direct product, abelian, monomial, 5-elementary
Aliases: C5×C95, SmallGroup(475,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C95 |
| C1 — C5×C95 |
| C1 — C5×C95 |
Generators and relations for C5×C95
G = < a,b | a5=b95=1, ab=ba >
Smallest permutation representation of C5×C95
►Regular action on 475 points
Generators in S475
(1 412 369 238 143)(2 413 370 239 144)(3 414 371 240 145)(4 415 372 241 146)(5 416 373 242 147)(6 417 374 243 148)(7 418 375 244 149)(8 419 376 245 150)(9 420 377 246 151)(10 421 378 247 152)(11 422 379 248 153)(12 423 380 249 154)(13 424 286 250 155)(14 425 287 251 156)(15 426 288 252 157)(16 427 289 253 158)(17 428 290 254 159)(18 429 291 255 160)(19 430 292 256 161)(20 431 293 257 162)(21 432 294 258 163)(22 433 295 259 164)(23 434 296 260 165)(24 435 297 261 166)(25 436 298 262 167)(26 437 299 263 168)(27 438 300 264 169)(28 439 301 265 170)(29 440 302 266 171)(30 441 303 267 172)(31 442 304 268 173)(32 443 305 269 174)(33 444 306 270 175)(34 445 307 271 176)(35 446 308 272 177)(36 447 309 273 178)(37 448 310 274 179)(38 449 311 275 180)(39 450 312 276 181)(40 451 313 277 182)(41 452 314 278 183)(42 453 315 279 184)(43 454 316 280 185)(44 455 317 281 186)(45 456 318 282 187)(46 457 319 283 188)(47 458 320 284 189)(48 459 321 285 190)(49 460 322 191 96)(50 461 323 192 97)(51 462 324 193 98)(52 463 325 194 99)(53 464 326 195 100)(54 465 327 196 101)(55 466 328 197 102)(56 467 329 198 103)(57 468 330 199 104)(58 469 331 200 105)(59 470 332 201 106)(60 471 333 202 107)(61 472 334 203 108)(62 473 335 204 109)(63 474 336 205 110)(64 475 337 206 111)(65 381 338 207 112)(66 382 339 208 113)(67 383 340 209 114)(68 384 341 210 115)(69 385 342 211 116)(70 386 343 212 117)(71 387 344 213 118)(72 388 345 214 119)(73 389 346 215 120)(74 390 347 216 121)(75 391 348 217 122)(76 392 349 218 123)(77 393 350 219 124)(78 394 351 220 125)(79 395 352 221 126)(80 396 353 222 127)(81 397 354 223 128)(82 398 355 224 129)(83 399 356 225 130)(84 400 357 226 131)(85 401 358 227 132)(86 402 359 228 133)(87 403 360 229 134)(88 404 361 230 135)(89 405 362 231 136)(90 406 363 232 137)(91 407 364 233 138)(92 408 365 234 139)(93 409 366 235 140)(94 410 367 236 141)(95 411 368 237 142)
(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)(381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475)
G:=sub<Sym(475)| (1,412,369,238,143)(2,413,370,239,144)(3,414,371,240,145)(4,415,372,241,146)(5,416,373,242,147)(6,417,374,243,148)(7,418,375,244,149)(8,419,376,245,150)(9,420,377,246,151)(10,421,378,247,152)(11,422,379,248,153)(12,423,380,249,154)(13,424,286,250,155)(14,425,287,251,156)(15,426,288,252,157)(16,427,289,253,158)(17,428,290,254,159)(18,429,291,255,160)(19,430,292,256,161)(20,431,293,257,162)(21,432,294,258,163)(22,433,295,259,164)(23,434,296,260,165)(24,435,297,261,166)(25,436,298,262,167)(26,437,299,263,168)(27,438,300,264,169)(28,439,301,265,170)(29,440,302,266,171)(30,441,303,267,172)(31,442,304,268,173)(32,443,305,269,174)(33,444,306,270,175)(34,445,307,271,176)(35,446,308,272,177)(36,447,309,273,178)(37,448,310,274,179)(38,449,311,275,180)(39,450,312,276,181)(40,451,313,277,182)(41,452,314,278,183)(42,453,315,279,184)(43,454,316,280,185)(44,455,317,281,186)(45,456,318,282,187)(46,457,319,283,188)(47,458,320,284,189)(48,459,321,285,190)(49,460,322,191,96)(50,461,323,192,97)(51,462,324,193,98)(52,463,325,194,99)(53,464,326,195,100)(54,465,327,196,101)(55,466,328,197,102)(56,467,329,198,103)(57,468,330,199,104)(58,469,331,200,105)(59,470,332,201,106)(60,471,333,202,107)(61,472,334,203,108)(62,473,335,204,109)(63,474,336,205,110)(64,475,337,206,111)(65,381,338,207,112)(66,382,339,208,113)(67,383,340,209,114)(68,384,341,210,115)(69,385,342,211,116)(70,386,343,212,117)(71,387,344,213,118)(72,388,345,214,119)(73,389,346,215,120)(74,390,347,216,121)(75,391,348,217,122)(76,392,349,218,123)(77,393,350,219,124)(78,394,351,220,125)(79,395,352,221,126)(80,396,353,222,127)(81,397,354,223,128)(82,398,355,224,129)(83,399,356,225,130)(84,400,357,226,131)(85,401,358,227,132)(86,402,359,228,133)(87,403,360,229,134)(88,404,361,230,135)(89,405,362,231,136)(90,406,363,232,137)(91,407,364,233,138)(92,408,365,234,139)(93,409,366,235,140)(94,410,367,236,141)(95,411,368,237,142), (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)(381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475)>;
G:=Group( (1,412,369,238,143)(2,413,370,239,144)(3,414,371,240,145)(4,415,372,241,146)(5,416,373,242,147)(6,417,374,243,148)(7,418,375,244,149)(8,419,376,245,150)(9,420,377,246,151)(10,421,378,247,152)(11,422,379,248,153)(12,423,380,249,154)(13,424,286,250,155)(14,425,287,251,156)(15,426,288,252,157)(16,427,289,253,158)(17,428,290,254,159)(18,429,291,255,160)(19,430,292,256,161)(20,431,293,257,162)(21,432,294,258,163)(22,433,295,259,164)(23,434,296,260,165)(24,435,297,261,166)(25,436,298,262,167)(26,437,299,263,168)(27,438,300,264,169)(28,439,301,265,170)(29,440,302,266,171)(30,441,303,267,172)(31,442,304,268,173)(32,443,305,269,174)(33,444,306,270,175)(34,445,307,271,176)(35,446,308,272,177)(36,447,309,273,178)(37,448,310,274,179)(38,449,311,275,180)(39,450,312,276,181)(40,451,313,277,182)(41,452,314,278,183)(42,453,315,279,184)(43,454,316,280,185)(44,455,317,281,186)(45,456,318,282,187)(46,457,319,283,188)(47,458,320,284,189)(48,459,321,285,190)(49,460,322,191,96)(50,461,323,192,97)(51,462,324,193,98)(52,463,325,194,99)(53,464,326,195,100)(54,465,327,196,101)(55,466,328,197,102)(56,467,329,198,103)(57,468,330,199,104)(58,469,331,200,105)(59,470,332,201,106)(60,471,333,202,107)(61,472,334,203,108)(62,473,335,204,109)(63,474,336,205,110)(64,475,337,206,111)(65,381,338,207,112)(66,382,339,208,113)(67,383,340,209,114)(68,384,341,210,115)(69,385,342,211,116)(70,386,343,212,117)(71,387,344,213,118)(72,388,345,214,119)(73,389,346,215,120)(74,390,347,216,121)(75,391,348,217,122)(76,392,349,218,123)(77,393,350,219,124)(78,394,351,220,125)(79,395,352,221,126)(80,396,353,222,127)(81,397,354,223,128)(82,398,355,224,129)(83,399,356,225,130)(84,400,357,226,131)(85,401,358,227,132)(86,402,359,228,133)(87,403,360,229,134)(88,404,361,230,135)(89,405,362,231,136)(90,406,363,232,137)(91,407,364,233,138)(92,408,365,234,139)(93,409,366,235,140)(94,410,367,236,141)(95,411,368,237,142), (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)(381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475) );
G=PermutationGroup([[(1,412,369,238,143),(2,413,370,239,144),(3,414,371,240,145),(4,415,372,241,146),(5,416,373,242,147),(6,417,374,243,148),(7,418,375,244,149),(8,419,376,245,150),(9,420,377,246,151),(10,421,378,247,152),(11,422,379,248,153),(12,423,380,249,154),(13,424,286,250,155),(14,425,287,251,156),(15,426,288,252,157),(16,427,289,253,158),(17,428,290,254,159),(18,429,291,255,160),(19,430,292,256,161),(20,431,293,257,162),(21,432,294,258,163),(22,433,295,259,164),(23,434,296,260,165),(24,435,297,261,166),(25,436,298,262,167),(26,437,299,263,168),(27,438,300,264,169),(28,439,301,265,170),(29,440,302,266,171),(30,441,303,267,172),(31,442,304,268,173),(32,443,305,269,174),(33,444,306,270,175),(34,445,307,271,176),(35,446,308,272,177),(36,447,309,273,178),(37,448,310,274,179),(38,449,311,275,180),(39,450,312,276,181),(40,451,313,277,182),(41,452,314,278,183),(42,453,315,279,184),(43,454,316,280,185),(44,455,317,281,186),(45,456,318,282,187),(46,457,319,283,188),(47,458,320,284,189),(48,459,321,285,190),(49,460,322,191,96),(50,461,323,192,97),(51,462,324,193,98),(52,463,325,194,99),(53,464,326,195,100),(54,465,327,196,101),(55,466,328,197,102),(56,467,329,198,103),(57,468,330,199,104),(58,469,331,200,105),(59,470,332,201,106),(60,471,333,202,107),(61,472,334,203,108),(62,473,335,204,109),(63,474,336,205,110),(64,475,337,206,111),(65,381,338,207,112),(66,382,339,208,113),(67,383,340,209,114),(68,384,341,210,115),(69,385,342,211,116),(70,386,343,212,117),(71,387,344,213,118),(72,388,345,214,119),(73,389,346,215,120),(74,390,347,216,121),(75,391,348,217,122),(76,392,349,218,123),(77,393,350,219,124),(78,394,351,220,125),(79,395,352,221,126),(80,396,353,222,127),(81,397,354,223,128),(82,398,355,224,129),(83,399,356,225,130),(84,400,357,226,131),(85,401,358,227,132),(86,402,359,228,133),(87,403,360,229,134),(88,404,361,230,135),(89,405,362,231,136),(90,406,363,232,137),(91,407,364,233,138),(92,408,365,234,139),(93,409,366,235,140),(94,410,367,236,141),(95,411,368,237,142)], [(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),(381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475)]])
475 conjugacy classes
| class | 1 | 5A | ··· | 5X | 19A | ··· | 19R | 95A | ··· | 95PP |
| order | 1 | 5 | ··· | 5 | 19 | ··· | 19 | 95 | ··· | 95 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
475 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C5 | C19 | C95 |
| kernel | C5×C95 | C95 | C52 | C5 |
| # reps | 1 | 24 | 18 | 432 |
Matrix representation of C5×C95 ►in GL2(𝔽191) generated by
| 39 | 0 |
| 0 | 39 |
| 130 | 0 |
| 0 | 65 |
G:=sub<GL(2,GF(191))| [39,0,0,39],[130,0,0,65] >;
C5×C95 in GAP, Magma, Sage, TeX
C_5\times C_{95} % in TeX
G:=Group("C5xC95"); // GroupNames label
G:=SmallGroup(475,2);
// by ID
G=gap.SmallGroup(475,2);
# by ID
G:=PCGroup([3,-5,-5,-19]);
// Polycyclic
G:=Group<a,b|a^5=b^95=1,a*b=b*a>;
// generators/relations
Export