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G = C5×C85order 425 = 52·17

Abelian group of type [5,85]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C85, SmallGroup(425,2)

Series: Derived Chief Lower central Upper central

C1 — C5×C85
C1C17C85 — 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
(1 402 321 193 170)(2 403 322 194 86)(3 404 323 195 87)(4 405 324 196 88)(5 406 325 197 89)(6 407 326 198 90)(7 408 327 199 91)(8 409 328 200 92)(9 410 329 201 93)(10 411 330 202 94)(11 412 331 203 95)(12 413 332 204 96)(13 414 333 205 97)(14 415 334 206 98)(15 416 335 207 99)(16 417 336 208 100)(17 418 337 209 101)(18 419 338 210 102)(19 420 339 211 103)(20 421 340 212 104)(21 422 256 213 105)(22 423 257 214 106)(23 424 258 215 107)(24 425 259 216 108)(25 341 260 217 109)(26 342 261 218 110)(27 343 262 219 111)(28 344 263 220 112)(29 345 264 221 113)(30 346 265 222 114)(31 347 266 223 115)(32 348 267 224 116)(33 349 268 225 117)(34 350 269 226 118)(35 351 270 227 119)(36 352 271 228 120)(37 353 272 229 121)(38 354 273 230 122)(39 355 274 231 123)(40 356 275 232 124)(41 357 276 233 125)(42 358 277 234 126)(43 359 278 235 127)(44 360 279 236 128)(45 361 280 237 129)(46 362 281 238 130)(47 363 282 239 131)(48 364 283 240 132)(49 365 284 241 133)(50 366 285 242 134)(51 367 286 243 135)(52 368 287 244 136)(53 369 288 245 137)(54 370 289 246 138)(55 371 290 247 139)(56 372 291 248 140)(57 373 292 249 141)(58 374 293 250 142)(59 375 294 251 143)(60 376 295 252 144)(61 377 296 253 145)(62 378 297 254 146)(63 379 298 255 147)(64 380 299 171 148)(65 381 300 172 149)(66 382 301 173 150)(67 383 302 174 151)(68 384 303 175 152)(69 385 304 176 153)(70 386 305 177 154)(71 387 306 178 155)(72 388 307 179 156)(73 389 308 180 157)(74 390 309 181 158)(75 391 310 182 159)(76 392 311 183 160)(77 393 312 184 161)(78 394 313 185 162)(79 395 314 186 163)(80 396 315 187 164)(81 397 316 188 165)(82 398 317 189 166)(83 399 318 190 167)(84 400 319 191 168)(85 401 320 192 169)
(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)| (1,402,321,193,170)(2,403,322,194,86)(3,404,323,195,87)(4,405,324,196,88)(5,406,325,197,89)(6,407,326,198,90)(7,408,327,199,91)(8,409,328,200,92)(9,410,329,201,93)(10,411,330,202,94)(11,412,331,203,95)(12,413,332,204,96)(13,414,333,205,97)(14,415,334,206,98)(15,416,335,207,99)(16,417,336,208,100)(17,418,337,209,101)(18,419,338,210,102)(19,420,339,211,103)(20,421,340,212,104)(21,422,256,213,105)(22,423,257,214,106)(23,424,258,215,107)(24,425,259,216,108)(25,341,260,217,109)(26,342,261,218,110)(27,343,262,219,111)(28,344,263,220,112)(29,345,264,221,113)(30,346,265,222,114)(31,347,266,223,115)(32,348,267,224,116)(33,349,268,225,117)(34,350,269,226,118)(35,351,270,227,119)(36,352,271,228,120)(37,353,272,229,121)(38,354,273,230,122)(39,355,274,231,123)(40,356,275,232,124)(41,357,276,233,125)(42,358,277,234,126)(43,359,278,235,127)(44,360,279,236,128)(45,361,280,237,129)(46,362,281,238,130)(47,363,282,239,131)(48,364,283,240,132)(49,365,284,241,133)(50,366,285,242,134)(51,367,286,243,135)(52,368,287,244,136)(53,369,288,245,137)(54,370,289,246,138)(55,371,290,247,139)(56,372,291,248,140)(57,373,292,249,141)(58,374,293,250,142)(59,375,294,251,143)(60,376,295,252,144)(61,377,296,253,145)(62,378,297,254,146)(63,379,298,255,147)(64,380,299,171,148)(65,381,300,172,149)(66,382,301,173,150)(67,383,302,174,151)(68,384,303,175,152)(69,385,304,176,153)(70,386,305,177,154)(71,387,306,178,155)(72,388,307,179,156)(73,389,308,180,157)(74,390,309,181,158)(75,391,310,182,159)(76,392,311,183,160)(77,393,312,184,161)(78,394,313,185,162)(79,395,314,186,163)(80,396,315,187,164)(81,397,316,188,165)(82,398,317,189,166)(83,399,318,190,167)(84,400,319,191,168)(85,401,320,192,169), (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( (1,402,321,193,170)(2,403,322,194,86)(3,404,323,195,87)(4,405,324,196,88)(5,406,325,197,89)(6,407,326,198,90)(7,408,327,199,91)(8,409,328,200,92)(9,410,329,201,93)(10,411,330,202,94)(11,412,331,203,95)(12,413,332,204,96)(13,414,333,205,97)(14,415,334,206,98)(15,416,335,207,99)(16,417,336,208,100)(17,418,337,209,101)(18,419,338,210,102)(19,420,339,211,103)(20,421,340,212,104)(21,422,256,213,105)(22,423,257,214,106)(23,424,258,215,107)(24,425,259,216,108)(25,341,260,217,109)(26,342,261,218,110)(27,343,262,219,111)(28,344,263,220,112)(29,345,264,221,113)(30,346,265,222,114)(31,347,266,223,115)(32,348,267,224,116)(33,349,268,225,117)(34,350,269,226,118)(35,351,270,227,119)(36,352,271,228,120)(37,353,272,229,121)(38,354,273,230,122)(39,355,274,231,123)(40,356,275,232,124)(41,357,276,233,125)(42,358,277,234,126)(43,359,278,235,127)(44,360,279,236,128)(45,361,280,237,129)(46,362,281,238,130)(47,363,282,239,131)(48,364,283,240,132)(49,365,284,241,133)(50,366,285,242,134)(51,367,286,243,135)(52,368,287,244,136)(53,369,288,245,137)(54,370,289,246,138)(55,371,290,247,139)(56,372,291,248,140)(57,373,292,249,141)(58,374,293,250,142)(59,375,294,251,143)(60,376,295,252,144)(61,377,296,253,145)(62,378,297,254,146)(63,379,298,255,147)(64,380,299,171,148)(65,381,300,172,149)(66,382,301,173,150)(67,383,302,174,151)(68,384,303,175,152)(69,385,304,176,153)(70,386,305,177,154)(71,387,306,178,155)(72,388,307,179,156)(73,389,308,180,157)(74,390,309,181,158)(75,391,310,182,159)(76,392,311,183,160)(77,393,312,184,161)(78,394,313,185,162)(79,395,314,186,163)(80,396,315,187,164)(81,397,316,188,165)(82,398,317,189,166)(83,399,318,190,167)(84,400,319,191,168)(85,401,320,192,169), (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([(1,402,321,193,170),(2,403,322,194,86),(3,404,323,195,87),(4,405,324,196,88),(5,406,325,197,89),(6,407,326,198,90),(7,408,327,199,91),(8,409,328,200,92),(9,410,329,201,93),(10,411,330,202,94),(11,412,331,203,95),(12,413,332,204,96),(13,414,333,205,97),(14,415,334,206,98),(15,416,335,207,99),(16,417,336,208,100),(17,418,337,209,101),(18,419,338,210,102),(19,420,339,211,103),(20,421,340,212,104),(21,422,256,213,105),(22,423,257,214,106),(23,424,258,215,107),(24,425,259,216,108),(25,341,260,217,109),(26,342,261,218,110),(27,343,262,219,111),(28,344,263,220,112),(29,345,264,221,113),(30,346,265,222,114),(31,347,266,223,115),(32,348,267,224,116),(33,349,268,225,117),(34,350,269,226,118),(35,351,270,227,119),(36,352,271,228,120),(37,353,272,229,121),(38,354,273,230,122),(39,355,274,231,123),(40,356,275,232,124),(41,357,276,233,125),(42,358,277,234,126),(43,359,278,235,127),(44,360,279,236,128),(45,361,280,237,129),(46,362,281,238,130),(47,363,282,239,131),(48,364,283,240,132),(49,365,284,241,133),(50,366,285,242,134),(51,367,286,243,135),(52,368,287,244,136),(53,369,288,245,137),(54,370,289,246,138),(55,371,290,247,139),(56,372,291,248,140),(57,373,292,249,141),(58,374,293,250,142),(59,375,294,251,143),(60,376,295,252,144),(61,377,296,253,145),(62,378,297,254,146),(63,379,298,255,147),(64,380,299,171,148),(65,381,300,172,149),(66,382,301,173,150),(67,383,302,174,151),(68,384,303,175,152),(69,385,304,176,153),(70,386,305,177,154),(71,387,306,178,155),(72,388,307,179,156),(73,389,308,180,157),(74,390,309,181,158),(75,391,310,182,159),(76,392,311,183,160),(77,393,312,184,161),(78,394,313,185,162),(79,395,314,186,163),(80,396,315,187,164),(81,397,316,188,165),(82,398,317,189,166),(83,399,318,190,167),(84,400,319,191,168),(85,401,320,192,169)], [(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···5X17A···17P85A···85NT
order15···517···1785···85
size11···11···11···1

425 irreducible representations

dim1111
type+
imageC1C5C17C85
kernelC5×C85C85C52C5
# reps12416384

Matrix representation of C5×C85 in GL2(𝔽1021) generated by

5890
01
,
2350
0676
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

Subgroup lattice of C5×C85 in TeX

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