direct product, abelian, monomial, 2-elementary
Aliases: C22×C124, SmallGroup(496,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C124 |
| C1 — C22×C124 |
| C1 — C22×C124 |
Generators and relations for C22×C124
G = < a,b,c | a2=b2=c124=1, ab=ba, ac=ca, bc=cb >
Subgroups: 54, all normal (8 characteristic)
C1, C2, C2, C4, C22, C2×C4, C23, C22×C4, C31, C62, C62, C124, C2×C62, C2×C124, C22×C62, C22×C124
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C31, C62, C124, C2×C62, C2×C124, C22×C62, C22×C124
►Regular action on 496 points
Generators in S496
(1 280)(2 281)(3 282)(4 283)(5 284)(6 285)(7 286)(8 287)(9 288)(10 289)(11 290)(12 291)(13 292)(14 293)(15 294)(16 295)(17 296)(18 297)(19 298)(20 299)(21 300)(22 301)(23 302)(24 303)(25 304)(26 305)(27 306)(28 307)(29 308)(30 309)(31 310)(32 311)(33 312)(34 313)(35 314)(36 315)(37 316)(38 317)(39 318)(40 319)(41 320)(42 321)(43 322)(44 323)(45 324)(46 325)(47 326)(48 327)(49 328)(50 329)(51 330)(52 331)(53 332)(54 333)(55 334)(56 335)(57 336)(58 337)(59 338)(60 339)(61 340)(62 341)(63 342)(64 343)(65 344)(66 345)(67 346)(68 347)(69 348)(70 349)(71 350)(72 351)(73 352)(74 353)(75 354)(76 355)(77 356)(78 357)(79 358)(80 359)(81 360)(82 361)(83 362)(84 363)(85 364)(86 365)(87 366)(88 367)(89 368)(90 369)(91 370)(92 371)(93 372)(94 249)(95 250)(96 251)(97 252)(98 253)(99 254)(100 255)(101 256)(102 257)(103 258)(104 259)(105 260)(106 261)(107 262)(108 263)(109 264)(110 265)(111 266)(112 267)(113 268)(114 269)(115 270)(116 271)(117 272)(118 273)(119 274)(120 275)(121 276)(122 277)(123 278)(124 279)(125 396)(126 397)(127 398)(128 399)(129 400)(130 401)(131 402)(132 403)(133 404)(134 405)(135 406)(136 407)(137 408)(138 409)(139 410)(140 411)(141 412)(142 413)(143 414)(144 415)(145 416)(146 417)(147 418)(148 419)(149 420)(150 421)(151 422)(152 423)(153 424)(154 425)(155 426)(156 427)(157 428)(158 429)(159 430)(160 431)(161 432)(162 433)(163 434)(164 435)(165 436)(166 437)(167 438)(168 439)(169 440)(170 441)(171 442)(172 443)(173 444)(174 445)(175 446)(176 447)(177 448)(178 449)(179 450)(180 451)(181 452)(182 453)(183 454)(184 455)(185 456)(186 457)(187 458)(188 459)(189 460)(190 461)(191 462)(192 463)(193 464)(194 465)(195 466)(196 467)(197 468)(198 469)(199 470)(200 471)(201 472)(202 473)(203 474)(204 475)(205 476)(206 477)(207 478)(208 479)(209 480)(210 481)(211 482)(212 483)(213 484)(214 485)(215 486)(216 487)(217 488)(218 489)(219 490)(220 491)(221 492)(222 493)(223 494)(224 495)(225 496)(226 373)(227 374)(228 375)(229 376)(230 377)(231 378)(232 379)(233 380)(234 381)(235 382)(236 383)(237 384)(238 385)(239 386)(240 387)(241 388)(242 389)(243 390)(244 391)(245 392)(246 393)(247 394)(248 395)
(1 198)(2 199)(3 200)(4 201)(5 202)(6 203)(7 204)(8 205)(9 206)(10 207)(11 208)(12 209)(13 210)(14 211)(15 212)(16 213)(17 214)(18 215)(19 216)(20 217)(21 218)(22 219)(23 220)(24 221)(25 222)(26 223)(27 224)(28 225)(29 226)(30 227)(31 228)(32 229)(33 230)(34 231)(35 232)(36 233)(37 234)(38 235)(39 236)(40 237)(41 238)(42 239)(43 240)(44 241)(45 242)(46 243)(47 244)(48 245)(49 246)(50 247)(51 248)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 141)(69 142)(70 143)(71 144)(72 145)(73 146)(74 147)(75 148)(76 149)(77 150)(78 151)(79 152)(80 153)(81 154)(82 155)(83 156)(84 157)(85 158)(86 159)(87 160)(88 161)(89 162)(90 163)(91 164)(92 165)(93 166)(94 167)(95 168)(96 169)(97 170)(98 171)(99 172)(100 173)(101 174)(102 175)(103 176)(104 177)(105 178)(106 179)(107 180)(108 181)(109 182)(110 183)(111 184)(112 185)(113 186)(114 187)(115 188)(116 189)(117 190)(118 191)(119 192)(120 193)(121 194)(122 195)(123 196)(124 197)(249 438)(250 439)(251 440)(252 441)(253 442)(254 443)(255 444)(256 445)(257 446)(258 447)(259 448)(260 449)(261 450)(262 451)(263 452)(264 453)(265 454)(266 455)(267 456)(268 457)(269 458)(270 459)(271 460)(272 461)(273 462)(274 463)(275 464)(276 465)(277 466)(278 467)(279 468)(280 469)(281 470)(282 471)(283 472)(284 473)(285 474)(286 475)(287 476)(288 477)(289 478)(290 479)(291 480)(292 481)(293 482)(294 483)(295 484)(296 485)(297 486)(298 487)(299 488)(300 489)(301 490)(302 491)(303 492)(304 493)(305 494)(306 495)(307 496)(308 373)(309 374)(310 375)(311 376)(312 377)(313 378)(314 379)(315 380)(316 381)(317 382)(318 383)(319 384)(320 385)(321 386)(322 387)(323 388)(324 389)(325 390)(326 391)(327 392)(328 393)(329 394)(330 395)(331 396)(332 397)(333 398)(334 399)(335 400)(336 401)(337 402)(338 403)(339 404)(340 405)(341 406)(342 407)(343 408)(344 409)(345 410)(346 411)(347 412)(348 413)(349 414)(350 415)(351 416)(352 417)(353 418)(354 419)(355 420)(356 421)(357 422)(358 423)(359 424)(360 425)(361 426)(362 427)(363 428)(364 429)(365 430)(366 431)(367 432)(368 433)(369 434)(370 435)(371 436)(372 437)
(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)| (1,280)(2,281)(3,282)(4,283)(5,284)(6,285)(7,286)(8,287)(9,288)(10,289)(11,290)(12,291)(13,292)(14,293)(15,294)(16,295)(17,296)(18,297)(19,298)(20,299)(21,300)(22,301)(23,302)(24,303)(25,304)(26,305)(27,306)(28,307)(29,308)(30,309)(31,310)(32,311)(33,312)(34,313)(35,314)(36,315)(37,316)(38,317)(39,318)(40,319)(41,320)(42,321)(43,322)(44,323)(45,324)(46,325)(47,326)(48,327)(49,328)(50,329)(51,330)(52,331)(53,332)(54,333)(55,334)(56,335)(57,336)(58,337)(59,338)(60,339)(61,340)(62,341)(63,342)(64,343)(65,344)(66,345)(67,346)(68,347)(69,348)(70,349)(71,350)(72,351)(73,352)(74,353)(75,354)(76,355)(77,356)(78,357)(79,358)(80,359)(81,360)(82,361)(83,362)(84,363)(85,364)(86,365)(87,366)(88,367)(89,368)(90,369)(91,370)(92,371)(93,372)(94,249)(95,250)(96,251)(97,252)(98,253)(99,254)(100,255)(101,256)(102,257)(103,258)(104,259)(105,260)(106,261)(107,262)(108,263)(109,264)(110,265)(111,266)(112,267)(113,268)(114,269)(115,270)(116,271)(117,272)(118,273)(119,274)(120,275)(121,276)(122,277)(123,278)(124,279)(125,396)(126,397)(127,398)(128,399)(129,400)(130,401)(131,402)(132,403)(133,404)(134,405)(135,406)(136,407)(137,408)(138,409)(139,410)(140,411)(141,412)(142,413)(143,414)(144,415)(145,416)(146,417)(147,418)(148,419)(149,420)(150,421)(151,422)(152,423)(153,424)(154,425)(155,426)(156,427)(157,428)(158,429)(159,430)(160,431)(161,432)(162,433)(163,434)(164,435)(165,436)(166,437)(167,438)(168,439)(169,440)(170,441)(171,442)(172,443)(173,444)(174,445)(175,446)(176,447)(177,448)(178,449)(179,450)(180,451)(181,452)(182,453)(183,454)(184,455)(185,456)(186,457)(187,458)(188,459)(189,460)(190,461)(191,462)(192,463)(193,464)(194,465)(195,466)(196,467)(197,468)(198,469)(199,470)(200,471)(201,472)(202,473)(203,474)(204,475)(205,476)(206,477)(207,478)(208,479)(209,480)(210,481)(211,482)(212,483)(213,484)(214,485)(215,486)(216,487)(217,488)(218,489)(219,490)(220,491)(221,492)(222,493)(223,494)(224,495)(225,496)(226,373)(227,374)(228,375)(229,376)(230,377)(231,378)(232,379)(233,380)(234,381)(235,382)(236,383)(237,384)(238,385)(239,386)(240,387)(241,388)(242,389)(243,390)(244,391)(245,392)(246,393)(247,394)(248,395), (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,211)(15,212)(16,213)(17,214)(18,215)(19,216)(20,217)(21,218)(22,219)(23,220)(24,221)(25,222)(26,223)(27,224)(28,225)(29,226)(30,227)(31,228)(32,229)(33,230)(34,231)(35,232)(36,233)(37,234)(38,235)(39,236)(40,237)(41,238)(42,239)(43,240)(44,241)(45,242)(46,243)(47,244)(48,245)(49,246)(50,247)(51,248)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,151)(79,152)(80,153)(81,154)(82,155)(83,156)(84,157)(85,158)(86,159)(87,160)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,169)(97,170)(98,171)(99,172)(100,173)(101,174)(102,175)(103,176)(104,177)(105,178)(106,179)(107,180)(108,181)(109,182)(110,183)(111,184)(112,185)(113,186)(114,187)(115,188)(116,189)(117,190)(118,191)(119,192)(120,193)(121,194)(122,195)(123,196)(124,197)(249,438)(250,439)(251,440)(252,441)(253,442)(254,443)(255,444)(256,445)(257,446)(258,447)(259,448)(260,449)(261,450)(262,451)(263,452)(264,453)(265,454)(266,455)(267,456)(268,457)(269,458)(270,459)(271,460)(272,461)(273,462)(274,463)(275,464)(276,465)(277,466)(278,467)(279,468)(280,469)(281,470)(282,471)(283,472)(284,473)(285,474)(286,475)(287,476)(288,477)(289,478)(290,479)(291,480)(292,481)(293,482)(294,483)(295,484)(296,485)(297,486)(298,487)(299,488)(300,489)(301,490)(302,491)(303,492)(304,493)(305,494)(306,495)(307,496)(308,373)(309,374)(310,375)(311,376)(312,377)(313,378)(314,379)(315,380)(316,381)(317,382)(318,383)(319,384)(320,385)(321,386)(322,387)(323,388)(324,389)(325,390)(326,391)(327,392)(328,393)(329,394)(330,395)(331,396)(332,397)(333,398)(334,399)(335,400)(336,401)(337,402)(338,403)(339,404)(340,405)(341,406)(342,407)(343,408)(344,409)(345,410)(346,411)(347,412)(348,413)(349,414)(350,415)(351,416)(352,417)(353,418)(354,419)(355,420)(356,421)(357,422)(358,423)(359,424)(360,425)(361,426)(362,427)(363,428)(364,429)(365,430)(366,431)(367,432)(368,433)(369,434)(370,435)(371,436)(372,437), (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( (1,280)(2,281)(3,282)(4,283)(5,284)(6,285)(7,286)(8,287)(9,288)(10,289)(11,290)(12,291)(13,292)(14,293)(15,294)(16,295)(17,296)(18,297)(19,298)(20,299)(21,300)(22,301)(23,302)(24,303)(25,304)(26,305)(27,306)(28,307)(29,308)(30,309)(31,310)(32,311)(33,312)(34,313)(35,314)(36,315)(37,316)(38,317)(39,318)(40,319)(41,320)(42,321)(43,322)(44,323)(45,324)(46,325)(47,326)(48,327)(49,328)(50,329)(51,330)(52,331)(53,332)(54,333)(55,334)(56,335)(57,336)(58,337)(59,338)(60,339)(61,340)(62,341)(63,342)(64,343)(65,344)(66,345)(67,346)(68,347)(69,348)(70,349)(71,350)(72,351)(73,352)(74,353)(75,354)(76,355)(77,356)(78,357)(79,358)(80,359)(81,360)(82,361)(83,362)(84,363)(85,364)(86,365)(87,366)(88,367)(89,368)(90,369)(91,370)(92,371)(93,372)(94,249)(95,250)(96,251)(97,252)(98,253)(99,254)(100,255)(101,256)(102,257)(103,258)(104,259)(105,260)(106,261)(107,262)(108,263)(109,264)(110,265)(111,266)(112,267)(113,268)(114,269)(115,270)(116,271)(117,272)(118,273)(119,274)(120,275)(121,276)(122,277)(123,278)(124,279)(125,396)(126,397)(127,398)(128,399)(129,400)(130,401)(131,402)(132,403)(133,404)(134,405)(135,406)(136,407)(137,408)(138,409)(139,410)(140,411)(141,412)(142,413)(143,414)(144,415)(145,416)(146,417)(147,418)(148,419)(149,420)(150,421)(151,422)(152,423)(153,424)(154,425)(155,426)(156,427)(157,428)(158,429)(159,430)(160,431)(161,432)(162,433)(163,434)(164,435)(165,436)(166,437)(167,438)(168,439)(169,440)(170,441)(171,442)(172,443)(173,444)(174,445)(175,446)(176,447)(177,448)(178,449)(179,450)(180,451)(181,452)(182,453)(183,454)(184,455)(185,456)(186,457)(187,458)(188,459)(189,460)(190,461)(191,462)(192,463)(193,464)(194,465)(195,466)(196,467)(197,468)(198,469)(199,470)(200,471)(201,472)(202,473)(203,474)(204,475)(205,476)(206,477)(207,478)(208,479)(209,480)(210,481)(211,482)(212,483)(213,484)(214,485)(215,486)(216,487)(217,488)(218,489)(219,490)(220,491)(221,492)(222,493)(223,494)(224,495)(225,496)(226,373)(227,374)(228,375)(229,376)(230,377)(231,378)(232,379)(233,380)(234,381)(235,382)(236,383)(237,384)(238,385)(239,386)(240,387)(241,388)(242,389)(243,390)(244,391)(245,392)(246,393)(247,394)(248,395), (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,211)(15,212)(16,213)(17,214)(18,215)(19,216)(20,217)(21,218)(22,219)(23,220)(24,221)(25,222)(26,223)(27,224)(28,225)(29,226)(30,227)(31,228)(32,229)(33,230)(34,231)(35,232)(36,233)(37,234)(38,235)(39,236)(40,237)(41,238)(42,239)(43,240)(44,241)(45,242)(46,243)(47,244)(48,245)(49,246)(50,247)(51,248)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,151)(79,152)(80,153)(81,154)(82,155)(83,156)(84,157)(85,158)(86,159)(87,160)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,169)(97,170)(98,171)(99,172)(100,173)(101,174)(102,175)(103,176)(104,177)(105,178)(106,179)(107,180)(108,181)(109,182)(110,183)(111,184)(112,185)(113,186)(114,187)(115,188)(116,189)(117,190)(118,191)(119,192)(120,193)(121,194)(122,195)(123,196)(124,197)(249,438)(250,439)(251,440)(252,441)(253,442)(254,443)(255,444)(256,445)(257,446)(258,447)(259,448)(260,449)(261,450)(262,451)(263,452)(264,453)(265,454)(266,455)(267,456)(268,457)(269,458)(270,459)(271,460)(272,461)(273,462)(274,463)(275,464)(276,465)(277,466)(278,467)(279,468)(280,469)(281,470)(282,471)(283,472)(284,473)(285,474)(286,475)(287,476)(288,477)(289,478)(290,479)(291,480)(292,481)(293,482)(294,483)(295,484)(296,485)(297,486)(298,487)(299,488)(300,489)(301,490)(302,491)(303,492)(304,493)(305,494)(306,495)(307,496)(308,373)(309,374)(310,375)(311,376)(312,377)(313,378)(314,379)(315,380)(316,381)(317,382)(318,383)(319,384)(320,385)(321,386)(322,387)(323,388)(324,389)(325,390)(326,391)(327,392)(328,393)(329,394)(330,395)(331,396)(332,397)(333,398)(334,399)(335,400)(336,401)(337,402)(338,403)(339,404)(340,405)(341,406)(342,407)(343,408)(344,409)(345,410)(346,411)(347,412)(348,413)(349,414)(350,415)(351,416)(352,417)(353,418)(354,419)(355,420)(356,421)(357,422)(358,423)(359,424)(360,425)(361,426)(362,427)(363,428)(364,429)(365,430)(366,431)(367,432)(368,433)(369,434)(370,435)(371,436)(372,437), (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([[(1,280),(2,281),(3,282),(4,283),(5,284),(6,285),(7,286),(8,287),(9,288),(10,289),(11,290),(12,291),(13,292),(14,293),(15,294),(16,295),(17,296),(18,297),(19,298),(20,299),(21,300),(22,301),(23,302),(24,303),(25,304),(26,305),(27,306),(28,307),(29,308),(30,309),(31,310),(32,311),(33,312),(34,313),(35,314),(36,315),(37,316),(38,317),(39,318),(40,319),(41,320),(42,321),(43,322),(44,323),(45,324),(46,325),(47,326),(48,327),(49,328),(50,329),(51,330),(52,331),(53,332),(54,333),(55,334),(56,335),(57,336),(58,337),(59,338),(60,339),(61,340),(62,341),(63,342),(64,343),(65,344),(66,345),(67,346),(68,347),(69,348),(70,349),(71,350),(72,351),(73,352),(74,353),(75,354),(76,355),(77,356),(78,357),(79,358),(80,359),(81,360),(82,361),(83,362),(84,363),(85,364),(86,365),(87,366),(88,367),(89,368),(90,369),(91,370),(92,371),(93,372),(94,249),(95,250),(96,251),(97,252),(98,253),(99,254),(100,255),(101,256),(102,257),(103,258),(104,259),(105,260),(106,261),(107,262),(108,263),(109,264),(110,265),(111,266),(112,267),(113,268),(114,269),(115,270),(116,271),(117,272),(118,273),(119,274),(120,275),(121,276),(122,277),(123,278),(124,279),(125,396),(126,397),(127,398),(128,399),(129,400),(130,401),(131,402),(132,403),(133,404),(134,405),(135,406),(136,407),(137,408),(138,409),(139,410),(140,411),(141,412),(142,413),(143,414),(144,415),(145,416),(146,417),(147,418),(148,419),(149,420),(150,421),(151,422),(152,423),(153,424),(154,425),(155,426),(156,427),(157,428),(158,429),(159,430),(160,431),(161,432),(162,433),(163,434),(164,435),(165,436),(166,437),(167,438),(168,439),(169,440),(170,441),(171,442),(172,443),(173,444),(174,445),(175,446),(176,447),(177,448),(178,449),(179,450),(180,451),(181,452),(182,453),(183,454),(184,455),(185,456),(186,457),(187,458),(188,459),(189,460),(190,461),(191,462),(192,463),(193,464),(194,465),(195,466),(196,467),(197,468),(198,469),(199,470),(200,471),(201,472),(202,473),(203,474),(204,475),(205,476),(206,477),(207,478),(208,479),(209,480),(210,481),(211,482),(212,483),(213,484),(214,485),(215,486),(216,487),(217,488),(218,489),(219,490),(220,491),(221,492),(222,493),(223,494),(224,495),(225,496),(226,373),(227,374),(228,375),(229,376),(230,377),(231,378),(232,379),(233,380),(234,381),(235,382),(236,383),(237,384),(238,385),(239,386),(240,387),(241,388),(242,389),(243,390),(244,391),(245,392),(246,393),(247,394),(248,395)], [(1,198),(2,199),(3,200),(4,201),(5,202),(6,203),(7,204),(8,205),(9,206),(10,207),(11,208),(12,209),(13,210),(14,211),(15,212),(16,213),(17,214),(18,215),(19,216),(20,217),(21,218),(22,219),(23,220),(24,221),(25,222),(26,223),(27,224),(28,225),(29,226),(30,227),(31,228),(32,229),(33,230),(34,231),(35,232),(36,233),(37,234),(38,235),(39,236),(40,237),(41,238),(42,239),(43,240),(44,241),(45,242),(46,243),(47,244),(48,245),(49,246),(50,247),(51,248),(52,125),(53,126),(54,127),(55,128),(56,129),(57,130),(58,131),(59,132),(60,133),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,141),(69,142),(70,143),(71,144),(72,145),(73,146),(74,147),(75,148),(76,149),(77,150),(78,151),(79,152),(80,153),(81,154),(82,155),(83,156),(84,157),(85,158),(86,159),(87,160),(88,161),(89,162),(90,163),(91,164),(92,165),(93,166),(94,167),(95,168),(96,169),(97,170),(98,171),(99,172),(100,173),(101,174),(102,175),(103,176),(104,177),(105,178),(106,179),(107,180),(108,181),(109,182),(110,183),(111,184),(112,185),(113,186),(114,187),(115,188),(116,189),(117,190),(118,191),(119,192),(120,193),(121,194),(122,195),(123,196),(124,197),(249,438),(250,439),(251,440),(252,441),(253,442),(254,443),(255,444),(256,445),(257,446),(258,447),(259,448),(260,449),(261,450),(262,451),(263,452),(264,453),(265,454),(266,455),(267,456),(268,457),(269,458),(270,459),(271,460),(272,461),(273,462),(274,463),(275,464),(276,465),(277,466),(278,467),(279,468),(280,469),(281,470),(282,471),(283,472),(284,473),(285,474),(286,475),(287,476),(288,477),(289,478),(290,479),(291,480),(292,481),(293,482),(294,483),(295,484),(296,485),(297,486),(298,487),(299,488),(300,489),(301,490),(302,491),(303,492),(304,493),(305,494),(306,495),(307,496),(308,373),(309,374),(310,375),(311,376),(312,377),(313,378),(314,379),(315,380),(316,381),(317,382),(318,383),(319,384),(320,385),(321,386),(322,387),(323,388),(324,389),(325,390),(326,391),(327,392),(328,393),(329,394),(330,395),(331,396),(332,397),(333,398),(334,399),(335,400),(336,401),(337,402),(338,403),(339,404),(340,405),(341,406),(342,407),(343,408),(344,409),(345,410),(346,411),(347,412),(348,413),(349,414),(350,415),(351,416),(352,417),(353,418),(354,419),(355,420),(356,421),(357,422),(358,423),(359,424),(360,425),(361,426),(362,427),(363,428),(364,429),(365,430),(366,431),(367,432),(368,433),(369,434),(370,435),(371,436),(372,437)], [(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 | ··· | 2G | 4A | ··· | 4H | 31A | ··· | 31AD | 62A | ··· | 62HB | 124A | ··· | 124IF |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
496 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C31 | C62 | C62 | C124 |
| kernel | C22×C124 | C2×C124 | C22×C62 | C2×C62 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 8 | 30 | 180 | 30 | 240 |
Matrix representation of C22×C124 ►in GL3(𝔽373) generated by
| 1 | 0 | 0 |
| 0 | 372 | 0 |
| 0 | 0 | 372 |
| 372 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 317 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 27 |
G:=sub<GL(3,GF(373))| [1,0,0,0,372,0,0,0,372],[372,0,0,0,1,0,0,0,1],[317,0,0,0,228,0,0,0,27] >;
C22×C124 in GAP, Magma, Sage, TeX
C_2^2\times C_{124} % in TeX
G:=Group("C2^2xC124"); // GroupNames label
G:=SmallGroup(496,37);
// by ID
G=gap.SmallGroup(496,37);
# by ID
G:=PCGroup([5,-2,-2,-2,-31,-2,1240]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^124=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations