direct product, abelian, monomial, 5-elementary
Aliases: C5×C90, SmallGroup(450,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C90 |
| C1 — C5×C90 |
| C1 — C5×C90 |
Generators and relations for C5×C90
G = < a,b | a5=b90=1, ab=ba >
Smallest permutation representation of C5×C90
►Regular action on 450 points
Generators in S450
(1 230 369 131 309)(2 231 370 132 310)(3 232 371 133 311)(4 233 372 134 312)(5 234 373 135 313)(6 235 374 136 314)(7 236 375 137 315)(8 237 376 138 316)(9 238 377 139 317)(10 239 378 140 318)(11 240 379 141 319)(12 241 380 142 320)(13 242 381 143 321)(14 243 382 144 322)(15 244 383 145 323)(16 245 384 146 324)(17 246 385 147 325)(18 247 386 148 326)(19 248 387 149 327)(20 249 388 150 328)(21 250 389 151 329)(22 251 390 152 330)(23 252 391 153 331)(24 253 392 154 332)(25 254 393 155 333)(26 255 394 156 334)(27 256 395 157 335)(28 257 396 158 336)(29 258 397 159 337)(30 259 398 160 338)(31 260 399 161 339)(32 261 400 162 340)(33 262 401 163 341)(34 263 402 164 342)(35 264 403 165 343)(36 265 404 166 344)(37 266 405 167 345)(38 267 406 168 346)(39 268 407 169 347)(40 269 408 170 348)(41 270 409 171 349)(42 181 410 172 350)(43 182 411 173 351)(44 183 412 174 352)(45 184 413 175 353)(46 185 414 176 354)(47 186 415 177 355)(48 187 416 178 356)(49 188 417 179 357)(50 189 418 180 358)(51 190 419 91 359)(52 191 420 92 360)(53 192 421 93 271)(54 193 422 94 272)(55 194 423 95 273)(56 195 424 96 274)(57 196 425 97 275)(58 197 426 98 276)(59 198 427 99 277)(60 199 428 100 278)(61 200 429 101 279)(62 201 430 102 280)(63 202 431 103 281)(64 203 432 104 282)(65 204 433 105 283)(66 205 434 106 284)(67 206 435 107 285)(68 207 436 108 286)(69 208 437 109 287)(70 209 438 110 288)(71 210 439 111 289)(72 211 440 112 290)(73 212 441 113 291)(74 213 442 114 292)(75 214 443 115 293)(76 215 444 116 294)(77 216 445 117 295)(78 217 446 118 296)(79 218 447 119 297)(80 219 448 120 298)(81 220 449 121 299)(82 221 450 122 300)(83 222 361 123 301)(84 223 362 124 302)(85 224 363 125 303)(86 225 364 126 304)(87 226 365 127 305)(88 227 366 128 306)(89 228 367 129 307)(90 229 368 130 308)
(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)| (1,230,369,131,309)(2,231,370,132,310)(3,232,371,133,311)(4,233,372,134,312)(5,234,373,135,313)(6,235,374,136,314)(7,236,375,137,315)(8,237,376,138,316)(9,238,377,139,317)(10,239,378,140,318)(11,240,379,141,319)(12,241,380,142,320)(13,242,381,143,321)(14,243,382,144,322)(15,244,383,145,323)(16,245,384,146,324)(17,246,385,147,325)(18,247,386,148,326)(19,248,387,149,327)(20,249,388,150,328)(21,250,389,151,329)(22,251,390,152,330)(23,252,391,153,331)(24,253,392,154,332)(25,254,393,155,333)(26,255,394,156,334)(27,256,395,157,335)(28,257,396,158,336)(29,258,397,159,337)(30,259,398,160,338)(31,260,399,161,339)(32,261,400,162,340)(33,262,401,163,341)(34,263,402,164,342)(35,264,403,165,343)(36,265,404,166,344)(37,266,405,167,345)(38,267,406,168,346)(39,268,407,169,347)(40,269,408,170,348)(41,270,409,171,349)(42,181,410,172,350)(43,182,411,173,351)(44,183,412,174,352)(45,184,413,175,353)(46,185,414,176,354)(47,186,415,177,355)(48,187,416,178,356)(49,188,417,179,357)(50,189,418,180,358)(51,190,419,91,359)(52,191,420,92,360)(53,192,421,93,271)(54,193,422,94,272)(55,194,423,95,273)(56,195,424,96,274)(57,196,425,97,275)(58,197,426,98,276)(59,198,427,99,277)(60,199,428,100,278)(61,200,429,101,279)(62,201,430,102,280)(63,202,431,103,281)(64,203,432,104,282)(65,204,433,105,283)(66,205,434,106,284)(67,206,435,107,285)(68,207,436,108,286)(69,208,437,109,287)(70,209,438,110,288)(71,210,439,111,289)(72,211,440,112,290)(73,212,441,113,291)(74,213,442,114,292)(75,214,443,115,293)(76,215,444,116,294)(77,216,445,117,295)(78,217,446,118,296)(79,218,447,119,297)(80,219,448,120,298)(81,220,449,121,299)(82,221,450,122,300)(83,222,361,123,301)(84,223,362,124,302)(85,224,363,125,303)(86,225,364,126,304)(87,226,365,127,305)(88,227,366,128,306)(89,228,367,129,307)(90,229,368,130,308), (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( (1,230,369,131,309)(2,231,370,132,310)(3,232,371,133,311)(4,233,372,134,312)(5,234,373,135,313)(6,235,374,136,314)(7,236,375,137,315)(8,237,376,138,316)(9,238,377,139,317)(10,239,378,140,318)(11,240,379,141,319)(12,241,380,142,320)(13,242,381,143,321)(14,243,382,144,322)(15,244,383,145,323)(16,245,384,146,324)(17,246,385,147,325)(18,247,386,148,326)(19,248,387,149,327)(20,249,388,150,328)(21,250,389,151,329)(22,251,390,152,330)(23,252,391,153,331)(24,253,392,154,332)(25,254,393,155,333)(26,255,394,156,334)(27,256,395,157,335)(28,257,396,158,336)(29,258,397,159,337)(30,259,398,160,338)(31,260,399,161,339)(32,261,400,162,340)(33,262,401,163,341)(34,263,402,164,342)(35,264,403,165,343)(36,265,404,166,344)(37,266,405,167,345)(38,267,406,168,346)(39,268,407,169,347)(40,269,408,170,348)(41,270,409,171,349)(42,181,410,172,350)(43,182,411,173,351)(44,183,412,174,352)(45,184,413,175,353)(46,185,414,176,354)(47,186,415,177,355)(48,187,416,178,356)(49,188,417,179,357)(50,189,418,180,358)(51,190,419,91,359)(52,191,420,92,360)(53,192,421,93,271)(54,193,422,94,272)(55,194,423,95,273)(56,195,424,96,274)(57,196,425,97,275)(58,197,426,98,276)(59,198,427,99,277)(60,199,428,100,278)(61,200,429,101,279)(62,201,430,102,280)(63,202,431,103,281)(64,203,432,104,282)(65,204,433,105,283)(66,205,434,106,284)(67,206,435,107,285)(68,207,436,108,286)(69,208,437,109,287)(70,209,438,110,288)(71,210,439,111,289)(72,211,440,112,290)(73,212,441,113,291)(74,213,442,114,292)(75,214,443,115,293)(76,215,444,116,294)(77,216,445,117,295)(78,217,446,118,296)(79,218,447,119,297)(80,219,448,120,298)(81,220,449,121,299)(82,221,450,122,300)(83,222,361,123,301)(84,223,362,124,302)(85,224,363,125,303)(86,225,364,126,304)(87,226,365,127,305)(88,227,366,128,306)(89,228,367,129,307)(90,229,368,130,308), (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([[(1,230,369,131,309),(2,231,370,132,310),(3,232,371,133,311),(4,233,372,134,312),(5,234,373,135,313),(6,235,374,136,314),(7,236,375,137,315),(8,237,376,138,316),(9,238,377,139,317),(10,239,378,140,318),(11,240,379,141,319),(12,241,380,142,320),(13,242,381,143,321),(14,243,382,144,322),(15,244,383,145,323),(16,245,384,146,324),(17,246,385,147,325),(18,247,386,148,326),(19,248,387,149,327),(20,249,388,150,328),(21,250,389,151,329),(22,251,390,152,330),(23,252,391,153,331),(24,253,392,154,332),(25,254,393,155,333),(26,255,394,156,334),(27,256,395,157,335),(28,257,396,158,336),(29,258,397,159,337),(30,259,398,160,338),(31,260,399,161,339),(32,261,400,162,340),(33,262,401,163,341),(34,263,402,164,342),(35,264,403,165,343),(36,265,404,166,344),(37,266,405,167,345),(38,267,406,168,346),(39,268,407,169,347),(40,269,408,170,348),(41,270,409,171,349),(42,181,410,172,350),(43,182,411,173,351),(44,183,412,174,352),(45,184,413,175,353),(46,185,414,176,354),(47,186,415,177,355),(48,187,416,178,356),(49,188,417,179,357),(50,189,418,180,358),(51,190,419,91,359),(52,191,420,92,360),(53,192,421,93,271),(54,193,422,94,272),(55,194,423,95,273),(56,195,424,96,274),(57,196,425,97,275),(58,197,426,98,276),(59,198,427,99,277),(60,199,428,100,278),(61,200,429,101,279),(62,201,430,102,280),(63,202,431,103,281),(64,203,432,104,282),(65,204,433,105,283),(66,205,434,106,284),(67,206,435,107,285),(68,207,436,108,286),(69,208,437,109,287),(70,209,438,110,288),(71,210,439,111,289),(72,211,440,112,290),(73,212,441,113,291),(74,213,442,114,292),(75,214,443,115,293),(76,215,444,116,294),(77,216,445,117,295),(78,217,446,118,296),(79,218,447,119,297),(80,219,448,120,298),(81,220,449,121,299),(82,221,450,122,300),(83,222,361,123,301),(84,223,362,124,302),(85,224,363,125,303),(86,225,364,126,304),(87,226,365,127,305),(88,227,366,128,306),(89,228,367,129,307),(90,229,368,130,308)], [(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 | 3B | 5A | ··· | 5X | 6A | 6B | 9A | ··· | 9F | 10A | ··· | 10X | 15A | ··· | 15AV | 18A | ··· | 18F | 30A | ··· | 30AV | 45A | ··· | 45EN | 90A | ··· | 90EN |
| order | 1 | 2 | 3 | 3 | 5 | ··· | 5 | 6 | 6 | 9 | ··· | 9 | 10 | ··· | 10 | 15 | ··· | 15 | 18 | ··· | 18 | 30 | ··· | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 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 | C9 | C10 | C15 | C18 | C30 | C45 | C90 |
| kernel | C5×C90 | C5×C45 | C5×C30 | C90 | C5×C15 | C5×C10 | C45 | C30 | C52 | C15 | C10 | C5 |
| # reps | 1 | 1 | 2 | 24 | 2 | 6 | 24 | 48 | 6 | 48 | 144 | 144 |
Matrix representation of C5×C90 ►in GL2(𝔽181) generated by
| 59 | 0 |
| 0 | 59 |
| 94 | 0 |
| 0 | 11 |
G:=sub<GL(2,GF(181))| [59,0,0,59],[94,0,0,11] >;
C5×C90 in GAP, Magma, Sage, TeX
C_5\times C_{90} % in TeX
G:=Group("C5xC90"); // GroupNames label
G:=SmallGroup(450,19);
// by ID
G=gap.SmallGroup(450,19);
# by ID
G:=PCGroup([5,-2,-3,-5,-5,-3,756]);
// Polycyclic
G:=Group<a,b|a^5=b^90=1,a*b=b*a>;
// generators/relations
Export