direct product, abelian, monomial, 3-elementary
Aliases: C3×C144, SmallGroup(432,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C144 |
| C1 — C3×C144 |
| C1 — C3×C144 |
Generators and relations for C3×C144
G = < a,b | a3=b144=1, ab=ba >
Smallest permutation representation of C3×C144
►Regular action on 432 points
Generators in S432
(1 224 388)(2 225 389)(3 226 390)(4 227 391)(5 228 392)(6 229 393)(7 230 394)(8 231 395)(9 232 396)(10 233 397)(11 234 398)(12 235 399)(13 236 400)(14 237 401)(15 238 402)(16 239 403)(17 240 404)(18 241 405)(19 242 406)(20 243 407)(21 244 408)(22 245 409)(23 246 410)(24 247 411)(25 248 412)(26 249 413)(27 250 414)(28 251 415)(29 252 416)(30 253 417)(31 254 418)(32 255 419)(33 256 420)(34 257 421)(35 258 422)(36 259 423)(37 260 424)(38 261 425)(39 262 426)(40 263 427)(41 264 428)(42 265 429)(43 266 430)(44 267 431)(45 268 432)(46 269 289)(47 270 290)(48 271 291)(49 272 292)(50 273 293)(51 274 294)(52 275 295)(53 276 296)(54 277 297)(55 278 298)(56 279 299)(57 280 300)(58 281 301)(59 282 302)(60 283 303)(61 284 304)(62 285 305)(63 286 306)(64 287 307)(65 288 308)(66 145 309)(67 146 310)(68 147 311)(69 148 312)(70 149 313)(71 150 314)(72 151 315)(73 152 316)(74 153 317)(75 154 318)(76 155 319)(77 156 320)(78 157 321)(79 158 322)(80 159 323)(81 160 324)(82 161 325)(83 162 326)(84 163 327)(85 164 328)(86 165 329)(87 166 330)(88 167 331)(89 168 332)(90 169 333)(91 170 334)(92 171 335)(93 172 336)(94 173 337)(95 174 338)(96 175 339)(97 176 340)(98 177 341)(99 178 342)(100 179 343)(101 180 344)(102 181 345)(103 182 346)(104 183 347)(105 184 348)(106 185 349)(107 186 350)(108 187 351)(109 188 352)(110 189 353)(111 190 354)(112 191 355)(113 192 356)(114 193 357)(115 194 358)(116 195 359)(117 196 360)(118 197 361)(119 198 362)(120 199 363)(121 200 364)(122 201 365)(123 202 366)(124 203 367)(125 204 368)(126 205 369)(127 206 370)(128 207 371)(129 208 372)(130 209 373)(131 210 374)(132 211 375)(133 212 376)(134 213 377)(135 214 378)(136 215 379)(137 216 380)(138 217 381)(139 218 382)(140 219 383)(141 220 384)(142 221 385)(143 222 386)(144 223 387)
(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)
G:=sub<Sym(432)| (1,224,388)(2,225,389)(3,226,390)(4,227,391)(5,228,392)(6,229,393)(7,230,394)(8,231,395)(9,232,396)(10,233,397)(11,234,398)(12,235,399)(13,236,400)(14,237,401)(15,238,402)(16,239,403)(17,240,404)(18,241,405)(19,242,406)(20,243,407)(21,244,408)(22,245,409)(23,246,410)(24,247,411)(25,248,412)(26,249,413)(27,250,414)(28,251,415)(29,252,416)(30,253,417)(31,254,418)(32,255,419)(33,256,420)(34,257,421)(35,258,422)(36,259,423)(37,260,424)(38,261,425)(39,262,426)(40,263,427)(41,264,428)(42,265,429)(43,266,430)(44,267,431)(45,268,432)(46,269,289)(47,270,290)(48,271,291)(49,272,292)(50,273,293)(51,274,294)(52,275,295)(53,276,296)(54,277,297)(55,278,298)(56,279,299)(57,280,300)(58,281,301)(59,282,302)(60,283,303)(61,284,304)(62,285,305)(63,286,306)(64,287,307)(65,288,308)(66,145,309)(67,146,310)(68,147,311)(69,148,312)(70,149,313)(71,150,314)(72,151,315)(73,152,316)(74,153,317)(75,154,318)(76,155,319)(77,156,320)(78,157,321)(79,158,322)(80,159,323)(81,160,324)(82,161,325)(83,162,326)(84,163,327)(85,164,328)(86,165,329)(87,166,330)(88,167,331)(89,168,332)(90,169,333)(91,170,334)(92,171,335)(93,172,336)(94,173,337)(95,174,338)(96,175,339)(97,176,340)(98,177,341)(99,178,342)(100,179,343)(101,180,344)(102,181,345)(103,182,346)(104,183,347)(105,184,348)(106,185,349)(107,186,350)(108,187,351)(109,188,352)(110,189,353)(111,190,354)(112,191,355)(113,192,356)(114,193,357)(115,194,358)(116,195,359)(117,196,360)(118,197,361)(119,198,362)(120,199,363)(121,200,364)(122,201,365)(123,202,366)(124,203,367)(125,204,368)(126,205,369)(127,206,370)(128,207,371)(129,208,372)(130,209,373)(131,210,374)(132,211,375)(133,212,376)(134,213,377)(135,214,378)(136,215,379)(137,216,380)(138,217,381)(139,218,382)(140,219,383)(141,220,384)(142,221,385)(143,222,386)(144,223,387), (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)>;
G:=Group( (1,224,388)(2,225,389)(3,226,390)(4,227,391)(5,228,392)(6,229,393)(7,230,394)(8,231,395)(9,232,396)(10,233,397)(11,234,398)(12,235,399)(13,236,400)(14,237,401)(15,238,402)(16,239,403)(17,240,404)(18,241,405)(19,242,406)(20,243,407)(21,244,408)(22,245,409)(23,246,410)(24,247,411)(25,248,412)(26,249,413)(27,250,414)(28,251,415)(29,252,416)(30,253,417)(31,254,418)(32,255,419)(33,256,420)(34,257,421)(35,258,422)(36,259,423)(37,260,424)(38,261,425)(39,262,426)(40,263,427)(41,264,428)(42,265,429)(43,266,430)(44,267,431)(45,268,432)(46,269,289)(47,270,290)(48,271,291)(49,272,292)(50,273,293)(51,274,294)(52,275,295)(53,276,296)(54,277,297)(55,278,298)(56,279,299)(57,280,300)(58,281,301)(59,282,302)(60,283,303)(61,284,304)(62,285,305)(63,286,306)(64,287,307)(65,288,308)(66,145,309)(67,146,310)(68,147,311)(69,148,312)(70,149,313)(71,150,314)(72,151,315)(73,152,316)(74,153,317)(75,154,318)(76,155,319)(77,156,320)(78,157,321)(79,158,322)(80,159,323)(81,160,324)(82,161,325)(83,162,326)(84,163,327)(85,164,328)(86,165,329)(87,166,330)(88,167,331)(89,168,332)(90,169,333)(91,170,334)(92,171,335)(93,172,336)(94,173,337)(95,174,338)(96,175,339)(97,176,340)(98,177,341)(99,178,342)(100,179,343)(101,180,344)(102,181,345)(103,182,346)(104,183,347)(105,184,348)(106,185,349)(107,186,350)(108,187,351)(109,188,352)(110,189,353)(111,190,354)(112,191,355)(113,192,356)(114,193,357)(115,194,358)(116,195,359)(117,196,360)(118,197,361)(119,198,362)(120,199,363)(121,200,364)(122,201,365)(123,202,366)(124,203,367)(125,204,368)(126,205,369)(127,206,370)(128,207,371)(129,208,372)(130,209,373)(131,210,374)(132,211,375)(133,212,376)(134,213,377)(135,214,378)(136,215,379)(137,216,380)(138,217,381)(139,218,382)(140,219,383)(141,220,384)(142,221,385)(143,222,386)(144,223,387), (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) );
G=PermutationGroup([[(1,224,388),(2,225,389),(3,226,390),(4,227,391),(5,228,392),(6,229,393),(7,230,394),(8,231,395),(9,232,396),(10,233,397),(11,234,398),(12,235,399),(13,236,400),(14,237,401),(15,238,402),(16,239,403),(17,240,404),(18,241,405),(19,242,406),(20,243,407),(21,244,408),(22,245,409),(23,246,410),(24,247,411),(25,248,412),(26,249,413),(27,250,414),(28,251,415),(29,252,416),(30,253,417),(31,254,418),(32,255,419),(33,256,420),(34,257,421),(35,258,422),(36,259,423),(37,260,424),(38,261,425),(39,262,426),(40,263,427),(41,264,428),(42,265,429),(43,266,430),(44,267,431),(45,268,432),(46,269,289),(47,270,290),(48,271,291),(49,272,292),(50,273,293),(51,274,294),(52,275,295),(53,276,296),(54,277,297),(55,278,298),(56,279,299),(57,280,300),(58,281,301),(59,282,302),(60,283,303),(61,284,304),(62,285,305),(63,286,306),(64,287,307),(65,288,308),(66,145,309),(67,146,310),(68,147,311),(69,148,312),(70,149,313),(71,150,314),(72,151,315),(73,152,316),(74,153,317),(75,154,318),(76,155,319),(77,156,320),(78,157,321),(79,158,322),(80,159,323),(81,160,324),(82,161,325),(83,162,326),(84,163,327),(85,164,328),(86,165,329),(87,166,330),(88,167,331),(89,168,332),(90,169,333),(91,170,334),(92,171,335),(93,172,336),(94,173,337),(95,174,338),(96,175,339),(97,176,340),(98,177,341),(99,178,342),(100,179,343),(101,180,344),(102,181,345),(103,182,346),(104,183,347),(105,184,348),(106,185,349),(107,186,350),(108,187,351),(109,188,352),(110,189,353),(111,190,354),(112,191,355),(113,192,356),(114,193,357),(115,194,358),(116,195,359),(117,196,360),(118,197,361),(119,198,362),(120,199,363),(121,200,364),(122,201,365),(123,202,366),(124,203,367),(125,204,368),(126,205,369),(127,206,370),(128,207,371),(129,208,372),(130,209,373),(131,210,374),(132,211,375),(133,212,376),(134,213,377),(135,214,378),(136,215,379),(137,216,380),(138,217,381),(139,218,382),(140,219,383),(141,220,384),(142,221,385),(143,222,386),(144,223,387)], [(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)]])
432 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 8A | 8B | 8C | 8D | 9A | ··· | 9R | 12A | ··· | 12P | 16A | ··· | 16H | 18A | ··· | 18R | 24A | ··· | 24AF | 36A | ··· | 36AJ | 48A | ··· | 48BL | 72A | ··· | 72BT | 144A | ··· | 144EN |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 16 | ··· | 16 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 144 | ··· | 144 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
432 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C8 | C9 | C12 | C12 | C16 | C18 | C24 | C24 | C36 | C48 | C48 | C72 | C144 |
| kernel | C3×C144 | C3×C72 | C144 | C3×C48 | C3×C36 | C72 | C3×C24 | C3×C18 | C48 | C36 | C3×C12 | C3×C9 | C24 | C18 | C3×C6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 4 | 18 | 12 | 4 | 8 | 18 | 24 | 8 | 36 | 48 | 16 | 72 | 144 |
Matrix representation of C3×C144 ►in GL2(𝔽433) generated by
| 198 | 0 |
| 0 | 234 |
| 194 | 0 |
| 0 | 300 |
G:=sub<GL(2,GF(433))| [198,0,0,234],[194,0,0,300] >;
C3×C144 in GAP, Magma, Sage, TeX
C_3\times C_{144} % in TeX
G:=Group("C3xC144"); // GroupNames label
G:=SmallGroup(432,34);
// by ID
G=gap.SmallGroup(432,34);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,-3,-2,-2,126,260,192,124]);
// Polycyclic
G:=Group<a,b|a^3=b^144=1,a*b=b*a>;
// generators/relations
Export