Copied to
clipboard

G = C2×C230order 460 = 22·5·23

Abelian group of type [2,230]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C230, SmallGroup(460,11)

Series: Derived Chief Lower central Upper central

C1 — C2×C230
C1C23C115C230 — C2×C230
C1 — C2×C230
C1 — C2×C230

Generators and relations for C2×C230
 G = < a,b | a2=b230=1, ab=ba >


Smallest permutation representation of C2×C230
Regular action on 460 points
Generators in S460
(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)(60 429)(61 430)(62 431)(63 432)(64 433)(65 434)(66 435)(67 436)(68 437)(69 438)(70 439)(71 440)(72 441)(73 442)(74 443)(75 444)(76 445)(77 446)(78 447)(79 448)(80 449)(81 450)(82 451)(83 452)(84 453)(85 454)(86 455)(87 456)(88 457)(89 458)(90 459)(91 460)(92 231)(93 232)(94 233)(95 234)(96 235)(97 236)(98 237)(99 238)(100 239)(101 240)(102 241)(103 242)(104 243)(105 244)(106 245)(107 246)(108 247)(109 248)(110 249)(111 250)(112 251)(113 252)(114 253)(115 254)(116 255)(117 256)(118 257)(119 258)(120 259)(121 260)(122 261)(123 262)(124 263)(125 264)(126 265)(127 266)(128 267)(129 268)(130 269)(131 270)(132 271)(133 272)(134 273)(135 274)(136 275)(137 276)(138 277)(139 278)(140 279)(141 280)(142 281)(143 282)(144 283)(145 284)(146 285)(147 286)(148 287)(149 288)(150 289)(151 290)(152 291)(153 292)(154 293)(155 294)(156 295)(157 296)(158 297)(159 298)(160 299)(161 300)(162 301)(163 302)(164 303)(165 304)(166 305)(167 306)(168 307)(169 308)(170 309)(171 310)(172 311)(173 312)(174 313)(175 314)(176 315)(177 316)(178 317)(179 318)(180 319)(181 320)(182 321)(183 322)(184 323)(185 324)(186 325)(187 326)(188 327)(189 328)(190 329)(191 330)(192 331)(193 332)(194 333)(195 334)(196 335)(197 336)(198 337)(199 338)(200 339)(201 340)(202 341)(203 342)(204 343)(205 344)(206 345)(207 346)(208 347)(209 348)(210 349)(211 350)(212 351)(213 352)(214 353)(215 354)(216 355)(217 356)(218 357)(219 358)(220 359)(221 360)(222 361)(223 362)(224 363)(225 364)(226 365)(227 366)(228 367)(229 368)(230 369)
(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)(231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460)

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

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

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

460 conjugacy classes

class 1 2A2B2C5A5B5C5D10A···10L23A···23V46A···46BN115A···115CJ230A···230JD
order1222555510···1023···2346···46115···115230···230
size111111111···11···11···11···11···1

460 irreducible representations

dim11111111
type++
imageC1C2C5C10C23C46C115C230
kernelC2×C230C230C2×C46C46C2×C10C10C22C2
# reps13412226688264

Matrix representation of C2×C230 in GL2(𝔽461) generated by

4600
0460
,
1650
085
G:=sub<GL(2,GF(461))| [460,0,0,460],[165,0,0,85] >;

C2×C230 in GAP, Magma, Sage, TeX

C_2\times C_{230}
% in TeX

G:=Group("C2xC230");
// GroupNames label

G:=SmallGroup(460,11);
// by ID

G=gap.SmallGroup(460,11);
# by ID

G:=PCGroup([4,-2,-2,-5,-23]);
// Polycyclic

G:=Group<a,b|a^2=b^230=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C230 in TeX

׿
×
𝔽