metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.1Q8, C30.37D8, C30.16Q16, C20.5Dic6, C4.1Dic30, C12.5Dic10, C15⋊3C8⋊7C4, C4⋊C4.1D15, C15⋊7(C2.D8), C20.45(C4×S3), C60.75(C2×C4), C12.13(C4×D5), (C2×C4).32D30, (C2×C20).64D6, C4.11(C4×D15), C5⋊4(C6.Q16), C30.36(C4⋊C4), C6.15(D4⋊D5), C2.1(D4⋊D15), (C2×C12).65D10, (C2×C30).135D4, C3⋊3(C10.D8), C60⋊5C4.10C2, C6.7(C5⋊Q16), C10.15(D4⋊S3), (C2×C60).50C22, C2.1(C15⋊7Q16), C10.7(C3⋊Q16), C2.3(C30.4Q8), C10.17(Dic3⋊C4), C6.10(C10.D4), C22.12(C15⋊7D4), (C5×C4⋊C4).1S3, (C3×C4⋊C4).1D5, (C15×C4⋊C4).1C2, (C2×C15⋊3C8).1C2, (C2×C6).67(C5⋊D4), (C2×C10).67(C3⋊D4), SmallGroup(480,167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.1Q8
G = < a,b,c | a60=b4=1, c2=a45b2, bab-1=a31, cac-1=a29, cbc-1=a45b-1 >
Subgroups: 324 in 72 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C4⋊C4, C2×C8, Dic5, C20, C20, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C30, C2.D8, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, Dic15, C60, C60, C2×C30, C2×C5⋊2C8, C4⋊Dic5, C5×C4⋊C4, C6.Q16, C15⋊3C8, C2×Dic15, C2×C60, C2×C60, C10.D8, C2×C15⋊3C8, C60⋊5C4, C15×C4⋊C4, C60.1Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, D8, Q16, D10, Dic6, C4×S3, C3⋊D4, D15, C2.D8, Dic10, C4×D5, C5⋊D4, Dic3⋊C4, D4⋊S3, C3⋊Q16, D30, C10.D4, D4⋊D5, C5⋊Q16, C6.Q16, Dic30, C4×D15, C15⋊7D4, C10.D8, C30.4Q8, D4⋊D15, C15⋊7Q16, C60.1Q8
►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)
(1 386 230 281)(2 417 231 252)(3 388 232 283)(4 419 233 254)(5 390 234 285)(6 361 235 256)(7 392 236 287)(8 363 237 258)(9 394 238 289)(10 365 239 260)(11 396 240 291)(12 367 181 262)(13 398 182 293)(14 369 183 264)(15 400 184 295)(16 371 185 266)(17 402 186 297)(18 373 187 268)(19 404 188 299)(20 375 189 270)(21 406 190 241)(22 377 191 272)(23 408 192 243)(24 379 193 274)(25 410 194 245)(26 381 195 276)(27 412 196 247)(28 383 197 278)(29 414 198 249)(30 385 199 280)(31 416 200 251)(32 387 201 282)(33 418 202 253)(34 389 203 284)(35 420 204 255)(36 391 205 286)(37 362 206 257)(38 393 207 288)(39 364 208 259)(40 395 209 290)(41 366 210 261)(42 397 211 292)(43 368 212 263)(44 399 213 294)(45 370 214 265)(46 401 215 296)(47 372 216 267)(48 403 217 298)(49 374 218 269)(50 405 219 300)(51 376 220 271)(52 407 221 242)(53 378 222 273)(54 409 223 244)(55 380 224 275)(56 411 225 246)(57 382 226 277)(58 413 227 248)(59 384 228 279)(60 415 229 250)(61 435 312 146)(62 466 313 177)(63 437 314 148)(64 468 315 179)(65 439 316 150)(66 470 317 121)(67 441 318 152)(68 472 319 123)(69 443 320 154)(70 474 321 125)(71 445 322 156)(72 476 323 127)(73 447 324 158)(74 478 325 129)(75 449 326 160)(76 480 327 131)(77 451 328 162)(78 422 329 133)(79 453 330 164)(80 424 331 135)(81 455 332 166)(82 426 333 137)(83 457 334 168)(84 428 335 139)(85 459 336 170)(86 430 337 141)(87 461 338 172)(88 432 339 143)(89 463 340 174)(90 434 341 145)(91 465 342 176)(92 436 343 147)(93 467 344 178)(94 438 345 149)(95 469 346 180)(96 440 347 151)(97 471 348 122)(98 442 349 153)(99 473 350 124)(100 444 351 155)(101 475 352 126)(102 446 353 157)(103 477 354 128)(104 448 355 159)(105 479 356 130)(106 450 357 161)(107 421 358 132)(108 452 359 163)(109 423 360 134)(110 454 301 165)(111 425 302 136)(112 456 303 167)(113 427 304 138)(114 458 305 169)(115 429 306 140)(116 460 307 171)(117 431 308 142)(118 462 309 173)(119 433 310 144)(120 464 311 175)
(1 310 215 104 31 340 185 74)(2 339 216 73 32 309 186 103)(3 308 217 102 33 338 187 72)(4 337 218 71 34 307 188 101)(5 306 219 100 35 336 189 70)(6 335 220 69 36 305 190 99)(7 304 221 98 37 334 191 68)(8 333 222 67 38 303 192 97)(9 302 223 96 39 332 193 66)(10 331 224 65 40 301 194 95)(11 360 225 94 41 330 195 64)(12 329 226 63 42 359 196 93)(13 358 227 92 43 328 197 62)(14 327 228 61 44 357 198 91)(15 356 229 90 45 326 199 120)(16 325 230 119 46 355 200 89)(17 354 231 88 47 324 201 118)(18 323 232 117 48 353 202 87)(19 352 233 86 49 322 203 116)(20 321 234 115 50 351 204 85)(21 350 235 84 51 320 205 114)(22 319 236 113 52 349 206 83)(23 348 237 82 53 318 207 112)(24 317 238 111 54 347 208 81)(25 346 239 80 55 316 209 110)(26 315 240 109 56 345 210 79)(27 344 181 78 57 314 211 108)(28 313 182 107 58 343 212 77)(29 342 183 76 59 312 213 106)(30 311 184 105 60 341 214 75)(121 379 455 259 151 409 425 289)(122 408 456 288 152 378 426 258)(123 377 457 257 153 407 427 287)(124 406 458 286 154 376 428 256)(125 375 459 255 155 405 429 285)(126 404 460 284 156 374 430 254)(127 373 461 253 157 403 431 283)(128 402 462 282 158 372 432 252)(129 371 463 251 159 401 433 281)(130 400 464 280 160 370 434 250)(131 369 465 249 161 399 435 279)(132 398 466 278 162 368 436 248)(133 367 467 247 163 397 437 277)(134 396 468 276 164 366 438 246)(135 365 469 245 165 395 439 275)(136 394 470 274 166 364 440 244)(137 363 471 243 167 393 441 273)(138 392 472 272 168 362 442 242)(139 361 473 241 169 391 443 271)(140 390 474 270 170 420 444 300)(141 419 475 299 171 389 445 269)(142 388 476 268 172 418 446 298)(143 417 477 297 173 387 447 267)(144 386 478 266 174 416 448 296)(145 415 479 295 175 385 449 265)(146 384 480 264 176 414 450 294)(147 413 421 293 177 383 451 263)(148 382 422 262 178 412 452 292)(149 411 423 291 179 381 453 261)(150 380 424 260 180 410 454 290)
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), (1,386,230,281)(2,417,231,252)(3,388,232,283)(4,419,233,254)(5,390,234,285)(6,361,235,256)(7,392,236,287)(8,363,237,258)(9,394,238,289)(10,365,239,260)(11,396,240,291)(12,367,181,262)(13,398,182,293)(14,369,183,264)(15,400,184,295)(16,371,185,266)(17,402,186,297)(18,373,187,268)(19,404,188,299)(20,375,189,270)(21,406,190,241)(22,377,191,272)(23,408,192,243)(24,379,193,274)(25,410,194,245)(26,381,195,276)(27,412,196,247)(28,383,197,278)(29,414,198,249)(30,385,199,280)(31,416,200,251)(32,387,201,282)(33,418,202,253)(34,389,203,284)(35,420,204,255)(36,391,205,286)(37,362,206,257)(38,393,207,288)(39,364,208,259)(40,395,209,290)(41,366,210,261)(42,397,211,292)(43,368,212,263)(44,399,213,294)(45,370,214,265)(46,401,215,296)(47,372,216,267)(48,403,217,298)(49,374,218,269)(50,405,219,300)(51,376,220,271)(52,407,221,242)(53,378,222,273)(54,409,223,244)(55,380,224,275)(56,411,225,246)(57,382,226,277)(58,413,227,248)(59,384,228,279)(60,415,229,250)(61,435,312,146)(62,466,313,177)(63,437,314,148)(64,468,315,179)(65,439,316,150)(66,470,317,121)(67,441,318,152)(68,472,319,123)(69,443,320,154)(70,474,321,125)(71,445,322,156)(72,476,323,127)(73,447,324,158)(74,478,325,129)(75,449,326,160)(76,480,327,131)(77,451,328,162)(78,422,329,133)(79,453,330,164)(80,424,331,135)(81,455,332,166)(82,426,333,137)(83,457,334,168)(84,428,335,139)(85,459,336,170)(86,430,337,141)(87,461,338,172)(88,432,339,143)(89,463,340,174)(90,434,341,145)(91,465,342,176)(92,436,343,147)(93,467,344,178)(94,438,345,149)(95,469,346,180)(96,440,347,151)(97,471,348,122)(98,442,349,153)(99,473,350,124)(100,444,351,155)(101,475,352,126)(102,446,353,157)(103,477,354,128)(104,448,355,159)(105,479,356,130)(106,450,357,161)(107,421,358,132)(108,452,359,163)(109,423,360,134)(110,454,301,165)(111,425,302,136)(112,456,303,167)(113,427,304,138)(114,458,305,169)(115,429,306,140)(116,460,307,171)(117,431,308,142)(118,462,309,173)(119,433,310,144)(120,464,311,175), (1,310,215,104,31,340,185,74)(2,339,216,73,32,309,186,103)(3,308,217,102,33,338,187,72)(4,337,218,71,34,307,188,101)(5,306,219,100,35,336,189,70)(6,335,220,69,36,305,190,99)(7,304,221,98,37,334,191,68)(8,333,222,67,38,303,192,97)(9,302,223,96,39,332,193,66)(10,331,224,65,40,301,194,95)(11,360,225,94,41,330,195,64)(12,329,226,63,42,359,196,93)(13,358,227,92,43,328,197,62)(14,327,228,61,44,357,198,91)(15,356,229,90,45,326,199,120)(16,325,230,119,46,355,200,89)(17,354,231,88,47,324,201,118)(18,323,232,117,48,353,202,87)(19,352,233,86,49,322,203,116)(20,321,234,115,50,351,204,85)(21,350,235,84,51,320,205,114)(22,319,236,113,52,349,206,83)(23,348,237,82,53,318,207,112)(24,317,238,111,54,347,208,81)(25,346,239,80,55,316,209,110)(26,315,240,109,56,345,210,79)(27,344,181,78,57,314,211,108)(28,313,182,107,58,343,212,77)(29,342,183,76,59,312,213,106)(30,311,184,105,60,341,214,75)(121,379,455,259,151,409,425,289)(122,408,456,288,152,378,426,258)(123,377,457,257,153,407,427,287)(124,406,458,286,154,376,428,256)(125,375,459,255,155,405,429,285)(126,404,460,284,156,374,430,254)(127,373,461,253,157,403,431,283)(128,402,462,282,158,372,432,252)(129,371,463,251,159,401,433,281)(130,400,464,280,160,370,434,250)(131,369,465,249,161,399,435,279)(132,398,466,278,162,368,436,248)(133,367,467,247,163,397,437,277)(134,396,468,276,164,366,438,246)(135,365,469,245,165,395,439,275)(136,394,470,274,166,364,440,244)(137,363,471,243,167,393,441,273)(138,392,472,272,168,362,442,242)(139,361,473,241,169,391,443,271)(140,390,474,270,170,420,444,300)(141,419,475,299,171,389,445,269)(142,388,476,268,172,418,446,298)(143,417,477,297,173,387,447,267)(144,386,478,266,174,416,448,296)(145,415,479,295,175,385,449,265)(146,384,480,264,176,414,450,294)(147,413,421,293,177,383,451,263)(148,382,422,262,178,412,452,292)(149,411,423,291,179,381,453,261)(150,380,424,260,180,410,454,290)>;
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), (1,386,230,281)(2,417,231,252)(3,388,232,283)(4,419,233,254)(5,390,234,285)(6,361,235,256)(7,392,236,287)(8,363,237,258)(9,394,238,289)(10,365,239,260)(11,396,240,291)(12,367,181,262)(13,398,182,293)(14,369,183,264)(15,400,184,295)(16,371,185,266)(17,402,186,297)(18,373,187,268)(19,404,188,299)(20,375,189,270)(21,406,190,241)(22,377,191,272)(23,408,192,243)(24,379,193,274)(25,410,194,245)(26,381,195,276)(27,412,196,247)(28,383,197,278)(29,414,198,249)(30,385,199,280)(31,416,200,251)(32,387,201,282)(33,418,202,253)(34,389,203,284)(35,420,204,255)(36,391,205,286)(37,362,206,257)(38,393,207,288)(39,364,208,259)(40,395,209,290)(41,366,210,261)(42,397,211,292)(43,368,212,263)(44,399,213,294)(45,370,214,265)(46,401,215,296)(47,372,216,267)(48,403,217,298)(49,374,218,269)(50,405,219,300)(51,376,220,271)(52,407,221,242)(53,378,222,273)(54,409,223,244)(55,380,224,275)(56,411,225,246)(57,382,226,277)(58,413,227,248)(59,384,228,279)(60,415,229,250)(61,435,312,146)(62,466,313,177)(63,437,314,148)(64,468,315,179)(65,439,316,150)(66,470,317,121)(67,441,318,152)(68,472,319,123)(69,443,320,154)(70,474,321,125)(71,445,322,156)(72,476,323,127)(73,447,324,158)(74,478,325,129)(75,449,326,160)(76,480,327,131)(77,451,328,162)(78,422,329,133)(79,453,330,164)(80,424,331,135)(81,455,332,166)(82,426,333,137)(83,457,334,168)(84,428,335,139)(85,459,336,170)(86,430,337,141)(87,461,338,172)(88,432,339,143)(89,463,340,174)(90,434,341,145)(91,465,342,176)(92,436,343,147)(93,467,344,178)(94,438,345,149)(95,469,346,180)(96,440,347,151)(97,471,348,122)(98,442,349,153)(99,473,350,124)(100,444,351,155)(101,475,352,126)(102,446,353,157)(103,477,354,128)(104,448,355,159)(105,479,356,130)(106,450,357,161)(107,421,358,132)(108,452,359,163)(109,423,360,134)(110,454,301,165)(111,425,302,136)(112,456,303,167)(113,427,304,138)(114,458,305,169)(115,429,306,140)(116,460,307,171)(117,431,308,142)(118,462,309,173)(119,433,310,144)(120,464,311,175), (1,310,215,104,31,340,185,74)(2,339,216,73,32,309,186,103)(3,308,217,102,33,338,187,72)(4,337,218,71,34,307,188,101)(5,306,219,100,35,336,189,70)(6,335,220,69,36,305,190,99)(7,304,221,98,37,334,191,68)(8,333,222,67,38,303,192,97)(9,302,223,96,39,332,193,66)(10,331,224,65,40,301,194,95)(11,360,225,94,41,330,195,64)(12,329,226,63,42,359,196,93)(13,358,227,92,43,328,197,62)(14,327,228,61,44,357,198,91)(15,356,229,90,45,326,199,120)(16,325,230,119,46,355,200,89)(17,354,231,88,47,324,201,118)(18,323,232,117,48,353,202,87)(19,352,233,86,49,322,203,116)(20,321,234,115,50,351,204,85)(21,350,235,84,51,320,205,114)(22,319,236,113,52,349,206,83)(23,348,237,82,53,318,207,112)(24,317,238,111,54,347,208,81)(25,346,239,80,55,316,209,110)(26,315,240,109,56,345,210,79)(27,344,181,78,57,314,211,108)(28,313,182,107,58,343,212,77)(29,342,183,76,59,312,213,106)(30,311,184,105,60,341,214,75)(121,379,455,259,151,409,425,289)(122,408,456,288,152,378,426,258)(123,377,457,257,153,407,427,287)(124,406,458,286,154,376,428,256)(125,375,459,255,155,405,429,285)(126,404,460,284,156,374,430,254)(127,373,461,253,157,403,431,283)(128,402,462,282,158,372,432,252)(129,371,463,251,159,401,433,281)(130,400,464,280,160,370,434,250)(131,369,465,249,161,399,435,279)(132,398,466,278,162,368,436,248)(133,367,467,247,163,397,437,277)(134,396,468,276,164,366,438,246)(135,365,469,245,165,395,439,275)(136,394,470,274,166,364,440,244)(137,363,471,243,167,393,441,273)(138,392,472,272,168,362,442,242)(139,361,473,241,169,391,443,271)(140,390,474,270,170,420,444,300)(141,419,475,299,171,389,445,269)(142,388,476,268,172,418,446,298)(143,417,477,297,173,387,447,267)(144,386,478,266,174,416,448,296)(145,415,479,295,175,385,449,265)(146,384,480,264,176,414,450,294)(147,413,421,293,177,383,451,263)(148,382,422,262,178,412,452,292)(149,411,423,291,179,381,453,261)(150,380,424,260,180,410,454,290) );
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)], [(1,386,230,281),(2,417,231,252),(3,388,232,283),(4,419,233,254),(5,390,234,285),(6,361,235,256),(7,392,236,287),(8,363,237,258),(9,394,238,289),(10,365,239,260),(11,396,240,291),(12,367,181,262),(13,398,182,293),(14,369,183,264),(15,400,184,295),(16,371,185,266),(17,402,186,297),(18,373,187,268),(19,404,188,299),(20,375,189,270),(21,406,190,241),(22,377,191,272),(23,408,192,243),(24,379,193,274),(25,410,194,245),(26,381,195,276),(27,412,196,247),(28,383,197,278),(29,414,198,249),(30,385,199,280),(31,416,200,251),(32,387,201,282),(33,418,202,253),(34,389,203,284),(35,420,204,255),(36,391,205,286),(37,362,206,257),(38,393,207,288),(39,364,208,259),(40,395,209,290),(41,366,210,261),(42,397,211,292),(43,368,212,263),(44,399,213,294),(45,370,214,265),(46,401,215,296),(47,372,216,267),(48,403,217,298),(49,374,218,269),(50,405,219,300),(51,376,220,271),(52,407,221,242),(53,378,222,273),(54,409,223,244),(55,380,224,275),(56,411,225,246),(57,382,226,277),(58,413,227,248),(59,384,228,279),(60,415,229,250),(61,435,312,146),(62,466,313,177),(63,437,314,148),(64,468,315,179),(65,439,316,150),(66,470,317,121),(67,441,318,152),(68,472,319,123),(69,443,320,154),(70,474,321,125),(71,445,322,156),(72,476,323,127),(73,447,324,158),(74,478,325,129),(75,449,326,160),(76,480,327,131),(77,451,328,162),(78,422,329,133),(79,453,330,164),(80,424,331,135),(81,455,332,166),(82,426,333,137),(83,457,334,168),(84,428,335,139),(85,459,336,170),(86,430,337,141),(87,461,338,172),(88,432,339,143),(89,463,340,174),(90,434,341,145),(91,465,342,176),(92,436,343,147),(93,467,344,178),(94,438,345,149),(95,469,346,180),(96,440,347,151),(97,471,348,122),(98,442,349,153),(99,473,350,124),(100,444,351,155),(101,475,352,126),(102,446,353,157),(103,477,354,128),(104,448,355,159),(105,479,356,130),(106,450,357,161),(107,421,358,132),(108,452,359,163),(109,423,360,134),(110,454,301,165),(111,425,302,136),(112,456,303,167),(113,427,304,138),(114,458,305,169),(115,429,306,140),(116,460,307,171),(117,431,308,142),(118,462,309,173),(119,433,310,144),(120,464,311,175)], [(1,310,215,104,31,340,185,74),(2,339,216,73,32,309,186,103),(3,308,217,102,33,338,187,72),(4,337,218,71,34,307,188,101),(5,306,219,100,35,336,189,70),(6,335,220,69,36,305,190,99),(7,304,221,98,37,334,191,68),(8,333,222,67,38,303,192,97),(9,302,223,96,39,332,193,66),(10,331,224,65,40,301,194,95),(11,360,225,94,41,330,195,64),(12,329,226,63,42,359,196,93),(13,358,227,92,43,328,197,62),(14,327,228,61,44,357,198,91),(15,356,229,90,45,326,199,120),(16,325,230,119,46,355,200,89),(17,354,231,88,47,324,201,118),(18,323,232,117,48,353,202,87),(19,352,233,86,49,322,203,116),(20,321,234,115,50,351,204,85),(21,350,235,84,51,320,205,114),(22,319,236,113,52,349,206,83),(23,348,237,82,53,318,207,112),(24,317,238,111,54,347,208,81),(25,346,239,80,55,316,209,110),(26,315,240,109,56,345,210,79),(27,344,181,78,57,314,211,108),(28,313,182,107,58,343,212,77),(29,342,183,76,59,312,213,106),(30,311,184,105,60,341,214,75),(121,379,455,259,151,409,425,289),(122,408,456,288,152,378,426,258),(123,377,457,257,153,407,427,287),(124,406,458,286,154,376,428,256),(125,375,459,255,155,405,429,285),(126,404,460,284,156,374,430,254),(127,373,461,253,157,403,431,283),(128,402,462,282,158,372,432,252),(129,371,463,251,159,401,433,281),(130,400,464,280,160,370,434,250),(131,369,465,249,161,399,435,279),(132,398,466,278,162,368,436,248),(133,367,467,247,163,397,437,277),(134,396,468,276,164,366,438,246),(135,365,469,245,165,395,439,275),(136,394,470,274,166,364,440,244),(137,363,471,243,167,393,441,273),(138,392,472,272,168,362,442,242),(139,361,473,241,169,391,443,271),(140,390,474,270,170,420,444,300),(141,419,475,299,171,389,445,269),(142,388,476,268,172,418,446,298),(143,417,477,297,173,387,447,267),(144,386,478,266,174,416,448,296),(145,415,479,295,175,385,449,265),(146,384,480,264,176,414,450,294),(147,413,421,293,177,383,451,263),(148,382,422,262,178,412,452,292),(149,411,423,291,179,381,453,261),(150,380,424,260,180,410,454,290)]])
84 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 | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| 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 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | + | - | + | - | + | - | + | - | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D5 | D6 | D8 | Q16 | D10 | Dic6 | C4×S3 | C3⋊D4 | D15 | Dic10 | C4×D5 | C5⋊D4 | D30 | Dic30 | C4×D15 | C15⋊7D4 | D4⋊S3 | C3⋊Q16 | D4⋊D5 | C5⋊Q16 | D4⋊D15 | C15⋊7Q16 |
| kernel | C60.1Q8 | C2×C15⋊3C8 | C60⋊5C4 | C15×C4⋊C4 | C15⋊3C8 | C5×C4⋊C4 | C60 | C2×C30 | C3×C4⋊C4 | C2×C20 | C30 | C30 | C2×C12 | C20 | C20 | C2×C10 | C4⋊C4 | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C60.1Q8 ►in GL4(𝔽241) generated by
| 80 | 110 | 0 | 0 |
| 131 | 147 | 0 | 0 |
| 0 | 0 | 240 | 2 |
| 0 | 0 | 240 | 1 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 76 | 138 |
| 0 | 0 | 145 | 165 |
| 201 | 20 | 0 | 0 |
| 149 | 40 | 0 | 0 |
| 0 | 0 | 22 | 219 |
| 0 | 0 | 11 | 0 |
G:=sub<GL(4,GF(241))| [80,131,0,0,110,147,0,0,0,0,240,240,0,0,2,1],[64,0,0,0,0,64,0,0,0,0,76,145,0,0,138,165],[201,149,0,0,20,40,0,0,0,0,22,11,0,0,219,0] >;
C60.1Q8 in GAP, Magma, Sage, TeX
C_{60}._1Q_8 % in TeX
G:=Group("C60.1Q8"); // GroupNames label
G:=SmallGroup(480,167);
// by ID
G=gap.SmallGroup(480,167);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,36,346,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^45*b^2,b*a*b^-1=a^31,c*a*c^-1=a^29,c*b*c^-1=a^45*b^-1>;
// generators/relations