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G = C22×C120order 480 = 25·3·5

Abelian group of type [2,2,120]

Aliases: C22×C120, SmallGroup(480,934)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C120
 Chief series C1 — C2 — C4 — C20 — C60 — C120 — C2×C120 — C22×C120
 Lower central C1 — C22×C120
 Upper central C1 — C22×C120

Generators and relations for C22×C120
G = < a,b,c | a2=b2=c120=1, ab=ba, ac=ca, bc=cb >

Subgroups: 152, all normal (24 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×3], C22 [×7], C5, C6, C6 [×6], C8 [×4], C2×C4 [×6], C23, C10, C10 [×6], C12, C12 [×3], C2×C6 [×7], C15, C2×C8 [×6], C22×C4, C20, C20 [×3], C2×C10 [×7], C24 [×4], C2×C12 [×6], C22×C6, C30, C30 [×6], C22×C8, C40 [×4], C2×C20 [×6], C22×C10, C2×C24 [×6], C22×C12, C60, C60 [×3], C2×C30 [×7], C2×C40 [×6], C22×C20, C22×C24, C120 [×4], C2×C60 [×6], C22×C30, C22×C40, C2×C120 [×6], C22×C60, C22×C120
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C5, C6 [×7], C8 [×4], C2×C4 [×6], C23, C10 [×7], C12 [×4], C2×C6 [×7], C15, C2×C8 [×6], C22×C4, C20 [×4], C2×C10 [×7], C24 [×4], C2×C12 [×6], C22×C6, C30 [×7], C22×C8, C40 [×4], C2×C20 [×6], C22×C10, C2×C24 [×6], C22×C12, C60 [×4], C2×C30 [×7], C2×C40 [×6], C22×C20, C22×C24, C120 [×4], C2×C60 [×6], C22×C30, C22×C40, C2×C120 [×6], C22×C60, C22×C120

Smallest permutation representation of C22×C120
Regular action on 480 points
Generators in S480
(1 322)(2 323)(3 324)(4 325)(5 326)(6 327)(7 328)(8 329)(9 330)(10 331)(11 332)(12 333)(13 334)(14 335)(15 336)(16 337)(17 338)(18 339)(19 340)(20 341)(21 342)(22 343)(23 344)(24 345)(25 346)(26 347)(27 348)(28 349)(29 350)(30 351)(31 352)(32 353)(33 354)(34 355)(35 356)(36 357)(37 358)(38 359)(39 360)(40 241)(41 242)(42 243)(43 244)(44 245)(45 246)(46 247)(47 248)(48 249)(49 250)(50 251)(51 252)(52 253)(53 254)(54 255)(55 256)(56 257)(57 258)(58 259)(59 260)(60 261)(61 262)(62 263)(63 264)(64 265)(65 266)(66 267)(67 268)(68 269)(69 270)(70 271)(71 272)(72 273)(73 274)(74 275)(75 276)(76 277)(77 278)(78 279)(79 280)(80 281)(81 282)(82 283)(83 284)(84 285)(85 286)(86 287)(87 288)(88 289)(89 290)(90 291)(91 292)(92 293)(93 294)(94 295)(95 296)(96 297)(97 298)(98 299)(99 300)(100 301)(101 302)(102 303)(103 304)(104 305)(105 306)(106 307)(107 308)(108 309)(109 310)(110 311)(111 312)(112 313)(113 314)(114 315)(115 316)(116 317)(117 318)(118 319)(119 320)(120 321)(121 383)(122 384)(123 385)(124 386)(125 387)(126 388)(127 389)(128 390)(129 391)(130 392)(131 393)(132 394)(133 395)(134 396)(135 397)(136 398)(137 399)(138 400)(139 401)(140 402)(141 403)(142 404)(143 405)(144 406)(145 407)(146 408)(147 409)(148 410)(149 411)(150 412)(151 413)(152 414)(153 415)(154 416)(155 417)(156 418)(157 419)(158 420)(159 421)(160 422)(161 423)(162 424)(163 425)(164 426)(165 427)(166 428)(167 429)(168 430)(169 431)(170 432)(171 433)(172 434)(173 435)(174 436)(175 437)(176 438)(177 439)(178 440)(179 441)(180 442)(181 443)(182 444)(183 445)(184 446)(185 447)(186 448)(187 449)(188 450)(189 451)(190 452)(191 453)(192 454)(193 455)(194 456)(195 457)(196 458)(197 459)(198 460)(199 461)(200 462)(201 463)(202 464)(203 465)(204 466)(205 467)(206 468)(207 469)(208 470)(209 471)(210 472)(211 473)(212 474)(213 475)(214 476)(215 477)(216 478)(217 479)(218 480)(219 361)(220 362)(221 363)(222 364)(223 365)(224 366)(225 367)(226 368)(227 369)(228 370)(229 371)(230 372)(231 373)(232 374)(233 375)(234 376)(235 377)(236 378)(237 379)(238 380)(239 381)(240 382)
(1 236)(2 237)(3 238)(4 239)(5 240)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 136)(22 137)(23 138)(24 139)(25 140)(26 141)(27 142)(28 143)(29 144)(30 145)(31 146)(32 147)(33 148)(34 149)(35 150)(36 151)(37 152)(38 153)(39 154)(40 155)(41 156)(42 157)(43 158)(44 159)(45 160)(46 161)(47 162)(48 163)(49 164)(50 165)(51 166)(52 167)(53 168)(54 169)(55 170)(56 171)(57 172)(58 173)(59 174)(60 175)(61 176)(62 177)(63 178)(64 179)(65 180)(66 181)(67 182)(68 183)(69 184)(70 185)(71 186)(72 187)(73 188)(74 189)(75 190)(76 191)(77 192)(78 193)(79 194)(80 195)(81 196)(82 197)(83 198)(84 199)(85 200)(86 201)(87 202)(88 203)(89 204)(90 205)(91 206)(92 207)(93 208)(94 209)(95 210)(96 211)(97 212)(98 213)(99 214)(100 215)(101 216)(102 217)(103 218)(104 219)(105 220)(106 221)(107 222)(108 223)(109 224)(110 225)(111 226)(112 227)(113 228)(114 229)(115 230)(116 231)(117 232)(118 233)(119 234)(120 235)(241 417)(242 418)(243 419)(244 420)(245 421)(246 422)(247 423)(248 424)(249 425)(250 426)(251 427)(252 428)(253 429)(254 430)(255 431)(256 432)(257 433)(258 434)(259 435)(260 436)(261 437)(262 438)(263 439)(264 440)(265 441)(266 442)(267 443)(268 444)(269 445)(270 446)(271 447)(272 448)(273 449)(274 450)(275 451)(276 452)(277 453)(278 454)(279 455)(280 456)(281 457)(282 458)(283 459)(284 460)(285 461)(286 462)(287 463)(288 464)(289 465)(290 466)(291 467)(292 468)(293 469)(294 470)(295 471)(296 472)(297 473)(298 474)(299 475)(300 476)(301 477)(302 478)(303 479)(304 480)(305 361)(306 362)(307 363)(308 364)(309 365)(310 366)(311 367)(312 368)(313 369)(314 370)(315 371)(316 372)(317 373)(318 374)(319 375)(320 376)(321 377)(322 378)(323 379)(324 380)(325 381)(326 382)(327 383)(328 384)(329 385)(330 386)(331 387)(332 388)(333 389)(334 390)(335 391)(336 392)(337 393)(338 394)(339 395)(340 396)(341 397)(342 398)(343 399)(344 400)(345 401)(346 402)(347 403)(348 404)(349 405)(350 406)(351 407)(352 408)(353 409)(354 410)(355 411)(356 412)(357 413)(358 414)(359 415)(360 416)
(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)

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

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

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

480 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 5A 5B 5C 5D 6A ··· 6N 8A ··· 8P 10A ··· 10AB 12A ··· 12P 15A ··· 15H 20A ··· 20AF 24A ··· 24AF 30A ··· 30BD 40A ··· 40BL 60A ··· 60BL 120A ··· 120DX order 1 2 ··· 2 3 3 4 ··· 4 5 5 5 5 6 ··· 6 8 ··· 8 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 ··· 1 1 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

480 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C4 C5 C6 C6 C8 C10 C10 C12 C12 C15 C20 C20 C24 C30 C30 C40 C60 C60 C120 kernel C22×C120 C2×C120 C22×C60 C22×C40 C2×C60 C22×C30 C22×C24 C2×C40 C22×C20 C2×C30 C2×C24 C22×C12 C2×C20 C22×C10 C22×C8 C2×C12 C22×C6 C2×C10 C2×C8 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 6 1 2 6 2 4 12 2 16 24 4 12 4 8 24 8 32 48 8 64 48 16 128

Matrix representation of C22×C120 in GL3(𝔽241) generated by

 1 0 0 0 240 0 0 0 240
,
 240 0 0 0 240 0 0 0 1
,
 87 0 0 0 209 0 0 0 187
G:=sub<GL(3,GF(241))| [1,0,0,0,240,0,0,0,240],[240,0,0,0,240,0,0,0,1],[87,0,0,0,209,0,0,0,187] >;

C22×C120 in GAP, Magma, Sage, TeX

C_2^2\times C_{120}
% in TeX

G:=Group("C2^2xC120");
// GroupNames label

G:=SmallGroup(480,934);
// by ID

G=gap.SmallGroup(480,934);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,840,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^120=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽