444)(159,445)(160,446)(161,447)(162,448)(163,449)(164,450)(165,451)(166,452)(167,453)(168,454)(169,455)(170,456)(171,457)(172,458)(173,459)(174,460)(175,461)(176,462)(177,463)(178,464)(179,465)(180,466)(181,467)(182,468)(183,469)(184,470)(185,471)(186,472)(187,237)(188,238)(189,239)(190,240)(191,241)(192,242)(193,243)(194,244)(195,245)(196,246)(197,247)(198,248)(199,249)(200,250)(201,251)(202,252)(203,253)(204,254)(205,255)(206,256)(207,257)(208,258)(209,259)(210,260)(211,261)(212,262)(213,263)(214,264)(215,265)(216,266)(217,267)(218,268)(219,269)(220,270)(221,271)(222,272)(223,273)(224,274)(225,275)(226,276)(227,277)(228,278)(229,279)(230,280)(231,281)(232,282)(233,283)(234,284)(235,285)(236,286), (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)(237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472) );

G=PermutationGroup([(1,287),(2,288),(3,289),(4,290),(5,291),(6,292),(7,293),(8,294),(9,295),(10,296),(11,297),(12,298),(13,299),(14,300),(15,301),(16,302),(17,303),(18,304),(19,305),(20,306),(21,307),(22,308),(23,309),(24,310),(25,311),(26,312),(27,313),(28,314),(29,315),(30,316),(31,317),(32,318),(33,319),(34,320),(35,321),(36,322),(37,323),(38,324),(39,325),(40,326),(41,327),(42,328),(43,329),(44,330),(45,331),(46,332),(47,333),(48,334),(49,335),(50,336),(51,337),(52,338),(53,339),(54,340),(55,341),(56,342),(57,343),(58,344),(59,345),(60,346),(61,347),(62,348),(63,349),(64,350),(65,351),(66,352),(67,353),(68,354),(69,355),(70,356),(71,357),(72,358),(73,359),(74,360),(75,361),(76,362),(77,363),(78,364),(79,365),(80,366),(81,367),(82,368),(83,369),(84,370),(85,371),(86,372),(87,373),(88,374),(89,375),(90,376),(91,377),(92,378),(93,379),(94,380),(95,381),(96,382),(97,383),(98,384),(99,385),(100,386),(101,387),(102,388),(103,389),(104,390),(105,391),(106,392),(107,393),(108,394),(109,395),(110,396),(111,397),(112,398),(113,399),(114,400),(115,401),(116,402),(117,403),(118,404),(119,405),(120,406),(121,407),(122,408),(123,409),(124,410),(125,411),(126,412),(127,413),(128,414),(129,415),(130,416),(131,417),(132,418),(133,419),(134,420),(135,421),(136,422),(137,423),(138,424),(139,425),(140,426),(141,427),(142,428),(143,429),(144,430),(145,431),(146,432),(147,433),(148,434),(149,435),(150,436),(151,437),(152,438),(153,439),(154,440),(155,441),(156,442),(157,443),(158,444),(159,445),(160,446),(161,447),(162,448),(163,449),(164,450),(165,451),(166,452),(167,453),(168,454),(169,455),(170,456),(171,457),(172,458),(173,459),(174,460),(175,461),(176,462),(177,463),(178,464),(179,465),(180,466),(181,467),(182,468),(183,469),(184,470),(185,471),(186,472),(187,237),(188,238),(189,239),(190,240),(191,241),(192,242),(193,243),(194,244),(195,245),(196,246),(197,247),(198,248),(199,249),(200,250),(201,251),(202,252),(203,253),(204,254),(205,255),(206,256),(207,257),(208,258),(209,259),(210,260),(211,261),(212,262),(213,263),(214,264),(215,265),(216,266),(217,267),(218,268),(219,269),(220,270),(221,271),(222,272),(223,273),(224,274),(225,275),(226,276),(227,277),(228,278),(229,279),(230,280),(231,281),(232,282),(233,283),(234,284),(235,285),(236,286)], [(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),(237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472)])

472 conjugacy classes

class 1 2A2B2C4A4B4C4D59A···59BF118A···118FR236A···236HX
order1222444459···59118···118236···236
size111111111···11···11···1

472 irreducible representations

dim11111111
type+++
imageC1C2C2C4C59C118C118C236
kernelC2×C236C236C2×C118C118C2×C4C4C22C2
# reps12145811658232

Matrix representation of C2×C236 in GL2(𝔽709) generated by

7080
01
,
5020
0237
G:=sub<GL(2,GF(709))| [708,0,0,1],[502,0,0,237] >;

C2×C236 in GAP, Magma, Sage, TeX

C_2\times C_{236}
% in TeX

G:=Group("C2xC236");
// GroupNames label

G:=SmallGroup(472,8);
// by ID

G=gap.SmallGroup(472,8);
# by ID

G:=PCGroup([4,-2,-2,-59,-2,944]);
// Polycyclic

G:=Group<a,b|a^2=b^236=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C236 in TeX

׿
×
𝔽