metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.5Q8, C30.21D8, C6.12D40, C30.9Q16, C6.5Dic20, C12.4Dic10, C20.11Dic6, C3⋊C8⋊2Dic5, C3⋊1(C40⋊5C4), C15⋊6(C2.D8), C4.2(C15⋊Q8), C60.97(C2×C4), (C2×C30).34D4, (C2×C6).35D20, C5⋊2(C6.Q16), C4⋊Dic5.3S3, C30.30(C4⋊C4), C20.101(C4×S3), C10.7(D4⋊S3), (C2×C12).62D10, (C2×C20).284D6, C2.3(C3⋊D40), C6.3(C4⋊Dic5), C60⋊5C4.17C2, C4.12(S3×Dic5), C12.5(C2×Dic5), C10.3(C3⋊Q16), C2.3(C3⋊Dic20), (C2×C60).103C22, C2.4(C6.Dic10), C10.13(Dic3⋊C4), C22.17(C3⋊D20), (C5×C3⋊C8)⋊8C4, (C2×C3⋊C8).4D5, (C10×C3⋊C8).5C2, (C2×C4).89(S3×D5), (C3×C4⋊Dic5).3C2, (C2×C10).28(C3⋊D4), SmallGroup(480,66)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.5Q8
G = < a,b,c | a60=1, b4=a30, c2=a45b2, bab-1=a41, cac-1=a19, cbc-1=a30b3 >
Subgroups: 332 in 72 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C30, C2.D8, C40, C2×Dic5, C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C3×Dic5, Dic15, C60, C2×C30, C4⋊Dic5, C4⋊Dic5, C2×C40, C6.Q16, C5×C3⋊C8, C6×Dic5, C2×Dic15, C2×C60, C40⋊5C4, C3×C4⋊Dic5, C10×C3⋊C8, C60⋊5C4, C60.5Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, D8, Q16, Dic5, D10, Dic6, C4×S3, C3⋊D4, C2.D8, Dic10, D20, C2×Dic5, Dic3⋊C4, D4⋊S3, C3⋊Q16, S3×D5, D40, Dic20, C4⋊Dic5, C6.Q16, S3×Dic5, C3⋊D20, C15⋊Q8, C40⋊5C4, C3⋊D40, C3⋊Dic20, C6.Dic10, C60.5Q8
►Regular action on 480 points
Generators in S480
(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)
(1 162 104 370 31 132 74 400)(2 143 105 411 32 173 75 381)(3 124 106 392 33 154 76 362)(4 165 107 373 34 135 77 403)(5 146 108 414 35 176 78 384)(6 127 109 395 36 157 79 365)(7 168 110 376 37 138 80 406)(8 149 111 417 38 179 81 387)(9 130 112 398 39 160 82 368)(10 171 113 379 40 141 83 409)(11 152 114 420 41 122 84 390)(12 133 115 401 42 163 85 371)(13 174 116 382 43 144 86 412)(14 155 117 363 44 125 87 393)(15 136 118 404 45 166 88 374)(16 177 119 385 46 147 89 415)(17 158 120 366 47 128 90 396)(18 139 61 407 48 169 91 377)(19 180 62 388 49 150 92 418)(20 161 63 369 50 131 93 399)(21 142 64 410 51 172 94 380)(22 123 65 391 52 153 95 361)(23 164 66 372 53 134 96 402)(24 145 67 413 54 175 97 383)(25 126 68 394 55 156 98 364)(26 167 69 375 56 137 99 405)(27 148 70 416 57 178 100 386)(28 129 71 397 58 159 101 367)(29 170 72 378 59 140 102 408)(30 151 73 419 60 121 103 389)(181 340 288 463 211 310 258 433)(182 321 289 444 212 351 259 474)(183 302 290 425 213 332 260 455)(184 343 291 466 214 313 261 436)(185 324 292 447 215 354 262 477)(186 305 293 428 216 335 263 458)(187 346 294 469 217 316 264 439)(188 327 295 450 218 357 265 480)(189 308 296 431 219 338 266 461)(190 349 297 472 220 319 267 442)(191 330 298 453 221 360 268 423)(192 311 299 434 222 341 269 464)(193 352 300 475 223 322 270 445)(194 333 241 456 224 303 271 426)(195 314 242 437 225 344 272 467)(196 355 243 478 226 325 273 448)(197 336 244 459 227 306 274 429)(198 317 245 440 228 347 275 470)(199 358 246 421 229 328 276 451)(200 339 247 462 230 309 277 432)(201 320 248 443 231 350 278 473)(202 301 249 424 232 331 279 454)(203 342 250 465 233 312 280 435)(204 323 251 446 234 353 281 476)(205 304 252 427 235 334 282 457)(206 345 253 468 236 315 283 438)(207 326 254 449 237 356 284 479)(208 307 255 430 238 337 285 460)(209 348 256 471 239 318 286 441)(210 329 257 452 240 359 287 422)
(1 252 89 220)(2 271 90 239)(3 290 91 198)(4 249 92 217)(5 268 93 236)(6 287 94 195)(7 246 95 214)(8 265 96 233)(9 284 97 192)(10 243 98 211)(11 262 99 230)(12 281 100 189)(13 300 101 208)(14 259 102 227)(15 278 103 186)(16 297 104 205)(17 256 105 224)(18 275 106 183)(19 294 107 202)(20 253 108 221)(21 272 109 240)(22 291 110 199)(23 250 111 218)(24 269 112 237)(25 288 113 196)(26 247 114 215)(27 266 115 234)(28 285 116 193)(29 244 117 212)(30 263 118 231)(31 282 119 190)(32 241 120 209)(33 260 61 228)(34 279 62 187)(35 298 63 206)(36 257 64 225)(37 276 65 184)(38 295 66 203)(39 254 67 222)(40 273 68 181)(41 292 69 200)(42 251 70 219)(43 270 71 238)(44 289 72 197)(45 248 73 216)(46 267 74 235)(47 286 75 194)(48 245 76 213)(49 264 77 232)(50 283 78 191)(51 242 79 210)(52 261 80 229)(53 280 81 188)(54 299 82 207)(55 258 83 226)(56 277 84 185)(57 296 85 204)(58 255 86 223)(59 274 87 182)(60 293 88 201)(121 305 374 473)(122 324 375 432)(123 343 376 451)(124 302 377 470)(125 321 378 429)(126 340 379 448)(127 359 380 467)(128 318 381 426)(129 337 382 445)(130 356 383 464)(131 315 384 423)(132 334 385 442)(133 353 386 461)(134 312 387 480)(135 331 388 439)(136 350 389 458)(137 309 390 477)(138 328 391 436)(139 347 392 455)(140 306 393 474)(141 325 394 433)(142 344 395 452)(143 303 396 471)(144 322 397 430)(145 341 398 449)(146 360 399 468)(147 319 400 427)(148 338 401 446)(149 357 402 465)(150 316 403 424)(151 335 404 443)(152 354 405 462)(153 313 406 421)(154 332 407 440)(155 351 408 459)(156 310 409 478)(157 329 410 437)(158 348 411 456)(159 307 412 475)(160 326 413 434)(161 345 414 453)(162 304 415 472)(163 323 416 431)(164 342 417 450)(165 301 418 469)(166 320 419 428)(167 339 420 447)(168 358 361 466)(169 317 362 425)(170 336 363 444)(171 355 364 463)(172 314 365 422)(173 333 366 441)(174 352 367 460)(175 311 368 479)(176 330 369 438)(177 349 370 457)(178 308 371 476)(179 327 372 435)(180 346 373 454)
G:=sub<Sym(480)| (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), (1,162,104,370,31,132,74,400)(2,143,105,411,32,173,75,381)(3,124,106,392,33,154,76,362)(4,165,107,373,34,135,77,403)(5,146,108,414,35,176,78,384)(6,127,109,395,36,157,79,365)(7,168,110,376,37,138,80,406)(8,149,111,417,38,179,81,387)(9,130,112,398,39,160,82,368)(10,171,113,379,40,141,83,409)(11,152,114,420,41,122,84,390)(12,133,115,401,42,163,85,371)(13,174,116,382,43,144,86,412)(14,155,117,363,44,125,87,393)(15,136,118,404,45,166,88,374)(16,177,119,385,46,147,89,415)(17,158,120,366,47,128,90,396)(18,139,61,407,48,169,91,377)(19,180,62,388,49,150,92,418)(20,161,63,369,50,131,93,399)(21,142,64,410,51,172,94,380)(22,123,65,391,52,153,95,361)(23,164,66,372,53,134,96,402)(24,145,67,413,54,175,97,383)(25,126,68,394,55,156,98,364)(26,167,69,375,56,137,99,405)(27,148,70,416,57,178,100,386)(28,129,71,397,58,159,101,367)(29,170,72,378,59,140,102,408)(30,151,73,419,60,121,103,389)(181,340,288,463,211,310,258,433)(182,321,289,444,212,351,259,474)(183,302,290,425,213,332,260,455)(184,343,291,466,214,313,261,436)(185,324,292,447,215,354,262,477)(186,305,293,428,216,335,263,458)(187,346,294,469,217,316,264,439)(188,327,295,450,218,357,265,480)(189,308,296,431,219,338,266,461)(190,349,297,472,220,319,267,442)(191,330,298,453,221,360,268,423)(192,311,299,434,222,341,269,464)(193,352,300,475,223,322,270,445)(194,333,241,456,224,303,271,426)(195,314,242,437,225,344,272,467)(196,355,243,478,226,325,273,448)(197,336,244,459,227,306,274,429)(198,317,245,440,228,347,275,470)(199,358,246,421,229,328,276,451)(200,339,247,462,230,309,277,432)(201,320,248,443,231,350,278,473)(202,301,249,424,232,331,279,454)(203,342,250,465,233,312,280,435)(204,323,251,446,234,353,281,476)(205,304,252,427,235,334,282,457)(206,345,253,468,236,315,283,438)(207,326,254,449,237,356,284,479)(208,307,255,430,238,337,285,460)(209,348,256,471,239,318,286,441)(210,329,257,452,240,359,287,422), (1,252,89,220)(2,271,90,239)(3,290,91,198)(4,249,92,217)(5,268,93,236)(6,287,94,195)(7,246,95,214)(8,265,96,233)(9,284,97,192)(10,243,98,211)(11,262,99,230)(12,281,100,189)(13,300,101,208)(14,259,102,227)(15,278,103,186)(16,297,104,205)(17,256,105,224)(18,275,106,183)(19,294,107,202)(20,253,108,221)(21,272,109,240)(22,291,110,199)(23,250,111,218)(24,269,112,237)(25,288,113,196)(26,247,114,215)(27,266,115,234)(28,285,116,193)(29,244,117,212)(30,263,118,231)(31,282,119,190)(32,241,120,209)(33,260,61,228)(34,279,62,187)(35,298,63,206)(36,257,64,225)(37,276,65,184)(38,295,66,203)(39,254,67,222)(40,273,68,181)(41,292,69,200)(42,251,70,219)(43,270,71,238)(44,289,72,197)(45,248,73,216)(46,267,74,235)(47,286,75,194)(48,245,76,213)(49,264,77,232)(50,283,78,191)(51,242,79,210)(52,261,80,229)(53,280,81,188)(54,299,82,207)(55,258,83,226)(56,277,84,185)(57,296,85,204)(58,255,86,223)(59,274,87,182)(60,293,88,201)(121,305,374,473)(122,324,375,432)(123,343,376,451)(124,302,377,470)(125,321,378,429)(126,340,379,448)(127,359,380,467)(128,318,381,426)(129,337,382,445)(130,356,383,464)(131,315,384,423)(132,334,385,442)(133,353,386,461)(134,312,387,480)(135,331,388,439)(136,350,389,458)(137,309,390,477)(138,328,391,436)(139,347,392,455)(140,306,393,474)(141,325,394,433)(142,344,395,452)(143,303,396,471)(144,322,397,430)(145,341,398,449)(146,360,399,468)(147,319,400,427)(148,338,401,446)(149,357,402,465)(150,316,403,424)(151,335,404,443)(152,354,405,462)(153,313,406,421)(154,332,407,440)(155,351,408,459)(156,310,409,478)(157,329,410,437)(158,348,411,456)(159,307,412,475)(160,326,413,434)(161,345,414,453)(162,304,415,472)(163,323,416,431)(164,342,417,450)(165,301,418,469)(166,320,419,428)(167,339,420,447)(168,358,361,466)(169,317,362,425)(170,336,363,444)(171,355,364,463)(172,314,365,422)(173,333,366,441)(174,352,367,460)(175,311,368,479)(176,330,369,438)(177,349,370,457)(178,308,371,476)(179,327,372,435)(180,346,373,454)>;
G:=Group( (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), (1,162,104,370,31,132,74,400)(2,143,105,411,32,173,75,381)(3,124,106,392,33,154,76,362)(4,165,107,373,34,135,77,403)(5,146,108,414,35,176,78,384)(6,127,109,395,36,157,79,365)(7,168,110,376,37,138,80,406)(8,149,111,417,38,179,81,387)(9,130,112,398,39,160,82,368)(10,171,113,379,40,141,83,409)(11,152,114,420,41,122,84,390)(12,133,115,401,42,163,85,371)(13,174,116,382,43,144,86,412)(14,155,117,363,44,125,87,393)(15,136,118,404,45,166,88,374)(16,177,119,385,46,147,89,415)(17,158,120,366,47,128,90,396)(18,139,61,407,48,169,91,377)(19,180,62,388,49,150,92,418)(20,161,63,369,50,131,93,399)(21,142,64,410,51,172,94,380)(22,123,65,391,52,153,95,361)(23,164,66,372,53,134,96,402)(24,145,67,413,54,175,97,383)(25,126,68,394,55,156,98,364)(26,167,69,375,56,137,99,405)(27,148,70,416,57,178,100,386)(28,129,71,397,58,159,101,367)(29,170,72,378,59,140,102,408)(30,151,73,419,60,121,103,389)(181,340,288,463,211,310,258,433)(182,321,289,444,212,351,259,474)(183,302,290,425,213,332,260,455)(184,343,291,466,214,313,261,436)(185,324,292,447,215,354,262,477)(186,305,293,428,216,335,263,458)(187,346,294,469,217,316,264,439)(188,327,295,450,218,357,265,480)(189,308,296,431,219,338,266,461)(190,349,297,472,220,319,267,442)(191,330,298,453,221,360,268,423)(192,311,299,434,222,341,269,464)(193,352,300,475,223,322,270,445)(194,333,241,456,224,303,271,426)(195,314,242,437,225,344,272,467)(196,355,243,478,226,325,273,448)(197,336,244,459,227,306,274,429)(198,317,245,440,228,347,275,470)(199,358,246,421,229,328,276,451)(200,339,247,462,230,309,277,432)(201,320,248,443,231,350,278,473)(202,301,249,424,232,331,279,454)(203,342,250,465,233,312,280,435)(204,323,251,446,234,353,281,476)(205,304,252,427,235,334,282,457)(206,345,253,468,236,315,283,438)(207,326,254,449,237,356,284,479)(208,307,255,430,238,337,285,460)(209,348,256,471,239,318,286,441)(210,329,257,452,240,359,287,422), (1,252,89,220)(2,271,90,239)(3,290,91,198)(4,249,92,217)(5,268,93,236)(6,287,94,195)(7,246,95,214)(8,265,96,233)(9,284,97,192)(10,243,98,211)(11,262,99,230)(12,281,100,189)(13,300,101,208)(14,259,102,227)(15,278,103,186)(16,297,104,205)(17,256,105,224)(18,275,106,183)(19,294,107,202)(20,253,108,221)(21,272,109,240)(22,291,110,199)(23,250,111,218)(24,269,112,237)(25,288,113,196)(26,247,114,215)(27,266,115,234)(28,285,116,193)(29,244,117,212)(30,263,118,231)(31,282,119,190)(32,241,120,209)(33,260,61,228)(34,279,62,187)(35,298,63,206)(36,257,64,225)(37,276,65,184)(38,295,66,203)(39,254,67,222)(40,273,68,181)(41,292,69,200)(42,251,70,219)(43,270,71,238)(44,289,72,197)(45,248,73,216)(46,267,74,235)(47,286,75,194)(48,245,76,213)(49,264,77,232)(50,283,78,191)(51,242,79,210)(52,261,80,229)(53,280,81,188)(54,299,82,207)(55,258,83,226)(56,277,84,185)(57,296,85,204)(58,255,86,223)(59,274,87,182)(60,293,88,201)(121,305,374,473)(122,324,375,432)(123,343,376,451)(124,302,377,470)(125,321,378,429)(126,340,379,448)(127,359,380,467)(128,318,381,426)(129,337,382,445)(130,356,383,464)(131,315,384,423)(132,334,385,442)(133,353,386,461)(134,312,387,480)(135,331,388,439)(136,350,389,458)(137,309,390,477)(138,328,391,436)(139,347,392,455)(140,306,393,474)(141,325,394,433)(142,344,395,452)(143,303,396,471)(144,322,397,430)(145,341,398,449)(146,360,399,468)(147,319,400,427)(148,338,401,446)(149,357,402,465)(150,316,403,424)(151,335,404,443)(152,354,405,462)(153,313,406,421)(154,332,407,440)(155,351,408,459)(156,310,409,478)(157,329,410,437)(158,348,411,456)(159,307,412,475)(160,326,413,434)(161,345,414,453)(162,304,415,472)(163,323,416,431)(164,342,417,450)(165,301,418,469)(166,320,419,428)(167,339,420,447)(168,358,361,466)(169,317,362,425)(170,336,363,444)(171,355,364,463)(172,314,365,422)(173,333,366,441)(174,352,367,460)(175,311,368,479)(176,330,369,438)(177,349,370,457)(178,308,371,476)(179,327,372,435)(180,346,373,454) );
G=PermutationGroup([[(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)], [(1,162,104,370,31,132,74,400),(2,143,105,411,32,173,75,381),(3,124,106,392,33,154,76,362),(4,165,107,373,34,135,77,403),(5,146,108,414,35,176,78,384),(6,127,109,395,36,157,79,365),(7,168,110,376,37,138,80,406),(8,149,111,417,38,179,81,387),(9,130,112,398,39,160,82,368),(10,171,113,379,40,141,83,409),(11,152,114,420,41,122,84,390),(12,133,115,401,42,163,85,371),(13,174,116,382,43,144,86,412),(14,155,117,363,44,125,87,393),(15,136,118,404,45,166,88,374),(16,177,119,385,46,147,89,415),(17,158,120,366,47,128,90,396),(18,139,61,407,48,169,91,377),(19,180,62,388,49,150,92,418),(20,161,63,369,50,131,93,399),(21,142,64,410,51,172,94,380),(22,123,65,391,52,153,95,361),(23,164,66,372,53,134,96,402),(24,145,67,413,54,175,97,383),(25,126,68,394,55,156,98,364),(26,167,69,375,56,137,99,405),(27,148,70,416,57,178,100,386),(28,129,71,397,58,159,101,367),(29,170,72,378,59,140,102,408),(30,151,73,419,60,121,103,389),(181,340,288,463,211,310,258,433),(182,321,289,444,212,351,259,474),(183,302,290,425,213,332,260,455),(184,343,291,466,214,313,261,436),(185,324,292,447,215,354,262,477),(186,305,293,428,216,335,263,458),(187,346,294,469,217,316,264,439),(188,327,295,450,218,357,265,480),(189,308,296,431,219,338,266,461),(190,349,297,472,220,319,267,442),(191,330,298,453,221,360,268,423),(192,311,299,434,222,341,269,464),(193,352,300,475,223,322,270,445),(194,333,241,456,224,303,271,426),(195,314,242,437,225,344,272,467),(196,355,243,478,226,325,273,448),(197,336,244,459,227,306,274,429),(198,317,245,440,228,347,275,470),(199,358,246,421,229,328,276,451),(200,339,247,462,230,309,277,432),(201,320,248,443,231,350,278,473),(202,301,249,424,232,331,279,454),(203,342,250,465,233,312,280,435),(204,323,251,446,234,353,281,476),(205,304,252,427,235,334,282,457),(206,345,253,468,236,315,283,438),(207,326,254,449,237,356,284,479),(208,307,255,430,238,337,285,460),(209,348,256,471,239,318,286,441),(210,329,257,452,240,359,287,422)], [(1,252,89,220),(2,271,90,239),(3,290,91,198),(4,249,92,217),(5,268,93,236),(6,287,94,195),(7,246,95,214),(8,265,96,233),(9,284,97,192),(10,243,98,211),(11,262,99,230),(12,281,100,189),(13,300,101,208),(14,259,102,227),(15,278,103,186),(16,297,104,205),(17,256,105,224),(18,275,106,183),(19,294,107,202),(20,253,108,221),(21,272,109,240),(22,291,110,199),(23,250,111,218),(24,269,112,237),(25,288,113,196),(26,247,114,215),(27,266,115,234),(28,285,116,193),(29,244,117,212),(30,263,118,231),(31,282,119,190),(32,241,120,209),(33,260,61,228),(34,279,62,187),(35,298,63,206),(36,257,64,225),(37,276,65,184),(38,295,66,203),(39,254,67,222),(40,273,68,181),(41,292,69,200),(42,251,70,219),(43,270,71,238),(44,289,72,197),(45,248,73,216),(46,267,74,235),(47,286,75,194),(48,245,76,213),(49,264,77,232),(50,283,78,191),(51,242,79,210),(52,261,80,229),(53,280,81,188),(54,299,82,207),(55,258,83,226),(56,277,84,185),(57,296,85,204),(58,255,86,223),(59,274,87,182),(60,293,88,201),(121,305,374,473),(122,324,375,432),(123,343,376,451),(124,302,377,470),(125,321,378,429),(126,340,379,448),(127,359,380,467),(128,318,381,426),(129,337,382,445),(130,356,383,464),(131,315,384,423),(132,334,385,442),(133,353,386,461),(134,312,387,480),(135,331,388,439),(136,350,389,458),(137,309,390,477),(138,328,391,436),(139,347,392,455),(140,306,393,474),(141,325,394,433),(142,344,395,452),(143,303,396,471),(144,322,397,430),(145,341,398,449),(146,360,399,468),(147,319,400,427),(148,338,401,446),(149,357,402,465),(150,316,403,424),(151,335,404,443),(152,354,405,462),(153,313,406,421),(154,332,407,440),(155,351,408,459),(156,310,409,478),(157,329,410,437),(158,348,411,456),(159,307,412,475),(160,326,413,434),(161,345,414,453),(162,304,415,472),(163,323,416,431),(164,342,417,450),(165,301,418,469),(166,320,419,428),(167,339,420,447),(168,358,361,466),(169,317,362,425),(170,336,363,444),(171,355,364,463),(172,314,365,422),(173,333,366,441),(174,352,367,460),(175,311,368,479),(176,330,369,438),(177,349,370,457),(178,308,371,476),(179,327,372,435),(180,346,373,454)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | - | + | - | - | + | + | - | + | - | + | - | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D5 | D6 | D8 | Q16 | Dic5 | D10 | Dic6 | C4×S3 | C3⋊D4 | Dic10 | D20 | D40 | Dic20 | D4⋊S3 | C3⋊Q16 | S3×D5 | S3×Dic5 | C15⋊Q8 | C3⋊D20 | C3⋊D40 | C3⋊Dic20 |
| kernel | C60.5Q8 | C3×C4⋊Dic5 | C10×C3⋊C8 | C60⋊5C4 | C5×C3⋊C8 | C4⋊Dic5 | C60 | C2×C30 | C2×C3⋊C8 | C2×C20 | C30 | C30 | C3⋊C8 | C2×C12 | C20 | C20 | C2×C10 | C12 | C2×C6 | C6 | C6 | C10 | C10 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C60.5Q8 ►in GL6(𝔽241)
| 44 | 78 | 0 | 0 | 0 | 0 |
| 163 | 163 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 189 | 0 | 0 |
| 0 | 0 | 52 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 13 | 208 | 0 | 0 | 0 | 0 |
| 33 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 111 | 20 |
| 0 | 0 | 0 | 0 | 131 | 130 |
| 166 | 181 | 0 | 0 | 0 | 0 |
| 150 | 75 | 0 | 0 | 0 | 0 |
| 0 | 0 | 128 | 144 | 0 | 0 |
| 0 | 0 | 236 | 113 | 0 | 0 |
| 0 | 0 | 0 | 0 | 171 | 140 |
| 0 | 0 | 0 | 0 | 101 | 70 |
G:=sub<GL(6,GF(241))| [44,163,0,0,0,0,78,163,0,0,0,0,0,0,189,52,0,0,0,0,189,1,0,0,0,0,0,0,1,1,0,0,0,0,240,0],[13,33,0,0,0,0,208,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,111,131,0,0,0,0,20,130],[166,150,0,0,0,0,181,75,0,0,0,0,0,0,128,236,0,0,0,0,144,113,0,0,0,0,0,0,171,101,0,0,0,0,140,70] >;
C60.5Q8 in GAP, Magma, Sage, TeX
C_{60}._5Q_8 % in TeX
G:=Group("C60.5Q8"); // GroupNames label
G:=SmallGroup(480,66);
// by ID
G=gap.SmallGroup(480,66);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45*b^2,b*a*b^-1=a^41,c*a*c^-1=a^19,c*b*c^-1=a^30*b^3>;
// generators/relations