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G = C22×C124order 496 = 24·31

Abelian group of type [2,2,124]

Aliases: C22×C124, SmallGroup(496,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C124
 Chief series C1 — C2 — C62 — C124 — C2×C124 — C22×C124
 Lower central C1 — C22×C124
 Upper central 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 [×6], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C31, C62, C62 [×6], C124 [×4], C2×C62 [×7], C2×C124 [×6], C22×C62, C22×C124
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C31, C62 [×7], C124 [×4], C2×C62 [×7], C2×C124 [×6], C22×C62, C22×C124

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

׿
×
𝔽