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G = C5×C100order 500 = 22·53

Abelian group of type [5,100]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C100, SmallGroup(500,12)

Series: Derived Chief Lower central Upper central

C1 — C5×C100
C1C5C10C5×C10C5×C50 — C5×C100
C1 — C5×C100
C1 — C5×C100

Generators and relations for C5×C100
 G = < a,b | a5=b100=1, ab=ba >


Smallest permutation representation of C5×C100
Regular action on 500 points
Generators in S500
(1 127 340 465 207)(2 128 341 466 208)(3 129 342 467 209)(4 130 343 468 210)(5 131 344 469 211)(6 132 345 470 212)(7 133 346 471 213)(8 134 347 472 214)(9 135 348 473 215)(10 136 349 474 216)(11 137 350 475 217)(12 138 351 476 218)(13 139 352 477 219)(14 140 353 478 220)(15 141 354 479 221)(16 142 355 480 222)(17 143 356 481 223)(18 144 357 482 224)(19 145 358 483 225)(20 146 359 484 226)(21 147 360 485 227)(22 148 361 486 228)(23 149 362 487 229)(24 150 363 488 230)(25 151 364 489 231)(26 152 365 490 232)(27 153 366 491 233)(28 154 367 492 234)(29 155 368 493 235)(30 156 369 494 236)(31 157 370 495 237)(32 158 371 496 238)(33 159 372 497 239)(34 160 373 498 240)(35 161 374 499 241)(36 162 375 500 242)(37 163 376 401 243)(38 164 377 402 244)(39 165 378 403 245)(40 166 379 404 246)(41 167 380 405 247)(42 168 381 406 248)(43 169 382 407 249)(44 170 383 408 250)(45 171 384 409 251)(46 172 385 410 252)(47 173 386 411 253)(48 174 387 412 254)(49 175 388 413 255)(50 176 389 414 256)(51 177 390 415 257)(52 178 391 416 258)(53 179 392 417 259)(54 180 393 418 260)(55 181 394 419 261)(56 182 395 420 262)(57 183 396 421 263)(58 184 397 422 264)(59 185 398 423 265)(60 186 399 424 266)(61 187 400 425 267)(62 188 301 426 268)(63 189 302 427 269)(64 190 303 428 270)(65 191 304 429 271)(66 192 305 430 272)(67 193 306 431 273)(68 194 307 432 274)(69 195 308 433 275)(70 196 309 434 276)(71 197 310 435 277)(72 198 311 436 278)(73 199 312 437 279)(74 200 313 438 280)(75 101 314 439 281)(76 102 315 440 282)(77 103 316 441 283)(78 104 317 442 284)(79 105 318 443 285)(80 106 319 444 286)(81 107 320 445 287)(82 108 321 446 288)(83 109 322 447 289)(84 110 323 448 290)(85 111 324 449 291)(86 112 325 450 292)(87 113 326 451 293)(88 114 327 452 294)(89 115 328 453 295)(90 116 329 454 296)(91 117 330 455 297)(92 118 331 456 298)(93 119 332 457 299)(94 120 333 458 300)(95 121 334 459 201)(96 122 335 460 202)(97 123 336 461 203)(98 124 337 462 204)(99 125 338 463 205)(100 126 339 464 206)
(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)(401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500)

G:=sub<Sym(500)| (1,127,340,465,207)(2,128,341,466,208)(3,129,342,467,209)(4,130,343,468,210)(5,131,344,469,211)(6,132,345,470,212)(7,133,346,471,213)(8,134,347,472,214)(9,135,348,473,215)(10,136,349,474,216)(11,137,350,475,217)(12,138,351,476,218)(13,139,352,477,219)(14,140,353,478,220)(15,141,354,479,221)(16,142,355,480,222)(17,143,356,481,223)(18,144,357,482,224)(19,145,358,483,225)(20,146,359,484,226)(21,147,360,485,227)(22,148,361,486,228)(23,149,362,487,229)(24,150,363,488,230)(25,151,364,489,231)(26,152,365,490,232)(27,153,366,491,233)(28,154,367,492,234)(29,155,368,493,235)(30,156,369,494,236)(31,157,370,495,237)(32,158,371,496,238)(33,159,372,497,239)(34,160,373,498,240)(35,161,374,499,241)(36,162,375,500,242)(37,163,376,401,243)(38,164,377,402,244)(39,165,378,403,245)(40,166,379,404,246)(41,167,380,405,247)(42,168,381,406,248)(43,169,382,407,249)(44,170,383,408,250)(45,171,384,409,251)(46,172,385,410,252)(47,173,386,411,253)(48,174,387,412,254)(49,175,388,413,255)(50,176,389,414,256)(51,177,390,415,257)(52,178,391,416,258)(53,179,392,417,259)(54,180,393,418,260)(55,181,394,419,261)(56,182,395,420,262)(57,183,396,421,263)(58,184,397,422,264)(59,185,398,423,265)(60,186,399,424,266)(61,187,400,425,267)(62,188,301,426,268)(63,189,302,427,269)(64,190,303,428,270)(65,191,304,429,271)(66,192,305,430,272)(67,193,306,431,273)(68,194,307,432,274)(69,195,308,433,275)(70,196,309,434,276)(71,197,310,435,277)(72,198,311,436,278)(73,199,312,437,279)(74,200,313,438,280)(75,101,314,439,281)(76,102,315,440,282)(77,103,316,441,283)(78,104,317,442,284)(79,105,318,443,285)(80,106,319,444,286)(81,107,320,445,287)(82,108,321,446,288)(83,109,322,447,289)(84,110,323,448,290)(85,111,324,449,291)(86,112,325,450,292)(87,113,326,451,293)(88,114,327,452,294)(89,115,328,453,295)(90,116,329,454,296)(91,117,330,455,297)(92,118,331,456,298)(93,119,332,457,299)(94,120,333,458,300)(95,121,334,459,201)(96,122,335,460,202)(97,123,336,461,203)(98,124,337,462,204)(99,125,338,463,205)(100,126,339,464,206), (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)(401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500)>;

G:=Group( (1,127,340,465,207)(2,128,341,466,208)(3,129,342,467,209)(4,130,343,468,210)(5,131,344,469,211)(6,132,345,470,212)(7,133,346,471,213)(8,134,347,472,214)(9,135,348,473,215)(10,136,349,474,216)(11,137,350,475,217)(12,138,351,476,218)(13,139,352,477,219)(14,140,353,478,220)(15,141,354,479,221)(16,142,355,480,222)(17,143,356,481,223)(18,144,357,482,224)(19,145,358,483,225)(20,146,359,484,226)(21,147,360,485,227)(22,148,361,486,228)(23,149,362,487,229)(24,150,363,488,230)(25,151,364,489,231)(26,152,365,490,232)(27,153,366,491,233)(28,154,367,492,234)(29,155,368,493,235)(30,156,369,494,236)(31,157,370,495,237)(32,158,371,496,238)(33,159,372,497,239)(34,160,373,498,240)(35,161,374,499,241)(36,162,375,500,242)(37,163,376,401,243)(38,164,377,402,244)(39,165,378,403,245)(40,166,379,404,246)(41,167,380,405,247)(42,168,381,406,248)(43,169,382,407,249)(44,170,383,408,250)(45,171,384,409,251)(46,172,385,410,252)(47,173,386,411,253)(48,174,387,412,254)(49,175,388,413,255)(50,176,389,414,256)(51,177,390,415,257)(52,178,391,416,258)(53,179,392,417,259)(54,180,393,418,260)(55,181,394,419,261)(56,182,395,420,262)(57,183,396,421,263)(58,184,397,422,264)(59,185,398,423,265)(60,186,399,424,266)(61,187,400,425,267)(62,188,301,426,268)(63,189,302,427,269)(64,190,303,428,270)(65,191,304,429,271)(66,192,305,430,272)(67,193,306,431,273)(68,194,307,432,274)(69,195,308,433,275)(70,196,309,434,276)(71,197,310,435,277)(72,198,311,436,278)(73,199,312,437,279)(74,200,313,438,280)(75,101,314,439,281)(76,102,315,440,282)(77,103,316,441,283)(78,104,317,442,284)(79,105,318,443,285)(80,106,319,444,286)(81,107,320,445,287)(82,108,321,446,288)(83,109,322,447,289)(84,110,323,448,290)(85,111,324,449,291)(86,112,325,450,292)(87,113,326,451,293)(88,114,327,452,294)(89,115,328,453,295)(90,116,329,454,296)(91,117,330,455,297)(92,118,331,456,298)(93,119,332,457,299)(94,120,333,458,300)(95,121,334,459,201)(96,122,335,460,202)(97,123,336,461,203)(98,124,337,462,204)(99,125,338,463,205)(100,126,339,464,206), (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)(401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500) );

G=PermutationGroup([(1,127,340,465,207),(2,128,341,466,208),(3,129,342,467,209),(4,130,343,468,210),(5,131,344,469,211),(6,132,345,470,212),(7,133,346,471,213),(8,134,347,472,214),(9,135,348,473,215),(10,136,349,474,216),(11,137,350,475,217),(12,138,351,476,218),(13,139,352,477,219),(14,140,353,478,220),(15,141,354,479,221),(16,142,355,480,222),(17,143,356,481,223),(18,144,357,482,224),(19,145,358,483,225),(20,146,359,484,226),(21,147,360,485,227),(22,148,361,486,228),(23,149,362,487,229),(24,150,363,488,230),(25,151,364,489,231),(26,152,365,490,232),(27,153,366,491,233),(28,154,367,492,234),(29,155,368,493,235),(30,156,369,494,236),(31,157,370,495,237),(32,158,371,496,238),(33,159,372,497,239),(34,160,373,498,240),(35,161,374,499,241),(36,162,375,500,242),(37,163,376,401,243),(38,164,377,402,244),(39,165,378,403,245),(40,166,379,404,246),(41,167,380,405,247),(42,168,381,406,248),(43,169,382,407,249),(44,170,383,408,250),(45,171,384,409,251),(46,172,385,410,252),(47,173,386,411,253),(48,174,387,412,254),(49,175,388,413,255),(50,176,389,414,256),(51,177,390,415,257),(52,178,391,416,258),(53,179,392,417,259),(54,180,393,418,260),(55,181,394,419,261),(56,182,395,420,262),(57,183,396,421,263),(58,184,397,422,264),(59,185,398,423,265),(60,186,399,424,266),(61,187,400,425,267),(62,188,301,426,268),(63,189,302,427,269),(64,190,303,428,270),(65,191,304,429,271),(66,192,305,430,272),(67,193,306,431,273),(68,194,307,432,274),(69,195,308,433,275),(70,196,309,434,276),(71,197,310,435,277),(72,198,311,436,278),(73,199,312,437,279),(74,200,313,438,280),(75,101,314,439,281),(76,102,315,440,282),(77,103,316,441,283),(78,104,317,442,284),(79,105,318,443,285),(80,106,319,444,286),(81,107,320,445,287),(82,108,321,446,288),(83,109,322,447,289),(84,110,323,448,290),(85,111,324,449,291),(86,112,325,450,292),(87,113,326,451,293),(88,114,327,452,294),(89,115,328,453,295),(90,116,329,454,296),(91,117,330,455,297),(92,118,331,456,298),(93,119,332,457,299),(94,120,333,458,300),(95,121,334,459,201),(96,122,335,460,202),(97,123,336,461,203),(98,124,337,462,204),(99,125,338,463,205),(100,126,339,464,206)], [(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),(401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500)])

500 conjugacy classes

class 1  2 4A4B5A···5X10A···10X20A···20AV25A···25CV50A···50CV100A···100GR
order12445···510···1020···2025···2550···50100···100
size11111···11···11···11···11···11···1

500 irreducible representations

dim111111111111
type++
imageC1C2C4C5C5C10C10C20C20C25C50C100
kernelC5×C100C5×C50C5×C25C100C5×C20C50C5×C10C25C52C20C10C5
# reps112204204408100100200

Matrix representation of C5×C100 in GL2(𝔽101) generated by

950
095
,
140
08
G:=sub<GL(2,GF(101))| [95,0,0,95],[14,0,0,8] >;

C5×C100 in GAP, Magma, Sage, TeX

C_5\times C_{100}
% in TeX

G:=Group("C5xC100");
// GroupNames label

G:=SmallGroup(500,12);
// by ID

G=gap.SmallGroup(500,12);
# by ID

G:=PCGroup([5,-2,-5,-5,-2,-5,250,506]);
// Polycyclic

G:=Group<a,b|a^5=b^100=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C5×C100 in TeX

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