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G = C2×C4×C60order 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

C1 — C2×C4×C60
C1C2C22C2×C10C2×C30C2×C60C4×C60 — C2×C4×C60
C1 — C2×C4×C60
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 [×7], C3, C4 [×12], C22, C22 [×6], C5, C6 [×7], C2×C4 [×18], C23, C10 [×7], C12 [×12], C2×C6, C2×C6 [×6], C15, C42 [×4], C22×C4 [×3], C20 [×12], C2×C10, C2×C10 [×6], C2×C12 [×18], C22×C6, C30 [×7], C2×C42, C2×C20 [×18], C22×C10, C4×C12 [×4], C22×C12 [×3], C60 [×12], C2×C30, C2×C30 [×6], C4×C20 [×4], C22×C20 [×3], C2×C4×C12, C2×C60 [×18], C22×C30, C2×C4×C20, C4×C60 [×4], C22×C60 [×3], C2×C4×C60
Quotients: C1, C2 [×7], C3, C4 [×12], C22 [×7], C5, C6 [×7], C2×C4 [×18], C23, C10 [×7], C12 [×12], C2×C6 [×7], C15, C42 [×4], C22×C4 [×3], C20 [×12], C2×C10 [×7], C2×C12 [×18], C22×C6, C30 [×7], C2×C42, C2×C20 [×18], C22×C10, C4×C12 [×4], C22×C12 [×3], C60 [×12], C2×C30 [×7], C4×C20 [×4], C22×C20 [×3], C2×C4×C12, C2×C60 [×18], C22×C30, C2×C4×C20, C4×C60 [×4], C22×C60 [×3], C2×C4×C60

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

׿
×
𝔽