Copied to
clipboard

G = C22×C120  order 480 = 25·3·5

Abelian group of type [2,2,120]

direct product, abelian, monomial, 2-elementary

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, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C23, C10, C10, C12, C12, C2×C6, C15, C2×C8, C22×C4, C20, C20, C2×C10, C24, C2×C12, C22×C6, C30, C30, C22×C8, C40, C2×C20, C22×C10, C2×C24, C22×C12, C60, C60, C2×C30, C2×C40, C22×C20, C22×C24, C120, C2×C60, C22×C30, C22×C40, C2×C120, C22×C60, C22×C120
Quotients: C1, C2, C3, C4, C22, C5, C6, C8, C2×C4, C23, C10, C12, C2×C6, C15, C2×C8, C22×C4, C20, C2×C10, C24, C2×C12, C22×C6, C30, C22×C8, C40, C2×C20, C22×C10, C2×C24, C22×C12, C60, C2×C30, C2×C40, C22×C20, C22×C24, C120, C2×C60, C22×C30, C22×C40, C2×C120, C22×C60, C22×C120

Smallest permutation representation of C22×C120
►Regular action on 480 points
Generators in S480
(1 334)(2 335)(3 336)(4 337)(5 338)(6 339)(7 340)(8 341)(9 342)(10 343)(11 344)(12 345)(13 346)(14 347)(15 348)(16 349)(17 350)(18 351)(19 352)(20 353)(21 354)(22 355)(23 356)(24 357)(25 358)(26 359)(27 360)(28 241)(29 242)(30 243)(31 244)(32 245)(33 246)(34 247)(35 248)(36 249)(37 250)(38 251)(39 252)(40 253)(41 254)(42 255)(43 256)(44 257)(45 258)(46 259)(47 260)(48 261)(49 262)(50 263)(51 264)(52 265)(53 266)(54 267)(55 268)(56 269)(57 270)(58 271)(59 272)(60 273)(61 274)(62 275)(63 276)(64 277)(65 278)(66 279)(67 280)(68 281)(69 282)(70 283)(71 284)(72 285)(73 286)(74 287)(75 288)(76 289)(77 290)(78 291)(79 292)(80 293)(81 294)(82 295)(83 296)(84 297)(85 298)(86 299)(87 300)(88 301)(89 302)(90 303)(91 304)(92 305)(93 306)(94 307)(95 308)(96 309)(97 310)(98 311)(99 312)(100 313)(101 314)(102 315)(103 316)(104 317)(105 318)(106 319)(107 320)(108 321)(109 322)(110 323)(111 324)(112 325)(113 326)(114 327)(115 328)(116 329)(117 330)(118 331)(119 332)(120 333)(121 392)(122 393)(123 394)(124 395)(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)(210 361)(211 362)(212 363)(213 364)(214 365)(215 366)(216 367)(217 368)(218 369)(219 370)(220 371)(221 372)(222 373)(223 374)(224 375)(225 376)(226 377)(227 378)(228 379)(229 380)(230 381)(231 382)(232 383)(233 384)(234 385)(235 386)(236 387)(237 388)(238 389)(239 390)(240 391)

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

(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)| (1,334)(2,335)(3,336)(4,337)(5,338)(6,339)(7,340)(8,341)(9,342)(10,343)(11,344)(12,345)(13,346)(14,347)(15,348)(16,349)(17,350)(18,351)(19,352)(20,353)(21,354)(22,355)(23,356)(24,357)(25,358)(26,359)(27,360)(28,241)(29,242)(30,243)(31,244)(32,245)(33,246)(34,247)(35,248)(36,249)(37,250)(38,251)(39,252)(40,253)(41,254)(42,255)(43,256)(44,257)(45,258)(46,259)(47,260)(48,261)(49,262)(50,263)(51,264)(52,265)(53,266)(54,267)(55,268)(56,269)(57,270)(58,271)(59,272)(60,273)(61,274)(62,275)(63,276)(64,277)(65,278)(66,279)(67,280)(68,281)(69,282)(70,283)(71,284)(72,285)(73,286)(74,287)(75,288)(76,289)(77,290)(78,291)(79,292)(80,293)(81,294)(82,295)(83,296)(84,297)(85,298)(86,299)(87,300)(88,301)(89,302)(90,303)(91,304)(92,305)(93,306)(94,307)(95,308)(96,309)(97,310)(98,311)(99,312)(100,313)(101,314)(102,315)(103,316)(104,317)(105,318)(106,319)(107,320)(108,321)(109,322)(110,323)(111,324)(112,325)(113,326)(114,327)(115,328)(116,329)(117,330)(118,331)(119,332)(120,333)(121,392)(122,393)(123,394)(124,395)(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)(210,361)(211,362)(212,363)(213,364)(214,365)(215,366)(216,367)(217,368)(218,369)(219,370)(220,371)(221,372)(222,373)(223,374)(224,375)(225,376)(226,377)(227,378)(228,379)(229,380)(230,381)(231,382)(232,383)(233,384)(234,385)(235,386)(236,387)(237,388)(238,389)(239,390)(240,391), (1,203)(2,204)(3,205)(4,206)(5,207)(6,208)(7,209)(8,210)(9,211)(10,212)(11,213)(12,214)(13,215)(14,216)(15,217)(16,218)(17,219)(18,220)(19,221)(20,222)(21,223)(22,224)(23,225)(24,226)(25,227)(26,228)(27,229)(28,230)(29,231)(30,232)(31,233)(32,234)(33,235)(34,236)(35,237)(36,238)(37,239)(38,240)(39,121)(40,122)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,161)(80,162)(81,163)(82,164)(83,165)(84,166)(85,167)(86,168)(87,169)(88,170)(89,171)(90,172)(91,173)(92,174)(93,175)(94,176)(95,177)(96,178)(97,179)(98,180)(99,181)(100,182)(101,183)(102,184)(103,185)(104,186)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,193)(112,194)(113,195)(114,196)(115,197)(116,198)(117,199)(118,200)(119,201)(120,202)(241,381)(242,382)(243,383)(244,384)(245,385)(246,386)(247,387)(248,388)(249,389)(250,390)(251,391)(252,392)(253,393)(254,394)(255,395)(256,396)(257,397)(258,398)(259,399)(260,400)(261,401)(262,402)(263,403)(264,404)(265,405)(266,406)(267,407)(268,408)(269,409)(270,410)(271,411)(272,412)(273,413)(274,414)(275,415)(276,416)(277,417)(278,418)(279,419)(280,420)(281,421)(282,422)(283,423)(284,424)(285,425)(286,426)(287,427)(288,428)(289,429)(290,430)(291,431)(292,432)(293,433)(294,434)(295,435)(296,436)(297,437)(298,438)(299,439)(300,440)(301,441)(302,442)(303,443)(304,444)(305,445)(306,446)(307,447)(308,448)(309,449)(310,450)(311,451)(312,452)(313,453)(314,454)(315,455)(316,456)(317,457)(318,458)(319,459)(320,460)(321,461)(322,462)(323,463)(324,464)(325,465)(326,466)(327,467)(328,468)(329,469)(330,470)(331,471)(332,472)(333,473)(334,474)(335,475)(336,476)(337,477)(338,478)(339,479)(340,480)(341,361)(342,362)(343,363)(344,364)(345,365)(346,366)(347,367)(348,368)(349,369)(350,370)(351,371)(352,372)(353,373)(354,374)(355,375)(356,376)(357,377)(358,378)(359,379)(360,380), (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( (1,334)(2,335)(3,336)(4,337)(5,338)(6,339)(7,340)(8,341)(9,342)(10,343)(11,344)(12,345)(13,346)(14,347)(15,348)(16,349)(17,350)(18,351)(19,352)(20,353)(21,354)(22,355)(23,356)(24,357)(25,358)(26,359)(27,360)(28,241)(29,242)(30,243)(31,244)(32,245)(33,246)(34,247)(35,248)(36,249)(37,250)(38,251)(39,252)(40,253)(41,254)(42,255)(43,256)(44,257)(45,258)(46,259)(47,260)(48,261)(49,262)(50,263)(51,264)(52,265)(53,266)(54,267)(55,268)(56,269)(57,270)(58,271)(59,272)(60,273)(61,274)(62,275)(63,276)(64,277)(65,278)(66,279)(67,280)(68,281)(69,282)(70,283)(71,284)(72,285)(73,286)(74,287)(75,288)(76,289)(77,290)(78,291)(79,292)(80,293)(81,294)(82,295)(83,296)(84,297)(85,298)(86,299)(87,300)(88,301)(89,302)(90,303)(91,304)(92,305)(93,306)(94,307)(95,308)(96,309)(97,310)(98,311)(99,312)(100,313)(101,314)(102,315)(103,316)(104,317)(105,318)(106,319)(107,320)(108,321)(109,322)(110,323)(111,324)(112,325)(113,326)(114,327)(115,328)(116,329)(117,330)(118,331)(119,332)(120,333)(121,392)(122,393)(123,394)(124,395)(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)(210,361)(211,362)(212,363)(213,364)(214,365)(215,366)(216,367)(217,368)(218,369)(219,370)(220,371)(221,372)(222,373)(223,374)(224,375)(225,376)(226,377)(227,378)(228,379)(229,380)(230,381)(231,382)(232,383)(233,384)(234,385)(235,386)(236,387)(237,388)(238,389)(239,390)(240,391), (1,203)(2,204)(3,205)(4,206)(5,207)(6,208)(7,209)(8,210)(9,211)(10,212)(11,213)(12,214)(13,215)(14,216)(15,217)(16,218)(17,219)(18,220)(19,221)(20,222)(21,223)(22,224)(23,225)(24,226)(25,227)(26,228)(27,229)(28,230)(29,231)(30,232)(31,233)(32,234)(33,235)(34,236)(35,237)(36,238)(37,239)(38,240)(39,121)(40,122)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,161)(80,162)(81,163)(82,164)(83,165)(84,166)(85,167)(86,168)(87,169)(88,170)(89,171)(90,172)(91,173)(92,174)(93,175)(94,176)(95,177)(96,178)(97,179)(98,180)(99,181)(100,182)(101,183)(102,184)(103,185)(104,186)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,193)(112,194)(113,195)(114,196)(115,197)(116,198)(117,199)(118,200)(119,201)(120,202)(241,381)(242,382)(243,383)(244,384)(245,385)(246,386)(247,387)(248,388)(249,389)(250,390)(251,391)(252,392)(253,393)(254,394)(255,395)(256,396)(257,397)(258,398)(259,399)(260,400)(261,401)(262,402)(263,403)(264,404)(265,405)(266,406)(267,407)(268,408)(269,409)(270,410)(271,411)(272,412)(273,413)(274,414)(275,415)(276,416)(277,417)(278,418)(279,419)(280,420)(281,421)(282,422)(283,423)(284,424)(285,425)(286,426)(287,427)(288,428)(289,429)(290,430)(291,431)(292,432)(293,433)(294,434)(295,435)(296,436)(297,437)(298,438)(299,439)(300,440)(301,441)(302,442)(303,443)(304,444)(305,445)(306,446)(307,447)(308,448)(309,449)(310,450)(311,451)(312,452)(313,453)(314,454)(315,455)(316,456)(317,457)(318,458)(319,459)(320,460)(321,461)(322,462)(323,463)(324,464)(325,465)(326,466)(327,467)(328,468)(329,469)(330,470)(331,471)(332,472)(333,473)(334,474)(335,475)(336,476)(337,477)(338,478)(339,479)(340,480)(341,361)(342,362)(343,363)(344,364)(345,365)(346,366)(347,367)(348,368)(349,369)(350,370)(351,371)(352,372)(353,373)(354,374)(355,375)(356,376)(357,377)(358,378)(359,379)(360,380), (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([[(1,334),(2,335),(3,336),(4,337),(5,338),(6,339),(7,340),(8,341),(9,342),(10,343),(11,344),(12,345),(13,346),(14,347),(15,348),(16,349),(17,350),(18,351),(19,352),(20,353),(21,354),(22,355),(23,356),(24,357),(25,358),(26,359),(27,360),(28,241),(29,242),(30,243),(31,244),(32,245),(33,246),(34,247),(35,248),(36,249),(37,250),(38,251),(39,252),(40,253),(41,254),(42,255),(43,256),(44,257),(45,258),(46,259),(47,260),(48,261),(49,262),(50,263),(51,264),(52,265),(53,266),(54,267),(55,268),(56,269),(57,270),(58,271),(59,272),(60,273),(61,274),(62,275),(63,276),(64,277),(65,278),(66,279),(67,280),(68,281),(69,282),(70,283),(71,284),(72,285),(73,286),(74,287),(75,288),(76,289),(77,290),(78,291),(79,292),(80,293),(81,294),(82,295),(83,296),(84,297),(85,298),(86,299),(87,300),(88,301),(89,302),(90,303),(91,304),(92,305),(93,306),(94,307),(95,308),(96,309),(97,310),(98,311),(99,312),(100,313),(101,314),(102,315),(103,316),(104,317),(105,318),(106,319),(107,320),(108,321),(109,322),(110,323),(111,324),(112,325),(113,326),(114,327),(115,328),(116,329),(117,330),(118,331),(119,332),(120,333),(121,392),(122,393),(123,394),(124,395),(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),(210,361),(211,362),(212,363),(213,364),(214,365),(215,366),(216,367),(217,368),(218,369),(219,370),(220,371),(221,372),(222,373),(223,374),(224,375),(225,376),(226,377),(227,378),(228,379),(229,380),(230,381),(231,382),(232,383),(233,384),(234,385),(235,386),(236,387),(237,388),(238,389),(239,390),(240,391)], [(1,203),(2,204),(3,205),(4,206),(5,207),(6,208),(7,209),(8,210),(9,211),(10,212),(11,213),(12,214),(13,215),(14,216),(15,217),(16,218),(17,219),(18,220),(19,221),(20,222),(21,223),(22,224),(23,225),(24,226),(25,227),(26,228),(27,229),(28,230),(29,231),(30,232),(31,233),(32,234),(33,235),(34,236),(35,237),(36,238),(37,239),(38,240),(39,121),(40,122),(41,123),(42,124),(43,125),(44,126),(45,127),(46,128),(47,129),(48,130),(49,131),(50,132),(51,133),(52,134),(53,135),(54,136),(55,137),(56,138),(57,139),(58,140),(59,141),(60,142),(61,143),(62,144),(63,145),(64,146),(65,147),(66,148),(67,149),(68,150),(69,151),(70,152),(71,153),(72,154),(73,155),(74,156),(75,157),(76,158),(77,159),(78,160),(79,161),(80,162),(81,163),(82,164),(83,165),(84,166),(85,167),(86,168),(87,169),(88,170),(89,171),(90,172),(91,173),(92,174),(93,175),(94,176),(95,177),(96,178),(97,179),(98,180),(99,181),(100,182),(101,183),(102,184),(103,185),(104,186),(105,187),(106,188),(107,189),(108,190),(109,191),(110,192),(111,193),(112,194),(113,195),(114,196),(115,197),(116,198),(117,199),(118,200),(119,201),(120,202),(241,381),(242,382),(243,383),(244,384),(245,385),(246,386),(247,387),(248,388),(249,389),(250,390),(251,391),(252,392),(253,393),(254,394),(255,395),(256,396),(257,397),(258,398),(259,399),(260,400),(261,401),(262,402),(263,403),(264,404),(265,405),(266,406),(267,407),(268,408),(269,409),(270,410),(271,411),(272,412),(273,413),(274,414),(275,415),(276,416),(277,417),(278,418),(279,419),(280,420),(281,421),(282,422),(283,423),(284,424),(285,425),(286,426),(287,427),(288,428),(289,429),(290,430),(291,431),(292,432),(293,433),(294,434),(295,435),(296,436),(297,437),(298,438),(299,439),(300,440),(301,441),(302,442),(303,443),(304,444),(305,445),(306,446),(307,447),(308,448),(309,449),(310,450),(311,451),(312,452),(313,453),(314,454),(315,455),(316,456),(317,457),(318,458),(319,459),(320,460),(321,461),(322,462),(323,463),(324,464),(325,465),(326,466),(327,467),(328,468),(329,469),(330,470),(331,471),(332,472),(333,473),(334,474),(335,475),(336,476),(337,477),(338,478),(339,479),(340,480),(341,361),(342,362),(343,363),(344,364),(345,365),(346,366),(347,367),(348,368),(349,369),(350,370),(351,371),(352,372),(353,373),(354,374),(355,375),(356,376),(357,377),(358,378),(359,379),(360,380)], [(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···2G3A3B4A···4H5A5B5C5D6A···6N8A···8P10A···10AB12A···12P15A···15H20A···20AF24A···24AF30A···30BD40A···40BL60A···60BL120A···120DX
order12···2334···455556···68···810···1012···1215···1520···2024···2430···3040···4060···60120···120
size11···1111···111111···11···11···11···11···11···11···11···11···11···11···1

480 irreducible representations

dim111111111111111111111111
type+++
imageC1C2C2C3C4C4C5C6C6C8C10C10C12C12C15C20C20C24C30C30C40C60C60C120
kernelC22×C120C2×C120C22×C60C22×C40C2×C60C22×C30C22×C24C2×C40C22×C20C2×C30C2×C24C22×C12C2×C20C22×C10C22×C8C2×C12C22×C6C2×C10C2×C8C22×C4C2×C6C2×C4C23C22
# reps161262412216244124824832488644816128

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

100
02400
00240
,
24000
02400
001
,
8700
02090
00187
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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁