direct product, abelian, monomial, 2-elementary
Aliases: C23×C60, SmallGroup(480,1180)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C23×C60 |
| C1 — C23×C60 |
| C1 — C23×C60 |
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
Generators and relations
G = < a,b,c,d | a2=b2=c2=d60=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
►Regular action on 480 points
Generators in S480
(1 465)(2 466)(3 467)(4 468)(5 469)(6 470)(7 471)(8 472)(9 473)(10 474)(11 475)(12 476)(13 477)(14 478)(15 479)(16 480)(17 421)(18 422)(19 423)(20 424)(21 425)(22 426)(23 427)(24 428)(25 429)(26 430)(27 431)(28 432)(29 433)(30 434)(31 435)(32 436)(33 437)(34 438)(35 439)(36 440)(37 441)(38 442)(39 443)(40 444)(41 445)(42 446)(43 447)(44 448)(45 449)(46 450)(47 451)(48 452)(49 453)(50 454)(51 455)(52 456)(53 457)(54 458)(55 459)(56 460)(57 461)(58 462)(59 463)(60 464)(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)(181 332)(182 333)(183 334)(184 335)(185 336)(186 337)(187 338)(188 339)(189 340)(190 341)(191 342)(192 343)(193 344)(194 345)(195 346)(196 347)(197 348)(198 349)(199 350)(200 351)(201 352)(202 353)(203 354)(204 355)(205 356)(206 357)(207 358)(208 359)(209 360)(210 301)(211 302)(212 303)(213 304)(214 305)(215 306)(216 307)(217 308)(218 309)(219 310)(220 311)(221 312)(222 313)(223 314)(224 315)(225 316)(226 317)(227 318)(228 319)(229 320)(230 321)(231 322)(232 323)(233 324)(234 325)(235 326)(236 327)(237 328)(238 329)(239 330)(240 331)(241 395)(242 396)(243 397)(244 398)(245 399)(246 400)(247 401)(248 402)(249 403)(250 404)(251 405)(252 406)(253 407)(254 408)(255 409)(256 410)(257 411)(258 412)(259 413)(260 414)(261 415)(262 416)(263 417)(264 418)(265 419)(266 420)(267 361)(268 362)(269 363)(270 364)(271 365)(272 366)(273 367)(274 368)(275 369)(276 370)(277 371)(278 372)(279 373)(280 374)(281 375)(282 376)(283 377)(284 378)(285 379)(286 380)(287 381)(288 382)(289 383)(290 384)(291 385)(292 386)(293 387)(294 388)(295 389)(296 390)(297 391)(298 392)(299 393)(300 394)
(1 393)(2 394)(3 395)(4 396)(5 397)(6 398)(7 399)(8 400)(9 401)(10 402)(11 403)(12 404)(13 405)(14 406)(15 407)(16 408)(17 409)(18 410)(19 411)(20 412)(21 413)(22 414)(23 415)(24 416)(25 417)(26 418)(27 419)(28 420)(29 361)(30 362)(31 363)(32 364)(33 365)(34 366)(35 367)(36 368)(37 369)(38 370)(39 371)(40 372)(41 373)(42 374)(43 375)(44 376)(45 377)(46 378)(47 379)(48 380)(49 381)(50 382)(51 383)(52 384)(53 385)(54 386)(55 387)(56 388)(57 389)(58 390)(59 391)(60 392)(61 225)(62 226)(63 227)(64 228)(65 229)(66 230)(67 231)(68 232)(69 233)(70 234)(71 235)(72 236)(73 237)(74 238)(75 239)(76 240)(77 181)(78 182)(79 183)(80 184)(81 185)(82 186)(83 187)(84 188)(85 189)(86 190)(87 191)(88 192)(89 193)(90 194)(91 195)(92 196)(93 197)(94 198)(95 199)(96 200)(97 201)(98 202)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)(105 209)(106 210)(107 211)(108 212)(109 213)(110 214)(111 215)(112 216)(113 217)(114 218)(115 219)(116 220)(117 221)(118 222)(119 223)(120 224)(121 351)(122 352)(123 353)(124 354)(125 355)(126 356)(127 357)(128 358)(129 359)(130 360)(131 301)(132 302)(133 303)(134 304)(135 305)(136 306)(137 307)(138 308)(139 309)(140 310)(141 311)(142 312)(143 313)(144 314)(145 315)(146 316)(147 317)(148 318)(149 319)(150 320)(151 321)(152 322)(153 323)(154 324)(155 325)(156 326)(157 327)(158 328)(159 329)(160 330)(161 331)(162 332)(163 333)(164 334)(165 335)(166 336)(167 337)(168 338)(169 339)(170 340)(171 341)(172 342)(173 343)(174 344)(175 345)(176 346)(177 347)(178 348)(179 349)(180 350)(241 467)(242 468)(243 469)(244 470)(245 471)(246 472)(247 473)(248 474)(249 475)(250 476)(251 477)(252 478)(253 479)(254 480)(255 421)(256 422)(257 423)(258 424)(259 425)(260 426)(261 427)(262 428)(263 429)(264 430)(265 431)(266 432)(267 433)(268 434)(269 435)(270 436)(271 437)(272 438)(273 439)(274 440)(275 441)(276 442)(277 443)(278 444)(279 445)(280 446)(281 447)(282 448)(283 449)(284 450)(285 451)(286 452)(287 453)(288 454)(289 455)(290 456)(291 457)(292 458)(293 459)(294 460)(295 461)(296 462)(297 463)(298 464)(299 465)(300 466)
(1 225)(2 226)(3 227)(4 228)(5 229)(6 230)(7 231)(8 232)(9 233)(10 234)(11 235)(12 236)(13 237)(14 238)(15 239)(16 240)(17 181)(18 182)(19 183)(20 184)(21 185)(22 186)(23 187)(24 188)(25 189)(26 190)(27 191)(28 192)(29 193)(30 194)(31 195)(32 196)(33 197)(34 198)(35 199)(36 200)(37 201)(38 202)(39 203)(40 204)(41 205)(42 206)(43 207)(44 208)(45 209)(46 210)(47 211)(48 212)(49 213)(50 214)(51 215)(52 216)(53 217)(54 218)(55 219)(56 220)(57 221)(58 222)(59 223)(60 224)(61 393)(62 394)(63 395)(64 396)(65 397)(66 398)(67 399)(68 400)(69 401)(70 402)(71 403)(72 404)(73 405)(74 406)(75 407)(76 408)(77 409)(78 410)(79 411)(80 412)(81 413)(82 414)(83 415)(84 416)(85 417)(86 418)(87 419)(88 420)(89 361)(90 362)(91 363)(92 364)(93 365)(94 366)(95 367)(96 368)(97 369)(98 370)(99 371)(100 372)(101 373)(102 374)(103 375)(104 376)(105 377)(106 378)(107 379)(108 380)(109 381)(110 382)(111 383)(112 384)(113 385)(114 386)(115 387)(116 388)(117 389)(118 390)(119 391)(120 392)(121 274)(122 275)(123 276)(124 277)(125 278)(126 279)(127 280)(128 281)(129 282)(130 283)(131 284)(132 285)(133 286)(134 287)(135 288)(136 289)(137 290)(138 291)(139 292)(140 293)(141 294)(142 295)(143 296)(144 297)(145 298)(146 299)(147 300)(148 241)(149 242)(150 243)(151 244)(152 245)(153 246)(154 247)(155 248)(156 249)(157 250)(158 251)(159 252)(160 253)(161 254)(162 255)(163 256)(164 257)(165 258)(166 259)(167 260)(168 261)(169 262)(170 263)(171 264)(172 265)(173 266)(174 267)(175 268)(176 269)(177 270)(178 271)(179 272)(180 273)(301 450)(302 451)(303 452)(304 453)(305 454)(306 455)(307 456)(308 457)(309 458)(310 459)(311 460)(312 461)(313 462)(314 463)(315 464)(316 465)(317 466)(318 467)(319 468)(320 469)(321 470)(322 471)(323 472)(324 473)(325 474)(326 475)(327 476)(328 477)(329 478)(330 479)(331 480)(332 421)(333 422)(334 423)(335 424)(336 425)(337 426)(338 427)(339 428)(340 429)(341 430)(342 431)(343 432)(344 433)(345 434)(346 435)(347 436)(348 437)(349 438)(350 439)(351 440)(352 441)(353 442)(354 443)(355 444)(356 445)(357 446)(358 447)(359 448)(360 449)
(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)| (1,465)(2,466)(3,467)(4,468)(5,469)(6,470)(7,471)(8,472)(9,473)(10,474)(11,475)(12,476)(13,477)(14,478)(15,479)(16,480)(17,421)(18,422)(19,423)(20,424)(21,425)(22,426)(23,427)(24,428)(25,429)(26,430)(27,431)(28,432)(29,433)(30,434)(31,435)(32,436)(33,437)(34,438)(35,439)(36,440)(37,441)(38,442)(39,443)(40,444)(41,445)(42,446)(43,447)(44,448)(45,449)(46,450)(47,451)(48,452)(49,453)(50,454)(51,455)(52,456)(53,457)(54,458)(55,459)(56,460)(57,461)(58,462)(59,463)(60,464)(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)(181,332)(182,333)(183,334)(184,335)(185,336)(186,337)(187,338)(188,339)(189,340)(190,341)(191,342)(192,343)(193,344)(194,345)(195,346)(196,347)(197,348)(198,349)(199,350)(200,351)(201,352)(202,353)(203,354)(204,355)(205,356)(206,357)(207,358)(208,359)(209,360)(210,301)(211,302)(212,303)(213,304)(214,305)(215,306)(216,307)(217,308)(218,309)(219,310)(220,311)(221,312)(222,313)(223,314)(224,315)(225,316)(226,317)(227,318)(228,319)(229,320)(230,321)(231,322)(232,323)(233,324)(234,325)(235,326)(236,327)(237,328)(238,329)(239,330)(240,331)(241,395)(242,396)(243,397)(244,398)(245,399)(246,400)(247,401)(248,402)(249,403)(250,404)(251,405)(252,406)(253,407)(254,408)(255,409)(256,410)(257,411)(258,412)(259,413)(260,414)(261,415)(262,416)(263,417)(264,418)(265,419)(266,420)(267,361)(268,362)(269,363)(270,364)(271,365)(272,366)(273,367)(274,368)(275,369)(276,370)(277,371)(278,372)(279,373)(280,374)(281,375)(282,376)(283,377)(284,378)(285,379)(286,380)(287,381)(288,382)(289,383)(290,384)(291,385)(292,386)(293,387)(294,388)(295,389)(296,390)(297,391)(298,392)(299,393)(300,394), (1,393)(2,394)(3,395)(4,396)(5,397)(6,398)(7,399)(8,400)(9,401)(10,402)(11,403)(12,404)(13,405)(14,406)(15,407)(16,408)(17,409)(18,410)(19,411)(20,412)(21,413)(22,414)(23,415)(24,416)(25,417)(26,418)(27,419)(28,420)(29,361)(30,362)(31,363)(32,364)(33,365)(34,366)(35,367)(36,368)(37,369)(38,370)(39,371)(40,372)(41,373)(42,374)(43,375)(44,376)(45,377)(46,378)(47,379)(48,380)(49,381)(50,382)(51,383)(52,384)(53,385)(54,386)(55,387)(56,388)(57,389)(58,390)(59,391)(60,392)(61,225)(62,226)(63,227)(64,228)(65,229)(66,230)(67,231)(68,232)(69,233)(70,234)(71,235)(72,236)(73,237)(74,238)(75,239)(76,240)(77,181)(78,182)(79,183)(80,184)(81,185)(82,186)(83,187)(84,188)(85,189)(86,190)(87,191)(88,192)(89,193)(90,194)(91,195)(92,196)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208)(105,209)(106,210)(107,211)(108,212)(109,213)(110,214)(111,215)(112,216)(113,217)(114,218)(115,219)(116,220)(117,221)(118,222)(119,223)(120,224)(121,351)(122,352)(123,353)(124,354)(125,355)(126,356)(127,357)(128,358)(129,359)(130,360)(131,301)(132,302)(133,303)(134,304)(135,305)(136,306)(137,307)(138,308)(139,309)(140,310)(141,311)(142,312)(143,313)(144,314)(145,315)(146,316)(147,317)(148,318)(149,319)(150,320)(151,321)(152,322)(153,323)(154,324)(155,325)(156,326)(157,327)(158,328)(159,329)(160,330)(161,331)(162,332)(163,333)(164,334)(165,335)(166,336)(167,337)(168,338)(169,339)(170,340)(171,341)(172,342)(173,343)(174,344)(175,345)(176,346)(177,347)(178,348)(179,349)(180,350)(241,467)(242,468)(243,469)(244,470)(245,471)(246,472)(247,473)(248,474)(249,475)(250,476)(251,477)(252,478)(253,479)(254,480)(255,421)(256,422)(257,423)(258,424)(259,425)(260,426)(261,427)(262,428)(263,429)(264,430)(265,431)(266,432)(267,433)(268,434)(269,435)(270,436)(271,437)(272,438)(273,439)(274,440)(275,441)(276,442)(277,443)(278,444)(279,445)(280,446)(281,447)(282,448)(283,449)(284,450)(285,451)(286,452)(287,453)(288,454)(289,455)(290,456)(291,457)(292,458)(293,459)(294,460)(295,461)(296,462)(297,463)(298,464)(299,465)(300,466), (1,225)(2,226)(3,227)(4,228)(5,229)(6,230)(7,231)(8,232)(9,233)(10,234)(11,235)(12,236)(13,237)(14,238)(15,239)(16,240)(17,181)(18,182)(19,183)(20,184)(21,185)(22,186)(23,187)(24,188)(25,189)(26,190)(27,191)(28,192)(29,193)(30,194)(31,195)(32,196)(33,197)(34,198)(35,199)(36,200)(37,201)(38,202)(39,203)(40,204)(41,205)(42,206)(43,207)(44,208)(45,209)(46,210)(47,211)(48,212)(49,213)(50,214)(51,215)(52,216)(53,217)(54,218)(55,219)(56,220)(57,221)(58,222)(59,223)(60,224)(61,393)(62,394)(63,395)(64,396)(65,397)(66,398)(67,399)(68,400)(69,401)(70,402)(71,403)(72,404)(73,405)(74,406)(75,407)(76,408)(77,409)(78,410)(79,411)(80,412)(81,413)(82,414)(83,415)(84,416)(85,417)(86,418)(87,419)(88,420)(89,361)(90,362)(91,363)(92,364)(93,365)(94,366)(95,367)(96,368)(97,369)(98,370)(99,371)(100,372)(101,373)(102,374)(103,375)(104,376)(105,377)(106,378)(107,379)(108,380)(109,381)(110,382)(111,383)(112,384)(113,385)(114,386)(115,387)(116,388)(117,389)(118,390)(119,391)(120,392)(121,274)(122,275)(123,276)(124,277)(125,278)(126,279)(127,280)(128,281)(129,282)(130,283)(131,284)(132,285)(133,286)(134,287)(135,288)(136,289)(137,290)(138,291)(139,292)(140,293)(141,294)(142,295)(143,296)(144,297)(145,298)(146,299)(147,300)(148,241)(149,242)(150,243)(151,244)(152,245)(153,246)(154,247)(155,248)(156,249)(157,250)(158,251)(159,252)(160,253)(161,254)(162,255)(163,256)(164,257)(165,258)(166,259)(167,260)(168,261)(169,262)(170,263)(171,264)(172,265)(173,266)(174,267)(175,268)(176,269)(177,270)(178,271)(179,272)(180,273)(301,450)(302,451)(303,452)(304,453)(305,454)(306,455)(307,456)(308,457)(309,458)(310,459)(311,460)(312,461)(313,462)(314,463)(315,464)(316,465)(317,466)(318,467)(319,468)(320,469)(321,470)(322,471)(323,472)(324,473)(325,474)(326,475)(327,476)(328,477)(329,478)(330,479)(331,480)(332,421)(333,422)(334,423)(335,424)(336,425)(337,426)(338,427)(339,428)(340,429)(341,430)(342,431)(343,432)(344,433)(345,434)(346,435)(347,436)(348,437)(349,438)(350,439)(351,440)(352,441)(353,442)(354,443)(355,444)(356,445)(357,446)(358,447)(359,448)(360,449), (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( (1,465)(2,466)(3,467)(4,468)(5,469)(6,470)(7,471)(8,472)(9,473)(10,474)(11,475)(12,476)(13,477)(14,478)(15,479)(16,480)(17,421)(18,422)(19,423)(20,424)(21,425)(22,426)(23,427)(24,428)(25,429)(26,430)(27,431)(28,432)(29,433)(30,434)(31,435)(32,436)(33,437)(34,438)(35,439)(36,440)(37,441)(38,442)(39,443)(40,444)(41,445)(42,446)(43,447)(44,448)(45,449)(46,450)(47,451)(48,452)(49,453)(50,454)(51,455)(52,456)(53,457)(54,458)(55,459)(56,460)(57,461)(58,462)(59,463)(60,464)(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)(181,332)(182,333)(183,334)(184,335)(185,336)(186,337)(187,338)(188,339)(189,340)(190,341)(191,342)(192,343)(193,344)(194,345)(195,346)(196,347)(197,348)(198,349)(199,350)(200,351)(201,352)(202,353)(203,354)(204,355)(205,356)(206,357)(207,358)(208,359)(209,360)(210,301)(211,302)(212,303)(213,304)(214,305)(215,306)(216,307)(217,308)(218,309)(219,310)(220,311)(221,312)(222,313)(223,314)(224,315)(225,316)(226,317)(227,318)(228,319)(229,320)(230,321)(231,322)(232,323)(233,324)(234,325)(235,326)(236,327)(237,328)(238,329)(239,330)(240,331)(241,395)(242,396)(243,397)(244,398)(245,399)(246,400)(247,401)(248,402)(249,403)(250,404)(251,405)(252,406)(253,407)(254,408)(255,409)(256,410)(257,411)(258,412)(259,413)(260,414)(261,415)(262,416)(263,417)(264,418)(265,419)(266,420)(267,361)(268,362)(269,363)(270,364)(271,365)(272,366)(273,367)(274,368)(275,369)(276,370)(277,371)(278,372)(279,373)(280,374)(281,375)(282,376)(283,377)(284,378)(285,379)(286,380)(287,381)(288,382)(289,383)(290,384)(291,385)(292,386)(293,387)(294,388)(295,389)(296,390)(297,391)(298,392)(299,393)(300,394), (1,393)(2,394)(3,395)(4,396)(5,397)(6,398)(7,399)(8,400)(9,401)(10,402)(11,403)(12,404)(13,405)(14,406)(15,407)(16,408)(17,409)(18,410)(19,411)(20,412)(21,413)(22,414)(23,415)(24,416)(25,417)(26,418)(27,419)(28,420)(29,361)(30,362)(31,363)(32,364)(33,365)(34,366)(35,367)(36,368)(37,369)(38,370)(39,371)(40,372)(41,373)(42,374)(43,375)(44,376)(45,377)(46,378)(47,379)(48,380)(49,381)(50,382)(51,383)(52,384)(53,385)(54,386)(55,387)(56,388)(57,389)(58,390)(59,391)(60,392)(61,225)(62,226)(63,227)(64,228)(65,229)(66,230)(67,231)(68,232)(69,233)(70,234)(71,235)(72,236)(73,237)(74,238)(75,239)(76,240)(77,181)(78,182)(79,183)(80,184)(81,185)(82,186)(83,187)(84,188)(85,189)(86,190)(87,191)(88,192)(89,193)(90,194)(91,195)(92,196)(93,197)(94,198)(95,199)(96,200)(97,201)(98,202)(99,203)(100,204)(101,205)(102,206)(103,207)(104,208)(105,209)(106,210)(107,211)(108,212)(109,213)(110,214)(111,215)(112,216)(113,217)(114,218)(115,219)(116,220)(117,221)(118,222)(119,223)(120,224)(121,351)(122,352)(123,353)(124,354)(125,355)(126,356)(127,357)(128,358)(129,359)(130,360)(131,301)(132,302)(133,303)(134,304)(135,305)(136,306)(137,307)(138,308)(139,309)(140,310)(141,311)(142,312)(143,313)(144,314)(145,315)(146,316)(147,317)(148,318)(149,319)(150,320)(151,321)(152,322)(153,323)(154,324)(155,325)(156,326)(157,327)(158,328)(159,329)(160,330)(161,331)(162,332)(163,333)(164,334)(165,335)(166,336)(167,337)(168,338)(169,339)(170,340)(171,341)(172,342)(173,343)(174,344)(175,345)(176,346)(177,347)(178,348)(179,349)(180,350)(241,467)(242,468)(243,469)(244,470)(245,471)(246,472)(247,473)(248,474)(249,475)(250,476)(251,477)(252,478)(253,479)(254,480)(255,421)(256,422)(257,423)(258,424)(259,425)(260,426)(261,427)(262,428)(263,429)(264,430)(265,431)(266,432)(267,433)(268,434)(269,435)(270,436)(271,437)(272,438)(273,439)(274,440)(275,441)(276,442)(277,443)(278,444)(279,445)(280,446)(281,447)(282,448)(283,449)(284,450)(285,451)(286,452)(287,453)(288,454)(289,455)(290,456)(291,457)(292,458)(293,459)(294,460)(295,461)(296,462)(297,463)(298,464)(299,465)(300,466), (1,225)(2,226)(3,227)(4,228)(5,229)(6,230)(7,231)(8,232)(9,233)(10,234)(11,235)(12,236)(13,237)(14,238)(15,239)(16,240)(17,181)(18,182)(19,183)(20,184)(21,185)(22,186)(23,187)(24,188)(25,189)(26,190)(27,191)(28,192)(29,193)(30,194)(31,195)(32,196)(33,197)(34,198)(35,199)(36,200)(37,201)(38,202)(39,203)(40,204)(41,205)(42,206)(43,207)(44,208)(45,209)(46,210)(47,211)(48,212)(49,213)(50,214)(51,215)(52,216)(53,217)(54,218)(55,219)(56,220)(57,221)(58,222)(59,223)(60,224)(61,393)(62,394)(63,395)(64,396)(65,397)(66,398)(67,399)(68,400)(69,401)(70,402)(71,403)(72,404)(73,405)(74,406)(75,407)(76,408)(77,409)(78,410)(79,411)(80,412)(81,413)(82,414)(83,415)(84,416)(85,417)(86,418)(87,419)(88,420)(89,361)(90,362)(91,363)(92,364)(93,365)(94,366)(95,367)(96,368)(97,369)(98,370)(99,371)(100,372)(101,373)(102,374)(103,375)(104,376)(105,377)(106,378)(107,379)(108,380)(109,381)(110,382)(111,383)(112,384)(113,385)(114,386)(115,387)(116,388)(117,389)(118,390)(119,391)(120,392)(121,274)(122,275)(123,276)(124,277)(125,278)(126,279)(127,280)(128,281)(129,282)(130,283)(131,284)(132,285)(133,286)(134,287)(135,288)(136,289)(137,290)(138,291)(139,292)(140,293)(141,294)(142,295)(143,296)(144,297)(145,298)(146,299)(147,300)(148,241)(149,242)(150,243)(151,244)(152,245)(153,246)(154,247)(155,248)(156,249)(157,250)(158,251)(159,252)(160,253)(161,254)(162,255)(163,256)(164,257)(165,258)(166,259)(167,260)(168,261)(169,262)(170,263)(171,264)(172,265)(173,266)(174,267)(175,268)(176,269)(177,270)(178,271)(179,272)(180,273)(301,450)(302,451)(303,452)(304,453)(305,454)(306,455)(307,456)(308,457)(309,458)(310,459)(311,460)(312,461)(313,462)(314,463)(315,464)(316,465)(317,466)(318,467)(319,468)(320,469)(321,470)(322,471)(323,472)(324,473)(325,474)(326,475)(327,476)(328,477)(329,478)(330,479)(331,480)(332,421)(333,422)(334,423)(335,424)(336,425)(337,426)(338,427)(339,428)(340,429)(341,430)(342,431)(343,432)(344,433)(345,434)(346,435)(347,436)(348,437)(349,438)(350,439)(351,440)(352,441)(353,442)(354,443)(355,444)(356,445)(357,446)(358,447)(359,448)(360,449), (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([(1,465),(2,466),(3,467),(4,468),(5,469),(6,470),(7,471),(8,472),(9,473),(10,474),(11,475),(12,476),(13,477),(14,478),(15,479),(16,480),(17,421),(18,422),(19,423),(20,424),(21,425),(22,426),(23,427),(24,428),(25,429),(26,430),(27,431),(28,432),(29,433),(30,434),(31,435),(32,436),(33,437),(34,438),(35,439),(36,440),(37,441),(38,442),(39,443),(40,444),(41,445),(42,446),(43,447),(44,448),(45,449),(46,450),(47,451),(48,452),(49,453),(50,454),(51,455),(52,456),(53,457),(54,458),(55,459),(56,460),(57,461),(58,462),(59,463),(60,464),(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),(181,332),(182,333),(183,334),(184,335),(185,336),(186,337),(187,338),(188,339),(189,340),(190,341),(191,342),(192,343),(193,344),(194,345),(195,346),(196,347),(197,348),(198,349),(199,350),(200,351),(201,352),(202,353),(203,354),(204,355),(205,356),(206,357),(207,358),(208,359),(209,360),(210,301),(211,302),(212,303),(213,304),(214,305),(215,306),(216,307),(217,308),(218,309),(219,310),(220,311),(221,312),(222,313),(223,314),(224,315),(225,316),(226,317),(227,318),(228,319),(229,320),(230,321),(231,322),(232,323),(233,324),(234,325),(235,326),(236,327),(237,328),(238,329),(239,330),(240,331),(241,395),(242,396),(243,397),(244,398),(245,399),(246,400),(247,401),(248,402),(249,403),(250,404),(251,405),(252,406),(253,407),(254,408),(255,409),(256,410),(257,411),(258,412),(259,413),(260,414),(261,415),(262,416),(263,417),(264,418),(265,419),(266,420),(267,361),(268,362),(269,363),(270,364),(271,365),(272,366),(273,367),(274,368),(275,369),(276,370),(277,371),(278,372),(279,373),(280,374),(281,375),(282,376),(283,377),(284,378),(285,379),(286,380),(287,381),(288,382),(289,383),(290,384),(291,385),(292,386),(293,387),(294,388),(295,389),(296,390),(297,391),(298,392),(299,393),(300,394)], [(1,393),(2,394),(3,395),(4,396),(5,397),(6,398),(7,399),(8,400),(9,401),(10,402),(11,403),(12,404),(13,405),(14,406),(15,407),(16,408),(17,409),(18,410),(19,411),(20,412),(21,413),(22,414),(23,415),(24,416),(25,417),(26,418),(27,419),(28,420),(29,361),(30,362),(31,363),(32,364),(33,365),(34,366),(35,367),(36,368),(37,369),(38,370),(39,371),(40,372),(41,373),(42,374),(43,375),(44,376),(45,377),(46,378),(47,379),(48,380),(49,381),(50,382),(51,383),(52,384),(53,385),(54,386),(55,387),(56,388),(57,389),(58,390),(59,391),(60,392),(61,225),(62,226),(63,227),(64,228),(65,229),(66,230),(67,231),(68,232),(69,233),(70,234),(71,235),(72,236),(73,237),(74,238),(75,239),(76,240),(77,181),(78,182),(79,183),(80,184),(81,185),(82,186),(83,187),(84,188),(85,189),(86,190),(87,191),(88,192),(89,193),(90,194),(91,195),(92,196),(93,197),(94,198),(95,199),(96,200),(97,201),(98,202),(99,203),(100,204),(101,205),(102,206),(103,207),(104,208),(105,209),(106,210),(107,211),(108,212),(109,213),(110,214),(111,215),(112,216),(113,217),(114,218),(115,219),(116,220),(117,221),(118,222),(119,223),(120,224),(121,351),(122,352),(123,353),(124,354),(125,355),(126,356),(127,357),(128,358),(129,359),(130,360),(131,301),(132,302),(133,303),(134,304),(135,305),(136,306),(137,307),(138,308),(139,309),(140,310),(141,311),(142,312),(143,313),(144,314),(145,315),(146,316),(147,317),(148,318),(149,319),(150,320),(151,321),(152,322),(153,323),(154,324),(155,325),(156,326),(157,327),(158,328),(159,329),(160,330),(161,331),(162,332),(163,333),(164,334),(165,335),(166,336),(167,337),(168,338),(169,339),(170,340),(171,341),(172,342),(173,343),(174,344),(175,345),(176,346),(177,347),(178,348),(179,349),(180,350),(241,467),(242,468),(243,469),(244,470),(245,471),(246,472),(247,473),(248,474),(249,475),(250,476),(251,477),(252,478),(253,479),(254,480),(255,421),(256,422),(257,423),(258,424),(259,425),(260,426),(261,427),(262,428),(263,429),(264,430),(265,431),(266,432),(267,433),(268,434),(269,435),(270,436),(271,437),(272,438),(273,439),(274,440),(275,441),(276,442),(277,443),(278,444),(279,445),(280,446),(281,447),(282,448),(283,449),(284,450),(285,451),(286,452),(287,453),(288,454),(289,455),(290,456),(291,457),(292,458),(293,459),(294,460),(295,461),(296,462),(297,463),(298,464),(299,465),(300,466)], [(1,225),(2,226),(3,227),(4,228),(5,229),(6,230),(7,231),(8,232),(9,233),(10,234),(11,235),(12,236),(13,237),(14,238),(15,239),(16,240),(17,181),(18,182),(19,183),(20,184),(21,185),(22,186),(23,187),(24,188),(25,189),(26,190),(27,191),(28,192),(29,193),(30,194),(31,195),(32,196),(33,197),(34,198),(35,199),(36,200),(37,201),(38,202),(39,203),(40,204),(41,205),(42,206),(43,207),(44,208),(45,209),(46,210),(47,211),(48,212),(49,213),(50,214),(51,215),(52,216),(53,217),(54,218),(55,219),(56,220),(57,221),(58,222),(59,223),(60,224),(61,393),(62,394),(63,395),(64,396),(65,397),(66,398),(67,399),(68,400),(69,401),(70,402),(71,403),(72,404),(73,405),(74,406),(75,407),(76,408),(77,409),(78,410),(79,411),(80,412),(81,413),(82,414),(83,415),(84,416),(85,417),(86,418),(87,419),(88,420),(89,361),(90,362),(91,363),(92,364),(93,365),(94,366),(95,367),(96,368),(97,369),(98,370),(99,371),(100,372),(101,373),(102,374),(103,375),(104,376),(105,377),(106,378),(107,379),(108,380),(109,381),(110,382),(111,383),(112,384),(113,385),(114,386),(115,387),(116,388),(117,389),(118,390),(119,391),(120,392),(121,274),(122,275),(123,276),(124,277),(125,278),(126,279),(127,280),(128,281),(129,282),(130,283),(131,284),(132,285),(133,286),(134,287),(135,288),(136,289),(137,290),(138,291),(139,292),(140,293),(141,294),(142,295),(143,296),(144,297),(145,298),(146,299),(147,300),(148,241),(149,242),(150,243),(151,244),(152,245),(153,246),(154,247),(155,248),(156,249),(157,250),(158,251),(159,252),(160,253),(161,254),(162,255),(163,256),(164,257),(165,258),(166,259),(167,260),(168,261),(169,262),(170,263),(171,264),(172,265),(173,266),(174,267),(175,268),(176,269),(177,270),(178,271),(179,272),(180,273),(301,450),(302,451),(303,452),(304,453),(305,454),(306,455),(307,456),(308,457),(309,458),(310,459),(311,460),(312,461),(313,462),(314,463),(315,464),(316,465),(317,466),(318,467),(319,468),(320,469),(321,470),(322,471),(323,472),(324,473),(325,474),(326,475),(327,476),(328,477),(329,478),(330,479),(331,480),(332,421),(333,422),(334,423),(335,424),(336,425),(337,426),(338,427),(339,428),(340,429),(341,430),(342,431),(343,432),(344,433),(345,434),(346,435),(347,436),(348,437),(349,438),(350,439),(351,440),(352,441),(353,442),(354,443),(355,444),(356,445),(357,446),(358,447),(359,448),(360,449)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
○
≀