direct product, abelian, monomial, 3-elementary
Aliases: C3×C153, SmallGroup(459,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C153 |
| C1 — C3×C153 |
| C1 — C3×C153 |
Generators and relations for C3×C153
G = < a,b | a3=b153=1, ab=ba >
Smallest permutation representation of C3×C153
►Regular action on 459 points
Generators in S459
(1 251 416)(2 252 417)(3 253 418)(4 254 419)(5 255 420)(6 256 421)(7 257 422)(8 258 423)(9 259 424)(10 260 425)(11 261 426)(12 262 427)(13 263 428)(14 264 429)(15 265 430)(16 266 431)(17 267 432)(18 268 433)(19 269 434)(20 270 435)(21 271 436)(22 272 437)(23 273 438)(24 274 439)(25 275 440)(26 276 441)(27 277 442)(28 278 443)(29 279 444)(30 280 445)(31 281 446)(32 282 447)(33 283 448)(34 284 449)(35 285 450)(36 286 451)(37 287 452)(38 288 453)(39 289 454)(40 290 455)(41 291 456)(42 292 457)(43 293 458)(44 294 459)(45 295 307)(46 296 308)(47 297 309)(48 298 310)(49 299 311)(50 300 312)(51 301 313)(52 302 314)(53 303 315)(54 304 316)(55 305 317)(56 306 318)(57 154 319)(58 155 320)(59 156 321)(60 157 322)(61 158 323)(62 159 324)(63 160 325)(64 161 326)(65 162 327)(66 163 328)(67 164 329)(68 165 330)(69 166 331)(70 167 332)(71 168 333)(72 169 334)(73 170 335)(74 171 336)(75 172 337)(76 173 338)(77 174 339)(78 175 340)(79 176 341)(80 177 342)(81 178 343)(82 179 344)(83 180 345)(84 181 346)(85 182 347)(86 183 348)(87 184 349)(88 185 350)(89 186 351)(90 187 352)(91 188 353)(92 189 354)(93 190 355)(94 191 356)(95 192 357)(96 193 358)(97 194 359)(98 195 360)(99 196 361)(100 197 362)(101 198 363)(102 199 364)(103 200 365)(104 201 366)(105 202 367)(106 203 368)(107 204 369)(108 205 370)(109 206 371)(110 207 372)(111 208 373)(112 209 374)(113 210 375)(114 211 376)(115 212 377)(116 213 378)(117 214 379)(118 215 380)(119 216 381)(120 217 382)(121 218 383)(122 219 384)(123 220 385)(124 221 386)(125 222 387)(126 223 388)(127 224 389)(128 225 390)(129 226 391)(130 227 392)(131 228 393)(132 229 394)(133 230 395)(134 231 396)(135 232 397)(136 233 398)(137 234 399)(138 235 400)(139 236 401)(140 237 402)(141 238 403)(142 239 404)(143 240 405)(144 241 406)(145 242 407)(146 243 408)(147 244 409)(148 245 410)(149 246 411)(150 247 412)(151 248 413)(152 249 414)(153 250 415)
(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)
G:=sub<Sym(459)| (1,251,416)(2,252,417)(3,253,418)(4,254,419)(5,255,420)(6,256,421)(7,257,422)(8,258,423)(9,259,424)(10,260,425)(11,261,426)(12,262,427)(13,263,428)(14,264,429)(15,265,430)(16,266,431)(17,267,432)(18,268,433)(19,269,434)(20,270,435)(21,271,436)(22,272,437)(23,273,438)(24,274,439)(25,275,440)(26,276,441)(27,277,442)(28,278,443)(29,279,444)(30,280,445)(31,281,446)(32,282,447)(33,283,448)(34,284,449)(35,285,450)(36,286,451)(37,287,452)(38,288,453)(39,289,454)(40,290,455)(41,291,456)(42,292,457)(43,293,458)(44,294,459)(45,295,307)(46,296,308)(47,297,309)(48,298,310)(49,299,311)(50,300,312)(51,301,313)(52,302,314)(53,303,315)(54,304,316)(55,305,317)(56,306,318)(57,154,319)(58,155,320)(59,156,321)(60,157,322)(61,158,323)(62,159,324)(63,160,325)(64,161,326)(65,162,327)(66,163,328)(67,164,329)(68,165,330)(69,166,331)(70,167,332)(71,168,333)(72,169,334)(73,170,335)(74,171,336)(75,172,337)(76,173,338)(77,174,339)(78,175,340)(79,176,341)(80,177,342)(81,178,343)(82,179,344)(83,180,345)(84,181,346)(85,182,347)(86,183,348)(87,184,349)(88,185,350)(89,186,351)(90,187,352)(91,188,353)(92,189,354)(93,190,355)(94,191,356)(95,192,357)(96,193,358)(97,194,359)(98,195,360)(99,196,361)(100,197,362)(101,198,363)(102,199,364)(103,200,365)(104,201,366)(105,202,367)(106,203,368)(107,204,369)(108,205,370)(109,206,371)(110,207,372)(111,208,373)(112,209,374)(113,210,375)(114,211,376)(115,212,377)(116,213,378)(117,214,379)(118,215,380)(119,216,381)(120,217,382)(121,218,383)(122,219,384)(123,220,385)(124,221,386)(125,222,387)(126,223,388)(127,224,389)(128,225,390)(129,226,391)(130,227,392)(131,228,393)(132,229,394)(133,230,395)(134,231,396)(135,232,397)(136,233,398)(137,234,399)(138,235,400)(139,236,401)(140,237,402)(141,238,403)(142,239,404)(143,240,405)(144,241,406)(145,242,407)(146,243,408)(147,244,409)(148,245,410)(149,246,411)(150,247,412)(151,248,413)(152,249,414)(153,250,415), (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)>;
G:=Group( (1,251,416)(2,252,417)(3,253,418)(4,254,419)(5,255,420)(6,256,421)(7,257,422)(8,258,423)(9,259,424)(10,260,425)(11,261,426)(12,262,427)(13,263,428)(14,264,429)(15,265,430)(16,266,431)(17,267,432)(18,268,433)(19,269,434)(20,270,435)(21,271,436)(22,272,437)(23,273,438)(24,274,439)(25,275,440)(26,276,441)(27,277,442)(28,278,443)(29,279,444)(30,280,445)(31,281,446)(32,282,447)(33,283,448)(34,284,449)(35,285,450)(36,286,451)(37,287,452)(38,288,453)(39,289,454)(40,290,455)(41,291,456)(42,292,457)(43,293,458)(44,294,459)(45,295,307)(46,296,308)(47,297,309)(48,298,310)(49,299,311)(50,300,312)(51,301,313)(52,302,314)(53,303,315)(54,304,316)(55,305,317)(56,306,318)(57,154,319)(58,155,320)(59,156,321)(60,157,322)(61,158,323)(62,159,324)(63,160,325)(64,161,326)(65,162,327)(66,163,328)(67,164,329)(68,165,330)(69,166,331)(70,167,332)(71,168,333)(72,169,334)(73,170,335)(74,171,336)(75,172,337)(76,173,338)(77,174,339)(78,175,340)(79,176,341)(80,177,342)(81,178,343)(82,179,344)(83,180,345)(84,181,346)(85,182,347)(86,183,348)(87,184,349)(88,185,350)(89,186,351)(90,187,352)(91,188,353)(92,189,354)(93,190,355)(94,191,356)(95,192,357)(96,193,358)(97,194,359)(98,195,360)(99,196,361)(100,197,362)(101,198,363)(102,199,364)(103,200,365)(104,201,366)(105,202,367)(106,203,368)(107,204,369)(108,205,370)(109,206,371)(110,207,372)(111,208,373)(112,209,374)(113,210,375)(114,211,376)(115,212,377)(116,213,378)(117,214,379)(118,215,380)(119,216,381)(120,217,382)(121,218,383)(122,219,384)(123,220,385)(124,221,386)(125,222,387)(126,223,388)(127,224,389)(128,225,390)(129,226,391)(130,227,392)(131,228,393)(132,229,394)(133,230,395)(134,231,396)(135,232,397)(136,233,398)(137,234,399)(138,235,400)(139,236,401)(140,237,402)(141,238,403)(142,239,404)(143,240,405)(144,241,406)(145,242,407)(146,243,408)(147,244,409)(148,245,410)(149,246,411)(150,247,412)(151,248,413)(152,249,414)(153,250,415), (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) );
G=PermutationGroup([[(1,251,416),(2,252,417),(3,253,418),(4,254,419),(5,255,420),(6,256,421),(7,257,422),(8,258,423),(9,259,424),(10,260,425),(11,261,426),(12,262,427),(13,263,428),(14,264,429),(15,265,430),(16,266,431),(17,267,432),(18,268,433),(19,269,434),(20,270,435),(21,271,436),(22,272,437),(23,273,438),(24,274,439),(25,275,440),(26,276,441),(27,277,442),(28,278,443),(29,279,444),(30,280,445),(31,281,446),(32,282,447),(33,283,448),(34,284,449),(35,285,450),(36,286,451),(37,287,452),(38,288,453),(39,289,454),(40,290,455),(41,291,456),(42,292,457),(43,293,458),(44,294,459),(45,295,307),(46,296,308),(47,297,309),(48,298,310),(49,299,311),(50,300,312),(51,301,313),(52,302,314),(53,303,315),(54,304,316),(55,305,317),(56,306,318),(57,154,319),(58,155,320),(59,156,321),(60,157,322),(61,158,323),(62,159,324),(63,160,325),(64,161,326),(65,162,327),(66,163,328),(67,164,329),(68,165,330),(69,166,331),(70,167,332),(71,168,333),(72,169,334),(73,170,335),(74,171,336),(75,172,337),(76,173,338),(77,174,339),(78,175,340),(79,176,341),(80,177,342),(81,178,343),(82,179,344),(83,180,345),(84,181,346),(85,182,347),(86,183,348),(87,184,349),(88,185,350),(89,186,351),(90,187,352),(91,188,353),(92,189,354),(93,190,355),(94,191,356),(95,192,357),(96,193,358),(97,194,359),(98,195,360),(99,196,361),(100,197,362),(101,198,363),(102,199,364),(103,200,365),(104,201,366),(105,202,367),(106,203,368),(107,204,369),(108,205,370),(109,206,371),(110,207,372),(111,208,373),(112,209,374),(113,210,375),(114,211,376),(115,212,377),(116,213,378),(117,214,379),(118,215,380),(119,216,381),(120,217,382),(121,218,383),(122,219,384),(123,220,385),(124,221,386),(125,222,387),(126,223,388),(127,224,389),(128,225,390),(129,226,391),(130,227,392),(131,228,393),(132,229,394),(133,230,395),(134,231,396),(135,232,397),(136,233,398),(137,234,399),(138,235,400),(139,236,401),(140,237,402),(141,238,403),(142,239,404),(143,240,405),(144,241,406),(145,242,407),(146,243,408),(147,244,409),(148,245,410),(149,246,411),(150,247,412),(151,248,413),(152,249,414),(153,250,415)], [(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)]])
459 conjugacy classes
| class | 1 | 3A | ··· | 3H | 9A | ··· | 9R | 17A | ··· | 17P | 51A | ··· | 51DX | 153A | ··· | 153KB |
| order | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 17 | ··· | 17 | 51 | ··· | 51 | 153 | ··· | 153 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
459 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||||
| image | C1 | C3 | C3 | C9 | C17 | C51 | C51 | C153 |
| kernel | C3×C153 | C153 | C3×C51 | C51 | C3×C9 | C9 | C32 | C3 |
| # reps | 1 | 6 | 2 | 18 | 16 | 96 | 32 | 288 |
Matrix representation of C3×C153 ►in GL2(𝔽307) generated by
| 17 | 0 |
| 0 | 1 |
| 262 | 0 |
| 0 | 65 |
G:=sub<GL(2,GF(307))| [17,0,0,1],[262,0,0,65] >;
C3×C153 in GAP, Magma, Sage, TeX
C_3\times C_{153} % in TeX
G:=Group("C3xC153"); // GroupNames label
G:=SmallGroup(459,2);
// by ID
G=gap.SmallGroup(459,2);
# by ID
G:=PCGroup([4,-3,-3,-17,-3,612]);
// Polycyclic
G:=Group<a,b|a^3=b^153=1,a*b=b*a>;
// generators/relations
Export