direct product, abelian, monomial, 5-elementary
Aliases: C5×C85, SmallGroup(425,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C85 |
| C1 — C5×C85 |
| C1 — C5×C85 |
Generators and relations for C5×C85
G = < a,b | a5=b85=1, ab=ba >
Smallest permutation representation of C5×C85
►Regular action on 425 points
Generators in S425
(1 364 308 190 105)(2 365 309 191 106)(3 366 310 192 107)(4 367 311 193 108)(5 368 312 194 109)(6 369 313 195 110)(7 370 314 196 111)(8 371 315 197 112)(9 372 316 198 113)(10 373 317 199 114)(11 374 318 200 115)(12 375 319 201 116)(13 376 320 202 117)(14 377 321 203 118)(15 378 322 204 119)(16 379 323 205 120)(17 380 324 206 121)(18 381 325 207 122)(19 382 326 208 123)(20 383 327 209 124)(21 384 328 210 125)(22 385 329 211 126)(23 386 330 212 127)(24 387 331 213 128)(25 388 332 214 129)(26 389 333 215 130)(27 390 334 216 131)(28 391 335 217 132)(29 392 336 218 133)(30 393 337 219 134)(31 394 338 220 135)(32 395 339 221 136)(33 396 340 222 137)(34 397 256 223 138)(35 398 257 224 139)(36 399 258 225 140)(37 400 259 226 141)(38 401 260 227 142)(39 402 261 228 143)(40 403 262 229 144)(41 404 263 230 145)(42 405 264 231 146)(43 406 265 232 147)(44 407 266 233 148)(45 408 267 234 149)(46 409 268 235 150)(47 410 269 236 151)(48 411 270 237 152)(49 412 271 238 153)(50 413 272 239 154)(51 414 273 240 155)(52 415 274 241 156)(53 416 275 242 157)(54 417 276 243 158)(55 418 277 244 159)(56 419 278 245 160)(57 420 279 246 161)(58 421 280 247 162)(59 422 281 248 163)(60 423 282 249 164)(61 424 283 250 165)(62 425 284 251 166)(63 341 285 252 167)(64 342 286 253 168)(65 343 287 254 169)(66 344 288 255 170)(67 345 289 171 86)(68 346 290 172 87)(69 347 291 173 88)(70 348 292 174 89)(71 349 293 175 90)(72 350 294 176 91)(73 351 295 177 92)(74 352 296 178 93)(75 353 297 179 94)(76 354 298 180 95)(77 355 299 181 96)(78 356 300 182 97)(79 357 301 183 98)(80 358 302 184 99)(81 359 303 185 100)(82 360 304 186 101)(83 361 305 187 102)(84 362 306 188 103)(85 363 307 189 104)
(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)
G:=sub<Sym(425)| (1,364,308,190,105)(2,365,309,191,106)(3,366,310,192,107)(4,367,311,193,108)(5,368,312,194,109)(6,369,313,195,110)(7,370,314,196,111)(8,371,315,197,112)(9,372,316,198,113)(10,373,317,199,114)(11,374,318,200,115)(12,375,319,201,116)(13,376,320,202,117)(14,377,321,203,118)(15,378,322,204,119)(16,379,323,205,120)(17,380,324,206,121)(18,381,325,207,122)(19,382,326,208,123)(20,383,327,209,124)(21,384,328,210,125)(22,385,329,211,126)(23,386,330,212,127)(24,387,331,213,128)(25,388,332,214,129)(26,389,333,215,130)(27,390,334,216,131)(28,391,335,217,132)(29,392,336,218,133)(30,393,337,219,134)(31,394,338,220,135)(32,395,339,221,136)(33,396,340,222,137)(34,397,256,223,138)(35,398,257,224,139)(36,399,258,225,140)(37,400,259,226,141)(38,401,260,227,142)(39,402,261,228,143)(40,403,262,229,144)(41,404,263,230,145)(42,405,264,231,146)(43,406,265,232,147)(44,407,266,233,148)(45,408,267,234,149)(46,409,268,235,150)(47,410,269,236,151)(48,411,270,237,152)(49,412,271,238,153)(50,413,272,239,154)(51,414,273,240,155)(52,415,274,241,156)(53,416,275,242,157)(54,417,276,243,158)(55,418,277,244,159)(56,419,278,245,160)(57,420,279,246,161)(58,421,280,247,162)(59,422,281,248,163)(60,423,282,249,164)(61,424,283,250,165)(62,425,284,251,166)(63,341,285,252,167)(64,342,286,253,168)(65,343,287,254,169)(66,344,288,255,170)(67,345,289,171,86)(68,346,290,172,87)(69,347,291,173,88)(70,348,292,174,89)(71,349,293,175,90)(72,350,294,176,91)(73,351,295,177,92)(74,352,296,178,93)(75,353,297,179,94)(76,354,298,180,95)(77,355,299,181,96)(78,356,300,182,97)(79,357,301,183,98)(80,358,302,184,99)(81,359,303,185,100)(82,360,304,186,101)(83,361,305,187,102)(84,362,306,188,103)(85,363,307,189,104), (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)>;
G:=Group( (1,364,308,190,105)(2,365,309,191,106)(3,366,310,192,107)(4,367,311,193,108)(5,368,312,194,109)(6,369,313,195,110)(7,370,314,196,111)(8,371,315,197,112)(9,372,316,198,113)(10,373,317,199,114)(11,374,318,200,115)(12,375,319,201,116)(13,376,320,202,117)(14,377,321,203,118)(15,378,322,204,119)(16,379,323,205,120)(17,380,324,206,121)(18,381,325,207,122)(19,382,326,208,123)(20,383,327,209,124)(21,384,328,210,125)(22,385,329,211,126)(23,386,330,212,127)(24,387,331,213,128)(25,388,332,214,129)(26,389,333,215,130)(27,390,334,216,131)(28,391,335,217,132)(29,392,336,218,133)(30,393,337,219,134)(31,394,338,220,135)(32,395,339,221,136)(33,396,340,222,137)(34,397,256,223,138)(35,398,257,224,139)(36,399,258,225,140)(37,400,259,226,141)(38,401,260,227,142)(39,402,261,228,143)(40,403,262,229,144)(41,404,263,230,145)(42,405,264,231,146)(43,406,265,232,147)(44,407,266,233,148)(45,408,267,234,149)(46,409,268,235,150)(47,410,269,236,151)(48,411,270,237,152)(49,412,271,238,153)(50,413,272,239,154)(51,414,273,240,155)(52,415,274,241,156)(53,416,275,242,157)(54,417,276,243,158)(55,418,277,244,159)(56,419,278,245,160)(57,420,279,246,161)(58,421,280,247,162)(59,422,281,248,163)(60,423,282,249,164)(61,424,283,250,165)(62,425,284,251,166)(63,341,285,252,167)(64,342,286,253,168)(65,343,287,254,169)(66,344,288,255,170)(67,345,289,171,86)(68,346,290,172,87)(69,347,291,173,88)(70,348,292,174,89)(71,349,293,175,90)(72,350,294,176,91)(73,351,295,177,92)(74,352,296,178,93)(75,353,297,179,94)(76,354,298,180,95)(77,355,299,181,96)(78,356,300,182,97)(79,357,301,183,98)(80,358,302,184,99)(81,359,303,185,100)(82,360,304,186,101)(83,361,305,187,102)(84,362,306,188,103)(85,363,307,189,104), (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) );
G=PermutationGroup([[(1,364,308,190,105),(2,365,309,191,106),(3,366,310,192,107),(4,367,311,193,108),(5,368,312,194,109),(6,369,313,195,110),(7,370,314,196,111),(8,371,315,197,112),(9,372,316,198,113),(10,373,317,199,114),(11,374,318,200,115),(12,375,319,201,116),(13,376,320,202,117),(14,377,321,203,118),(15,378,322,204,119),(16,379,323,205,120),(17,380,324,206,121),(18,381,325,207,122),(19,382,326,208,123),(20,383,327,209,124),(21,384,328,210,125),(22,385,329,211,126),(23,386,330,212,127),(24,387,331,213,128),(25,388,332,214,129),(26,389,333,215,130),(27,390,334,216,131),(28,391,335,217,132),(29,392,336,218,133),(30,393,337,219,134),(31,394,338,220,135),(32,395,339,221,136),(33,396,340,222,137),(34,397,256,223,138),(35,398,257,224,139),(36,399,258,225,140),(37,400,259,226,141),(38,401,260,227,142),(39,402,261,228,143),(40,403,262,229,144),(41,404,263,230,145),(42,405,264,231,146),(43,406,265,232,147),(44,407,266,233,148),(45,408,267,234,149),(46,409,268,235,150),(47,410,269,236,151),(48,411,270,237,152),(49,412,271,238,153),(50,413,272,239,154),(51,414,273,240,155),(52,415,274,241,156),(53,416,275,242,157),(54,417,276,243,158),(55,418,277,244,159),(56,419,278,245,160),(57,420,279,246,161),(58,421,280,247,162),(59,422,281,248,163),(60,423,282,249,164),(61,424,283,250,165),(62,425,284,251,166),(63,341,285,252,167),(64,342,286,253,168),(65,343,287,254,169),(66,344,288,255,170),(67,345,289,171,86),(68,346,290,172,87),(69,347,291,173,88),(70,348,292,174,89),(71,349,293,175,90),(72,350,294,176,91),(73,351,295,177,92),(74,352,296,178,93),(75,353,297,179,94),(76,354,298,180,95),(77,355,299,181,96),(78,356,300,182,97),(79,357,301,183,98),(80,358,302,184,99),(81,359,303,185,100),(82,360,304,186,101),(83,361,305,187,102),(84,362,306,188,103),(85,363,307,189,104)], [(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)]])
425 conjugacy classes
| class | 1 | 5A | ··· | 5X | 17A | ··· | 17P | 85A | ··· | 85NT |
| order | 1 | 5 | ··· | 5 | 17 | ··· | 17 | 85 | ··· | 85 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
425 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C5 | C17 | C85 |
| kernel | C5×C85 | C85 | C52 | C5 |
| # reps | 1 | 24 | 16 | 384 |
Matrix representation of C5×C85 ►in GL2(𝔽1021) generated by
| 589 | 0 |
| 0 | 1 |
| 235 | 0 |
| 0 | 676 |
G:=sub<GL(2,GF(1021))| [589,0,0,1],[235,0,0,676] >;
C5×C85 in GAP, Magma, Sage, TeX
C_5\times C_{85} % in TeX
G:=Group("C5xC85"); // GroupNames label
G:=SmallGroup(425,2);
// by ID
G=gap.SmallGroup(425,2);
# by ID
G:=PCGroup([3,-5,-5,-17]);
// Polycyclic
G:=Group<a,b|a^5=b^85=1,a*b=b*a>;
// generators/relations
Export