direct product, abelian, monomial, 3-elementary
Aliases: C3×C147, SmallGroup(441,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C147 |
| C1 — C3×C147 |
| C1 — C3×C147 |
Generators and relations for C3×C147
G = < a,b | a3=b147=1, ab=ba >
Smallest permutation representation of C3×C147
►Regular action on 441 points
Generators in S441
(1 427 164)(2 428 165)(3 429 166)(4 430 167)(5 431 168)(6 432 169)(7 433 170)(8 434 171)(9 435 172)(10 436 173)(11 437 174)(12 438 175)(13 439 176)(14 440 177)(15 441 178)(16 295 179)(17 296 180)(18 297 181)(19 298 182)(20 299 183)(21 300 184)(22 301 185)(23 302 186)(24 303 187)(25 304 188)(26 305 189)(27 306 190)(28 307 191)(29 308 192)(30 309 193)(31 310 194)(32 311 195)(33 312 196)(34 313 197)(35 314 198)(36 315 199)(37 316 200)(38 317 201)(39 318 202)(40 319 203)(41 320 204)(42 321 205)(43 322 206)(44 323 207)(45 324 208)(46 325 209)(47 326 210)(48 327 211)(49 328 212)(50 329 213)(51 330 214)(52 331 215)(53 332 216)(54 333 217)(55 334 218)(56 335 219)(57 336 220)(58 337 221)(59 338 222)(60 339 223)(61 340 224)(62 341 225)(63 342 226)(64 343 227)(65 344 228)(66 345 229)(67 346 230)(68 347 231)(69 348 232)(70 349 233)(71 350 234)(72 351 235)(73 352 236)(74 353 237)(75 354 238)(76 355 239)(77 356 240)(78 357 241)(79 358 242)(80 359 243)(81 360 244)(82 361 245)(83 362 246)(84 363 247)(85 364 248)(86 365 249)(87 366 250)(88 367 251)(89 368 252)(90 369 253)(91 370 254)(92 371 255)(93 372 256)(94 373 257)(95 374 258)(96 375 259)(97 376 260)(98 377 261)(99 378 262)(100 379 263)(101 380 264)(102 381 265)(103 382 266)(104 383 267)(105 384 268)(106 385 269)(107 386 270)(108 387 271)(109 388 272)(110 389 273)(111 390 274)(112 391 275)(113 392 276)(114 393 277)(115 394 278)(116 395 279)(117 396 280)(118 397 281)(119 398 282)(120 399 283)(121 400 284)(122 401 285)(123 402 286)(124 403 287)(125 404 288)(126 405 289)(127 406 290)(128 407 291)(129 408 292)(130 409 293)(131 410 294)(132 411 148)(133 412 149)(134 413 150)(135 414 151)(136 415 152)(137 416 153)(138 417 154)(139 418 155)(140 419 156)(141 420 157)(142 421 158)(143 422 159)(144 423 160)(145 424 161)(146 425 162)(147 426 163)
(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)
G:=sub<Sym(441)| (1,427,164)(2,428,165)(3,429,166)(4,430,167)(5,431,168)(6,432,169)(7,433,170)(8,434,171)(9,435,172)(10,436,173)(11,437,174)(12,438,175)(13,439,176)(14,440,177)(15,441,178)(16,295,179)(17,296,180)(18,297,181)(19,298,182)(20,299,183)(21,300,184)(22,301,185)(23,302,186)(24,303,187)(25,304,188)(26,305,189)(27,306,190)(28,307,191)(29,308,192)(30,309,193)(31,310,194)(32,311,195)(33,312,196)(34,313,197)(35,314,198)(36,315,199)(37,316,200)(38,317,201)(39,318,202)(40,319,203)(41,320,204)(42,321,205)(43,322,206)(44,323,207)(45,324,208)(46,325,209)(47,326,210)(48,327,211)(49,328,212)(50,329,213)(51,330,214)(52,331,215)(53,332,216)(54,333,217)(55,334,218)(56,335,219)(57,336,220)(58,337,221)(59,338,222)(60,339,223)(61,340,224)(62,341,225)(63,342,226)(64,343,227)(65,344,228)(66,345,229)(67,346,230)(68,347,231)(69,348,232)(70,349,233)(71,350,234)(72,351,235)(73,352,236)(74,353,237)(75,354,238)(76,355,239)(77,356,240)(78,357,241)(79,358,242)(80,359,243)(81,360,244)(82,361,245)(83,362,246)(84,363,247)(85,364,248)(86,365,249)(87,366,250)(88,367,251)(89,368,252)(90,369,253)(91,370,254)(92,371,255)(93,372,256)(94,373,257)(95,374,258)(96,375,259)(97,376,260)(98,377,261)(99,378,262)(100,379,263)(101,380,264)(102,381,265)(103,382,266)(104,383,267)(105,384,268)(106,385,269)(107,386,270)(108,387,271)(109,388,272)(110,389,273)(111,390,274)(112,391,275)(113,392,276)(114,393,277)(115,394,278)(116,395,279)(117,396,280)(118,397,281)(119,398,282)(120,399,283)(121,400,284)(122,401,285)(123,402,286)(124,403,287)(125,404,288)(126,405,289)(127,406,290)(128,407,291)(129,408,292)(130,409,293)(131,410,294)(132,411,148)(133,412,149)(134,413,150)(135,414,151)(136,415,152)(137,416,153)(138,417,154)(139,418,155)(140,419,156)(141,420,157)(142,421,158)(143,422,159)(144,423,160)(145,424,161)(146,425,162)(147,426,163), (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)>;
G:=Group( (1,427,164)(2,428,165)(3,429,166)(4,430,167)(5,431,168)(6,432,169)(7,433,170)(8,434,171)(9,435,172)(10,436,173)(11,437,174)(12,438,175)(13,439,176)(14,440,177)(15,441,178)(16,295,179)(17,296,180)(18,297,181)(19,298,182)(20,299,183)(21,300,184)(22,301,185)(23,302,186)(24,303,187)(25,304,188)(26,305,189)(27,306,190)(28,307,191)(29,308,192)(30,309,193)(31,310,194)(32,311,195)(33,312,196)(34,313,197)(35,314,198)(36,315,199)(37,316,200)(38,317,201)(39,318,202)(40,319,203)(41,320,204)(42,321,205)(43,322,206)(44,323,207)(45,324,208)(46,325,209)(47,326,210)(48,327,211)(49,328,212)(50,329,213)(51,330,214)(52,331,215)(53,332,216)(54,333,217)(55,334,218)(56,335,219)(57,336,220)(58,337,221)(59,338,222)(60,339,223)(61,340,224)(62,341,225)(63,342,226)(64,343,227)(65,344,228)(66,345,229)(67,346,230)(68,347,231)(69,348,232)(70,349,233)(71,350,234)(72,351,235)(73,352,236)(74,353,237)(75,354,238)(76,355,239)(77,356,240)(78,357,241)(79,358,242)(80,359,243)(81,360,244)(82,361,245)(83,362,246)(84,363,247)(85,364,248)(86,365,249)(87,366,250)(88,367,251)(89,368,252)(90,369,253)(91,370,254)(92,371,255)(93,372,256)(94,373,257)(95,374,258)(96,375,259)(97,376,260)(98,377,261)(99,378,262)(100,379,263)(101,380,264)(102,381,265)(103,382,266)(104,383,267)(105,384,268)(106,385,269)(107,386,270)(108,387,271)(109,388,272)(110,389,273)(111,390,274)(112,391,275)(113,392,276)(114,393,277)(115,394,278)(116,395,279)(117,396,280)(118,397,281)(119,398,282)(120,399,283)(121,400,284)(122,401,285)(123,402,286)(124,403,287)(125,404,288)(126,405,289)(127,406,290)(128,407,291)(129,408,292)(130,409,293)(131,410,294)(132,411,148)(133,412,149)(134,413,150)(135,414,151)(136,415,152)(137,416,153)(138,417,154)(139,418,155)(140,419,156)(141,420,157)(142,421,158)(143,422,159)(144,423,160)(145,424,161)(146,425,162)(147,426,163), (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) );
G=PermutationGroup([[(1,427,164),(2,428,165),(3,429,166),(4,430,167),(5,431,168),(6,432,169),(7,433,170),(8,434,171),(9,435,172),(10,436,173),(11,437,174),(12,438,175),(13,439,176),(14,440,177),(15,441,178),(16,295,179),(17,296,180),(18,297,181),(19,298,182),(20,299,183),(21,300,184),(22,301,185),(23,302,186),(24,303,187),(25,304,188),(26,305,189),(27,306,190),(28,307,191),(29,308,192),(30,309,193),(31,310,194),(32,311,195),(33,312,196),(34,313,197),(35,314,198),(36,315,199),(37,316,200),(38,317,201),(39,318,202),(40,319,203),(41,320,204),(42,321,205),(43,322,206),(44,323,207),(45,324,208),(46,325,209),(47,326,210),(48,327,211),(49,328,212),(50,329,213),(51,330,214),(52,331,215),(53,332,216),(54,333,217),(55,334,218),(56,335,219),(57,336,220),(58,337,221),(59,338,222),(60,339,223),(61,340,224),(62,341,225),(63,342,226),(64,343,227),(65,344,228),(66,345,229),(67,346,230),(68,347,231),(69,348,232),(70,349,233),(71,350,234),(72,351,235),(73,352,236),(74,353,237),(75,354,238),(76,355,239),(77,356,240),(78,357,241),(79,358,242),(80,359,243),(81,360,244),(82,361,245),(83,362,246),(84,363,247),(85,364,248),(86,365,249),(87,366,250),(88,367,251),(89,368,252),(90,369,253),(91,370,254),(92,371,255),(93,372,256),(94,373,257),(95,374,258),(96,375,259),(97,376,260),(98,377,261),(99,378,262),(100,379,263),(101,380,264),(102,381,265),(103,382,266),(104,383,267),(105,384,268),(106,385,269),(107,386,270),(108,387,271),(109,388,272),(110,389,273),(111,390,274),(112,391,275),(113,392,276),(114,393,277),(115,394,278),(116,395,279),(117,396,280),(118,397,281),(119,398,282),(120,399,283),(121,400,284),(122,401,285),(123,402,286),(124,403,287),(125,404,288),(126,405,289),(127,406,290),(128,407,291),(129,408,292),(130,409,293),(131,410,294),(132,411,148),(133,412,149),(134,413,150),(135,414,151),(136,415,152),(137,416,153),(138,417,154),(139,418,155),(140,419,156),(141,420,157),(142,421,158),(143,422,159),(144,423,160),(145,424,161),(146,425,162),(147,426,163)], [(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)]])
441 conjugacy classes
| class | 1 | 3A | ··· | 3H | 7A | ··· | 7F | 21A | ··· | 21AV | 49A | ··· | 49AP | 147A | ··· | 147LX |
| order | 1 | 3 | ··· | 3 | 7 | ··· | 7 | 21 | ··· | 21 | 49 | ··· | 49 | 147 | ··· | 147 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
441 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||
| image | C1 | C3 | C7 | C21 | C49 | C147 |
| kernel | C3×C147 | C147 | C3×C21 | C21 | C32 | C3 |
| # reps | 1 | 8 | 6 | 48 | 42 | 336 |
Matrix representation of C3×C147 ►in GL2(𝔽883) generated by
| 337 | 0 |
| 0 | 337 |
| 427 | 0 |
| 0 | 315 |
G:=sub<GL(2,GF(883))| [337,0,0,337],[427,0,0,315] >;
C3×C147 in GAP, Magma, Sage, TeX
C_3\times C_{147} % in TeX
G:=Group("C3xC147"); // GroupNames label
G:=SmallGroup(441,4);
// by ID
G=gap.SmallGroup(441,4);
# by ID
G:=PCGroup([4,-3,-3,-7,-7,94]);
// Polycyclic
G:=Group<a,b|a^3=b^147=1,a*b=b*a>;
// generators/relations
Export