direct product, abelian, monomial, 3-elementary
Aliases: C3×C156, SmallGroup(468,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C156 |
| C1 — C3×C156 |
| C1 — C3×C156 |
Generators and relations for C3×C156
G = < a,b | a3=b156=1, ab=ba >
Smallest permutation representation of C3×C156
►Regular action on 468 points
Generators in S468
(1 295 460)(2 296 461)(3 297 462)(4 298 463)(5 299 464)(6 300 465)(7 301 466)(8 302 467)(9 303 468)(10 304 313)(11 305 314)(12 306 315)(13 307 316)(14 308 317)(15 309 318)(16 310 319)(17 311 320)(18 312 321)(19 157 322)(20 158 323)(21 159 324)(22 160 325)(23 161 326)(24 162 327)(25 163 328)(26 164 329)(27 165 330)(28 166 331)(29 167 332)(30 168 333)(31 169 334)(32 170 335)(33 171 336)(34 172 337)(35 173 338)(36 174 339)(37 175 340)(38 176 341)(39 177 342)(40 178 343)(41 179 344)(42 180 345)(43 181 346)(44 182 347)(45 183 348)(46 184 349)(47 185 350)(48 186 351)(49 187 352)(50 188 353)(51 189 354)(52 190 355)(53 191 356)(54 192 357)(55 193 358)(56 194 359)(57 195 360)(58 196 361)(59 197 362)(60 198 363)(61 199 364)(62 200 365)(63 201 366)(64 202 367)(65 203 368)(66 204 369)(67 205 370)(68 206 371)(69 207 372)(70 208 373)(71 209 374)(72 210 375)(73 211 376)(74 212 377)(75 213 378)(76 214 379)(77 215 380)(78 216 381)(79 217 382)(80 218 383)(81 219 384)(82 220 385)(83 221 386)(84 222 387)(85 223 388)(86 224 389)(87 225 390)(88 226 391)(89 227 392)(90 228 393)(91 229 394)(92 230 395)(93 231 396)(94 232 397)(95 233 398)(96 234 399)(97 235 400)(98 236 401)(99 237 402)(100 238 403)(101 239 404)(102 240 405)(103 241 406)(104 242 407)(105 243 408)(106 244 409)(107 245 410)(108 246 411)(109 247 412)(110 248 413)(111 249 414)(112 250 415)(113 251 416)(114 252 417)(115 253 418)(116 254 419)(117 255 420)(118 256 421)(119 257 422)(120 258 423)(121 259 424)(122 260 425)(123 261 426)(124 262 427)(125 263 428)(126 264 429)(127 265 430)(128 266 431)(129 267 432)(130 268 433)(131 269 434)(132 270 435)(133 271 436)(134 272 437)(135 273 438)(136 274 439)(137 275 440)(138 276 441)(139 277 442)(140 278 443)(141 279 444)(142 280 445)(143 281 446)(144 282 447)(145 283 448)(146 284 449)(147 285 450)(148 286 451)(149 287 452)(150 288 453)(151 289 454)(152 290 455)(153 291 456)(154 292 457)(155 293 458)(156 294 459)
(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)
G:=sub<Sym(468)| (1,295,460)(2,296,461)(3,297,462)(4,298,463)(5,299,464)(6,300,465)(7,301,466)(8,302,467)(9,303,468)(10,304,313)(11,305,314)(12,306,315)(13,307,316)(14,308,317)(15,309,318)(16,310,319)(17,311,320)(18,312,321)(19,157,322)(20,158,323)(21,159,324)(22,160,325)(23,161,326)(24,162,327)(25,163,328)(26,164,329)(27,165,330)(28,166,331)(29,167,332)(30,168,333)(31,169,334)(32,170,335)(33,171,336)(34,172,337)(35,173,338)(36,174,339)(37,175,340)(38,176,341)(39,177,342)(40,178,343)(41,179,344)(42,180,345)(43,181,346)(44,182,347)(45,183,348)(46,184,349)(47,185,350)(48,186,351)(49,187,352)(50,188,353)(51,189,354)(52,190,355)(53,191,356)(54,192,357)(55,193,358)(56,194,359)(57,195,360)(58,196,361)(59,197,362)(60,198,363)(61,199,364)(62,200,365)(63,201,366)(64,202,367)(65,203,368)(66,204,369)(67,205,370)(68,206,371)(69,207,372)(70,208,373)(71,209,374)(72,210,375)(73,211,376)(74,212,377)(75,213,378)(76,214,379)(77,215,380)(78,216,381)(79,217,382)(80,218,383)(81,219,384)(82,220,385)(83,221,386)(84,222,387)(85,223,388)(86,224,389)(87,225,390)(88,226,391)(89,227,392)(90,228,393)(91,229,394)(92,230,395)(93,231,396)(94,232,397)(95,233,398)(96,234,399)(97,235,400)(98,236,401)(99,237,402)(100,238,403)(101,239,404)(102,240,405)(103,241,406)(104,242,407)(105,243,408)(106,244,409)(107,245,410)(108,246,411)(109,247,412)(110,248,413)(111,249,414)(112,250,415)(113,251,416)(114,252,417)(115,253,418)(116,254,419)(117,255,420)(118,256,421)(119,257,422)(120,258,423)(121,259,424)(122,260,425)(123,261,426)(124,262,427)(125,263,428)(126,264,429)(127,265,430)(128,266,431)(129,267,432)(130,268,433)(131,269,434)(132,270,435)(133,271,436)(134,272,437)(135,273,438)(136,274,439)(137,275,440)(138,276,441)(139,277,442)(140,278,443)(141,279,444)(142,280,445)(143,281,446)(144,282,447)(145,283,448)(146,284,449)(147,285,450)(148,286,451)(149,287,452)(150,288,453)(151,289,454)(152,290,455)(153,291,456)(154,292,457)(155,293,458)(156,294,459), (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)>;
G:=Group( (1,295,460)(2,296,461)(3,297,462)(4,298,463)(5,299,464)(6,300,465)(7,301,466)(8,302,467)(9,303,468)(10,304,313)(11,305,314)(12,306,315)(13,307,316)(14,308,317)(15,309,318)(16,310,319)(17,311,320)(18,312,321)(19,157,322)(20,158,323)(21,159,324)(22,160,325)(23,161,326)(24,162,327)(25,163,328)(26,164,329)(27,165,330)(28,166,331)(29,167,332)(30,168,333)(31,169,334)(32,170,335)(33,171,336)(34,172,337)(35,173,338)(36,174,339)(37,175,340)(38,176,341)(39,177,342)(40,178,343)(41,179,344)(42,180,345)(43,181,346)(44,182,347)(45,183,348)(46,184,349)(47,185,350)(48,186,351)(49,187,352)(50,188,353)(51,189,354)(52,190,355)(53,191,356)(54,192,357)(55,193,358)(56,194,359)(57,195,360)(58,196,361)(59,197,362)(60,198,363)(61,199,364)(62,200,365)(63,201,366)(64,202,367)(65,203,368)(66,204,369)(67,205,370)(68,206,371)(69,207,372)(70,208,373)(71,209,374)(72,210,375)(73,211,376)(74,212,377)(75,213,378)(76,214,379)(77,215,380)(78,216,381)(79,217,382)(80,218,383)(81,219,384)(82,220,385)(83,221,386)(84,222,387)(85,223,388)(86,224,389)(87,225,390)(88,226,391)(89,227,392)(90,228,393)(91,229,394)(92,230,395)(93,231,396)(94,232,397)(95,233,398)(96,234,399)(97,235,400)(98,236,401)(99,237,402)(100,238,403)(101,239,404)(102,240,405)(103,241,406)(104,242,407)(105,243,408)(106,244,409)(107,245,410)(108,246,411)(109,247,412)(110,248,413)(111,249,414)(112,250,415)(113,251,416)(114,252,417)(115,253,418)(116,254,419)(117,255,420)(118,256,421)(119,257,422)(120,258,423)(121,259,424)(122,260,425)(123,261,426)(124,262,427)(125,263,428)(126,264,429)(127,265,430)(128,266,431)(129,267,432)(130,268,433)(131,269,434)(132,270,435)(133,271,436)(134,272,437)(135,273,438)(136,274,439)(137,275,440)(138,276,441)(139,277,442)(140,278,443)(141,279,444)(142,280,445)(143,281,446)(144,282,447)(145,283,448)(146,284,449)(147,285,450)(148,286,451)(149,287,452)(150,288,453)(151,289,454)(152,290,455)(153,291,456)(154,292,457)(155,293,458)(156,294,459), (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) );
G=PermutationGroup([[(1,295,460),(2,296,461),(3,297,462),(4,298,463),(5,299,464),(6,300,465),(7,301,466),(8,302,467),(9,303,468),(10,304,313),(11,305,314),(12,306,315),(13,307,316),(14,308,317),(15,309,318),(16,310,319),(17,311,320),(18,312,321),(19,157,322),(20,158,323),(21,159,324),(22,160,325),(23,161,326),(24,162,327),(25,163,328),(26,164,329),(27,165,330),(28,166,331),(29,167,332),(30,168,333),(31,169,334),(32,170,335),(33,171,336),(34,172,337),(35,173,338),(36,174,339),(37,175,340),(38,176,341),(39,177,342),(40,178,343),(41,179,344),(42,180,345),(43,181,346),(44,182,347),(45,183,348),(46,184,349),(47,185,350),(48,186,351),(49,187,352),(50,188,353),(51,189,354),(52,190,355),(53,191,356),(54,192,357),(55,193,358),(56,194,359),(57,195,360),(58,196,361),(59,197,362),(60,198,363),(61,199,364),(62,200,365),(63,201,366),(64,202,367),(65,203,368),(66,204,369),(67,205,370),(68,206,371),(69,207,372),(70,208,373),(71,209,374),(72,210,375),(73,211,376),(74,212,377),(75,213,378),(76,214,379),(77,215,380),(78,216,381),(79,217,382),(80,218,383),(81,219,384),(82,220,385),(83,221,386),(84,222,387),(85,223,388),(86,224,389),(87,225,390),(88,226,391),(89,227,392),(90,228,393),(91,229,394),(92,230,395),(93,231,396),(94,232,397),(95,233,398),(96,234,399),(97,235,400),(98,236,401),(99,237,402),(100,238,403),(101,239,404),(102,240,405),(103,241,406),(104,242,407),(105,243,408),(106,244,409),(107,245,410),(108,246,411),(109,247,412),(110,248,413),(111,249,414),(112,250,415),(113,251,416),(114,252,417),(115,253,418),(116,254,419),(117,255,420),(118,256,421),(119,257,422),(120,258,423),(121,259,424),(122,260,425),(123,261,426),(124,262,427),(125,263,428),(126,264,429),(127,265,430),(128,266,431),(129,267,432),(130,268,433),(131,269,434),(132,270,435),(133,271,436),(134,272,437),(135,273,438),(136,274,439),(137,275,440),(138,276,441),(139,277,442),(140,278,443),(141,279,444),(142,280,445),(143,281,446),(144,282,447),(145,283,448),(146,284,449),(147,285,450),(148,286,451),(149,287,452),(150,288,453),(151,289,454),(152,290,455),(153,291,456),(154,292,457),(155,293,458),(156,294,459)], [(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)]])
468 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 12A | ··· | 12P | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39CR | 52A | ··· | 52X | 78A | ··· | 78CR | 156A | ··· | 156GJ |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
468 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | C13 | C26 | C39 | C52 | C78 | C156 |
| kernel | C3×C156 | C3×C78 | C156 | C3×C39 | C78 | C39 | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 16 | 12 | 12 | 96 | 24 | 96 | 192 |
Matrix representation of C3×C156 ►in GL2(𝔽157) generated by
| 1 | 0 |
| 0 | 144 |
| 70 | 0 |
| 0 | 32 |
G:=sub<GL(2,GF(157))| [1,0,0,144],[70,0,0,32] >;
C3×C156 in GAP, Magma, Sage, TeX
C_3\times C_{156} % in TeX
G:=Group("C3xC156"); // GroupNames label
G:=SmallGroup(468,28);
// by ID
G=gap.SmallGroup(468,28);
# by ID
G:=PCGroup([5,-2,-3,-3,-13,-2,1170]);
// Polycyclic
G:=Group<a,b|a^3=b^156=1,a*b=b*a>;
// generators/relations
Export