direct product, abelian, monomial, 3-elementary
Aliases: C3×C150, SmallGroup(450,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C150 |
| C1 — C3×C150 |
| C1 — C3×C150 |
Generators and relations for C3×C150
G = < a,b | a3=b150=1, ab=ba >
Smallest permutation representation of C3×C150
►Regular action on 450 points
Generators in S450
(1 307 236)(2 308 237)(3 309 238)(4 310 239)(5 311 240)(6 312 241)(7 313 242)(8 314 243)(9 315 244)(10 316 245)(11 317 246)(12 318 247)(13 319 248)(14 320 249)(15 321 250)(16 322 251)(17 323 252)(18 324 253)(19 325 254)(20 326 255)(21 327 256)(22 328 257)(23 329 258)(24 330 259)(25 331 260)(26 332 261)(27 333 262)(28 334 263)(29 335 264)(30 336 265)(31 337 266)(32 338 267)(33 339 268)(34 340 269)(35 341 270)(36 342 271)(37 343 272)(38 344 273)(39 345 274)(40 346 275)(41 347 276)(42 348 277)(43 349 278)(44 350 279)(45 351 280)(46 352 281)(47 353 282)(48 354 283)(49 355 284)(50 356 285)(51 357 286)(52 358 287)(53 359 288)(54 360 289)(55 361 290)(56 362 291)(57 363 292)(58 364 293)(59 365 294)(60 366 295)(61 367 296)(62 368 297)(63 369 298)(64 370 299)(65 371 300)(66 372 151)(67 373 152)(68 374 153)(69 375 154)(70 376 155)(71 377 156)(72 378 157)(73 379 158)(74 380 159)(75 381 160)(76 382 161)(77 383 162)(78 384 163)(79 385 164)(80 386 165)(81 387 166)(82 388 167)(83 389 168)(84 390 169)(85 391 170)(86 392 171)(87 393 172)(88 394 173)(89 395 174)(90 396 175)(91 397 176)(92 398 177)(93 399 178)(94 400 179)(95 401 180)(96 402 181)(97 403 182)(98 404 183)(99 405 184)(100 406 185)(101 407 186)(102 408 187)(103 409 188)(104 410 189)(105 411 190)(106 412 191)(107 413 192)(108 414 193)(109 415 194)(110 416 195)(111 417 196)(112 418 197)(113 419 198)(114 420 199)(115 421 200)(116 422 201)(117 423 202)(118 424 203)(119 425 204)(120 426 205)(121 427 206)(122 428 207)(123 429 208)(124 430 209)(125 431 210)(126 432 211)(127 433 212)(128 434 213)(129 435 214)(130 436 215)(131 437 216)(132 438 217)(133 439 218)(134 440 219)(135 441 220)(136 442 221)(137 443 222)(138 444 223)(139 445 224)(140 446 225)(141 447 226)(142 448 227)(143 449 228)(144 450 229)(145 301 230)(146 302 231)(147 303 232)(148 304 233)(149 305 234)(150 306 235)
(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)
G:=sub<Sym(450)| (1,307,236)(2,308,237)(3,309,238)(4,310,239)(5,311,240)(6,312,241)(7,313,242)(8,314,243)(9,315,244)(10,316,245)(11,317,246)(12,318,247)(13,319,248)(14,320,249)(15,321,250)(16,322,251)(17,323,252)(18,324,253)(19,325,254)(20,326,255)(21,327,256)(22,328,257)(23,329,258)(24,330,259)(25,331,260)(26,332,261)(27,333,262)(28,334,263)(29,335,264)(30,336,265)(31,337,266)(32,338,267)(33,339,268)(34,340,269)(35,341,270)(36,342,271)(37,343,272)(38,344,273)(39,345,274)(40,346,275)(41,347,276)(42,348,277)(43,349,278)(44,350,279)(45,351,280)(46,352,281)(47,353,282)(48,354,283)(49,355,284)(50,356,285)(51,357,286)(52,358,287)(53,359,288)(54,360,289)(55,361,290)(56,362,291)(57,363,292)(58,364,293)(59,365,294)(60,366,295)(61,367,296)(62,368,297)(63,369,298)(64,370,299)(65,371,300)(66,372,151)(67,373,152)(68,374,153)(69,375,154)(70,376,155)(71,377,156)(72,378,157)(73,379,158)(74,380,159)(75,381,160)(76,382,161)(77,383,162)(78,384,163)(79,385,164)(80,386,165)(81,387,166)(82,388,167)(83,389,168)(84,390,169)(85,391,170)(86,392,171)(87,393,172)(88,394,173)(89,395,174)(90,396,175)(91,397,176)(92,398,177)(93,399,178)(94,400,179)(95,401,180)(96,402,181)(97,403,182)(98,404,183)(99,405,184)(100,406,185)(101,407,186)(102,408,187)(103,409,188)(104,410,189)(105,411,190)(106,412,191)(107,413,192)(108,414,193)(109,415,194)(110,416,195)(111,417,196)(112,418,197)(113,419,198)(114,420,199)(115,421,200)(116,422,201)(117,423,202)(118,424,203)(119,425,204)(120,426,205)(121,427,206)(122,428,207)(123,429,208)(124,430,209)(125,431,210)(126,432,211)(127,433,212)(128,434,213)(129,435,214)(130,436,215)(131,437,216)(132,438,217)(133,439,218)(134,440,219)(135,441,220)(136,442,221)(137,443,222)(138,444,223)(139,445,224)(140,446,225)(141,447,226)(142,448,227)(143,449,228)(144,450,229)(145,301,230)(146,302,231)(147,303,232)(148,304,233)(149,305,234)(150,306,235), (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)>;
G:=Group( (1,307,236)(2,308,237)(3,309,238)(4,310,239)(5,311,240)(6,312,241)(7,313,242)(8,314,243)(9,315,244)(10,316,245)(11,317,246)(12,318,247)(13,319,248)(14,320,249)(15,321,250)(16,322,251)(17,323,252)(18,324,253)(19,325,254)(20,326,255)(21,327,256)(22,328,257)(23,329,258)(24,330,259)(25,331,260)(26,332,261)(27,333,262)(28,334,263)(29,335,264)(30,336,265)(31,337,266)(32,338,267)(33,339,268)(34,340,269)(35,341,270)(36,342,271)(37,343,272)(38,344,273)(39,345,274)(40,346,275)(41,347,276)(42,348,277)(43,349,278)(44,350,279)(45,351,280)(46,352,281)(47,353,282)(48,354,283)(49,355,284)(50,356,285)(51,357,286)(52,358,287)(53,359,288)(54,360,289)(55,361,290)(56,362,291)(57,363,292)(58,364,293)(59,365,294)(60,366,295)(61,367,296)(62,368,297)(63,369,298)(64,370,299)(65,371,300)(66,372,151)(67,373,152)(68,374,153)(69,375,154)(70,376,155)(71,377,156)(72,378,157)(73,379,158)(74,380,159)(75,381,160)(76,382,161)(77,383,162)(78,384,163)(79,385,164)(80,386,165)(81,387,166)(82,388,167)(83,389,168)(84,390,169)(85,391,170)(86,392,171)(87,393,172)(88,394,173)(89,395,174)(90,396,175)(91,397,176)(92,398,177)(93,399,178)(94,400,179)(95,401,180)(96,402,181)(97,403,182)(98,404,183)(99,405,184)(100,406,185)(101,407,186)(102,408,187)(103,409,188)(104,410,189)(105,411,190)(106,412,191)(107,413,192)(108,414,193)(109,415,194)(110,416,195)(111,417,196)(112,418,197)(113,419,198)(114,420,199)(115,421,200)(116,422,201)(117,423,202)(118,424,203)(119,425,204)(120,426,205)(121,427,206)(122,428,207)(123,429,208)(124,430,209)(125,431,210)(126,432,211)(127,433,212)(128,434,213)(129,435,214)(130,436,215)(131,437,216)(132,438,217)(133,439,218)(134,440,219)(135,441,220)(136,442,221)(137,443,222)(138,444,223)(139,445,224)(140,446,225)(141,447,226)(142,448,227)(143,449,228)(144,450,229)(145,301,230)(146,302,231)(147,303,232)(148,304,233)(149,305,234)(150,306,235), (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) );
G=PermutationGroup([[(1,307,236),(2,308,237),(3,309,238),(4,310,239),(5,311,240),(6,312,241),(7,313,242),(8,314,243),(9,315,244),(10,316,245),(11,317,246),(12,318,247),(13,319,248),(14,320,249),(15,321,250),(16,322,251),(17,323,252),(18,324,253),(19,325,254),(20,326,255),(21,327,256),(22,328,257),(23,329,258),(24,330,259),(25,331,260),(26,332,261),(27,333,262),(28,334,263),(29,335,264),(30,336,265),(31,337,266),(32,338,267),(33,339,268),(34,340,269),(35,341,270),(36,342,271),(37,343,272),(38,344,273),(39,345,274),(40,346,275),(41,347,276),(42,348,277),(43,349,278),(44,350,279),(45,351,280),(46,352,281),(47,353,282),(48,354,283),(49,355,284),(50,356,285),(51,357,286),(52,358,287),(53,359,288),(54,360,289),(55,361,290),(56,362,291),(57,363,292),(58,364,293),(59,365,294),(60,366,295),(61,367,296),(62,368,297),(63,369,298),(64,370,299),(65,371,300),(66,372,151),(67,373,152),(68,374,153),(69,375,154),(70,376,155),(71,377,156),(72,378,157),(73,379,158),(74,380,159),(75,381,160),(76,382,161),(77,383,162),(78,384,163),(79,385,164),(80,386,165),(81,387,166),(82,388,167),(83,389,168),(84,390,169),(85,391,170),(86,392,171),(87,393,172),(88,394,173),(89,395,174),(90,396,175),(91,397,176),(92,398,177),(93,399,178),(94,400,179),(95,401,180),(96,402,181),(97,403,182),(98,404,183),(99,405,184),(100,406,185),(101,407,186),(102,408,187),(103,409,188),(104,410,189),(105,411,190),(106,412,191),(107,413,192),(108,414,193),(109,415,194),(110,416,195),(111,417,196),(112,418,197),(113,419,198),(114,420,199),(115,421,200),(116,422,201),(117,423,202),(118,424,203),(119,425,204),(120,426,205),(121,427,206),(122,428,207),(123,429,208),(124,430,209),(125,431,210),(126,432,211),(127,433,212),(128,434,213),(129,435,214),(130,436,215),(131,437,216),(132,438,217),(133,439,218),(134,440,219),(135,441,220),(136,442,221),(137,443,222),(138,444,223),(139,445,224),(140,446,225),(141,447,226),(142,448,227),(143,449,228),(144,450,229),(145,301,230),(146,302,231),(147,303,232),(148,304,233),(149,305,234),(150,306,235)], [(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)]])
450 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 6A | ··· | 6H | 10A | 10B | 10C | 10D | 15A | ··· | 15AF | 25A | ··· | 25T | 30A | ··· | 30AF | 50A | ··· | 50T | 75A | ··· | 75FD | 150A | ··· | 150FD |
| order | 1 | 2 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 25 | ··· | 25 | 30 | ··· | 30 | 50 | ··· | 50 | 75 | ··· | 75 | 150 | ··· | 150 |
| 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 |
450 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C25 | C30 | C50 | C75 | C150 |
| kernel | C3×C150 | C3×C75 | C150 | C3×C30 | C75 | C3×C15 | C30 | C3×C6 | C15 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 4 | 8 | 4 | 32 | 20 | 32 | 20 | 160 | 160 |
Matrix representation of C3×C150 ►in GL2(𝔽151) generated by
| 118 | 0 |
| 0 | 32 |
| 95 | 0 |
| 0 | 107 |
G:=sub<GL(2,GF(151))| [118,0,0,32],[95,0,0,107] >;
C3×C150 in GAP, Magma, Sage, TeX
C_3\times C_{150} % in TeX
G:=Group("C3xC150"); // GroupNames label
G:=SmallGroup(450,10);
// by ID
G=gap.SmallGroup(450,10);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,-5,118]);
// Polycyclic
G:=Group<a,b|a^3=b^150=1,a*b=b*a>;
// generators/relations
Export