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G = C5×C90order 450 = 2·32·52

Abelian group of type [5,90]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C90, SmallGroup(450,19)

Series: Derived Chief Lower central Upper central

C1 — C5×C90
C1C3C15C5×C15C5×C45 — 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
(1 202 387 288 153)(2 203 388 289 154)(3 204 389 290 155)(4 205 390 291 156)(5 206 391 292 157)(6 207 392 293 158)(7 208 393 294 159)(8 209 394 295 160)(9 210 395 296 161)(10 211 396 297 162)(11 212 397 298 163)(12 213 398 299 164)(13 214 399 300 165)(14 215 400 301 166)(15 216 401 302 167)(16 217 402 303 168)(17 218 403 304 169)(18 219 404 305 170)(19 220 405 306 171)(20 221 406 307 172)(21 222 407 308 173)(22 223 408 309 174)(23 224 409 310 175)(24 225 410 311 176)(25 226 411 312 177)(26 227 412 313 178)(27 228 413 314 179)(28 229 414 315 180)(29 230 415 316 91)(30 231 416 317 92)(31 232 417 318 93)(32 233 418 319 94)(33 234 419 320 95)(34 235 420 321 96)(35 236 421 322 97)(36 237 422 323 98)(37 238 423 324 99)(38 239 424 325 100)(39 240 425 326 101)(40 241 426 327 102)(41 242 427 328 103)(42 243 428 329 104)(43 244 429 330 105)(44 245 430 331 106)(45 246 431 332 107)(46 247 432 333 108)(47 248 433 334 109)(48 249 434 335 110)(49 250 435 336 111)(50 251 436 337 112)(51 252 437 338 113)(52 253 438 339 114)(53 254 439 340 115)(54 255 440 341 116)(55 256 441 342 117)(56 257 442 343 118)(57 258 443 344 119)(58 259 444 345 120)(59 260 445 346 121)(60 261 446 347 122)(61 262 447 348 123)(62 263 448 349 124)(63 264 449 350 125)(64 265 450 351 126)(65 266 361 352 127)(66 267 362 353 128)(67 268 363 354 129)(68 269 364 355 130)(69 270 365 356 131)(70 181 366 357 132)(71 182 367 358 133)(72 183 368 359 134)(73 184 369 360 135)(74 185 370 271 136)(75 186 371 272 137)(76 187 372 273 138)(77 188 373 274 139)(78 189 374 275 140)(79 190 375 276 141)(80 191 376 277 142)(81 192 377 278 143)(82 193 378 279 144)(83 194 379 280 145)(84 195 380 281 146)(85 196 381 282 147)(86 197 382 283 148)(87 198 383 284 149)(88 199 384 285 150)(89 200 385 286 151)(90 201 386 287 152)
(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)| (1,202,387,288,153)(2,203,388,289,154)(3,204,389,290,155)(4,205,390,291,156)(5,206,391,292,157)(6,207,392,293,158)(7,208,393,294,159)(8,209,394,295,160)(9,210,395,296,161)(10,211,396,297,162)(11,212,397,298,163)(12,213,398,299,164)(13,214,399,300,165)(14,215,400,301,166)(15,216,401,302,167)(16,217,402,303,168)(17,218,403,304,169)(18,219,404,305,170)(19,220,405,306,171)(20,221,406,307,172)(21,222,407,308,173)(22,223,408,309,174)(23,224,409,310,175)(24,225,410,311,176)(25,226,411,312,177)(26,227,412,313,178)(27,228,413,314,179)(28,229,414,315,180)(29,230,415,316,91)(30,231,416,317,92)(31,232,417,318,93)(32,233,418,319,94)(33,234,419,320,95)(34,235,420,321,96)(35,236,421,322,97)(36,237,422,323,98)(37,238,423,324,99)(38,239,424,325,100)(39,240,425,326,101)(40,241,426,327,102)(41,242,427,328,103)(42,243,428,329,104)(43,244,429,330,105)(44,245,430,331,106)(45,246,431,332,107)(46,247,432,333,108)(47,248,433,334,109)(48,249,434,335,110)(49,250,435,336,111)(50,251,436,337,112)(51,252,437,338,113)(52,253,438,339,114)(53,254,439,340,115)(54,255,440,341,116)(55,256,441,342,117)(56,257,442,343,118)(57,258,443,344,119)(58,259,444,345,120)(59,260,445,346,121)(60,261,446,347,122)(61,262,447,348,123)(62,263,448,349,124)(63,264,449,350,125)(64,265,450,351,126)(65,266,361,352,127)(66,267,362,353,128)(67,268,363,354,129)(68,269,364,355,130)(69,270,365,356,131)(70,181,366,357,132)(71,182,367,358,133)(72,183,368,359,134)(73,184,369,360,135)(74,185,370,271,136)(75,186,371,272,137)(76,187,372,273,138)(77,188,373,274,139)(78,189,374,275,140)(79,190,375,276,141)(80,191,376,277,142)(81,192,377,278,143)(82,193,378,279,144)(83,194,379,280,145)(84,195,380,281,146)(85,196,381,282,147)(86,197,382,283,148)(87,198,383,284,149)(88,199,384,285,150)(89,200,385,286,151)(90,201,386,287,152), (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( (1,202,387,288,153)(2,203,388,289,154)(3,204,389,290,155)(4,205,390,291,156)(5,206,391,292,157)(6,207,392,293,158)(7,208,393,294,159)(8,209,394,295,160)(9,210,395,296,161)(10,211,396,297,162)(11,212,397,298,163)(12,213,398,299,164)(13,214,399,300,165)(14,215,400,301,166)(15,216,401,302,167)(16,217,402,303,168)(17,218,403,304,169)(18,219,404,305,170)(19,220,405,306,171)(20,221,406,307,172)(21,222,407,308,173)(22,223,408,309,174)(23,224,409,310,175)(24,225,410,311,176)(25,226,411,312,177)(26,227,412,313,178)(27,228,413,314,179)(28,229,414,315,180)(29,230,415,316,91)(30,231,416,317,92)(31,232,417,318,93)(32,233,418,319,94)(33,234,419,320,95)(34,235,420,321,96)(35,236,421,322,97)(36,237,422,323,98)(37,238,423,324,99)(38,239,424,325,100)(39,240,425,326,101)(40,241,426,327,102)(41,242,427,328,103)(42,243,428,329,104)(43,244,429,330,105)(44,245,430,331,106)(45,246,431,332,107)(46,247,432,333,108)(47,248,433,334,109)(48,249,434,335,110)(49,250,435,336,111)(50,251,436,337,112)(51,252,437,338,113)(52,253,438,339,114)(53,254,439,340,115)(54,255,440,341,116)(55,256,441,342,117)(56,257,442,343,118)(57,258,443,344,119)(58,259,444,345,120)(59,260,445,346,121)(60,261,446,347,122)(61,262,447,348,123)(62,263,448,349,124)(63,264,449,350,125)(64,265,450,351,126)(65,266,361,352,127)(66,267,362,353,128)(67,268,363,354,129)(68,269,364,355,130)(69,270,365,356,131)(70,181,366,357,132)(71,182,367,358,133)(72,183,368,359,134)(73,184,369,360,135)(74,185,370,271,136)(75,186,371,272,137)(76,187,372,273,138)(77,188,373,274,139)(78,189,374,275,140)(79,190,375,276,141)(80,191,376,277,142)(81,192,377,278,143)(82,193,378,279,144)(83,194,379,280,145)(84,195,380,281,146)(85,196,381,282,147)(86,197,382,283,148)(87,198,383,284,149)(88,199,384,285,150)(89,200,385,286,151)(90,201,386,287,152), (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([(1,202,387,288,153),(2,203,388,289,154),(3,204,389,290,155),(4,205,390,291,156),(5,206,391,292,157),(6,207,392,293,158),(7,208,393,294,159),(8,209,394,295,160),(9,210,395,296,161),(10,211,396,297,162),(11,212,397,298,163),(12,213,398,299,164),(13,214,399,300,165),(14,215,400,301,166),(15,216,401,302,167),(16,217,402,303,168),(17,218,403,304,169),(18,219,404,305,170),(19,220,405,306,171),(20,221,406,307,172),(21,222,407,308,173),(22,223,408,309,174),(23,224,409,310,175),(24,225,410,311,176),(25,226,411,312,177),(26,227,412,313,178),(27,228,413,314,179),(28,229,414,315,180),(29,230,415,316,91),(30,231,416,317,92),(31,232,417,318,93),(32,233,418,319,94),(33,234,419,320,95),(34,235,420,321,96),(35,236,421,322,97),(36,237,422,323,98),(37,238,423,324,99),(38,239,424,325,100),(39,240,425,326,101),(40,241,426,327,102),(41,242,427,328,103),(42,243,428,329,104),(43,244,429,330,105),(44,245,430,331,106),(45,246,431,332,107),(46,247,432,333,108),(47,248,433,334,109),(48,249,434,335,110),(49,250,435,336,111),(50,251,436,337,112),(51,252,437,338,113),(52,253,438,339,114),(53,254,439,340,115),(54,255,440,341,116),(55,256,441,342,117),(56,257,442,343,118),(57,258,443,344,119),(58,259,444,345,120),(59,260,445,346,121),(60,261,446,347,122),(61,262,447,348,123),(62,263,448,349,124),(63,264,449,350,125),(64,265,450,351,126),(65,266,361,352,127),(66,267,362,353,128),(67,268,363,354,129),(68,269,364,355,130),(69,270,365,356,131),(70,181,366,357,132),(71,182,367,358,133),(72,183,368,359,134),(73,184,369,360,135),(74,185,370,271,136),(75,186,371,272,137),(76,187,372,273,138),(77,188,373,274,139),(78,189,374,275,140),(79,190,375,276,141),(80,191,376,277,142),(81,192,377,278,143),(82,193,378,279,144),(83,194,379,280,145),(84,195,380,281,146),(85,196,381,282,147),(86,197,382,283,148),(87,198,383,284,149),(88,199,384,285,150),(89,200,385,286,151),(90,201,386,287,152)], [(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 3A3B5A···5X6A6B9A···9F10A···10X15A···15AV18A···18F30A···30AV45A···45EN90A···90EN
order12335···5669···910···1015···1518···1830···3045···4590···90
size11111···1111···11···11···11···11···11···11···1

450 irreducible representations

dim111111111111
type++
imageC1C2C3C5C6C9C10C15C18C30C45C90
kernelC5×C90C5×C45C5×C30C90C5×C15C5×C10C45C30C52C15C10C5
# reps11224262448648144144

Matrix representation of C5×C90 in GL2(𝔽181) generated by

590
059
,
940
011
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

Subgroup lattice of C5×C90 in TeX

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