direct product, abelian, monomial, 2-elementary
Aliases: C4×C124, SmallGroup(496,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C124 |
| C1 — C4×C124 |
| C1 — C4×C124 |
Generators and relations for C4×C124
G = < a,b | a4=b124=1, ab=ba >
Smallest permutation representation of C4×C124
►Regular action on 496 points
Generators in S496
(1 253 380 219)(2 254 381 220)(3 255 382 221)(4 256 383 222)(5 257 384 223)(6 258 385 224)(7 259 386 225)(8 260 387 226)(9 261 388 227)(10 262 389 228)(11 263 390 229)(12 264 391 230)(13 265 392 231)(14 266 393 232)(15 267 394 233)(16 268 395 234)(17 269 396 235)(18 270 397 236)(19 271 398 237)(20 272 399 238)(21 273 400 239)(22 274 401 240)(23 275 402 241)(24 276 403 242)(25 277 404 243)(26 278 405 244)(27 279 406 245)(28 280 407 246)(29 281 408 247)(30 282 409 248)(31 283 410 125)(32 284 411 126)(33 285 412 127)(34 286 413 128)(35 287 414 129)(36 288 415 130)(37 289 416 131)(38 290 417 132)(39 291 418 133)(40 292 419 134)(41 293 420 135)(42 294 421 136)(43 295 422 137)(44 296 423 138)(45 297 424 139)(46 298 425 140)(47 299 426 141)(48 300 427 142)(49 301 428 143)(50 302 429 144)(51 303 430 145)(52 304 431 146)(53 305 432 147)(54 306 433 148)(55 307 434 149)(56 308 435 150)(57 309 436 151)(58 310 437 152)(59 311 438 153)(60 312 439 154)(61 313 440 155)(62 314 441 156)(63 315 442 157)(64 316 443 158)(65 317 444 159)(66 318 445 160)(67 319 446 161)(68 320 447 162)(69 321 448 163)(70 322 449 164)(71 323 450 165)(72 324 451 166)(73 325 452 167)(74 326 453 168)(75 327 454 169)(76 328 455 170)(77 329 456 171)(78 330 457 172)(79 331 458 173)(80 332 459 174)(81 333 460 175)(82 334 461 176)(83 335 462 177)(84 336 463 178)(85 337 464 179)(86 338 465 180)(87 339 466 181)(88 340 467 182)(89 341 468 183)(90 342 469 184)(91 343 470 185)(92 344 471 186)(93 345 472 187)(94 346 473 188)(95 347 474 189)(96 348 475 190)(97 349 476 191)(98 350 477 192)(99 351 478 193)(100 352 479 194)(101 353 480 195)(102 354 481 196)(103 355 482 197)(104 356 483 198)(105 357 484 199)(106 358 485 200)(107 359 486 201)(108 360 487 202)(109 361 488 203)(110 362 489 204)(111 363 490 205)(112 364 491 206)(113 365 492 207)(114 366 493 208)(115 367 494 209)(116 368 495 210)(117 369 496 211)(118 370 373 212)(119 371 374 213)(120 372 375 214)(121 249 376 215)(122 250 377 216)(123 251 378 217)(124 252 379 218)
(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)
G:=sub<Sym(496)| (1,253,380,219)(2,254,381,220)(3,255,382,221)(4,256,383,222)(5,257,384,223)(6,258,385,224)(7,259,386,225)(8,260,387,226)(9,261,388,227)(10,262,389,228)(11,263,390,229)(12,264,391,230)(13,265,392,231)(14,266,393,232)(15,267,394,233)(16,268,395,234)(17,269,396,235)(18,270,397,236)(19,271,398,237)(20,272,399,238)(21,273,400,239)(22,274,401,240)(23,275,402,241)(24,276,403,242)(25,277,404,243)(26,278,405,244)(27,279,406,245)(28,280,407,246)(29,281,408,247)(30,282,409,248)(31,283,410,125)(32,284,411,126)(33,285,412,127)(34,286,413,128)(35,287,414,129)(36,288,415,130)(37,289,416,131)(38,290,417,132)(39,291,418,133)(40,292,419,134)(41,293,420,135)(42,294,421,136)(43,295,422,137)(44,296,423,138)(45,297,424,139)(46,298,425,140)(47,299,426,141)(48,300,427,142)(49,301,428,143)(50,302,429,144)(51,303,430,145)(52,304,431,146)(53,305,432,147)(54,306,433,148)(55,307,434,149)(56,308,435,150)(57,309,436,151)(58,310,437,152)(59,311,438,153)(60,312,439,154)(61,313,440,155)(62,314,441,156)(63,315,442,157)(64,316,443,158)(65,317,444,159)(66,318,445,160)(67,319,446,161)(68,320,447,162)(69,321,448,163)(70,322,449,164)(71,323,450,165)(72,324,451,166)(73,325,452,167)(74,326,453,168)(75,327,454,169)(76,328,455,170)(77,329,456,171)(78,330,457,172)(79,331,458,173)(80,332,459,174)(81,333,460,175)(82,334,461,176)(83,335,462,177)(84,336,463,178)(85,337,464,179)(86,338,465,180)(87,339,466,181)(88,340,467,182)(89,341,468,183)(90,342,469,184)(91,343,470,185)(92,344,471,186)(93,345,472,187)(94,346,473,188)(95,347,474,189)(96,348,475,190)(97,349,476,191)(98,350,477,192)(99,351,478,193)(100,352,479,194)(101,353,480,195)(102,354,481,196)(103,355,482,197)(104,356,483,198)(105,357,484,199)(106,358,485,200)(107,359,486,201)(108,360,487,202)(109,361,488,203)(110,362,489,204)(111,363,490,205)(112,364,491,206)(113,365,492,207)(114,366,493,208)(115,367,494,209)(116,368,495,210)(117,369,496,211)(118,370,373,212)(119,371,374,213)(120,372,375,214)(121,249,376,215)(122,250,377,216)(123,251,378,217)(124,252,379,218), (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)>;
G:=Group( (1,253,380,219)(2,254,381,220)(3,255,382,221)(4,256,383,222)(5,257,384,223)(6,258,385,224)(7,259,386,225)(8,260,387,226)(9,261,388,227)(10,262,389,228)(11,263,390,229)(12,264,391,230)(13,265,392,231)(14,266,393,232)(15,267,394,233)(16,268,395,234)(17,269,396,235)(18,270,397,236)(19,271,398,237)(20,272,399,238)(21,273,400,239)(22,274,401,240)(23,275,402,241)(24,276,403,242)(25,277,404,243)(26,278,405,244)(27,279,406,245)(28,280,407,246)(29,281,408,247)(30,282,409,248)(31,283,410,125)(32,284,411,126)(33,285,412,127)(34,286,413,128)(35,287,414,129)(36,288,415,130)(37,289,416,131)(38,290,417,132)(39,291,418,133)(40,292,419,134)(41,293,420,135)(42,294,421,136)(43,295,422,137)(44,296,423,138)(45,297,424,139)(46,298,425,140)(47,299,426,141)(48,300,427,142)(49,301,428,143)(50,302,429,144)(51,303,430,145)(52,304,431,146)(53,305,432,147)(54,306,433,148)(55,307,434,149)(56,308,435,150)(57,309,436,151)(58,310,437,152)(59,311,438,153)(60,312,439,154)(61,313,440,155)(62,314,441,156)(63,315,442,157)(64,316,443,158)(65,317,444,159)(66,318,445,160)(67,319,446,161)(68,320,447,162)(69,321,448,163)(70,322,449,164)(71,323,450,165)(72,324,451,166)(73,325,452,167)(74,326,453,168)(75,327,454,169)(76,328,455,170)(77,329,456,171)(78,330,457,172)(79,331,458,173)(80,332,459,174)(81,333,460,175)(82,334,461,176)(83,335,462,177)(84,336,463,178)(85,337,464,179)(86,338,465,180)(87,339,466,181)(88,340,467,182)(89,341,468,183)(90,342,469,184)(91,343,470,185)(92,344,471,186)(93,345,472,187)(94,346,473,188)(95,347,474,189)(96,348,475,190)(97,349,476,191)(98,350,477,192)(99,351,478,193)(100,352,479,194)(101,353,480,195)(102,354,481,196)(103,355,482,197)(104,356,483,198)(105,357,484,199)(106,358,485,200)(107,359,486,201)(108,360,487,202)(109,361,488,203)(110,362,489,204)(111,363,490,205)(112,364,491,206)(113,365,492,207)(114,366,493,208)(115,367,494,209)(116,368,495,210)(117,369,496,211)(118,370,373,212)(119,371,374,213)(120,372,375,214)(121,249,376,215)(122,250,377,216)(123,251,378,217)(124,252,379,218), (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) );
G=PermutationGroup([[(1,253,380,219),(2,254,381,220),(3,255,382,221),(4,256,383,222),(5,257,384,223),(6,258,385,224),(7,259,386,225),(8,260,387,226),(9,261,388,227),(10,262,389,228),(11,263,390,229),(12,264,391,230),(13,265,392,231),(14,266,393,232),(15,267,394,233),(16,268,395,234),(17,269,396,235),(18,270,397,236),(19,271,398,237),(20,272,399,238),(21,273,400,239),(22,274,401,240),(23,275,402,241),(24,276,403,242),(25,277,404,243),(26,278,405,244),(27,279,406,245),(28,280,407,246),(29,281,408,247),(30,282,409,248),(31,283,410,125),(32,284,411,126),(33,285,412,127),(34,286,413,128),(35,287,414,129),(36,288,415,130),(37,289,416,131),(38,290,417,132),(39,291,418,133),(40,292,419,134),(41,293,420,135),(42,294,421,136),(43,295,422,137),(44,296,423,138),(45,297,424,139),(46,298,425,140),(47,299,426,141),(48,300,427,142),(49,301,428,143),(50,302,429,144),(51,303,430,145),(52,304,431,146),(53,305,432,147),(54,306,433,148),(55,307,434,149),(56,308,435,150),(57,309,436,151),(58,310,437,152),(59,311,438,153),(60,312,439,154),(61,313,440,155),(62,314,441,156),(63,315,442,157),(64,316,443,158),(65,317,444,159),(66,318,445,160),(67,319,446,161),(68,320,447,162),(69,321,448,163),(70,322,449,164),(71,323,450,165),(72,324,451,166),(73,325,452,167),(74,326,453,168),(75,327,454,169),(76,328,455,170),(77,329,456,171),(78,330,457,172),(79,331,458,173),(80,332,459,174),(81,333,460,175),(82,334,461,176),(83,335,462,177),(84,336,463,178),(85,337,464,179),(86,338,465,180),(87,339,466,181),(88,340,467,182),(89,341,468,183),(90,342,469,184),(91,343,470,185),(92,344,471,186),(93,345,472,187),(94,346,473,188),(95,347,474,189),(96,348,475,190),(97,349,476,191),(98,350,477,192),(99,351,478,193),(100,352,479,194),(101,353,480,195),(102,354,481,196),(103,355,482,197),(104,356,483,198),(105,357,484,199),(106,358,485,200),(107,359,486,201),(108,360,487,202),(109,361,488,203),(110,362,489,204),(111,363,490,205),(112,364,491,206),(113,365,492,207),(114,366,493,208),(115,367,494,209),(116,368,495,210),(117,369,496,211),(118,370,373,212),(119,371,374,213),(120,372,375,214),(121,249,376,215),(122,250,377,216),(123,251,378,217),(124,252,379,218)], [(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)]])
496 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 31A | ··· | 31AD | 62A | ··· | 62CL | 124A | ··· | 124MV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
496 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C31 | C62 | C124 |
| kernel | C4×C124 | C2×C124 | C124 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 12 | 30 | 90 | 360 |
Matrix representation of C4×C124 ►in GL2(𝔽373) generated by
| 269 | 0 |
| 0 | 269 |
| 323 | 0 |
| 0 | 84 |
G:=sub<GL(2,GF(373))| [269,0,0,269],[323,0,0,84] >;
C4×C124 in GAP, Magma, Sage, TeX
C_4\times C_{124} % in TeX
G:=Group("C4xC124"); // GroupNames label
G:=SmallGroup(496,19);
// by ID
G=gap.SmallGroup(496,19);
# by ID
G:=PCGroup([5,-2,-2,-31,-2,-2,620,1246]);
// Polycyclic
G:=Group<a,b|a^4=b^124=1,a*b=b*a>;
// generators/relations
Export