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G = C4×C124order 496 = 24·31

Abelian group of type [4,124]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C124, SmallGroup(496,19)

Series: Derived Chief Lower central Upper central

C1 — C4×C124
C1C2C22C2×C62C2×C124 — 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
(1 251 165 481)(2 252 166 482)(3 253 167 483)(4 254 168 484)(5 255 169 485)(6 256 170 486)(7 257 171 487)(8 258 172 488)(9 259 173 489)(10 260 174 490)(11 261 175 491)(12 262 176 492)(13 263 177 493)(14 264 178 494)(15 265 179 495)(16 266 180 496)(17 267 181 373)(18 268 182 374)(19 269 183 375)(20 270 184 376)(21 271 185 377)(22 272 186 378)(23 273 187 379)(24 274 188 380)(25 275 189 381)(26 276 190 382)(27 277 191 383)(28 278 192 384)(29 279 193 385)(30 280 194 386)(31 281 195 387)(32 282 196 388)(33 283 197 389)(34 284 198 390)(35 285 199 391)(36 286 200 392)(37 287 201 393)(38 288 202 394)(39 289 203 395)(40 290 204 396)(41 291 205 397)(42 292 206 398)(43 293 207 399)(44 294 208 400)(45 295 209 401)(46 296 210 402)(47 297 211 403)(48 298 212 404)(49 299 213 405)(50 300 214 406)(51 301 215 407)(52 302 216 408)(53 303 217 409)(54 304 218 410)(55 305 219 411)(56 306 220 412)(57 307 221 413)(58 308 222 414)(59 309 223 415)(60 310 224 416)(61 311 225 417)(62 312 226 418)(63 313 227 419)(64 314 228 420)(65 315 229 421)(66 316 230 422)(67 317 231 423)(68 318 232 424)(69 319 233 425)(70 320 234 426)(71 321 235 427)(72 322 236 428)(73 323 237 429)(74 324 238 430)(75 325 239 431)(76 326 240 432)(77 327 241 433)(78 328 242 434)(79 329 243 435)(80 330 244 436)(81 331 245 437)(82 332 246 438)(83 333 247 439)(84 334 248 440)(85 335 125 441)(86 336 126 442)(87 337 127 443)(88 338 128 444)(89 339 129 445)(90 340 130 446)(91 341 131 447)(92 342 132 448)(93 343 133 449)(94 344 134 450)(95 345 135 451)(96 346 136 452)(97 347 137 453)(98 348 138 454)(99 349 139 455)(100 350 140 456)(101 351 141 457)(102 352 142 458)(103 353 143 459)(104 354 144 460)(105 355 145 461)(106 356 146 462)(107 357 147 463)(108 358 148 464)(109 359 149 465)(110 360 150 466)(111 361 151 467)(112 362 152 468)(113 363 153 469)(114 364 154 470)(115 365 155 471)(116 366 156 472)(117 367 157 473)(118 368 158 474)(119 369 159 475)(120 370 160 476)(121 371 161 477)(122 372 162 478)(123 249 163 479)(124 250 164 480)
(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)| (1,251,165,481)(2,252,166,482)(3,253,167,483)(4,254,168,484)(5,255,169,485)(6,256,170,486)(7,257,171,487)(8,258,172,488)(9,259,173,489)(10,260,174,490)(11,261,175,491)(12,262,176,492)(13,263,177,493)(14,264,178,494)(15,265,179,495)(16,266,180,496)(17,267,181,373)(18,268,182,374)(19,269,183,375)(20,270,184,376)(21,271,185,377)(22,272,186,378)(23,273,187,379)(24,274,188,380)(25,275,189,381)(26,276,190,382)(27,277,191,383)(28,278,192,384)(29,279,193,385)(30,280,194,386)(31,281,195,387)(32,282,196,388)(33,283,197,389)(34,284,198,390)(35,285,199,391)(36,286,200,392)(37,287,201,393)(38,288,202,394)(39,289,203,395)(40,290,204,396)(41,291,205,397)(42,292,206,398)(43,293,207,399)(44,294,208,400)(45,295,209,401)(46,296,210,402)(47,297,211,403)(48,298,212,404)(49,299,213,405)(50,300,214,406)(51,301,215,407)(52,302,216,408)(53,303,217,409)(54,304,218,410)(55,305,219,411)(56,306,220,412)(57,307,221,413)(58,308,222,414)(59,309,223,415)(60,310,224,416)(61,311,225,417)(62,312,226,418)(63,313,227,419)(64,314,228,420)(65,315,229,421)(66,316,230,422)(67,317,231,423)(68,318,232,424)(69,319,233,425)(70,320,234,426)(71,321,235,427)(72,322,236,428)(73,323,237,429)(74,324,238,430)(75,325,239,431)(76,326,240,432)(77,327,241,433)(78,328,242,434)(79,329,243,435)(80,330,244,436)(81,331,245,437)(82,332,246,438)(83,333,247,439)(84,334,248,440)(85,335,125,441)(86,336,126,442)(87,337,127,443)(88,338,128,444)(89,339,129,445)(90,340,130,446)(91,341,131,447)(92,342,132,448)(93,343,133,449)(94,344,134,450)(95,345,135,451)(96,346,136,452)(97,347,137,453)(98,348,138,454)(99,349,139,455)(100,350,140,456)(101,351,141,457)(102,352,142,458)(103,353,143,459)(104,354,144,460)(105,355,145,461)(106,356,146,462)(107,357,147,463)(108,358,148,464)(109,359,149,465)(110,360,150,466)(111,361,151,467)(112,362,152,468)(113,363,153,469)(114,364,154,470)(115,365,155,471)(116,366,156,472)(117,367,157,473)(118,368,158,474)(119,369,159,475)(120,370,160,476)(121,371,161,477)(122,372,162,478)(123,249,163,479)(124,250,164,480), (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( (1,251,165,481)(2,252,166,482)(3,253,167,483)(4,254,168,484)(5,255,169,485)(6,256,170,486)(7,257,171,487)(8,258,172,488)(9,259,173,489)(10,260,174,490)(11,261,175,491)(12,262,176,492)(13,263,177,493)(14,264,178,494)(15,265,179,495)(16,266,180,496)(17,267,181,373)(18,268,182,374)(19,269,183,375)(20,270,184,376)(21,271,185,377)(22,272,186,378)(23,273,187,379)(24,274,188,380)(25,275,189,381)(26,276,190,382)(27,277,191,383)(28,278,192,384)(29,279,193,385)(30,280,194,386)(31,281,195,387)(32,282,196,388)(33,283,197,389)(34,284,198,390)(35,285,199,391)(36,286,200,392)(37,287,201,393)(38,288,202,394)(39,289,203,395)(40,290,204,396)(41,291,205,397)(42,292,206,398)(43,293,207,399)(44,294,208,400)(45,295,209,401)(46,296,210,402)(47,297,211,403)(48,298,212,404)(49,299,213,405)(50,300,214,406)(51,301,215,407)(52,302,216,408)(53,303,217,409)(54,304,218,410)(55,305,219,411)(56,306,220,412)(57,307,221,413)(58,308,222,414)(59,309,223,415)(60,310,224,416)(61,311,225,417)(62,312,226,418)(63,313,227,419)(64,314,228,420)(65,315,229,421)(66,316,230,422)(67,317,231,423)(68,318,232,424)(69,319,233,425)(70,320,234,426)(71,321,235,427)(72,322,236,428)(73,323,237,429)(74,324,238,430)(75,325,239,431)(76,326,240,432)(77,327,241,433)(78,328,242,434)(79,329,243,435)(80,330,244,436)(81,331,245,437)(82,332,246,438)(83,333,247,439)(84,334,248,440)(85,335,125,441)(86,336,126,442)(87,337,127,443)(88,338,128,444)(89,339,129,445)(90,340,130,446)(91,341,131,447)(92,342,132,448)(93,343,133,449)(94,344,134,450)(95,345,135,451)(96,346,136,452)(97,347,137,453)(98,348,138,454)(99,349,139,455)(100,350,140,456)(101,351,141,457)(102,352,142,458)(103,353,143,459)(104,354,144,460)(105,355,145,461)(106,356,146,462)(107,357,147,463)(108,358,148,464)(109,359,149,465)(110,360,150,466)(111,361,151,467)(112,362,152,468)(113,363,153,469)(114,364,154,470)(115,365,155,471)(116,366,156,472)(117,367,157,473)(118,368,158,474)(119,369,159,475)(120,370,160,476)(121,371,161,477)(122,372,162,478)(123,249,163,479)(124,250,164,480), (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([(1,251,165,481),(2,252,166,482),(3,253,167,483),(4,254,168,484),(5,255,169,485),(6,256,170,486),(7,257,171,487),(8,258,172,488),(9,259,173,489),(10,260,174,490),(11,261,175,491),(12,262,176,492),(13,263,177,493),(14,264,178,494),(15,265,179,495),(16,266,180,496),(17,267,181,373),(18,268,182,374),(19,269,183,375),(20,270,184,376),(21,271,185,377),(22,272,186,378),(23,273,187,379),(24,274,188,380),(25,275,189,381),(26,276,190,382),(27,277,191,383),(28,278,192,384),(29,279,193,385),(30,280,194,386),(31,281,195,387),(32,282,196,388),(33,283,197,389),(34,284,198,390),(35,285,199,391),(36,286,200,392),(37,287,201,393),(38,288,202,394),(39,289,203,395),(40,290,204,396),(41,291,205,397),(42,292,206,398),(43,293,207,399),(44,294,208,400),(45,295,209,401),(46,296,210,402),(47,297,211,403),(48,298,212,404),(49,299,213,405),(50,300,214,406),(51,301,215,407),(52,302,216,408),(53,303,217,409),(54,304,218,410),(55,305,219,411),(56,306,220,412),(57,307,221,413),(58,308,222,414),(59,309,223,415),(60,310,224,416),(61,311,225,417),(62,312,226,418),(63,313,227,419),(64,314,228,420),(65,315,229,421),(66,316,230,422),(67,317,231,423),(68,318,232,424),(69,319,233,425),(70,320,234,426),(71,321,235,427),(72,322,236,428),(73,323,237,429),(74,324,238,430),(75,325,239,431),(76,326,240,432),(77,327,241,433),(78,328,242,434),(79,329,243,435),(80,330,244,436),(81,331,245,437),(82,332,246,438),(83,333,247,439),(84,334,248,440),(85,335,125,441),(86,336,126,442),(87,337,127,443),(88,338,128,444),(89,339,129,445),(90,340,130,446),(91,341,131,447),(92,342,132,448),(93,343,133,449),(94,344,134,450),(95,345,135,451),(96,346,136,452),(97,347,137,453),(98,348,138,454),(99,349,139,455),(100,350,140,456),(101,351,141,457),(102,352,142,458),(103,353,143,459),(104,354,144,460),(105,355,145,461),(106,356,146,462),(107,357,147,463),(108,358,148,464),(109,359,149,465),(110,360,150,466),(111,361,151,467),(112,362,152,468),(113,363,153,469),(114,364,154,470),(115,365,155,471),(116,366,156,472),(117,367,157,473),(118,368,158,474),(119,369,159,475),(120,370,160,476),(121,371,161,477),(122,372,162,478),(123,249,163,479),(124,250,164,480)], [(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 2A2B2C4A···4L31A···31AD62A···62CL124A···124MV
order12224···431···3162···62124···124
size11111···11···11···11···1

496 irreducible representations

dim111111
type++
imageC1C2C4C31C62C124
kernelC4×C124C2×C124C124C42C2×C4C4
# reps13123090360

Matrix representation of C4×C124 in GL2(𝔽373) generated by

2690
0269
,
3230
084
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

Subgroup lattice of C4×C124 in TeX

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