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G = C3×C150order 450 = 2·32·52

Abelian group of type [3,150]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C150, SmallGroup(450,10)

Series: Derived Chief Lower central Upper central

C1 — C3×C150
C1C5C25C75C3×C75 — 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
(1 206 315)(2 207 316)(3 208 317)(4 209 318)(5 210 319)(6 211 320)(7 212 321)(8 213 322)(9 214 323)(10 215 324)(11 216 325)(12 217 326)(13 218 327)(14 219 328)(15 220 329)(16 221 330)(17 222 331)(18 223 332)(19 224 333)(20 225 334)(21 226 335)(22 227 336)(23 228 337)(24 229 338)(25 230 339)(26 231 340)(27 232 341)(28 233 342)(29 234 343)(30 235 344)(31 236 345)(32 237 346)(33 238 347)(34 239 348)(35 240 349)(36 241 350)(37 242 351)(38 243 352)(39 244 353)(40 245 354)(41 246 355)(42 247 356)(43 248 357)(44 249 358)(45 250 359)(46 251 360)(47 252 361)(48 253 362)(49 254 363)(50 255 364)(51 256 365)(52 257 366)(53 258 367)(54 259 368)(55 260 369)(56 261 370)(57 262 371)(58 263 372)(59 264 373)(60 265 374)(61 266 375)(62 267 376)(63 268 377)(64 269 378)(65 270 379)(66 271 380)(67 272 381)(68 273 382)(69 274 383)(70 275 384)(71 276 385)(72 277 386)(73 278 387)(74 279 388)(75 280 389)(76 281 390)(77 282 391)(78 283 392)(79 284 393)(80 285 394)(81 286 395)(82 287 396)(83 288 397)(84 289 398)(85 290 399)(86 291 400)(87 292 401)(88 293 402)(89 294 403)(90 295 404)(91 296 405)(92 297 406)(93 298 407)(94 299 408)(95 300 409)(96 151 410)(97 152 411)(98 153 412)(99 154 413)(100 155 414)(101 156 415)(102 157 416)(103 158 417)(104 159 418)(105 160 419)(106 161 420)(107 162 421)(108 163 422)(109 164 423)(110 165 424)(111 166 425)(112 167 426)(113 168 427)(114 169 428)(115 170 429)(116 171 430)(117 172 431)(118 173 432)(119 174 433)(120 175 434)(121 176 435)(122 177 436)(123 178 437)(124 179 438)(125 180 439)(126 181 440)(127 182 441)(128 183 442)(129 184 443)(130 185 444)(131 186 445)(132 187 446)(133 188 447)(134 189 448)(135 190 449)(136 191 450)(137 192 301)(138 193 302)(139 194 303)(140 195 304)(141 196 305)(142 197 306)(143 198 307)(144 199 308)(145 200 309)(146 201 310)(147 202 311)(148 203 312)(149 204 313)(150 205 314)
(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)| (1,206,315)(2,207,316)(3,208,317)(4,209,318)(5,210,319)(6,211,320)(7,212,321)(8,213,322)(9,214,323)(10,215,324)(11,216,325)(12,217,326)(13,218,327)(14,219,328)(15,220,329)(16,221,330)(17,222,331)(18,223,332)(19,224,333)(20,225,334)(21,226,335)(22,227,336)(23,228,337)(24,229,338)(25,230,339)(26,231,340)(27,232,341)(28,233,342)(29,234,343)(30,235,344)(31,236,345)(32,237,346)(33,238,347)(34,239,348)(35,240,349)(36,241,350)(37,242,351)(38,243,352)(39,244,353)(40,245,354)(41,246,355)(42,247,356)(43,248,357)(44,249,358)(45,250,359)(46,251,360)(47,252,361)(48,253,362)(49,254,363)(50,255,364)(51,256,365)(52,257,366)(53,258,367)(54,259,368)(55,260,369)(56,261,370)(57,262,371)(58,263,372)(59,264,373)(60,265,374)(61,266,375)(62,267,376)(63,268,377)(64,269,378)(65,270,379)(66,271,380)(67,272,381)(68,273,382)(69,274,383)(70,275,384)(71,276,385)(72,277,386)(73,278,387)(74,279,388)(75,280,389)(76,281,390)(77,282,391)(78,283,392)(79,284,393)(80,285,394)(81,286,395)(82,287,396)(83,288,397)(84,289,398)(85,290,399)(86,291,400)(87,292,401)(88,293,402)(89,294,403)(90,295,404)(91,296,405)(92,297,406)(93,298,407)(94,299,408)(95,300,409)(96,151,410)(97,152,411)(98,153,412)(99,154,413)(100,155,414)(101,156,415)(102,157,416)(103,158,417)(104,159,418)(105,160,419)(106,161,420)(107,162,421)(108,163,422)(109,164,423)(110,165,424)(111,166,425)(112,167,426)(113,168,427)(114,169,428)(115,170,429)(116,171,430)(117,172,431)(118,173,432)(119,174,433)(120,175,434)(121,176,435)(122,177,436)(123,178,437)(124,179,438)(125,180,439)(126,181,440)(127,182,441)(128,183,442)(129,184,443)(130,185,444)(131,186,445)(132,187,446)(133,188,447)(134,189,448)(135,190,449)(136,191,450)(137,192,301)(138,193,302)(139,194,303)(140,195,304)(141,196,305)(142,197,306)(143,198,307)(144,199,308)(145,200,309)(146,201,310)(147,202,311)(148,203,312)(149,204,313)(150,205,314), (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( (1,206,315)(2,207,316)(3,208,317)(4,209,318)(5,210,319)(6,211,320)(7,212,321)(8,213,322)(9,214,323)(10,215,324)(11,216,325)(12,217,326)(13,218,327)(14,219,328)(15,220,329)(16,221,330)(17,222,331)(18,223,332)(19,224,333)(20,225,334)(21,226,335)(22,227,336)(23,228,337)(24,229,338)(25,230,339)(26,231,340)(27,232,341)(28,233,342)(29,234,343)(30,235,344)(31,236,345)(32,237,346)(33,238,347)(34,239,348)(35,240,349)(36,241,350)(37,242,351)(38,243,352)(39,244,353)(40,245,354)(41,246,355)(42,247,356)(43,248,357)(44,249,358)(45,250,359)(46,251,360)(47,252,361)(48,253,362)(49,254,363)(50,255,364)(51,256,365)(52,257,366)(53,258,367)(54,259,368)(55,260,369)(56,261,370)(57,262,371)(58,263,372)(59,264,373)(60,265,374)(61,266,375)(62,267,376)(63,268,377)(64,269,378)(65,270,379)(66,271,380)(67,272,381)(68,273,382)(69,274,383)(70,275,384)(71,276,385)(72,277,386)(73,278,387)(74,279,388)(75,280,389)(76,281,390)(77,282,391)(78,283,392)(79,284,393)(80,285,394)(81,286,395)(82,287,396)(83,288,397)(84,289,398)(85,290,399)(86,291,400)(87,292,401)(88,293,402)(89,294,403)(90,295,404)(91,296,405)(92,297,406)(93,298,407)(94,299,408)(95,300,409)(96,151,410)(97,152,411)(98,153,412)(99,154,413)(100,155,414)(101,156,415)(102,157,416)(103,158,417)(104,159,418)(105,160,419)(106,161,420)(107,162,421)(108,163,422)(109,164,423)(110,165,424)(111,166,425)(112,167,426)(113,168,427)(114,169,428)(115,170,429)(116,171,430)(117,172,431)(118,173,432)(119,174,433)(120,175,434)(121,176,435)(122,177,436)(123,178,437)(124,179,438)(125,180,439)(126,181,440)(127,182,441)(128,183,442)(129,184,443)(130,185,444)(131,186,445)(132,187,446)(133,188,447)(134,189,448)(135,190,449)(136,191,450)(137,192,301)(138,193,302)(139,194,303)(140,195,304)(141,196,305)(142,197,306)(143,198,307)(144,199,308)(145,200,309)(146,201,310)(147,202,311)(148,203,312)(149,204,313)(150,205,314), (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([(1,206,315),(2,207,316),(3,208,317),(4,209,318),(5,210,319),(6,211,320),(7,212,321),(8,213,322),(9,214,323),(10,215,324),(11,216,325),(12,217,326),(13,218,327),(14,219,328),(15,220,329),(16,221,330),(17,222,331),(18,223,332),(19,224,333),(20,225,334),(21,226,335),(22,227,336),(23,228,337),(24,229,338),(25,230,339),(26,231,340),(27,232,341),(28,233,342),(29,234,343),(30,235,344),(31,236,345),(32,237,346),(33,238,347),(34,239,348),(35,240,349),(36,241,350),(37,242,351),(38,243,352),(39,244,353),(40,245,354),(41,246,355),(42,247,356),(43,248,357),(44,249,358),(45,250,359),(46,251,360),(47,252,361),(48,253,362),(49,254,363),(50,255,364),(51,256,365),(52,257,366),(53,258,367),(54,259,368),(55,260,369),(56,261,370),(57,262,371),(58,263,372),(59,264,373),(60,265,374),(61,266,375),(62,267,376),(63,268,377),(64,269,378),(65,270,379),(66,271,380),(67,272,381),(68,273,382),(69,274,383),(70,275,384),(71,276,385),(72,277,386),(73,278,387),(74,279,388),(75,280,389),(76,281,390),(77,282,391),(78,283,392),(79,284,393),(80,285,394),(81,286,395),(82,287,396),(83,288,397),(84,289,398),(85,290,399),(86,291,400),(87,292,401),(88,293,402),(89,294,403),(90,295,404),(91,296,405),(92,297,406),(93,298,407),(94,299,408),(95,300,409),(96,151,410),(97,152,411),(98,153,412),(99,154,413),(100,155,414),(101,156,415),(102,157,416),(103,158,417),(104,159,418),(105,160,419),(106,161,420),(107,162,421),(108,163,422),(109,164,423),(110,165,424),(111,166,425),(112,167,426),(113,168,427),(114,169,428),(115,170,429),(116,171,430),(117,172,431),(118,173,432),(119,174,433),(120,175,434),(121,176,435),(122,177,436),(123,178,437),(124,179,438),(125,180,439),(126,181,440),(127,182,441),(128,183,442),(129,184,443),(130,185,444),(131,186,445),(132,187,446),(133,188,447),(134,189,448),(135,190,449),(136,191,450),(137,192,301),(138,193,302),(139,194,303),(140,195,304),(141,196,305),(142,197,306),(143,198,307),(144,199,308),(145,200,309),(146,201,310),(147,202,311),(148,203,312),(149,204,313),(150,205,314)], [(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···3H5A5B5C5D6A···6H10A10B10C10D15A···15AF25A···25T30A···30AF50A···50T75A···75FD150A···150FD
order123···355556···61010101015···1525···2530···3050···5075···75150···150
size111···111111···111111···11···11···11···11···11···1

450 irreducible representations

dim111111111111
type++
imageC1C2C3C5C6C10C15C25C30C50C75C150
kernelC3×C150C3×C75C150C3×C30C75C3×C15C30C3×C6C15C32C6C3
# reps11848432203220160160

Matrix representation of C3×C150 in GL2(𝔽151) generated by

1180
032
,
950
0107
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

Subgroup lattice of C3×C150 in TeX

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