metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.6Q8, C12.7Dic10, C20.12Dic6, C4.3(C15⋊Q8), C4⋊Dic5.6S3, C30.52(C2×Q8), (C2×C20).296D6, (C4×Dic3).3D5, C60⋊5C4.19C2, C5⋊1(C4.Dic6), C3⋊2(C20.6Q8), C15⋊9(C42.C2), C30.45(C4○D4), C6.51(C4○D20), (C2×C12).118D10, (C2×C30).71C23, (C2×Dic5).22D6, (Dic3×C20).3C2, C10.20(C2×Dic6), C6.19(C2×Dic10), (C2×C60).115C22, C30.Q8.13C2, Dic15⋊5C4.12C2, C2.12(D20⋊5S3), C2.14(D60⋊C2), C10.24(D4⋊2S3), C10.11(Q8⋊3S3), (C2×Dic3).146D10, (C6×Dic5).41C22, (C2×Dic15).65C22, (C10×Dic3).170C22, C2.5(C2×C15⋊Q8), (C2×C4).106(S3×D5), (C3×C4⋊Dic5).5C2, C22.157(C2×S3×D5), (C2×C6).83(C22×D5), (C2×C10).83(C22×S3), SmallGroup(480,457)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.6Q8
G = < a,b,c | a60=b4=1, c2=a30b2, bab-1=a41, cac-1=a19, cbc-1=a30b-1 >
Subgroups: 460 in 112 conjugacy classes, 52 normal (34 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, Dic5, C20, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C10.D4, C4⋊Dic5, C4⋊Dic5, C4×C20, C4.Dic6, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C20.6Q8, C30.Q8, Dic15⋊5C4, C3×C4⋊Dic5, Dic3×C20, C60⋊5C4, C60.6Q8
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, Dic6, C22×S3, C42.C2, Dic10, C22×D5, C2×Dic6, D4⋊2S3, Q8⋊3S3, S3×D5, C2×Dic10, C4○D20, C4.Dic6, C15⋊Q8, C2×S3×D5, C20.6Q8, D20⋊5S3, D60⋊C2, C2×C15⋊Q8, C60.6Q8
►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)
(1 442 258 83)(2 423 259 64)(3 464 260 105)(4 445 261 86)(5 426 262 67)(6 467 263 108)(7 448 264 89)(8 429 265 70)(9 470 266 111)(10 451 267 92)(11 432 268 73)(12 473 269 114)(13 454 270 95)(14 435 271 76)(15 476 272 117)(16 457 273 98)(17 438 274 79)(18 479 275 120)(19 460 276 101)(20 441 277 82)(21 422 278 63)(22 463 279 104)(23 444 280 85)(24 425 281 66)(25 466 282 107)(26 447 283 88)(27 428 284 69)(28 469 285 110)(29 450 286 91)(30 431 287 72)(31 472 288 113)(32 453 289 94)(33 434 290 75)(34 475 291 116)(35 456 292 97)(36 437 293 78)(37 478 294 119)(38 459 295 100)(39 440 296 81)(40 421 297 62)(41 462 298 103)(42 443 299 84)(43 424 300 65)(44 465 241 106)(45 446 242 87)(46 427 243 68)(47 468 244 109)(48 449 245 90)(49 430 246 71)(50 471 247 112)(51 452 248 93)(52 433 249 74)(53 474 250 115)(54 455 251 96)(55 436 252 77)(56 477 253 118)(57 458 254 99)(58 439 255 80)(59 480 256 61)(60 461 257 102)(121 355 404 203)(122 336 405 184)(123 317 406 225)(124 358 407 206)(125 339 408 187)(126 320 409 228)(127 301 410 209)(128 342 411 190)(129 323 412 231)(130 304 413 212)(131 345 414 193)(132 326 415 234)(133 307 416 215)(134 348 417 196)(135 329 418 237)(136 310 419 218)(137 351 420 199)(138 332 361 240)(139 313 362 221)(140 354 363 202)(141 335 364 183)(142 316 365 224)(143 357 366 205)(144 338 367 186)(145 319 368 227)(146 360 369 208)(147 341 370 189)(148 322 371 230)(149 303 372 211)(150 344 373 192)(151 325 374 233)(152 306 375 214)(153 347 376 195)(154 328 377 236)(155 309 378 217)(156 350 379 198)(157 331 380 239)(158 312 381 220)(159 353 382 201)(160 334 383 182)(161 315 384 223)(162 356 385 204)(163 337 386 185)(164 318 387 226)(165 359 388 207)(166 340 389 188)(167 321 390 229)(168 302 391 210)(169 343 392 191)(170 324 393 232)(171 305 394 213)(172 346 395 194)(173 327 396 235)(174 308 397 216)(175 349 398 197)(176 330 399 238)(177 311 400 219)(178 352 401 200)(179 333 402 181)(180 314 403 222)
(1 150 288 403)(2 169 289 362)(3 128 290 381)(4 147 291 400)(5 166 292 419)(6 125 293 378)(7 144 294 397)(8 163 295 416)(9 122 296 375)(10 141 297 394)(11 160 298 413)(12 179 299 372)(13 138 300 391)(14 157 241 410)(15 176 242 369)(16 135 243 388)(17 154 244 407)(18 173 245 366)(19 132 246 385)(20 151 247 404)(21 170 248 363)(22 129 249 382)(23 148 250 401)(24 167 251 420)(25 126 252 379)(26 145 253 398)(27 164 254 417)(28 123 255 376)(29 142 256 395)(30 161 257 414)(31 180 258 373)(32 139 259 392)(33 158 260 411)(34 177 261 370)(35 136 262 389)(36 155 263 408)(37 174 264 367)(38 133 265 386)(39 152 266 405)(40 171 267 364)(41 130 268 383)(42 149 269 402)(43 168 270 361)(44 127 271 380)(45 146 272 399)(46 165 273 418)(47 124 274 377)(48 143 275 396)(49 162 276 415)(50 121 277 374)(51 140 278 393)(52 159 279 412)(53 178 280 371)(54 137 281 390)(55 156 282 409)(56 175 283 368)(57 134 284 387)(58 153 285 406)(59 172 286 365)(60 131 287 384)(61 316 450 194)(62 335 451 213)(63 354 452 232)(64 313 453 191)(65 332 454 210)(66 351 455 229)(67 310 456 188)(68 329 457 207)(69 348 458 226)(70 307 459 185)(71 326 460 204)(72 345 461 223)(73 304 462 182)(74 323 463 201)(75 342 464 220)(76 301 465 239)(77 320 466 198)(78 339 467 217)(79 358 468 236)(80 317 469 195)(81 336 470 214)(82 355 471 233)(83 314 472 192)(84 333 473 211)(85 352 474 230)(86 311 475 189)(87 330 476 208)(88 349 477 227)(89 308 478 186)(90 327 479 205)(91 346 480 224)(92 305 421 183)(93 324 422 202)(94 343 423 221)(95 302 424 240)(96 321 425 199)(97 340 426 218)(98 359 427 237)(99 318 428 196)(100 337 429 215)(101 356 430 234)(102 315 431 193)(103 334 432 212)(104 353 433 231)(105 312 434 190)(106 331 435 209)(107 350 436 228)(108 309 437 187)(109 328 438 206)(110 347 439 225)(111 306 440 184)(112 325 441 203)(113 344 442 222)(114 303 443 181)(115 322 444 200)(116 341 445 219)(117 360 446 238)(118 319 447 197)(119 338 448 216)(120 357 449 235)
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), (1,442,258,83)(2,423,259,64)(3,464,260,105)(4,445,261,86)(5,426,262,67)(6,467,263,108)(7,448,264,89)(8,429,265,70)(9,470,266,111)(10,451,267,92)(11,432,268,73)(12,473,269,114)(13,454,270,95)(14,435,271,76)(15,476,272,117)(16,457,273,98)(17,438,274,79)(18,479,275,120)(19,460,276,101)(20,441,277,82)(21,422,278,63)(22,463,279,104)(23,444,280,85)(24,425,281,66)(25,466,282,107)(26,447,283,88)(27,428,284,69)(28,469,285,110)(29,450,286,91)(30,431,287,72)(31,472,288,113)(32,453,289,94)(33,434,290,75)(34,475,291,116)(35,456,292,97)(36,437,293,78)(37,478,294,119)(38,459,295,100)(39,440,296,81)(40,421,297,62)(41,462,298,103)(42,443,299,84)(43,424,300,65)(44,465,241,106)(45,446,242,87)(46,427,243,68)(47,468,244,109)(48,449,245,90)(49,430,246,71)(50,471,247,112)(51,452,248,93)(52,433,249,74)(53,474,250,115)(54,455,251,96)(55,436,252,77)(56,477,253,118)(57,458,254,99)(58,439,255,80)(59,480,256,61)(60,461,257,102)(121,355,404,203)(122,336,405,184)(123,317,406,225)(124,358,407,206)(125,339,408,187)(126,320,409,228)(127,301,410,209)(128,342,411,190)(129,323,412,231)(130,304,413,212)(131,345,414,193)(132,326,415,234)(133,307,416,215)(134,348,417,196)(135,329,418,237)(136,310,419,218)(137,351,420,199)(138,332,361,240)(139,313,362,221)(140,354,363,202)(141,335,364,183)(142,316,365,224)(143,357,366,205)(144,338,367,186)(145,319,368,227)(146,360,369,208)(147,341,370,189)(148,322,371,230)(149,303,372,211)(150,344,373,192)(151,325,374,233)(152,306,375,214)(153,347,376,195)(154,328,377,236)(155,309,378,217)(156,350,379,198)(157,331,380,239)(158,312,381,220)(159,353,382,201)(160,334,383,182)(161,315,384,223)(162,356,385,204)(163,337,386,185)(164,318,387,226)(165,359,388,207)(166,340,389,188)(167,321,390,229)(168,302,391,210)(169,343,392,191)(170,324,393,232)(171,305,394,213)(172,346,395,194)(173,327,396,235)(174,308,397,216)(175,349,398,197)(176,330,399,238)(177,311,400,219)(178,352,401,200)(179,333,402,181)(180,314,403,222), (1,150,288,403)(2,169,289,362)(3,128,290,381)(4,147,291,400)(5,166,292,419)(6,125,293,378)(7,144,294,397)(8,163,295,416)(9,122,296,375)(10,141,297,394)(11,160,298,413)(12,179,299,372)(13,138,300,391)(14,157,241,410)(15,176,242,369)(16,135,243,388)(17,154,244,407)(18,173,245,366)(19,132,246,385)(20,151,247,404)(21,170,248,363)(22,129,249,382)(23,148,250,401)(24,167,251,420)(25,126,252,379)(26,145,253,398)(27,164,254,417)(28,123,255,376)(29,142,256,395)(30,161,257,414)(31,180,258,373)(32,139,259,392)(33,158,260,411)(34,177,261,370)(35,136,262,389)(36,155,263,408)(37,174,264,367)(38,133,265,386)(39,152,266,405)(40,171,267,364)(41,130,268,383)(42,149,269,402)(43,168,270,361)(44,127,271,380)(45,146,272,399)(46,165,273,418)(47,124,274,377)(48,143,275,396)(49,162,276,415)(50,121,277,374)(51,140,278,393)(52,159,279,412)(53,178,280,371)(54,137,281,390)(55,156,282,409)(56,175,283,368)(57,134,284,387)(58,153,285,406)(59,172,286,365)(60,131,287,384)(61,316,450,194)(62,335,451,213)(63,354,452,232)(64,313,453,191)(65,332,454,210)(66,351,455,229)(67,310,456,188)(68,329,457,207)(69,348,458,226)(70,307,459,185)(71,326,460,204)(72,345,461,223)(73,304,462,182)(74,323,463,201)(75,342,464,220)(76,301,465,239)(77,320,466,198)(78,339,467,217)(79,358,468,236)(80,317,469,195)(81,336,470,214)(82,355,471,233)(83,314,472,192)(84,333,473,211)(85,352,474,230)(86,311,475,189)(87,330,476,208)(88,349,477,227)(89,308,478,186)(90,327,479,205)(91,346,480,224)(92,305,421,183)(93,324,422,202)(94,343,423,221)(95,302,424,240)(96,321,425,199)(97,340,426,218)(98,359,427,237)(99,318,428,196)(100,337,429,215)(101,356,430,234)(102,315,431,193)(103,334,432,212)(104,353,433,231)(105,312,434,190)(106,331,435,209)(107,350,436,228)(108,309,437,187)(109,328,438,206)(110,347,439,225)(111,306,440,184)(112,325,441,203)(113,344,442,222)(114,303,443,181)(115,322,444,200)(116,341,445,219)(117,360,446,238)(118,319,447,197)(119,338,448,216)(120,357,449,235)>;
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), (1,442,258,83)(2,423,259,64)(3,464,260,105)(4,445,261,86)(5,426,262,67)(6,467,263,108)(7,448,264,89)(8,429,265,70)(9,470,266,111)(10,451,267,92)(11,432,268,73)(12,473,269,114)(13,454,270,95)(14,435,271,76)(15,476,272,117)(16,457,273,98)(17,438,274,79)(18,479,275,120)(19,460,276,101)(20,441,277,82)(21,422,278,63)(22,463,279,104)(23,444,280,85)(24,425,281,66)(25,466,282,107)(26,447,283,88)(27,428,284,69)(28,469,285,110)(29,450,286,91)(30,431,287,72)(31,472,288,113)(32,453,289,94)(33,434,290,75)(34,475,291,116)(35,456,292,97)(36,437,293,78)(37,478,294,119)(38,459,295,100)(39,440,296,81)(40,421,297,62)(41,462,298,103)(42,443,299,84)(43,424,300,65)(44,465,241,106)(45,446,242,87)(46,427,243,68)(47,468,244,109)(48,449,245,90)(49,430,246,71)(50,471,247,112)(51,452,248,93)(52,433,249,74)(53,474,250,115)(54,455,251,96)(55,436,252,77)(56,477,253,118)(57,458,254,99)(58,439,255,80)(59,480,256,61)(60,461,257,102)(121,355,404,203)(122,336,405,184)(123,317,406,225)(124,358,407,206)(125,339,408,187)(126,320,409,228)(127,301,410,209)(128,342,411,190)(129,323,412,231)(130,304,413,212)(131,345,414,193)(132,326,415,234)(133,307,416,215)(134,348,417,196)(135,329,418,237)(136,310,419,218)(137,351,420,199)(138,332,361,240)(139,313,362,221)(140,354,363,202)(141,335,364,183)(142,316,365,224)(143,357,366,205)(144,338,367,186)(145,319,368,227)(146,360,369,208)(147,341,370,189)(148,322,371,230)(149,303,372,211)(150,344,373,192)(151,325,374,233)(152,306,375,214)(153,347,376,195)(154,328,377,236)(155,309,378,217)(156,350,379,198)(157,331,380,239)(158,312,381,220)(159,353,382,201)(160,334,383,182)(161,315,384,223)(162,356,385,204)(163,337,386,185)(164,318,387,226)(165,359,388,207)(166,340,389,188)(167,321,390,229)(168,302,391,210)(169,343,392,191)(170,324,393,232)(171,305,394,213)(172,346,395,194)(173,327,396,235)(174,308,397,216)(175,349,398,197)(176,330,399,238)(177,311,400,219)(178,352,401,200)(179,333,402,181)(180,314,403,222), (1,150,288,403)(2,169,289,362)(3,128,290,381)(4,147,291,400)(5,166,292,419)(6,125,293,378)(7,144,294,397)(8,163,295,416)(9,122,296,375)(10,141,297,394)(11,160,298,413)(12,179,299,372)(13,138,300,391)(14,157,241,410)(15,176,242,369)(16,135,243,388)(17,154,244,407)(18,173,245,366)(19,132,246,385)(20,151,247,404)(21,170,248,363)(22,129,249,382)(23,148,250,401)(24,167,251,420)(25,126,252,379)(26,145,253,398)(27,164,254,417)(28,123,255,376)(29,142,256,395)(30,161,257,414)(31,180,258,373)(32,139,259,392)(33,158,260,411)(34,177,261,370)(35,136,262,389)(36,155,263,408)(37,174,264,367)(38,133,265,386)(39,152,266,405)(40,171,267,364)(41,130,268,383)(42,149,269,402)(43,168,270,361)(44,127,271,380)(45,146,272,399)(46,165,273,418)(47,124,274,377)(48,143,275,396)(49,162,276,415)(50,121,277,374)(51,140,278,393)(52,159,279,412)(53,178,280,371)(54,137,281,390)(55,156,282,409)(56,175,283,368)(57,134,284,387)(58,153,285,406)(59,172,286,365)(60,131,287,384)(61,316,450,194)(62,335,451,213)(63,354,452,232)(64,313,453,191)(65,332,454,210)(66,351,455,229)(67,310,456,188)(68,329,457,207)(69,348,458,226)(70,307,459,185)(71,326,460,204)(72,345,461,223)(73,304,462,182)(74,323,463,201)(75,342,464,220)(76,301,465,239)(77,320,466,198)(78,339,467,217)(79,358,468,236)(80,317,469,195)(81,336,470,214)(82,355,471,233)(83,314,472,192)(84,333,473,211)(85,352,474,230)(86,311,475,189)(87,330,476,208)(88,349,477,227)(89,308,478,186)(90,327,479,205)(91,346,480,224)(92,305,421,183)(93,324,422,202)(94,343,423,221)(95,302,424,240)(96,321,425,199)(97,340,426,218)(98,359,427,237)(99,318,428,196)(100,337,429,215)(101,356,430,234)(102,315,431,193)(103,334,432,212)(104,353,433,231)(105,312,434,190)(106,331,435,209)(107,350,436,228)(108,309,437,187)(109,328,438,206)(110,347,439,225)(111,306,440,184)(112,325,441,203)(113,344,442,222)(114,303,443,181)(115,322,444,200)(116,341,445,219)(117,360,446,238)(118,319,447,197)(119,338,448,216)(120,357,449,235) );
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)], [(1,442,258,83),(2,423,259,64),(3,464,260,105),(4,445,261,86),(5,426,262,67),(6,467,263,108),(7,448,264,89),(8,429,265,70),(9,470,266,111),(10,451,267,92),(11,432,268,73),(12,473,269,114),(13,454,270,95),(14,435,271,76),(15,476,272,117),(16,457,273,98),(17,438,274,79),(18,479,275,120),(19,460,276,101),(20,441,277,82),(21,422,278,63),(22,463,279,104),(23,444,280,85),(24,425,281,66),(25,466,282,107),(26,447,283,88),(27,428,284,69),(28,469,285,110),(29,450,286,91),(30,431,287,72),(31,472,288,113),(32,453,289,94),(33,434,290,75),(34,475,291,116),(35,456,292,97),(36,437,293,78),(37,478,294,119),(38,459,295,100),(39,440,296,81),(40,421,297,62),(41,462,298,103),(42,443,299,84),(43,424,300,65),(44,465,241,106),(45,446,242,87),(46,427,243,68),(47,468,244,109),(48,449,245,90),(49,430,246,71),(50,471,247,112),(51,452,248,93),(52,433,249,74),(53,474,250,115),(54,455,251,96),(55,436,252,77),(56,477,253,118),(57,458,254,99),(58,439,255,80),(59,480,256,61),(60,461,257,102),(121,355,404,203),(122,336,405,184),(123,317,406,225),(124,358,407,206),(125,339,408,187),(126,320,409,228),(127,301,410,209),(128,342,411,190),(129,323,412,231),(130,304,413,212),(131,345,414,193),(132,326,415,234),(133,307,416,215),(134,348,417,196),(135,329,418,237),(136,310,419,218),(137,351,420,199),(138,332,361,240),(139,313,362,221),(140,354,363,202),(141,335,364,183),(142,316,365,224),(143,357,366,205),(144,338,367,186),(145,319,368,227),(146,360,369,208),(147,341,370,189),(148,322,371,230),(149,303,372,211),(150,344,373,192),(151,325,374,233),(152,306,375,214),(153,347,376,195),(154,328,377,236),(155,309,378,217),(156,350,379,198),(157,331,380,239),(158,312,381,220),(159,353,382,201),(160,334,383,182),(161,315,384,223),(162,356,385,204),(163,337,386,185),(164,318,387,226),(165,359,388,207),(166,340,389,188),(167,321,390,229),(168,302,391,210),(169,343,392,191),(170,324,393,232),(171,305,394,213),(172,346,395,194),(173,327,396,235),(174,308,397,216),(175,349,398,197),(176,330,399,238),(177,311,400,219),(178,352,401,200),(179,333,402,181),(180,314,403,222)], [(1,150,288,403),(2,169,289,362),(3,128,290,381),(4,147,291,400),(5,166,292,419),(6,125,293,378),(7,144,294,397),(8,163,295,416),(9,122,296,375),(10,141,297,394),(11,160,298,413),(12,179,299,372),(13,138,300,391),(14,157,241,410),(15,176,242,369),(16,135,243,388),(17,154,244,407),(18,173,245,366),(19,132,246,385),(20,151,247,404),(21,170,248,363),(22,129,249,382),(23,148,250,401),(24,167,251,420),(25,126,252,379),(26,145,253,398),(27,164,254,417),(28,123,255,376),(29,142,256,395),(30,161,257,414),(31,180,258,373),(32,139,259,392),(33,158,260,411),(34,177,261,370),(35,136,262,389),(36,155,263,408),(37,174,264,367),(38,133,265,386),(39,152,266,405),(40,171,267,364),(41,130,268,383),(42,149,269,402),(43,168,270,361),(44,127,271,380),(45,146,272,399),(46,165,273,418),(47,124,274,377),(48,143,275,396),(49,162,276,415),(50,121,277,374),(51,140,278,393),(52,159,279,412),(53,178,280,371),(54,137,281,390),(55,156,282,409),(56,175,283,368),(57,134,284,387),(58,153,285,406),(59,172,286,365),(60,131,287,384),(61,316,450,194),(62,335,451,213),(63,354,452,232),(64,313,453,191),(65,332,454,210),(66,351,455,229),(67,310,456,188),(68,329,457,207),(69,348,458,226),(70,307,459,185),(71,326,460,204),(72,345,461,223),(73,304,462,182),(74,323,463,201),(75,342,464,220),(76,301,465,239),(77,320,466,198),(78,339,467,217),(79,358,468,236),(80,317,469,195),(81,336,470,214),(82,355,471,233),(83,314,472,192),(84,333,473,211),(85,352,474,230),(86,311,475,189),(87,330,476,208),(88,349,477,227),(89,308,478,186),(90,327,479,205),(91,346,480,224),(92,305,421,183),(93,324,422,202),(94,343,423,221),(95,302,424,240),(96,321,425,199),(97,340,426,218),(98,359,427,237),(99,318,428,196),(100,337,429,215),(101,356,430,234),(102,315,431,193),(103,334,432,212),(104,353,433,231),(105,312,434,190),(106,331,435,209),(107,350,436,228),(108,309,437,187),(109,328,438,206),(110,347,439,225),(111,306,440,184),(112,325,441,203),(113,344,442,222),(114,303,443,181),(115,322,444,200),(116,341,445,219),(117,360,446,238),(118,319,447,197),(119,338,448,216),(120,357,449,235)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 20 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | + | + | - | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | Dic6 | Dic10 | C4○D20 | D4⋊2S3 | Q8⋊3S3 | S3×D5 | C15⋊Q8 | C2×S3×D5 | D20⋊5S3 | D60⋊C2 |
| kernel | C60.6Q8 | C30.Q8 | Dic15⋊5C4 | C3×C4⋊Dic5 | Dic3×C20 | C60⋊5C4 | C4⋊Dic5 | C60 | C4×Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C20 | C12 | C6 | C10 | C10 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 16 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C60.6Q8 ►in GL4(𝔽61) generated by
| 60 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 7 | 27 |
| 0 | 0 | 5 | 2 |
| 9 | 9 | 0 | 0 |
| 18 | 52 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 38 | 46 | 0 | 0 |
| 15 | 23 | 0 | 0 |
| 0 | 0 | 20 | 39 |
| 0 | 0 | 32 | 41 |
G:=sub<GL(4,GF(61))| [60,60,0,0,1,0,0,0,0,0,7,5,0,0,27,2],[9,18,0,0,9,52,0,0,0,0,50,0,0,0,0,50],[38,15,0,0,46,23,0,0,0,0,20,32,0,0,39,41] >;
C60.6Q8 in GAP, Magma, Sage, TeX
C_{60}._6Q_8 % in TeX
G:=Group("C60.6Q8"); // GroupNames label
G:=SmallGroup(480,457);
// by ID
G=gap.SmallGroup(480,457);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,176,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^30*b^2,b*a*b^-1=a^41,c*a*c^-1=a^19,c*b*c^-1=a^30*b^-1>;
// generators/relations