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## G = C23×C60order 480 = 25·3·5

### Abelian group of type [2,2,2,60]

Aliases: C23×C60, SmallGroup(480,1180)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C60
 Chief series C1 — C2 — C10 — C30 — C60 — C2×C60 — C22×C60 — C23×C60
 Lower central C1 — C23×C60
 Upper central C1 — C23×C60

Generators and relations for C23×C60
G = < a,b,c,d | a2=b2=c2=d60=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 472, all normal (16 characteristic)
C1, C2, C2 [×14], C3, C4 [×8], C22 [×35], C5, C6, C6 [×14], C2×C4 [×28], C23 [×15], C10, C10 [×14], C12 [×8], C2×C6 [×35], C15, C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C2×C12 [×28], C22×C6 [×15], C30, C30 [×14], C23×C4, C2×C20 [×28], C22×C10 [×15], C22×C12 [×14], C23×C6, C60 [×8], C2×C30 [×35], C22×C20 [×14], C23×C10, C23×C12, C2×C60 [×28], C22×C30 [×15], C23×C20, C22×C60 [×14], C23×C30, C23×C60
Quotients: C1, C2 [×15], C3, C4 [×8], C22 [×35], C5, C6 [×15], C2×C4 [×28], C23 [×15], C10 [×15], C12 [×8], C2×C6 [×35], C15, C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C2×C12 [×28], C22×C6 [×15], C30 [×15], C23×C4, C2×C20 [×28], C22×C10 [×15], C22×C12 [×14], C23×C6, C60 [×8], C2×C30 [×35], C22×C20 [×14], C23×C10, C23×C12, C2×C60 [×28], C22×C30 [×15], C23×C20, C22×C60 [×14], C23×C30, C23×C60

Smallest permutation representation of C23×C60
Regular action on 480 points
Generators in S480
(1 289)(2 290)(3 291)(4 292)(5 293)(6 294)(7 295)(8 296)(9 297)(10 298)(11 299)(12 300)(13 241)(14 242)(15 243)(16 244)(17 245)(18 246)(19 247)(20 248)(21 249)(22 250)(23 251)(24 252)(25 253)(26 254)(27 255)(28 256)(29 257)(30 258)(31 259)(32 260)(33 261)(34 262)(35 263)(36 264)(37 265)(38 266)(39 267)(40 268)(41 269)(42 270)(43 271)(44 272)(45 273)(46 274)(47 275)(48 276)(49 277)(50 278)(51 279)(52 280)(53 281)(54 282)(55 283)(56 284)(57 285)(58 286)(59 287)(60 288)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(84 169)(85 170)(86 171)(87 172)(88 173)(89 174)(90 175)(91 176)(92 177)(93 178)(94 179)(95 180)(96 121)(97 122)(98 123)(99 124)(100 125)(101 126)(102 127)(103 128)(104 129)(105 130)(106 131)(107 132)(108 133)(109 134)(110 135)(111 136)(112 137)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 145)(181 342)(182 343)(183 344)(184 345)(185 346)(186 347)(187 348)(188 349)(189 350)(190 351)(191 352)(192 353)(193 354)(194 355)(195 356)(196 357)(197 358)(198 359)(199 360)(200 301)(201 302)(202 303)(203 304)(204 305)(205 306)(206 307)(207 308)(208 309)(209 310)(210 311)(211 312)(212 313)(213 314)(214 315)(215 316)(216 317)(217 318)(218 319)(219 320)(220 321)(221 322)(222 323)(223 324)(224 325)(225 326)(226 327)(227 328)(228 329)(229 330)(230 331)(231 332)(232 333)(233 334)(234 335)(235 336)(236 337)(237 338)(238 339)(239 340)(240 341)(361 480)(362 421)(363 422)(364 423)(365 424)(366 425)(367 426)(368 427)(369 428)(370 429)(371 430)(372 431)(373 432)(374 433)(375 434)(376 435)(377 436)(378 437)(379 438)(380 439)(381 440)(382 441)(383 442)(384 443)(385 444)(386 445)(387 446)(388 447)(389 448)(390 449)(391 450)(392 451)(393 452)(394 453)(395 454)(396 455)(397 456)(398 457)(399 458)(400 459)(401 460)(402 461)(403 462)(404 463)(405 464)(406 465)(407 466)(408 467)(409 468)(410 469)(411 470)(412 471)(413 472)(414 473)(415 474)(416 475)(417 476)(418 477)(419 478)(420 479)
(1 397)(2 398)(3 399)(4 400)(5 401)(6 402)(7 403)(8 404)(9 405)(10 406)(11 407)(12 408)(13 409)(14 410)(15 411)(16 412)(17 413)(18 414)(19 415)(20 416)(21 417)(22 418)(23 419)(24 420)(25 361)(26 362)(27 363)(28 364)(29 365)(30 366)(31 367)(32 368)(33 369)(34 370)(35 371)(36 372)(37 373)(38 374)(39 375)(40 376)(41 377)(42 378)(43 379)(44 380)(45 381)(46 382)(47 383)(48 384)(49 385)(50 386)(51 387)(52 388)(53 389)(54 390)(55 391)(56 392)(57 393)(58 394)(59 395)(60 396)(61 338)(62 339)(63 340)(64 341)(65 342)(66 343)(67 344)(68 345)(69 346)(70 347)(71 348)(72 349)(73 350)(74 351)(75 352)(76 353)(77 354)(78 355)(79 356)(80 357)(81 358)(82 359)(83 360)(84 301)(85 302)(86 303)(87 304)(88 305)(89 306)(90 307)(91 308)(92 309)(93 310)(94 311)(95 312)(96 313)(97 314)(98 315)(99 316)(100 317)(101 318)(102 319)(103 320)(104 321)(105 322)(106 323)(107 324)(108 325)(109 326)(110 327)(111 328)(112 329)(113 330)(114 331)(115 332)(116 333)(117 334)(118 335)(119 336)(120 337)(121 212)(122 213)(123 214)(124 215)(125 216)(126 217)(127 218)(128 219)(129 220)(130 221)(131 222)(132 223)(133 224)(134 225)(135 226)(136 227)(137 228)(138 229)(139 230)(140 231)(141 232)(142 233)(143 234)(144 235)(145 236)(146 237)(147 238)(148 239)(149 240)(150 181)(151 182)(152 183)(153 184)(154 185)(155 186)(156 187)(157 188)(158 189)(159 190)(160 191)(161 192)(162 193)(163 194)(164 195)(165 196)(166 197)(167 198)(168 199)(169 200)(170 201)(171 202)(172 203)(173 204)(174 205)(175 206)(176 207)(177 208)(178 209)(179 210)(180 211)(241 468)(242 469)(243 470)(244 471)(245 472)(246 473)(247 474)(248 475)(249 476)(250 477)(251 478)(252 479)(253 480)(254 421)(255 422)(256 423)(257 424)(258 425)(259 426)(260 427)(261 428)(262 429)(263 430)(264 431)(265 432)(266 433)(267 434)(268 435)(269 436)(270 437)(271 438)(272 439)(273 440)(274 441)(275 442)(276 443)(277 444)(278 445)(279 446)(280 447)(281 448)(282 449)(283 450)(284 451)(285 452)(286 453)(287 454)(288 455)(289 456)(290 457)(291 458)(292 459)(293 460)(294 461)(295 462)(296 463)(297 464)(298 465)(299 466)(300 467)
(1 228)(2 229)(3 230)(4 231)(5 232)(6 233)(7 234)(8 235)(9 236)(10 237)(11 238)(12 239)(13 240)(14 181)(15 182)(16 183)(17 184)(18 185)(19 186)(20 187)(21 188)(22 189)(23 190)(24 191)(25 192)(26 193)(27 194)(28 195)(29 196)(30 197)(31 198)(32 199)(33 200)(34 201)(35 202)(36 203)(37 204)(38 205)(39 206)(40 207)(41 208)(42 209)(43 210)(44 211)(45 212)(46 213)(47 214)(48 215)(49 216)(50 217)(51 218)(52 219)(53 220)(54 221)(55 222)(56 223)(57 224)(58 225)(59 226)(60 227)(61 465)(62 466)(63 467)(64 468)(65 469)(66 470)(67 471)(68 472)(69 473)(70 474)(71 475)(72 476)(73 477)(74 478)(75 479)(76 480)(77 421)(78 422)(79 423)(80 424)(81 425)(82 426)(83 427)(84 428)(85 429)(86 430)(87 431)(88 432)(89 433)(90 434)(91 435)(92 436)(93 437)(94 438)(95 439)(96 440)(97 441)(98 442)(99 443)(100 444)(101 445)(102 446)(103 447)(104 448)(105 449)(106 450)(107 451)(108 452)(109 453)(110 454)(111 455)(112 456)(113 457)(114 458)(115 459)(116 460)(117 461)(118 462)(119 463)(120 464)(121 381)(122 382)(123 383)(124 384)(125 385)(126 386)(127 387)(128 388)(129 389)(130 390)(131 391)(132 392)(133 393)(134 394)(135 395)(136 396)(137 397)(138 398)(139 399)(140 400)(141 401)(142 402)(143 403)(144 404)(145 405)(146 406)(147 407)(148 408)(149 409)(150 410)(151 411)(152 412)(153 413)(154 414)(155 415)(156 416)(157 417)(158 418)(159 419)(160 420)(161 361)(162 362)(163 363)(164 364)(165 365)(166 366)(167 367)(168 368)(169 369)(170 370)(171 371)(172 372)(173 373)(174 374)(175 375)(176 376)(177 377)(178 378)(179 379)(180 380)(241 341)(242 342)(243 343)(244 344)(245 345)(246 346)(247 347)(248 348)(249 349)(250 350)(251 351)(252 352)(253 353)(254 354)(255 355)(256 356)(257 357)(258 358)(259 359)(260 360)(261 301)(262 302)(263 303)(264 304)(265 305)(266 306)(267 307)(268 308)(269 309)(270 310)(271 311)(272 312)(273 313)(274 314)(275 315)(276 316)(277 317)(278 318)(279 319)(280 320)(281 321)(282 322)(283 323)(284 324)(285 325)(286 326)(287 327)(288 328)(289 329)(290 330)(291 331)(292 332)(293 333)(294 334)(295 335)(296 336)(297 337)(298 338)(299 339)(300 340)
(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)| (1,289)(2,290)(3,291)(4,292)(5,293)(6,294)(7,295)(8,296)(9,297)(10,298)(11,299)(12,300)(13,241)(14,242)(15,243)(16,244)(17,245)(18,246)(19,247)(20,248)(21,249)(22,250)(23,251)(24,252)(25,253)(26,254)(27,255)(28,256)(29,257)(30,258)(31,259)(32,260)(33,261)(34,262)(35,263)(36,264)(37,265)(38,266)(39,267)(40,268)(41,269)(42,270)(43,271)(44,272)(45,273)(46,274)(47,275)(48,276)(49,277)(50,278)(51,279)(52,280)(53,281)(54,282)(55,283)(56,284)(57,285)(58,286)(59,287)(60,288)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,169)(85,170)(86,171)(87,172)(88,173)(89,174)(90,175)(91,176)(92,177)(93,178)(94,179)(95,180)(96,121)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,145)(181,342)(182,343)(183,344)(184,345)(185,346)(186,347)(187,348)(188,349)(189,350)(190,351)(191,352)(192,353)(193,354)(194,355)(195,356)(196,357)(197,358)(198,359)(199,360)(200,301)(201,302)(202,303)(203,304)(204,305)(205,306)(206,307)(207,308)(208,309)(209,310)(210,311)(211,312)(212,313)(213,314)(214,315)(215,316)(216,317)(217,318)(218,319)(219,320)(220,321)(221,322)(222,323)(223,324)(224,325)(225,326)(226,327)(227,328)(228,329)(229,330)(230,331)(231,332)(232,333)(233,334)(234,335)(235,336)(236,337)(237,338)(238,339)(239,340)(240,341)(361,480)(362,421)(363,422)(364,423)(365,424)(366,425)(367,426)(368,427)(369,428)(370,429)(371,430)(372,431)(373,432)(374,433)(375,434)(376,435)(377,436)(378,437)(379,438)(380,439)(381,440)(382,441)(383,442)(384,443)(385,444)(386,445)(387,446)(388,447)(389,448)(390,449)(391,450)(392,451)(393,452)(394,453)(395,454)(396,455)(397,456)(398,457)(399,458)(400,459)(401,460)(402,461)(403,462)(404,463)(405,464)(406,465)(407,466)(408,467)(409,468)(410,469)(411,470)(412,471)(413,472)(414,473)(415,474)(416,475)(417,476)(418,477)(419,478)(420,479), (1,397)(2,398)(3,399)(4,400)(5,401)(6,402)(7,403)(8,404)(9,405)(10,406)(11,407)(12,408)(13,409)(14,410)(15,411)(16,412)(17,413)(18,414)(19,415)(20,416)(21,417)(22,418)(23,419)(24,420)(25,361)(26,362)(27,363)(28,364)(29,365)(30,366)(31,367)(32,368)(33,369)(34,370)(35,371)(36,372)(37,373)(38,374)(39,375)(40,376)(41,377)(42,378)(43,379)(44,380)(45,381)(46,382)(47,383)(48,384)(49,385)(50,386)(51,387)(52,388)(53,389)(54,390)(55,391)(56,392)(57,393)(58,394)(59,395)(60,396)(61,338)(62,339)(63,340)(64,341)(65,342)(66,343)(67,344)(68,345)(69,346)(70,347)(71,348)(72,349)(73,350)(74,351)(75,352)(76,353)(77,354)(78,355)(79,356)(80,357)(81,358)(82,359)(83,360)(84,301)(85,302)(86,303)(87,304)(88,305)(89,306)(90,307)(91,308)(92,309)(93,310)(94,311)(95,312)(96,313)(97,314)(98,315)(99,316)(100,317)(101,318)(102,319)(103,320)(104,321)(105,322)(106,323)(107,324)(108,325)(109,326)(110,327)(111,328)(112,329)(113,330)(114,331)(115,332)(116,333)(117,334)(118,335)(119,336)(120,337)(121,212)(122,213)(123,214)(124,215)(125,216)(126,217)(127,218)(128,219)(129,220)(130,221)(131,222)(132,223)(133,224)(134,225)(135,226)(136,227)(137,228)(138,229)(139,230)(140,231)(141,232)(142,233)(143,234)(144,235)(145,236)(146,237)(147,238)(148,239)(149,240)(150,181)(151,182)(152,183)(153,184)(154,185)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,197)(167,198)(168,199)(169,200)(170,201)(171,202)(172,203)(173,204)(174,205)(175,206)(176,207)(177,208)(178,209)(179,210)(180,211)(241,468)(242,469)(243,470)(244,471)(245,472)(246,473)(247,474)(248,475)(249,476)(250,477)(251,478)(252,479)(253,480)(254,421)(255,422)(256,423)(257,424)(258,425)(259,426)(260,427)(261,428)(262,429)(263,430)(264,431)(265,432)(266,433)(267,434)(268,435)(269,436)(270,437)(271,438)(272,439)(273,440)(274,441)(275,442)(276,443)(277,444)(278,445)(279,446)(280,447)(281,448)(282,449)(283,450)(284,451)(285,452)(286,453)(287,454)(288,455)(289,456)(290,457)(291,458)(292,459)(293,460)(294,461)(295,462)(296,463)(297,464)(298,465)(299,466)(300,467), (1,228)(2,229)(3,230)(4,231)(5,232)(6,233)(7,234)(8,235)(9,236)(10,237)(11,238)(12,239)(13,240)(14,181)(15,182)(16,183)(17,184)(18,185)(19,186)(20,187)(21,188)(22,189)(23,190)(24,191)(25,192)(26,193)(27,194)(28,195)(29,196)(30,197)(31,198)(32,199)(33,200)(34,201)(35,202)(36,203)(37,204)(38,205)(39,206)(40,207)(41,208)(42,209)(43,210)(44,211)(45,212)(46,213)(47,214)(48,215)(49,216)(50,217)(51,218)(52,219)(53,220)(54,221)(55,222)(56,223)(57,224)(58,225)(59,226)(60,227)(61,465)(62,466)(63,467)(64,468)(65,469)(66,470)(67,471)(68,472)(69,473)(70,474)(71,475)(72,476)(73,477)(74,478)(75,479)(76,480)(77,421)(78,422)(79,423)(80,424)(81,425)(82,426)(83,427)(84,428)(85,429)(86,430)(87,431)(88,432)(89,433)(90,434)(91,435)(92,436)(93,437)(94,438)(95,439)(96,440)(97,441)(98,442)(99,443)(100,444)(101,445)(102,446)(103,447)(104,448)(105,449)(106,450)(107,451)(108,452)(109,453)(110,454)(111,455)(112,456)(113,457)(114,458)(115,459)(116,460)(117,461)(118,462)(119,463)(120,464)(121,381)(122,382)(123,383)(124,384)(125,385)(126,386)(127,387)(128,388)(129,389)(130,390)(131,391)(132,392)(133,393)(134,394)(135,395)(136,396)(137,397)(138,398)(139,399)(140,400)(141,401)(142,402)(143,403)(144,404)(145,405)(146,406)(147,407)(148,408)(149,409)(150,410)(151,411)(152,412)(153,413)(154,414)(155,415)(156,416)(157,417)(158,418)(159,419)(160,420)(161,361)(162,362)(163,363)(164,364)(165,365)(166,366)(167,367)(168,368)(169,369)(170,370)(171,371)(172,372)(173,373)(174,374)(175,375)(176,376)(177,377)(178,378)(179,379)(180,380)(241,341)(242,342)(243,343)(244,344)(245,345)(246,346)(247,347)(248,348)(249,349)(250,350)(251,351)(252,352)(253,353)(254,354)(255,355)(256,356)(257,357)(258,358)(259,359)(260,360)(261,301)(262,302)(263,303)(264,304)(265,305)(266,306)(267,307)(268,308)(269,309)(270,310)(271,311)(272,312)(273,313)(274,314)(275,315)(276,316)(277,317)(278,318)(279,319)(280,320)(281,321)(282,322)(283,323)(284,324)(285,325)(286,326)(287,327)(288,328)(289,329)(290,330)(291,331)(292,332)(293,333)(294,334)(295,335)(296,336)(297,337)(298,338)(299,339)(300,340), (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( (1,289)(2,290)(3,291)(4,292)(5,293)(6,294)(7,295)(8,296)(9,297)(10,298)(11,299)(12,300)(13,241)(14,242)(15,243)(16,244)(17,245)(18,246)(19,247)(20,248)(21,249)(22,250)(23,251)(24,252)(25,253)(26,254)(27,255)(28,256)(29,257)(30,258)(31,259)(32,260)(33,261)(34,262)(35,263)(36,264)(37,265)(38,266)(39,267)(40,268)(41,269)(42,270)(43,271)(44,272)(45,273)(46,274)(47,275)(48,276)(49,277)(50,278)(51,279)(52,280)(53,281)(54,282)(55,283)(56,284)(57,285)(58,286)(59,287)(60,288)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,169)(85,170)(86,171)(87,172)(88,173)(89,174)(90,175)(91,176)(92,177)(93,178)(94,179)(95,180)(96,121)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,145)(181,342)(182,343)(183,344)(184,345)(185,346)(186,347)(187,348)(188,349)(189,350)(190,351)(191,352)(192,353)(193,354)(194,355)(195,356)(196,357)(197,358)(198,359)(199,360)(200,301)(201,302)(202,303)(203,304)(204,305)(205,306)(206,307)(207,308)(208,309)(209,310)(210,311)(211,312)(212,313)(213,314)(214,315)(215,316)(216,317)(217,318)(218,319)(219,320)(220,321)(221,322)(222,323)(223,324)(224,325)(225,326)(226,327)(227,328)(228,329)(229,330)(230,331)(231,332)(232,333)(233,334)(234,335)(235,336)(236,337)(237,338)(238,339)(239,340)(240,341)(361,480)(362,421)(363,422)(364,423)(365,424)(366,425)(367,426)(368,427)(369,428)(370,429)(371,430)(372,431)(373,432)(374,433)(375,434)(376,435)(377,436)(378,437)(379,438)(380,439)(381,440)(382,441)(383,442)(384,443)(385,444)(386,445)(387,446)(388,447)(389,448)(390,449)(391,450)(392,451)(393,452)(394,453)(395,454)(396,455)(397,456)(398,457)(399,458)(400,459)(401,460)(402,461)(403,462)(404,463)(405,464)(406,465)(407,466)(408,467)(409,468)(410,469)(411,470)(412,471)(413,472)(414,473)(415,474)(416,475)(417,476)(418,477)(419,478)(420,479), (1,397)(2,398)(3,399)(4,400)(5,401)(6,402)(7,403)(8,404)(9,405)(10,406)(11,407)(12,408)(13,409)(14,410)(15,411)(16,412)(17,413)(18,414)(19,415)(20,416)(21,417)(22,418)(23,419)(24,420)(25,361)(26,362)(27,363)(28,364)(29,365)(30,366)(31,367)(32,368)(33,369)(34,370)(35,371)(36,372)(37,373)(38,374)(39,375)(40,376)(41,377)(42,378)(43,379)(44,380)(45,381)(46,382)(47,383)(48,384)(49,385)(50,386)(51,387)(52,388)(53,389)(54,390)(55,391)(56,392)(57,393)(58,394)(59,395)(60,396)(61,338)(62,339)(63,340)(64,341)(65,342)(66,343)(67,344)(68,345)(69,346)(70,347)(71,348)(72,349)(73,350)(74,351)(75,352)(76,353)(77,354)(78,355)(79,356)(80,357)(81,358)(82,359)(83,360)(84,301)(85,302)(86,303)(87,304)(88,305)(89,306)(90,307)(91,308)(92,309)(93,310)(94,311)(95,312)(96,313)(97,314)(98,315)(99,316)(100,317)(101,318)(102,319)(103,320)(104,321)(105,322)(106,323)(107,324)(108,325)(109,326)(110,327)(111,328)(112,329)(113,330)(114,331)(115,332)(116,333)(117,334)(118,335)(119,336)(120,337)(121,212)(122,213)(123,214)(124,215)(125,216)(126,217)(127,218)(128,219)(129,220)(130,221)(131,222)(132,223)(133,224)(134,225)(135,226)(136,227)(137,228)(138,229)(139,230)(140,231)(141,232)(142,233)(143,234)(144,235)(145,236)(146,237)(147,238)(148,239)(149,240)(150,181)(151,182)(152,183)(153,184)(154,185)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,197)(167,198)(168,199)(169,200)(170,201)(171,202)(172,203)(173,204)(174,205)(175,206)(176,207)(177,208)(178,209)(179,210)(180,211)(241,468)(242,469)(243,470)(244,471)(245,472)(246,473)(247,474)(248,475)(249,476)(250,477)(251,478)(252,479)(253,480)(254,421)(255,422)(256,423)(257,424)(258,425)(259,426)(260,427)(261,428)(262,429)(263,430)(264,431)(265,432)(266,433)(267,434)(268,435)(269,436)(270,437)(271,438)(272,439)(273,440)(274,441)(275,442)(276,443)(277,444)(278,445)(279,446)(280,447)(281,448)(282,449)(283,450)(284,451)(285,452)(286,453)(287,454)(288,455)(289,456)(290,457)(291,458)(292,459)(293,460)(294,461)(295,462)(296,463)(297,464)(298,465)(299,466)(300,467), (1,228)(2,229)(3,230)(4,231)(5,232)(6,233)(7,234)(8,235)(9,236)(10,237)(11,238)(12,239)(13,240)(14,181)(15,182)(16,183)(17,184)(18,185)(19,186)(20,187)(21,188)(22,189)(23,190)(24,191)(25,192)(26,193)(27,194)(28,195)(29,196)(30,197)(31,198)(32,199)(33,200)(34,201)(35,202)(36,203)(37,204)(38,205)(39,206)(40,207)(41,208)(42,209)(43,210)(44,211)(45,212)(46,213)(47,214)(48,215)(49,216)(50,217)(51,218)(52,219)(53,220)(54,221)(55,222)(56,223)(57,224)(58,225)(59,226)(60,227)(61,465)(62,466)(63,467)(64,468)(65,469)(66,470)(67,471)(68,472)(69,473)(70,474)(71,475)(72,476)(73,477)(74,478)(75,479)(76,480)(77,421)(78,422)(79,423)(80,424)(81,425)(82,426)(83,427)(84,428)(85,429)(86,430)(87,431)(88,432)(89,433)(90,434)(91,435)(92,436)(93,437)(94,438)(95,439)(96,440)(97,441)(98,442)(99,443)(100,444)(101,445)(102,446)(103,447)(104,448)(105,449)(106,450)(107,451)(108,452)(109,453)(110,454)(111,455)(112,456)(113,457)(114,458)(115,459)(116,460)(117,461)(118,462)(119,463)(120,464)(121,381)(122,382)(123,383)(124,384)(125,385)(126,386)(127,387)(128,388)(129,389)(130,390)(131,391)(132,392)(133,393)(134,394)(135,395)(136,396)(137,397)(138,398)(139,399)(140,400)(141,401)(142,402)(143,403)(144,404)(145,405)(146,406)(147,407)(148,408)(149,409)(150,410)(151,411)(152,412)(153,413)(154,414)(155,415)(156,416)(157,417)(158,418)(159,419)(160,420)(161,361)(162,362)(163,363)(164,364)(165,365)(166,366)(167,367)(168,368)(169,369)(170,370)(171,371)(172,372)(173,373)(174,374)(175,375)(176,376)(177,377)(178,378)(179,379)(180,380)(241,341)(242,342)(243,343)(244,344)(245,345)(246,346)(247,347)(248,348)(249,349)(250,350)(251,351)(252,352)(253,353)(254,354)(255,355)(256,356)(257,357)(258,358)(259,359)(260,360)(261,301)(262,302)(263,303)(264,304)(265,305)(266,306)(267,307)(268,308)(269,309)(270,310)(271,311)(272,312)(273,313)(274,314)(275,315)(276,316)(277,317)(278,318)(279,319)(280,320)(281,321)(282,322)(283,323)(284,324)(285,325)(286,326)(287,327)(288,328)(289,329)(290,330)(291,331)(292,332)(293,333)(294,334)(295,335)(296,336)(297,337)(298,338)(299,339)(300,340), (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([(1,289),(2,290),(3,291),(4,292),(5,293),(6,294),(7,295),(8,296),(9,297),(10,298),(11,299),(12,300),(13,241),(14,242),(15,243),(16,244),(17,245),(18,246),(19,247),(20,248),(21,249),(22,250),(23,251),(24,252),(25,253),(26,254),(27,255),(28,256),(29,257),(30,258),(31,259),(32,260),(33,261),(34,262),(35,263),(36,264),(37,265),(38,266),(39,267),(40,268),(41,269),(42,270),(43,271),(44,272),(45,273),(46,274),(47,275),(48,276),(49,277),(50,278),(51,279),(52,280),(53,281),(54,282),(55,283),(56,284),(57,285),(58,286),(59,287),(60,288),(61,146),(62,147),(63,148),(64,149),(65,150),(66,151),(67,152),(68,153),(69,154),(70,155),(71,156),(72,157),(73,158),(74,159),(75,160),(76,161),(77,162),(78,163),(79,164),(80,165),(81,166),(82,167),(83,168),(84,169),(85,170),(86,171),(87,172),(88,173),(89,174),(90,175),(91,176),(92,177),(93,178),(94,179),(95,180),(96,121),(97,122),(98,123),(99,124),(100,125),(101,126),(102,127),(103,128),(104,129),(105,130),(106,131),(107,132),(108,133),(109,134),(110,135),(111,136),(112,137),(113,138),(114,139),(115,140),(116,141),(117,142),(118,143),(119,144),(120,145),(181,342),(182,343),(183,344),(184,345),(185,346),(186,347),(187,348),(188,349),(189,350),(190,351),(191,352),(192,353),(193,354),(194,355),(195,356),(196,357),(197,358),(198,359),(199,360),(200,301),(201,302),(202,303),(203,304),(204,305),(205,306),(206,307),(207,308),(208,309),(209,310),(210,311),(211,312),(212,313),(213,314),(214,315),(215,316),(216,317),(217,318),(218,319),(219,320),(220,321),(221,322),(222,323),(223,324),(224,325),(225,326),(226,327),(227,328),(228,329),(229,330),(230,331),(231,332),(232,333),(233,334),(234,335),(235,336),(236,337),(237,338),(238,339),(239,340),(240,341),(361,480),(362,421),(363,422),(364,423),(365,424),(366,425),(367,426),(368,427),(369,428),(370,429),(371,430),(372,431),(373,432),(374,433),(375,434),(376,435),(377,436),(378,437),(379,438),(380,439),(381,440),(382,441),(383,442),(384,443),(385,444),(386,445),(387,446),(388,447),(389,448),(390,449),(391,450),(392,451),(393,452),(394,453),(395,454),(396,455),(397,456),(398,457),(399,458),(400,459),(401,460),(402,461),(403,462),(404,463),(405,464),(406,465),(407,466),(408,467),(409,468),(410,469),(411,470),(412,471),(413,472),(414,473),(415,474),(416,475),(417,476),(418,477),(419,478),(420,479)], [(1,397),(2,398),(3,399),(4,400),(5,401),(6,402),(7,403),(8,404),(9,405),(10,406),(11,407),(12,408),(13,409),(14,410),(15,411),(16,412),(17,413),(18,414),(19,415),(20,416),(21,417),(22,418),(23,419),(24,420),(25,361),(26,362),(27,363),(28,364),(29,365),(30,366),(31,367),(32,368),(33,369),(34,370),(35,371),(36,372),(37,373),(38,374),(39,375),(40,376),(41,377),(42,378),(43,379),(44,380),(45,381),(46,382),(47,383),(48,384),(49,385),(50,386),(51,387),(52,388),(53,389),(54,390),(55,391),(56,392),(57,393),(58,394),(59,395),(60,396),(61,338),(62,339),(63,340),(64,341),(65,342),(66,343),(67,344),(68,345),(69,346),(70,347),(71,348),(72,349),(73,350),(74,351),(75,352),(76,353),(77,354),(78,355),(79,356),(80,357),(81,358),(82,359),(83,360),(84,301),(85,302),(86,303),(87,304),(88,305),(89,306),(90,307),(91,308),(92,309),(93,310),(94,311),(95,312),(96,313),(97,314),(98,315),(99,316),(100,317),(101,318),(102,319),(103,320),(104,321),(105,322),(106,323),(107,324),(108,325),(109,326),(110,327),(111,328),(112,329),(113,330),(114,331),(115,332),(116,333),(117,334),(118,335),(119,336),(120,337),(121,212),(122,213),(123,214),(124,215),(125,216),(126,217),(127,218),(128,219),(129,220),(130,221),(131,222),(132,223),(133,224),(134,225),(135,226),(136,227),(137,228),(138,229),(139,230),(140,231),(141,232),(142,233),(143,234),(144,235),(145,236),(146,237),(147,238),(148,239),(149,240),(150,181),(151,182),(152,183),(153,184),(154,185),(155,186),(156,187),(157,188),(158,189),(159,190),(160,191),(161,192),(162,193),(163,194),(164,195),(165,196),(166,197),(167,198),(168,199),(169,200),(170,201),(171,202),(172,203),(173,204),(174,205),(175,206),(176,207),(177,208),(178,209),(179,210),(180,211),(241,468),(242,469),(243,470),(244,471),(245,472),(246,473),(247,474),(248,475),(249,476),(250,477),(251,478),(252,479),(253,480),(254,421),(255,422),(256,423),(257,424),(258,425),(259,426),(260,427),(261,428),(262,429),(263,430),(264,431),(265,432),(266,433),(267,434),(268,435),(269,436),(270,437),(271,438),(272,439),(273,440),(274,441),(275,442),(276,443),(277,444),(278,445),(279,446),(280,447),(281,448),(282,449),(283,450),(284,451),(285,452),(286,453),(287,454),(288,455),(289,456),(290,457),(291,458),(292,459),(293,460),(294,461),(295,462),(296,463),(297,464),(298,465),(299,466),(300,467)], [(1,228),(2,229),(3,230),(4,231),(5,232),(6,233),(7,234),(8,235),(9,236),(10,237),(11,238),(12,239),(13,240),(14,181),(15,182),(16,183),(17,184),(18,185),(19,186),(20,187),(21,188),(22,189),(23,190),(24,191),(25,192),(26,193),(27,194),(28,195),(29,196),(30,197),(31,198),(32,199),(33,200),(34,201),(35,202),(36,203),(37,204),(38,205),(39,206),(40,207),(41,208),(42,209),(43,210),(44,211),(45,212),(46,213),(47,214),(48,215),(49,216),(50,217),(51,218),(52,219),(53,220),(54,221),(55,222),(56,223),(57,224),(58,225),(59,226),(60,227),(61,465),(62,466),(63,467),(64,468),(65,469),(66,470),(67,471),(68,472),(69,473),(70,474),(71,475),(72,476),(73,477),(74,478),(75,479),(76,480),(77,421),(78,422),(79,423),(80,424),(81,425),(82,426),(83,427),(84,428),(85,429),(86,430),(87,431),(88,432),(89,433),(90,434),(91,435),(92,436),(93,437),(94,438),(95,439),(96,440),(97,441),(98,442),(99,443),(100,444),(101,445),(102,446),(103,447),(104,448),(105,449),(106,450),(107,451),(108,452),(109,453),(110,454),(111,455),(112,456),(113,457),(114,458),(115,459),(116,460),(117,461),(118,462),(119,463),(120,464),(121,381),(122,382),(123,383),(124,384),(125,385),(126,386),(127,387),(128,388),(129,389),(130,390),(131,391),(132,392),(133,393),(134,394),(135,395),(136,396),(137,397),(138,398),(139,399),(140,400),(141,401),(142,402),(143,403),(144,404),(145,405),(146,406),(147,407),(148,408),(149,409),(150,410),(151,411),(152,412),(153,413),(154,414),(155,415),(156,416),(157,417),(158,418),(159,419),(160,420),(161,361),(162,362),(163,363),(164,364),(165,365),(166,366),(167,367),(168,368),(169,369),(170,370),(171,371),(172,372),(173,373),(174,374),(175,375),(176,376),(177,377),(178,378),(179,379),(180,380),(241,341),(242,342),(243,343),(244,344),(245,345),(246,346),(247,347),(248,348),(249,349),(250,350),(251,351),(252,352),(253,353),(254,354),(255,355),(256,356),(257,357),(258,358),(259,359),(260,360),(261,301),(262,302),(263,303),(264,304),(265,305),(266,306),(267,307),(268,308),(269,309),(270,310),(271,311),(272,312),(273,313),(274,314),(275,315),(276,316),(277,317),(278,318),(279,319),(280,320),(281,321),(282,322),(283,323),(284,324),(285,325),(286,326),(287,327),(288,328),(289,329),(290,330),(291,331),(292,332),(293,333),(294,334),(295,335),(296,336),(297,337),(298,338),(299,339),(300,340)], [(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 ··· 2O 3A 3B 4A ··· 4P 5A 5B 5C 5D 6A ··· 6AD 10A ··· 10BH 12A ··· 12AF 15A ··· 15H 20A ··· 20BL 30A ··· 30DP 60A ··· 60DX order 1 2 ··· 2 3 3 4 ··· 4 5 5 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 30 ··· 30 60 ··· 60 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

480 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C5 C6 C6 C10 C10 C12 C15 C20 C30 C30 C60 kernel C23×C60 C22×C60 C23×C30 C23×C20 C22×C30 C23×C12 C22×C20 C23×C10 C22×C12 C23×C6 C22×C10 C23×C4 C22×C6 C22×C4 C24 C23 # reps 1 14 1 2 16 4 28 2 56 4 32 8 64 112 8 128

Matrix representation of C23×C60 in GL4(𝔽61) generated by

 1 0 0 0 0 60 0 0 0 0 60 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 35 0 0 0 0 23 0 0 0 0 37 0 0 0 0 5
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,60,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[35,0,0,0,0,23,0,0,0,0,37,0,0,0,0,5] >;

C23×C60 in GAP, Magma, Sage, TeX

C_2^3\times C_{60}
% in TeX

G:=Group("C2^3xC60");
// GroupNames label

G:=SmallGroup(480,1180);
// by ID

G=gap.SmallGroup(480,1180);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-5,-2,1680]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=d^60=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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