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G = C2×C4×C60  order 480 = 25·3·5

Abelian group of type [2,4,60]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C4×C60, SmallGroup(480,919)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C4×C60
Chief series
C1 — C2 — C22 — C2×C10 — C2×C30 — C2×C60 — C4×C60 — C2×C4×C60
Lower central
C1 — C2×C4×C60
Upper central
C1 — C2×C4×C60

Generators and relations for C2×C4×C60
 G = < a,b,c | a2=b4=c60=1, ab=ba, ac=ca, bc=cb >

Subgroups: 216, all normal (16 characteristic)
C1, C2, C3, C4, C22, C22, C5, C6, C2×C4, C23, C10, C12, C2×C6, C2×C6, C15, C42, C22×C4, C20, C2×C10, C2×C10, C2×C12, C22×C6, C30, C2×C42, C2×C20, C22×C10, C4×C12, C22×C12, C60, C2×C30, C2×C30, C4×C20, C22×C20, C2×C4×C12, C2×C60, C22×C30, C2×C4×C20, C4×C60, C22×C60, C2×C4×C60
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, C23, C10, C12, C2×C6, C15, C42, C22×C4, C20, C2×C10, C2×C12, C22×C6, C30, C2×C42, C2×C20, C22×C10, C4×C12, C22×C12, C60, C2×C30, C4×C20, C22×C20, C2×C4×C12, C2×C60, C22×C30, C2×C4×C20, C4×C60, C22×C60, C2×C4×C60

Smallest permutation representation of C2×C4×C60
►Regular action on 480 points
Generators in S480
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 97)(47 98)(48 99)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)(57 108)(58 109)(59 110)(60 111)(121 451)(122 452)(123 453)(124 454)(125 455)(126 456)(127 457)(128 458)(129 459)(130 460)(131 461)(132 462)(133 463)(134 464)(135 465)(136 466)(137 467)(138 468)(139 469)(140 470)(141 471)(142 472)(143 473)(144 474)(145 475)(146 476)(147 477)(148 478)(149 479)(150 480)(151 421)(152 422)(153 423)(154 424)(155 425)(156 426)(157 427)(158 428)(159 429)(160 430)(161 431)(162 432)(163 433)(164 434)(165 435)(166 436)(167 437)(168 438)(169 439)(170 440)(171 441)(172 442)(173 443)(174 444)(175 445)(176 446)(177 447)(178 448)(179 449)(180 450)(181 397)(182 398)(183 399)(184 400)(185 401)(186 402)(187 403)(188 404)(189 405)(190 406)(191 407)(192 408)(193 409)(194 410)(195 411)(196 412)(197 413)(198 414)(199 415)(200 416)(201 417)(202 418)(203 419)(204 420)(205 361)(206 362)(207 363)(208 364)(209 365)(210 366)(211 367)(212 368)(213 369)(214 370)(215 371)(216 372)(217 373)(218 374)(219 375)(220 376)(221 377)(222 378)(223 379)(224 380)(225 381)(226 382)(227 383)(228 384)(229 385)(230 386)(231 387)(232 388)(233 389)(234 390)(235 391)(236 392)(237 393)(238 394)(239 395)(240 396)(241 332)(242 333)(243 334)(244 335)(245 336)(246 337)(247 338)(248 339)(249 340)(250 341)(251 342)(252 343)(253 344)(254 345)(255 346)(256 347)(257 348)(258 349)(259 350)(260 351)(261 352)(262 353)(263 354)(264 355)(265 356)(266 357)(267 358)(268 359)(269 360)(270 301)(271 302)(272 303)(273 304)(274 305)(275 306)(276 307)(277 308)(278 309)(279 310)(280 311)(281 312)(282 313)(283 314)(284 315)(285 316)(286 317)(287 318)(288 319)(289 320)(290 321)(291 322)(292 323)(293 324)(294 325)(295 326)(296 327)(297 328)(298 329)(299 330)(300 331)

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

(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)| (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(121,451)(122,452)(123,453)(124,454)(125,455)(126,456)(127,457)(128,458)(129,459)(130,460)(131,461)(132,462)(133,463)(134,464)(135,465)(136,466)(137,467)(138,468)(139,469)(140,470)(141,471)(142,472)(143,473)(144,474)(145,475)(146,476)(147,477)(148,478)(149,479)(150,480)(151,421)(152,422)(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)(160,430)(161,431)(162,432)(163,433)(164,434)(165,435)(166,436)(167,437)(168,438)(169,439)(170,440)(171,441)(172,442)(173,443)(174,444)(175,445)(176,446)(177,447)(178,448)(179,449)(180,450)(181,397)(182,398)(183,399)(184,400)(185,401)(186,402)(187,403)(188,404)(189,405)(190,406)(191,407)(192,408)(193,409)(194,410)(195,411)(196,412)(197,413)(198,414)(199,415)(200,416)(201,417)(202,418)(203,419)(204,420)(205,361)(206,362)(207,363)(208,364)(209,365)(210,366)(211,367)(212,368)(213,369)(214,370)(215,371)(216,372)(217,373)(218,374)(219,375)(220,376)(221,377)(222,378)(223,379)(224,380)(225,381)(226,382)(227,383)(228,384)(229,385)(230,386)(231,387)(232,388)(233,389)(234,390)(235,391)(236,392)(237,393)(238,394)(239,395)(240,396)(241,332)(242,333)(243,334)(244,335)(245,336)(246,337)(247,338)(248,339)(249,340)(250,341)(251,342)(252,343)(253,344)(254,345)(255,346)(256,347)(257,348)(258,349)(259,350)(260,351)(261,352)(262,353)(263,354)(264,355)(265,356)(266,357)(267,358)(268,359)(269,360)(270,301)(271,302)(272,303)(273,304)(274,305)(275,306)(276,307)(277,308)(278,309)(279,310)(280,311)(281,312)(282,313)(283,314)(284,315)(285,316)(286,317)(287,318)(288,319)(289,320)(290,321)(291,322)(292,323)(293,324)(294,325)(295,326)(296,327)(297,328)(298,329)(299,330)(300,331), (1,267,172,413)(2,268,173,414)(3,269,174,415)(4,270,175,416)(5,271,176,417)(6,272,177,418)(7,273,178,419)(8,274,179,420)(9,275,180,361)(10,276,121,362)(11,277,122,363)(12,278,123,364)(13,279,124,365)(14,280,125,366)(15,281,126,367)(16,282,127,368)(17,283,128,369)(18,284,129,370)(19,285,130,371)(20,286,131,372)(21,287,132,373)(22,288,133,374)(23,289,134,375)(24,290,135,376)(25,291,136,377)(26,292,137,378)(27,293,138,379)(28,294,139,380)(29,295,140,381)(30,296,141,382)(31,297,142,383)(32,298,143,384)(33,299,144,385)(34,300,145,386)(35,241,146,387)(36,242,147,388)(37,243,148,389)(38,244,149,390)(39,245,150,391)(40,246,151,392)(41,247,152,393)(42,248,153,394)(43,249,154,395)(44,250,155,396)(45,251,156,397)(46,252,157,398)(47,253,158,399)(48,254,159,400)(49,255,160,401)(50,256,161,402)(51,257,162,403)(52,258,163,404)(53,259,164,405)(54,260,165,406)(55,261,166,407)(56,262,167,408)(57,263,168,409)(58,264,169,410)(59,265,170,411)(60,266,171,412)(61,307,451,206)(62,308,452,207)(63,309,453,208)(64,310,454,209)(65,311,455,210)(66,312,456,211)(67,313,457,212)(68,314,458,213)(69,315,459,214)(70,316,460,215)(71,317,461,216)(72,318,462,217)(73,319,463,218)(74,320,464,219)(75,321,465,220)(76,322,466,221)(77,323,467,222)(78,324,468,223)(79,325,469,224)(80,326,470,225)(81,327,471,226)(82,328,472,227)(83,329,473,228)(84,330,474,229)(85,331,475,230)(86,332,476,231)(87,333,477,232)(88,334,478,233)(89,335,479,234)(90,336,480,235)(91,337,421,236)(92,338,422,237)(93,339,423,238)(94,340,424,239)(95,341,425,240)(96,342,426,181)(97,343,427,182)(98,344,428,183)(99,345,429,184)(100,346,430,185)(101,347,431,186)(102,348,432,187)(103,349,433,188)(104,350,434,189)(105,351,435,190)(106,352,436,191)(107,353,437,192)(108,354,438,193)(109,355,439,194)(110,356,440,195)(111,357,441,196)(112,358,442,197)(113,359,443,198)(114,360,444,199)(115,301,445,200)(116,302,446,201)(117,303,447,202)(118,304,448,203)(119,305,449,204)(120,306,450,205), (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( (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(121,451)(122,452)(123,453)(124,454)(125,455)(126,456)(127,457)(128,458)(129,459)(130,460)(131,461)(132,462)(133,463)(134,464)(135,465)(136,466)(137,467)(138,468)(139,469)(140,470)(141,471)(142,472)(143,473)(144,474)(145,475)(146,476)(147,477)(148,478)(149,479)(150,480)(151,421)(152,422)(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)(160,430)(161,431)(162,432)(163,433)(164,434)(165,435)(166,436)(167,437)(168,438)(169,439)(170,440)(171,441)(172,442)(173,443)(174,444)(175,445)(176,446)(177,447)(178,448)(179,449)(180,450)(181,397)(182,398)(183,399)(184,400)(185,401)(186,402)(187,403)(188,404)(189,405)(190,406)(191,407)(192,408)(193,409)(194,410)(195,411)(196,412)(197,413)(198,414)(199,415)(200,416)(201,417)(202,418)(203,419)(204,420)(205,361)(206,362)(207,363)(208,364)(209,365)(210,366)(211,367)(212,368)(213,369)(214,370)(215,371)(216,372)(217,373)(218,374)(219,375)(220,376)(221,377)(222,378)(223,379)(224,380)(225,381)(226,382)(227,383)(228,384)(229,385)(230,386)(231,387)(232,388)(233,389)(234,390)(235,391)(236,392)(237,393)(238,394)(239,395)(240,396)(241,332)(242,333)(243,334)(244,335)(245,336)(246,337)(247,338)(248,339)(249,340)(250,341)(251,342)(252,343)(253,344)(254,345)(255,346)(256,347)(257,348)(258,349)(259,350)(260,351)(261,352)(262,353)(263,354)(264,355)(265,356)(266,357)(267,358)(268,359)(269,360)(270,301)(271,302)(272,303)(273,304)(274,305)(275,306)(276,307)(277,308)(278,309)(279,310)(280,311)(281,312)(282,313)(283,314)(284,315)(285,316)(286,317)(287,318)(288,319)(289,320)(290,321)(291,322)(292,323)(293,324)(294,325)(295,326)(296,327)(297,328)(298,329)(299,330)(300,331), (1,267,172,413)(2,268,173,414)(3,269,174,415)(4,270,175,416)(5,271,176,417)(6,272,177,418)(7,273,178,419)(8,274,179,420)(9,275,180,361)(10,276,121,362)(11,277,122,363)(12,278,123,364)(13,279,124,365)(14,280,125,366)(15,281,126,367)(16,282,127,368)(17,283,128,369)(18,284,129,370)(19,285,130,371)(20,286,131,372)(21,287,132,373)(22,288,133,374)(23,289,134,375)(24,290,135,376)(25,291,136,377)(26,292,137,378)(27,293,138,379)(28,294,139,380)(29,295,140,381)(30,296,141,382)(31,297,142,383)(32,298,143,384)(33,299,144,385)(34,300,145,386)(35,241,146,387)(36,242,147,388)(37,243,148,389)(38,244,149,390)(39,245,150,391)(40,246,151,392)(41,247,152,393)(42,248,153,394)(43,249,154,395)(44,250,155,396)(45,251,156,397)(46,252,157,398)(47,253,158,399)(48,254,159,400)(49,255,160,401)(50,256,161,402)(51,257,162,403)(52,258,163,404)(53,259,164,405)(54,260,165,406)(55,261,166,407)(56,262,167,408)(57,263,168,409)(58,264,169,410)(59,265,170,411)(60,266,171,412)(61,307,451,206)(62,308,452,207)(63,309,453,208)(64,310,454,209)(65,311,455,210)(66,312,456,211)(67,313,457,212)(68,314,458,213)(69,315,459,214)(70,316,460,215)(71,317,461,216)(72,318,462,217)(73,319,463,218)(74,320,464,219)(75,321,465,220)(76,322,466,221)(77,323,467,222)(78,324,468,223)(79,325,469,224)(80,326,470,225)(81,327,471,226)(82,328,472,227)(83,329,473,228)(84,330,474,229)(85,331,475,230)(86,332,476,231)(87,333,477,232)(88,334,478,233)(89,335,479,234)(90,336,480,235)(91,337,421,236)(92,338,422,237)(93,339,423,238)(94,340,424,239)(95,341,425,240)(96,342,426,181)(97,343,427,182)(98,344,428,183)(99,345,429,184)(100,346,430,185)(101,347,431,186)(102,348,432,187)(103,349,433,188)(104,350,434,189)(105,351,435,190)(106,352,436,191)(107,353,437,192)(108,354,438,193)(109,355,439,194)(110,356,440,195)(111,357,441,196)(112,358,442,197)(113,359,443,198)(114,360,444,199)(115,301,445,200)(116,302,446,201)(117,303,447,202)(118,304,448,203)(119,305,449,204)(120,306,450,205), (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([[(1,112),(2,113),(3,114),(4,115),(5,116),(6,117),(7,118),(8,119),(9,120),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,81),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,97),(47,98),(48,99),(49,100),(50,101),(51,102),(52,103),(53,104),(54,105),(55,106),(56,107),(57,108),(58,109),(59,110),(60,111),(121,451),(122,452),(123,453),(124,454),(125,455),(126,456),(127,457),(128,458),(129,459),(130,460),(131,461),(132,462),(133,463),(134,464),(135,465),(136,466),(137,467),(138,468),(139,469),(140,470),(141,471),(142,472),(143,473),(144,474),(145,475),(146,476),(147,477),(148,478),(149,479),(150,480),(151,421),(152,422),(153,423),(154,424),(155,425),(156,426),(157,427),(158,428),(159,429),(160,430),(161,431),(162,432),(163,433),(164,434),(165,435),(166,436),(167,437),(168,438),(169,439),(170,440),(171,441),(172,442),(173,443),(174,444),(175,445),(176,446),(177,447),(178,448),(179,449),(180,450),(181,397),(182,398),(183,399),(184,400),(185,401),(186,402),(187,403),(188,404),(189,405),(190,406),(191,407),(192,408),(193,409),(194,410),(195,411),(196,412),(197,413),(198,414),(199,415),(200,416),(201,417),(202,418),(203,419),(204,420),(205,361),(206,362),(207,363),(208,364),(209,365),(210,366),(211,367),(212,368),(213,369),(214,370),(215,371),(216,372),(217,373),(218,374),(219,375),(220,376),(221,377),(222,378),(223,379),(224,380),(225,381),(226,382),(227,383),(228,384),(229,385),(230,386),(231,387),(232,388),(233,389),(234,390),(235,391),(236,392),(237,393),(238,394),(239,395),(240,396),(241,332),(242,333),(243,334),(244,335),(245,336),(246,337),(247,338),(248,339),(249,340),(250,341),(251,342),(252,343),(253,344),(254,345),(255,346),(256,347),(257,348),(258,349),(259,350),(260,351),(261,352),(262,353),(263,354),(264,355),(265,356),(266,357),(267,358),(268,359),(269,360),(270,301),(271,302),(272,303),(273,304),(274,305),(275,306),(276,307),(277,308),(278,309),(279,310),(280,311),(281,312),(282,313),(283,314),(284,315),(285,316),(286,317),(287,318),(288,319),(289,320),(290,321),(291,322),(292,323),(293,324),(294,325),(295,326),(296,327),(297,328),(298,329),(299,330),(300,331)], [(1,267,172,413),(2,268,173,414),(3,269,174,415),(4,270,175,416),(5,271,176,417),(6,272,177,418),(7,273,178,419),(8,274,179,420),(9,275,180,361),(10,276,121,362),(11,277,122,363),(12,278,123,364),(13,279,124,365),(14,280,125,366),(15,281,126,367),(16,282,127,368),(17,283,128,369),(18,284,129,370),(19,285,130,371),(20,286,131,372),(21,287,132,373),(22,288,133,374),(23,289,134,375),(24,290,135,376),(25,291,136,377),(26,292,137,378),(27,293,138,379),(28,294,139,380),(29,295,140,381),(30,296,141,382),(31,297,142,383),(32,298,143,384),(33,299,144,385),(34,300,145,386),(35,241,146,387),(36,242,147,388),(37,243,148,389),(38,244,149,390),(39,245,150,391),(40,246,151,392),(41,247,152,393),(42,248,153,394),(43,249,154,395),(44,250,155,396),(45,251,156,397),(46,252,157,398),(47,253,158,399),(48,254,159,400),(49,255,160,401),(50,256,161,402),(51,257,162,403),(52,258,163,404),(53,259,164,405),(54,260,165,406),(55,261,166,407),(56,262,167,408),(57,263,168,409),(58,264,169,410),(59,265,170,411),(60,266,171,412),(61,307,451,206),(62,308,452,207),(63,309,453,208),(64,310,454,209),(65,311,455,210),(66,312,456,211),(67,313,457,212),(68,314,458,213),(69,315,459,214),(70,316,460,215),(71,317,461,216),(72,318,462,217),(73,319,463,218),(74,320,464,219),(75,321,465,220),(76,322,466,221),(77,323,467,222),(78,324,468,223),(79,325,469,224),(80,326,470,225),(81,327,471,226),(82,328,472,227),(83,329,473,228),(84,330,474,229),(85,331,475,230),(86,332,476,231),(87,333,477,232),(88,334,478,233),(89,335,479,234),(90,336,480,235),(91,337,421,236),(92,338,422,237),(93,339,423,238),(94,340,424,239),(95,341,425,240),(96,342,426,181),(97,343,427,182),(98,344,428,183),(99,345,429,184),(100,346,430,185),(101,347,431,186),(102,348,432,187),(103,349,433,188),(104,350,434,189),(105,351,435,190),(106,352,436,191),(107,353,437,192),(108,354,438,193),(109,355,439,194),(110,356,440,195),(111,357,441,196),(112,358,442,197),(113,359,443,198),(114,360,444,199),(115,301,445,200),(116,302,446,201),(117,303,447,202),(118,304,448,203),(119,305,449,204),(120,306,450,205)], [(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···4X5A5B5C5D6A···6N10A···10AB12A···12AV15A···15H20A···20CR30A···30BD60A···60GJ
order12···2334···455556···610···1012···1215···1520···2030···3060···60
size11···1111···111111···11···11···11···11···11···11···1

480 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C4C5C6C6C10C10C12C15C20C30C30C60
kernelC2×C4×C60C4×C60C22×C60C2×C4×C20C2×C60C2×C4×C12C4×C20C22×C20C4×C12C22×C12C2×C20C2×C42C2×C12C42C22×C4C2×C4
# reps1432244861612488963224192

Matrix representation of C2×C4×C60 ►in GL3(𝔽61) generated by

6000
0600
001
,
1100
0500
0011
,
2400
030
0047
G:=sub<GL(3,GF(61))| [60,0,0,0,60,0,0,0,1],[11,0,0,0,50,0,0,0,11],[24,0,0,0,3,0,0,0,47] >;
 

C2×C4×C60 in GAP, Magma, Sage, TeX 

C_2\times C_4\times C_{60}
 
% in TeX
 
G:=Group("C2xC4xC60");
 
// GroupNames label
 
G:=SmallGroup(480,919);
 
// by ID
 
G=gap.SmallGroup(480,919);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,840,1688]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^4=c^60=1,a*b=b*a,a*c=c*a,b*c=c*b>;
 
// generators/relations
 

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