direct product, abelian, monomial, 3-elementary
Aliases: C3×C135, SmallGroup(405,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C135 |
| C1 — C3×C135 |
| C1 — C3×C135 |
Generators and relations for C3×C135
G = < a,b | a3=b135=1, ab=ba >
Smallest permutation representation of C3×C135
►Regular action on 405 points
Generators in S405
(1 212 304)(2 213 305)(3 214 306)(4 215 307)(5 216 308)(6 217 309)(7 218 310)(8 219 311)(9 220 312)(10 221 313)(11 222 314)(12 223 315)(13 224 316)(14 225 317)(15 226 318)(16 227 319)(17 228 320)(18 229 321)(19 230 322)(20 231 323)(21 232 324)(22 233 325)(23 234 326)(24 235 327)(25 236 328)(26 237 329)(27 238 330)(28 239 331)(29 240 332)(30 241 333)(31 242 334)(32 243 335)(33 244 336)(34 245 337)(35 246 338)(36 247 339)(37 248 340)(38 249 341)(39 250 342)(40 251 343)(41 252 344)(42 253 345)(43 254 346)(44 255 347)(45 256 348)(46 257 349)(47 258 350)(48 259 351)(49 260 352)(50 261 353)(51 262 354)(52 263 355)(53 264 356)(54 265 357)(55 266 358)(56 267 359)(57 268 360)(58 269 361)(59 270 362)(60 136 363)(61 137 364)(62 138 365)(63 139 366)(64 140 367)(65 141 368)(66 142 369)(67 143 370)(68 144 371)(69 145 372)(70 146 373)(71 147 374)(72 148 375)(73 149 376)(74 150 377)(75 151 378)(76 152 379)(77 153 380)(78 154 381)(79 155 382)(80 156 383)(81 157 384)(82 158 385)(83 159 386)(84 160 387)(85 161 388)(86 162 389)(87 163 390)(88 164 391)(89 165 392)(90 166 393)(91 167 394)(92 168 395)(93 169 396)(94 170 397)(95 171 398)(96 172 399)(97 173 400)(98 174 401)(99 175 402)(100 176 403)(101 177 404)(102 178 405)(103 179 271)(104 180 272)(105 181 273)(106 182 274)(107 183 275)(108 184 276)(109 185 277)(110 186 278)(111 187 279)(112 188 280)(113 189 281)(114 190 282)(115 191 283)(116 192 284)(117 193 285)(118 194 286)(119 195 287)(120 196 288)(121 197 289)(122 198 290)(123 199 291)(124 200 292)(125 201 293)(126 202 294)(127 203 295)(128 204 296)(129 205 297)(130 206 298)(131 207 299)(132 208 300)(133 209 301)(134 210 302)(135 211 303)
(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)
G:=sub<Sym(405)| (1,212,304)(2,213,305)(3,214,306)(4,215,307)(5,216,308)(6,217,309)(7,218,310)(8,219,311)(9,220,312)(10,221,313)(11,222,314)(12,223,315)(13,224,316)(14,225,317)(15,226,318)(16,227,319)(17,228,320)(18,229,321)(19,230,322)(20,231,323)(21,232,324)(22,233,325)(23,234,326)(24,235,327)(25,236,328)(26,237,329)(27,238,330)(28,239,331)(29,240,332)(30,241,333)(31,242,334)(32,243,335)(33,244,336)(34,245,337)(35,246,338)(36,247,339)(37,248,340)(38,249,341)(39,250,342)(40,251,343)(41,252,344)(42,253,345)(43,254,346)(44,255,347)(45,256,348)(46,257,349)(47,258,350)(48,259,351)(49,260,352)(50,261,353)(51,262,354)(52,263,355)(53,264,356)(54,265,357)(55,266,358)(56,267,359)(57,268,360)(58,269,361)(59,270,362)(60,136,363)(61,137,364)(62,138,365)(63,139,366)(64,140,367)(65,141,368)(66,142,369)(67,143,370)(68,144,371)(69,145,372)(70,146,373)(71,147,374)(72,148,375)(73,149,376)(74,150,377)(75,151,378)(76,152,379)(77,153,380)(78,154,381)(79,155,382)(80,156,383)(81,157,384)(82,158,385)(83,159,386)(84,160,387)(85,161,388)(86,162,389)(87,163,390)(88,164,391)(89,165,392)(90,166,393)(91,167,394)(92,168,395)(93,169,396)(94,170,397)(95,171,398)(96,172,399)(97,173,400)(98,174,401)(99,175,402)(100,176,403)(101,177,404)(102,178,405)(103,179,271)(104,180,272)(105,181,273)(106,182,274)(107,183,275)(108,184,276)(109,185,277)(110,186,278)(111,187,279)(112,188,280)(113,189,281)(114,190,282)(115,191,283)(116,192,284)(117,193,285)(118,194,286)(119,195,287)(120,196,288)(121,197,289)(122,198,290)(123,199,291)(124,200,292)(125,201,293)(126,202,294)(127,203,295)(128,204,296)(129,205,297)(130,206,298)(131,207,299)(132,208,300)(133,209,301)(134,210,302)(135,211,303), (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)>;
G:=Group( (1,212,304)(2,213,305)(3,214,306)(4,215,307)(5,216,308)(6,217,309)(7,218,310)(8,219,311)(9,220,312)(10,221,313)(11,222,314)(12,223,315)(13,224,316)(14,225,317)(15,226,318)(16,227,319)(17,228,320)(18,229,321)(19,230,322)(20,231,323)(21,232,324)(22,233,325)(23,234,326)(24,235,327)(25,236,328)(26,237,329)(27,238,330)(28,239,331)(29,240,332)(30,241,333)(31,242,334)(32,243,335)(33,244,336)(34,245,337)(35,246,338)(36,247,339)(37,248,340)(38,249,341)(39,250,342)(40,251,343)(41,252,344)(42,253,345)(43,254,346)(44,255,347)(45,256,348)(46,257,349)(47,258,350)(48,259,351)(49,260,352)(50,261,353)(51,262,354)(52,263,355)(53,264,356)(54,265,357)(55,266,358)(56,267,359)(57,268,360)(58,269,361)(59,270,362)(60,136,363)(61,137,364)(62,138,365)(63,139,366)(64,140,367)(65,141,368)(66,142,369)(67,143,370)(68,144,371)(69,145,372)(70,146,373)(71,147,374)(72,148,375)(73,149,376)(74,150,377)(75,151,378)(76,152,379)(77,153,380)(78,154,381)(79,155,382)(80,156,383)(81,157,384)(82,158,385)(83,159,386)(84,160,387)(85,161,388)(86,162,389)(87,163,390)(88,164,391)(89,165,392)(90,166,393)(91,167,394)(92,168,395)(93,169,396)(94,170,397)(95,171,398)(96,172,399)(97,173,400)(98,174,401)(99,175,402)(100,176,403)(101,177,404)(102,178,405)(103,179,271)(104,180,272)(105,181,273)(106,182,274)(107,183,275)(108,184,276)(109,185,277)(110,186,278)(111,187,279)(112,188,280)(113,189,281)(114,190,282)(115,191,283)(116,192,284)(117,193,285)(118,194,286)(119,195,287)(120,196,288)(121,197,289)(122,198,290)(123,199,291)(124,200,292)(125,201,293)(126,202,294)(127,203,295)(128,204,296)(129,205,297)(130,206,298)(131,207,299)(132,208,300)(133,209,301)(134,210,302)(135,211,303), (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) );
G=PermutationGroup([[(1,212,304),(2,213,305),(3,214,306),(4,215,307),(5,216,308),(6,217,309),(7,218,310),(8,219,311),(9,220,312),(10,221,313),(11,222,314),(12,223,315),(13,224,316),(14,225,317),(15,226,318),(16,227,319),(17,228,320),(18,229,321),(19,230,322),(20,231,323),(21,232,324),(22,233,325),(23,234,326),(24,235,327),(25,236,328),(26,237,329),(27,238,330),(28,239,331),(29,240,332),(30,241,333),(31,242,334),(32,243,335),(33,244,336),(34,245,337),(35,246,338),(36,247,339),(37,248,340),(38,249,341),(39,250,342),(40,251,343),(41,252,344),(42,253,345),(43,254,346),(44,255,347),(45,256,348),(46,257,349),(47,258,350),(48,259,351),(49,260,352),(50,261,353),(51,262,354),(52,263,355),(53,264,356),(54,265,357),(55,266,358),(56,267,359),(57,268,360),(58,269,361),(59,270,362),(60,136,363),(61,137,364),(62,138,365),(63,139,366),(64,140,367),(65,141,368),(66,142,369),(67,143,370),(68,144,371),(69,145,372),(70,146,373),(71,147,374),(72,148,375),(73,149,376),(74,150,377),(75,151,378),(76,152,379),(77,153,380),(78,154,381),(79,155,382),(80,156,383),(81,157,384),(82,158,385),(83,159,386),(84,160,387),(85,161,388),(86,162,389),(87,163,390),(88,164,391),(89,165,392),(90,166,393),(91,167,394),(92,168,395),(93,169,396),(94,170,397),(95,171,398),(96,172,399),(97,173,400),(98,174,401),(99,175,402),(100,176,403),(101,177,404),(102,178,405),(103,179,271),(104,180,272),(105,181,273),(106,182,274),(107,183,275),(108,184,276),(109,185,277),(110,186,278),(111,187,279),(112,188,280),(113,189,281),(114,190,282),(115,191,283),(116,192,284),(117,193,285),(118,194,286),(119,195,287),(120,196,288),(121,197,289),(122,198,290),(123,199,291),(124,200,292),(125,201,293),(126,202,294),(127,203,295),(128,204,296),(129,205,297),(130,206,298),(131,207,299),(132,208,300),(133,209,301),(134,210,302),(135,211,303)], [(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)]])
405 conjugacy classes
| class | 1 | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 9A | ··· | 9R | 15A | ··· | 15AF | 27A | ··· | 27BB | 45A | ··· | 45BT | 135A | ··· | 135HH |
| order | 1 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 15 | ··· | 15 | 27 | ··· | 27 | 45 | ··· | 45 | 135 | ··· | 135 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
405 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||||||||
| image | C1 | C3 | C3 | C5 | C9 | C9 | C15 | C15 | C27 | C45 | C45 | C135 |
| kernel | C3×C135 | C135 | C3×C45 | C3×C27 | C45 | C3×C15 | C27 | C3×C9 | C15 | C9 | C32 | C3 |
| # reps | 1 | 6 | 2 | 4 | 12 | 6 | 24 | 8 | 54 | 48 | 24 | 216 |
Matrix representation of C3×C135 ►in GL2(𝔽271) generated by
| 1 | 0 |
| 0 | 242 |
| 50 | 0 |
| 0 | 160 |
G:=sub<GL(2,GF(271))| [1,0,0,242],[50,0,0,160] >;
C3×C135 in GAP, Magma, Sage, TeX
C_3\times C_{135} % in TeX
G:=Group("C3xC135"); // GroupNames label
G:=SmallGroup(405,5);
// by ID
G=gap.SmallGroup(405,5);
# by ID
G:=PCGroup([5,-3,-3,-5,-3,-3,225,78]);
// Polycyclic
G:=Group<a,b|a^3=b^135=1,a*b=b*a>;
// generators/relations
Export