Copied to
clipboard

G = C23×C60  order 480 = 25·3·5

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

direct product, abelian, monomial, 2-elementary

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

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

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

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

(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)| (1,266)(2,267)(3,268)(4,269)(5,270)(6,271)(7,272)(8,273)(9,274)(10,275)(11,276)(12,277)(13,278)(14,279)(15,280)(16,281)(17,282)(18,283)(19,284)(20,285)(21,286)(22,287)(23,288)(24,289)(25,290)(26,291)(27,292)(28,293)(29,294)(30,295)(31,296)(32,297)(33,298)(34,299)(35,300)(36,241)(37,242)(38,243)(39,244)(40,245)(41,246)(42,247)(43,248)(44,249)(45,250)(46,251)(47,252)(48,253)(49,254)(50,255)(51,256)(52,257)(53,258)(54,259)(55,260)(56,261)(57,262)(58,263)(59,264)(60,265)(61,453)(62,454)(63,455)(64,456)(65,457)(66,458)(67,459)(68,460)(69,461)(70,462)(71,463)(72,464)(73,465)(74,466)(75,467)(76,468)(77,469)(78,470)(79,471)(80,472)(81,473)(82,474)(83,475)(84,476)(85,477)(86,478)(87,479)(88,480)(89,421)(90,422)(91,423)(92,424)(93,425)(94,426)(95,427)(96,428)(97,429)(98,430)(99,431)(100,432)(101,433)(102,434)(103,435)(104,436)(105,437)(106,438)(107,439)(108,440)(109,441)(110,442)(111,443)(112,444)(113,445)(114,446)(115,447)(116,448)(117,449)(118,450)(119,451)(120,452)(121,208)(122,209)(123,210)(124,211)(125,212)(126,213)(127,214)(128,215)(129,216)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,225)(139,226)(140,227)(141,228)(142,229)(143,230)(144,231)(145,232)(146,233)(147,234)(148,235)(149,236)(150,237)(151,238)(152,239)(153,240)(154,181)(155,182)(156,183)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)(171,198)(172,199)(173,200)(174,201)(175,202)(176,203)(177,204)(178,205)(179,206)(180,207)(301,405)(302,406)(303,407)(304,408)(305,409)(306,410)(307,411)(308,412)(309,413)(310,414)(311,415)(312,416)(313,417)(314,418)(315,419)(316,420)(317,361)(318,362)(319,363)(320,364)(321,365)(322,366)(323,367)(324,368)(325,369)(326,370)(327,371)(328,372)(329,373)(330,374)(331,375)(332,376)(333,377)(334,378)(335,379)(336,380)(337,381)(338,382)(339,383)(340,384)(341,385)(342,386)(343,387)(344,388)(345,389)(346,390)(347,391)(348,392)(349,393)(350,394)(351,395)(352,396)(353,397)(354,398)(355,399)(356,400)(357,401)(358,402)(359,403)(360,404), (1,432)(2,433)(3,434)(4,435)(5,436)(6,437)(7,438)(8,439)(9,440)(10,441)(11,442)(12,443)(13,444)(14,445)(15,446)(16,447)(17,448)(18,449)(19,450)(20,451)(21,452)(22,453)(23,454)(24,455)(25,456)(26,457)(27,458)(28,459)(29,460)(30,461)(31,462)(32,463)(33,464)(34,465)(35,466)(36,467)(37,468)(38,469)(39,470)(40,471)(41,472)(42,473)(43,474)(44,475)(45,476)(46,477)(47,478)(48,479)(49,480)(50,421)(51,422)(52,423)(53,424)(54,425)(55,426)(56,427)(57,428)(58,429)(59,430)(60,431)(61,287)(62,288)(63,289)(64,290)(65,291)(66,292)(67,293)(68,294)(69,295)(70,296)(71,297)(72,298)(73,299)(74,300)(75,241)(76,242)(77,243)(78,244)(79,245)(80,246)(81,247)(82,248)(83,249)(84,250)(85,251)(86,252)(87,253)(88,254)(89,255)(90,256)(91,257)(92,258)(93,259)(94,260)(95,261)(96,262)(97,263)(98,264)(99,265)(100,266)(101,267)(102,268)(103,269)(104,270)(105,271)(106,272)(107,273)(108,274)(109,275)(110,276)(111,277)(112,278)(113,279)(114,280)(115,281)(116,282)(117,283)(118,284)(119,285)(120,286)(121,344)(122,345)(123,346)(124,347)(125,348)(126,349)(127,350)(128,351)(129,352)(130,353)(131,354)(132,355)(133,356)(134,357)(135,358)(136,359)(137,360)(138,301)(139,302)(140,303)(141,304)(142,305)(143,306)(144,307)(145,308)(146,309)(147,310)(148,311)(149,312)(150,313)(151,314)(152,315)(153,316)(154,317)(155,318)(156,319)(157,320)(158,321)(159,322)(160,323)(161,324)(162,325)(163,326)(164,327)(165,328)(166,329)(167,330)(168,331)(169,332)(170,333)(171,334)(172,335)(173,336)(174,337)(175,338)(176,339)(177,340)(178,341)(179,342)(180,343)(181,361)(182,362)(183,363)(184,364)(185,365)(186,366)(187,367)(188,368)(189,369)(190,370)(191,371)(192,372)(193,373)(194,374)(195,375)(196,376)(197,377)(198,378)(199,379)(200,380)(201,381)(202,382)(203,383)(204,384)(205,385)(206,386)(207,387)(208,388)(209,389)(210,390)(211,391)(212,392)(213,393)(214,394)(215,395)(216,396)(217,397)(218,398)(219,399)(220,400)(221,401)(222,402)(223,403)(224,404)(225,405)(226,406)(227,407)(228,408)(229,409)(230,410)(231,411)(232,412)(233,413)(234,414)(235,415)(236,416)(237,417)(238,418)(239,419)(240,420), (1,373)(2,374)(3,375)(4,376)(5,377)(6,378)(7,379)(8,380)(9,381)(10,382)(11,383)(12,384)(13,385)(14,386)(15,387)(16,388)(17,389)(18,390)(19,391)(20,392)(21,393)(22,394)(23,395)(24,396)(25,397)(26,398)(27,399)(28,400)(29,401)(30,402)(31,403)(32,404)(33,405)(34,406)(35,407)(36,408)(37,409)(38,410)(39,411)(40,412)(41,413)(42,414)(43,415)(44,416)(45,417)(46,418)(47,419)(48,420)(49,361)(50,362)(51,363)(52,364)(53,365)(54,366)(55,367)(56,368)(57,369)(58,370)(59,371)(60,372)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,169)(104,170)(105,171)(106,172)(107,173)(108,174)(109,175)(110,176)(111,177)(112,178)(113,179)(114,180)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126)(181,480)(182,421)(183,422)(184,423)(185,424)(186,425)(187,426)(188,427)(189,428)(190,429)(191,430)(192,431)(193,432)(194,433)(195,434)(196,435)(197,436)(198,437)(199,438)(200,439)(201,440)(202,441)(203,442)(204,443)(205,444)(206,445)(207,446)(208,447)(209,448)(210,449)(211,450)(212,451)(213,452)(214,453)(215,454)(216,455)(217,456)(218,457)(219,458)(220,459)(221,460)(222,461)(223,462)(224,463)(225,464)(226,465)(227,466)(228,467)(229,468)(230,469)(231,470)(232,471)(233,472)(234,473)(235,474)(236,475)(237,476)(238,477)(239,478)(240,479)(241,304)(242,305)(243,306)(244,307)(245,308)(246,309)(247,310)(248,311)(249,312)(250,313)(251,314)(252,315)(253,316)(254,317)(255,318)(256,319)(257,320)(258,321)(259,322)(260,323)(261,324)(262,325)(263,326)(264,327)(265,328)(266,329)(267,330)(268,331)(269,332)(270,333)(271,334)(272,335)(273,336)(274,337)(275,338)(276,339)(277,340)(278,341)(279,342)(280,343)(281,344)(282,345)(283,346)(284,347)(285,348)(286,349)(287,350)(288,351)(289,352)(290,353)(291,354)(292,355)(293,356)(294,357)(295,358)(296,359)(297,360)(298,301)(299,302)(300,303), (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( (1,266)(2,267)(3,268)(4,269)(5,270)(6,271)(7,272)(8,273)(9,274)(10,275)(11,276)(12,277)(13,278)(14,279)(15,280)(16,281)(17,282)(18,283)(19,284)(20,285)(21,286)(22,287)(23,288)(24,289)(25,290)(26,291)(27,292)(28,293)(29,294)(30,295)(31,296)(32,297)(33,298)(34,299)(35,300)(36,241)(37,242)(38,243)(39,244)(40,245)(41,246)(42,247)(43,248)(44,249)(45,250)(46,251)(47,252)(48,253)(49,254)(50,255)(51,256)(52,257)(53,258)(54,259)(55,260)(56,261)(57,262)(58,263)(59,264)(60,265)(61,453)(62,454)(63,455)(64,456)(65,457)(66,458)(67,459)(68,460)(69,461)(70,462)(71,463)(72,464)(73,465)(74,466)(75,467)(76,468)(77,469)(78,470)(79,471)(80,472)(81,473)(82,474)(83,475)(84,476)(85,477)(86,478)(87,479)(88,480)(89,421)(90,422)(91,423)(92,424)(93,425)(94,426)(95,427)(96,428)(97,429)(98,430)(99,431)(100,432)(101,433)(102,434)(103,435)(104,436)(105,437)(106,438)(107,439)(108,440)(109,441)(110,442)(111,443)(112,444)(113,445)(114,446)(115,447)(116,448)(117,449)(118,450)(119,451)(120,452)(121,208)(122,209)(123,210)(124,211)(125,212)(126,213)(127,214)(128,215)(129,216)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,225)(139,226)(140,227)(141,228)(142,229)(143,230)(144,231)(145,232)(146,233)(147,234)(148,235)(149,236)(150,237)(151,238)(152,239)(153,240)(154,181)(155,182)(156,183)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)(171,198)(172,199)(173,200)(174,201)(175,202)(176,203)(177,204)(178,205)(179,206)(180,207)(301,405)(302,406)(303,407)(304,408)(305,409)(306,410)(307,411)(308,412)(309,413)(310,414)(311,415)(312,416)(313,417)(314,418)(315,419)(316,420)(317,361)(318,362)(319,363)(320,364)(321,365)(322,366)(323,367)(324,368)(325,369)(326,370)(327,371)(328,372)(329,373)(330,374)(331,375)(332,376)(333,377)(334,378)(335,379)(336,380)(337,381)(338,382)(339,383)(340,384)(341,385)(342,386)(343,387)(344,388)(345,389)(346,390)(347,391)(348,392)(349,393)(350,394)(351,395)(352,396)(353,397)(354,398)(355,399)(356,400)(357,401)(358,402)(359,403)(360,404), (1,432)(2,433)(3,434)(4,435)(5,436)(6,437)(7,438)(8,439)(9,440)(10,441)(11,442)(12,443)(13,444)(14,445)(15,446)(16,447)(17,448)(18,449)(19,450)(20,451)(21,452)(22,453)(23,454)(24,455)(25,456)(26,457)(27,458)(28,459)(29,460)(30,461)(31,462)(32,463)(33,464)(34,465)(35,466)(36,467)(37,468)(38,469)(39,470)(40,471)(41,472)(42,473)(43,474)(44,475)(45,476)(46,477)(47,478)(48,479)(49,480)(50,421)(51,422)(52,423)(53,424)(54,425)(55,426)(56,427)(57,428)(58,429)(59,430)(60,431)(61,287)(62,288)(63,289)(64,290)(65,291)(66,292)(67,293)(68,294)(69,295)(70,296)(71,297)(72,298)(73,299)(74,300)(75,241)(76,242)(77,243)(78,244)(79,245)(80,246)(81,247)(82,248)(83,249)(84,250)(85,251)(86,252)(87,253)(88,254)(89,255)(90,256)(91,257)(92,258)(93,259)(94,260)(95,261)(96,262)(97,263)(98,264)(99,265)(100,266)(101,267)(102,268)(103,269)(104,270)(105,271)(106,272)(107,273)(108,274)(109,275)(110,276)(111,277)(112,278)(113,279)(114,280)(115,281)(116,282)(117,283)(118,284)(119,285)(120,286)(121,344)(122,345)(123,346)(124,347)(125,348)(126,349)(127,350)(128,351)(129,352)(130,353)(131,354)(132,355)(133,356)(134,357)(135,358)(136,359)(137,360)(138,301)(139,302)(140,303)(141,304)(142,305)(143,306)(144,307)(145,308)(146,309)(147,310)(148,311)(149,312)(150,313)(151,314)(152,315)(153,316)(154,317)(155,318)(156,319)(157,320)(158,321)(159,322)(160,323)(161,324)(162,325)(163,326)(164,327)(165,328)(166,329)(167,330)(168,331)(169,332)(170,333)(171,334)(172,335)(173,336)(174,337)(175,338)(176,339)(177,340)(178,341)(179,342)(180,343)(181,361)(182,362)(183,363)(184,364)(185,365)(186,366)(187,367)(188,368)(189,369)(190,370)(191,371)(192,372)(193,373)(194,374)(195,375)(196,376)(197,377)(198,378)(199,379)(200,380)(201,381)(202,382)(203,383)(204,384)(205,385)(206,386)(207,387)(208,388)(209,389)(210,390)(211,391)(212,392)(213,393)(214,394)(215,395)(216,396)(217,397)(218,398)(219,399)(220,400)(221,401)(222,402)(223,403)(224,404)(225,405)(226,406)(227,407)(228,408)(229,409)(230,410)(231,411)(232,412)(233,413)(234,414)(235,415)(236,416)(237,417)(238,418)(239,419)(240,420), (1,373)(2,374)(3,375)(4,376)(5,377)(6,378)(7,379)(8,380)(9,381)(10,382)(11,383)(12,384)(13,385)(14,386)(15,387)(16,388)(17,389)(18,390)(19,391)(20,392)(21,393)(22,394)(23,395)(24,396)(25,397)(26,398)(27,399)(28,400)(29,401)(30,402)(31,403)(32,404)(33,405)(34,406)(35,407)(36,408)(37,409)(38,410)(39,411)(40,412)(41,413)(42,414)(43,415)(44,416)(45,417)(46,418)(47,419)(48,420)(49,361)(50,362)(51,363)(52,364)(53,365)(54,366)(55,367)(56,368)(57,369)(58,370)(59,371)(60,372)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,169)(104,170)(105,171)(106,172)(107,173)(108,174)(109,175)(110,176)(111,177)(112,178)(113,179)(114,180)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126)(181,480)(182,421)(183,422)(184,423)(185,424)(186,425)(187,426)(188,427)(189,428)(190,429)(191,430)(192,431)(193,432)(194,433)(195,434)(196,435)(197,436)(198,437)(199,438)(200,439)(201,440)(202,441)(203,442)(204,443)(205,444)(206,445)(207,446)(208,447)(209,448)(210,449)(211,450)(212,451)(213,452)(214,453)(215,454)(216,455)(217,456)(218,457)(219,458)(220,459)(221,460)(222,461)(223,462)(224,463)(225,464)(226,465)(227,466)(228,467)(229,468)(230,469)(231,470)(232,471)(233,472)(234,473)(235,474)(236,475)(237,476)(238,477)(239,478)(240,479)(241,304)(242,305)(243,306)(244,307)(245,308)(246,309)(247,310)(248,311)(249,312)(250,313)(251,314)(252,315)(253,316)(254,317)(255,318)(256,319)(257,320)(258,321)(259,322)(260,323)(261,324)(262,325)(263,326)(264,327)(265,328)(266,329)(267,330)(268,331)(269,332)(270,333)(271,334)(272,335)(273,336)(274,337)(275,338)(276,339)(277,340)(278,341)(279,342)(280,343)(281,344)(282,345)(283,346)(284,347)(285,348)(286,349)(287,350)(288,351)(289,352)(290,353)(291,354)(292,355)(293,356)(294,357)(295,358)(296,359)(297,360)(298,301)(299,302)(300,303), (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([[(1,266),(2,267),(3,268),(4,269),(5,270),(6,271),(7,272),(8,273),(9,274),(10,275),(11,276),(12,277),(13,278),(14,279),(15,280),(16,281),(17,282),(18,283),(19,284),(20,285),(21,286),(22,287),(23,288),(24,289),(25,290),(26,291),(27,292),(28,293),(29,294),(30,295),(31,296),(32,297),(33,298),(34,299),(35,300),(36,241),(37,242),(38,243),(39,244),(40,245),(41,246),(42,247),(43,248),(44,249),(45,250),(46,251),(47,252),(48,253),(49,254),(50,255),(51,256),(52,257),(53,258),(54,259),(55,260),(56,261),(57,262),(58,263),(59,264),(60,265),(61,453),(62,454),(63,455),(64,456),(65,457),(66,458),(67,459),(68,460),(69,461),(70,462),(71,463),(72,464),(73,465),(74,466),(75,467),(76,468),(77,469),(78,470),(79,471),(80,472),(81,473),(82,474),(83,475),(84,476),(85,477),(86,478),(87,479),(88,480),(89,421),(90,422),(91,423),(92,424),(93,425),(94,426),(95,427),(96,428),(97,429),(98,430),(99,431),(100,432),(101,433),(102,434),(103,435),(104,436),(105,437),(106,438),(107,439),(108,440),(109,441),(110,442),(111,443),(112,444),(113,445),(114,446),(115,447),(116,448),(117,449),(118,450),(119,451),(120,452),(121,208),(122,209),(123,210),(124,211),(125,212),(126,213),(127,214),(128,215),(129,216),(130,217),(131,218),(132,219),(133,220),(134,221),(135,222),(136,223),(137,224),(138,225),(139,226),(140,227),(141,228),(142,229),(143,230),(144,231),(145,232),(146,233),(147,234),(148,235),(149,236),(150,237),(151,238),(152,239),(153,240),(154,181),(155,182),(156,183),(157,184),(158,185),(159,186),(160,187),(161,188),(162,189),(163,190),(164,191),(165,192),(166,193),(167,194),(168,195),(169,196),(170,197),(171,198),(172,199),(173,200),(174,201),(175,202),(176,203),(177,204),(178,205),(179,206),(180,207),(301,405),(302,406),(303,407),(304,408),(305,409),(306,410),(307,411),(308,412),(309,413),(310,414),(311,415),(312,416),(313,417),(314,418),(315,419),(316,420),(317,361),(318,362),(319,363),(320,364),(321,365),(322,366),(323,367),(324,368),(325,369),(326,370),(327,371),(328,372),(329,373),(330,374),(331,375),(332,376),(333,377),(334,378),(335,379),(336,380),(337,381),(338,382),(339,383),(340,384),(341,385),(342,386),(343,387),(344,388),(345,389),(346,390),(347,391),(348,392),(349,393),(350,394),(351,395),(352,396),(353,397),(354,398),(355,399),(356,400),(357,401),(358,402),(359,403),(360,404)], [(1,432),(2,433),(3,434),(4,435),(5,436),(6,437),(7,438),(8,439),(9,440),(10,441),(11,442),(12,443),(13,444),(14,445),(15,446),(16,447),(17,448),(18,449),(19,450),(20,451),(21,452),(22,453),(23,454),(24,455),(25,456),(26,457),(27,458),(28,459),(29,460),(30,461),(31,462),(32,463),(33,464),(34,465),(35,466),(36,467),(37,468),(38,469),(39,470),(40,471),(41,472),(42,473),(43,474),(44,475),(45,476),(46,477),(47,478),(48,479),(49,480),(50,421),(51,422),(52,423),(53,424),(54,425),(55,426),(56,427),(57,428),(58,429),(59,430),(60,431),(61,287),(62,288),(63,289),(64,290),(65,291),(66,292),(67,293),(68,294),(69,295),(70,296),(71,297),(72,298),(73,299),(74,300),(75,241),(76,242),(77,243),(78,244),(79,245),(80,246),(81,247),(82,248),(83,249),(84,250),(85,251),(86,252),(87,253),(88,254),(89,255),(90,256),(91,257),(92,258),(93,259),(94,260),(95,261),(96,262),(97,263),(98,264),(99,265),(100,266),(101,267),(102,268),(103,269),(104,270),(105,271),(106,272),(107,273),(108,274),(109,275),(110,276),(111,277),(112,278),(113,279),(114,280),(115,281),(116,282),(117,283),(118,284),(119,285),(120,286),(121,344),(122,345),(123,346),(124,347),(125,348),(126,349),(127,350),(128,351),(129,352),(130,353),(131,354),(132,355),(133,356),(134,357),(135,358),(136,359),(137,360),(138,301),(139,302),(140,303),(141,304),(142,305),(143,306),(144,307),(145,308),(146,309),(147,310),(148,311),(149,312),(150,313),(151,314),(152,315),(153,316),(154,317),(155,318),(156,319),(157,320),(158,321),(159,322),(160,323),(161,324),(162,325),(163,326),(164,327),(165,328),(166,329),(167,330),(168,331),(169,332),(170,333),(171,334),(172,335),(173,336),(174,337),(175,338),(176,339),(177,340),(178,341),(179,342),(180,343),(181,361),(182,362),(183,363),(184,364),(185,365),(186,366),(187,367),(188,368),(189,369),(190,370),(191,371),(192,372),(193,373),(194,374),(195,375),(196,376),(197,377),(198,378),(199,379),(200,380),(201,381),(202,382),(203,383),(204,384),(205,385),(206,386),(207,387),(208,388),(209,389),(210,390),(211,391),(212,392),(213,393),(214,394),(215,395),(216,396),(217,397),(218,398),(219,399),(220,400),(221,401),(222,402),(223,403),(224,404),(225,405),(226,406),(227,407),(228,408),(229,409),(230,410),(231,411),(232,412),(233,413),(234,414),(235,415),(236,416),(237,417),(238,418),(239,419),(240,420)], [(1,373),(2,374),(3,375),(4,376),(5,377),(6,378),(7,379),(8,380),(9,381),(10,382),(11,383),(12,384),(13,385),(14,386),(15,387),(16,388),(17,389),(18,390),(19,391),(20,392),(21,393),(22,394),(23,395),(24,396),(25,397),(26,398),(27,399),(28,400),(29,401),(30,402),(31,403),(32,404),(33,405),(34,406),(35,407),(36,408),(37,409),(38,410),(39,411),(40,412),(41,413),(42,414),(43,415),(44,416),(45,417),(46,418),(47,419),(48,420),(49,361),(50,362),(51,363),(52,364),(53,365),(54,366),(55,367),(56,368),(57,369),(58,370),(59,371),(60,372),(61,127),(62,128),(63,129),(64,130),(65,131),(66,132),(67,133),(68,134),(69,135),(70,136),(71,137),(72,138),(73,139),(74,140),(75,141),(76,142),(77,143),(78,144),(79,145),(80,146),(81,147),(82,148),(83,149),(84,150),(85,151),(86,152),(87,153),(88,154),(89,155),(90,156),(91,157),(92,158),(93,159),(94,160),(95,161),(96,162),(97,163),(98,164),(99,165),(100,166),(101,167),(102,168),(103,169),(104,170),(105,171),(106,172),(107,173),(108,174),(109,175),(110,176),(111,177),(112,178),(113,179),(114,180),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126),(181,480),(182,421),(183,422),(184,423),(185,424),(186,425),(187,426),(188,427),(189,428),(190,429),(191,430),(192,431),(193,432),(194,433),(195,434),(196,435),(197,436),(198,437),(199,438),(200,439),(201,440),(202,441),(203,442),(204,443),(205,444),(206,445),(207,446),(208,447),(209,448),(210,449),(211,450),(212,451),(213,452),(214,453),(215,454),(216,455),(217,456),(218,457),(219,458),(220,459),(221,460),(222,461),(223,462),(224,463),(225,464),(226,465),(227,466),(228,467),(229,468),(230,469),(231,470),(232,471),(233,472),(234,473),(235,474),(236,475),(237,476),(238,477),(239,478),(240,479),(241,304),(242,305),(243,306),(244,307),(245,308),(246,309),(247,310),(248,311),(249,312),(250,313),(251,314),(252,315),(253,316),(254,317),(255,318),(256,319),(257,320),(258,321),(259,322),(260,323),(261,324),(262,325),(263,326),(264,327),(265,328),(266,329),(267,330),(268,331),(269,332),(270,333),(271,334),(272,335),(273,336),(274,337),(275,338),(276,339),(277,340),(278,341),(279,342),(280,343),(281,344),(282,345),(283,346),(284,347),(285,348),(286,349),(287,350),(288,351),(289,352),(290,353),(291,354),(292,355),(293,356),(294,357),(295,358),(296,359),(297,360),(298,301),(299,302),(300,303)], [(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···2O3A3B4A···4P5A5B5C5D6A···6AD10A···10BH12A···12AF15A···15H20A···20BL30A···30DP60A···60DX
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
kernelC23×C60C22×C60C23×C30C23×C20C22×C30C23×C12C22×C20C23×C10C22×C12C23×C6C22×C10C23×C4C22×C6C22×C4C24C23
# reps11412164282564328641128128

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

1000
06000
00600
0001
,
1000
0100
00600
00060
,
1000
06000
0010
0001
,
35000
02300
00370
0005
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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁