metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.2Q8, C20.6Dic6, C4.2Dic30, C12.6Dic10, C30.25SD16, C15⋊3C8⋊8C4, C4⋊C4.2D15, C15⋊7(C4.Q8), C20.46(C4×S3), C60.76(C2×C4), (C2×C20).65D6, C4.12(C4×D15), C12.14(C4×D5), (C2×C4).33D30, C6.7(Q8⋊D5), C30.37(C4⋊C4), (C2×C30).136D4, (C2×C12).66D10, C6.7(D4.D5), C60⋊5C4.11C2, C3⋊3(C20.Q8), C5⋊4(C12.Q8), C10.7(D4.S3), C2.1(D4.D15), (C2×C60).51C22, C10.7(Q8⋊2S3), C2.1(Q8⋊2D15), C2.4(C30.4Q8), C10.18(Dic3⋊C4), C6.11(C10.D4), C22.13(C15⋊7D4), (C5×C4⋊C4).2S3, (C3×C4⋊C4).2D5, (C15×C4⋊C4).2C2, (C2×C15⋊3C8).2C2, (C2×C6).68(C5⋊D4), (C2×C10).68(C3⋊D4), SmallGroup(480,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.2Q8
G = < a,b,c | a60=b4=1, c2=a45b2, bab-1=a31, cac-1=a29, cbc-1=a15b-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, C4.Q8, 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, C12.Q8, C15⋊3C8, C2×Dic15, C2×C60, C2×C60, C20.Q8, C2×C15⋊3C8, C60⋊5C4, C15×C4⋊C4, C60.2Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, SD16, D10, Dic6, C4×S3, C3⋊D4, D15, C4.Q8, Dic10, C4×D5, C5⋊D4, Dic3⋊C4, D4.S3, Q8⋊2S3, D30, C10.D4, D4.D5, Q8⋊D5, C12.Q8, Dic30, C4×D15, C15⋊7D4, C20.Q8, C30.4Q8, D4.D15, Q8⋊2D15, C60.2Q8
►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)
(1 287 367 232)(2 258 368 203)(3 289 369 234)(4 260 370 205)(5 291 371 236)(6 262 372 207)(7 293 373 238)(8 264 374 209)(9 295 375 240)(10 266 376 211)(11 297 377 182)(12 268 378 213)(13 299 379 184)(14 270 380 215)(15 241 381 186)(16 272 382 217)(17 243 383 188)(18 274 384 219)(19 245 385 190)(20 276 386 221)(21 247 387 192)(22 278 388 223)(23 249 389 194)(24 280 390 225)(25 251 391 196)(26 282 392 227)(27 253 393 198)(28 284 394 229)(29 255 395 200)(30 286 396 231)(31 257 397 202)(32 288 398 233)(33 259 399 204)(34 290 400 235)(35 261 401 206)(36 292 402 237)(37 263 403 208)(38 294 404 239)(39 265 405 210)(40 296 406 181)(41 267 407 212)(42 298 408 183)(43 269 409 214)(44 300 410 185)(45 271 411 216)(46 242 412 187)(47 273 413 218)(48 244 414 189)(49 275 415 220)(50 246 416 191)(51 277 417 222)(52 248 418 193)(53 279 419 224)(54 250 420 195)(55 281 361 226)(56 252 362 197)(57 283 363 228)(58 254 364 199)(59 285 365 230)(60 256 366 201)(61 322 437 164)(62 353 438 135)(63 324 439 166)(64 355 440 137)(65 326 441 168)(66 357 442 139)(67 328 443 170)(68 359 444 141)(69 330 445 172)(70 301 446 143)(71 332 447 174)(72 303 448 145)(73 334 449 176)(74 305 450 147)(75 336 451 178)(76 307 452 149)(77 338 453 180)(78 309 454 151)(79 340 455 122)(80 311 456 153)(81 342 457 124)(82 313 458 155)(83 344 459 126)(84 315 460 157)(85 346 461 128)(86 317 462 159)(87 348 463 130)(88 319 464 161)(89 350 465 132)(90 321 466 163)(91 352 467 134)(92 323 468 165)(93 354 469 136)(94 325 470 167)(95 356 471 138)(96 327 472 169)(97 358 473 140)(98 329 474 171)(99 360 475 142)(100 331 476 173)(101 302 477 144)(102 333 478 175)(103 304 479 146)(104 335 480 177)(105 306 421 148)(106 337 422 179)(107 308 423 150)(108 339 424 121)(109 310 425 152)(110 341 426 123)(111 312 427 154)(112 343 428 125)(113 314 429 156)(114 345 430 127)(115 316 431 158)(116 347 432 129)(117 318 433 160)(118 349 434 131)(119 320 435 162)(120 351 436 133)
(1 324 412 151 31 354 382 121)(2 353 413 180 32 323 383 150)(3 322 414 149 33 352 384 179)(4 351 415 178 34 321 385 148)(5 320 416 147 35 350 386 177)(6 349 417 176 36 319 387 146)(7 318 418 145 37 348 388 175)(8 347 419 174 38 317 389 144)(9 316 420 143 39 346 390 173)(10 345 361 172 40 315 391 142)(11 314 362 141 41 344 392 171)(12 343 363 170 42 313 393 140)(13 312 364 139 43 342 394 169)(14 341 365 168 44 311 395 138)(15 310 366 137 45 340 396 167)(16 339 367 166 46 309 397 136)(17 308 368 135 47 338 398 165)(18 337 369 164 48 307 399 134)(19 306 370 133 49 336 400 163)(20 335 371 162 50 305 401 132)(21 304 372 131 51 334 402 161)(22 333 373 160 52 303 403 130)(23 302 374 129 53 332 404 159)(24 331 375 158 54 301 405 128)(25 360 376 127 55 330 406 157)(26 329 377 156 56 359 407 126)(27 358 378 125 57 328 408 155)(28 327 379 154 58 357 409 124)(29 356 380 123 59 326 410 153)(30 325 381 152 60 355 411 122)(61 204 422 244 91 234 452 274)(62 233 423 273 92 203 453 243)(63 202 424 242 93 232 454 272)(64 231 425 271 94 201 455 241)(65 200 426 300 95 230 456 270)(66 229 427 269 96 199 457 299)(67 198 428 298 97 228 458 268)(68 227 429 267 98 197 459 297)(69 196 430 296 99 226 460 266)(70 225 431 265 100 195 461 295)(71 194 432 294 101 224 462 264)(72 223 433 263 102 193 463 293)(73 192 434 292 103 222 464 262)(74 221 435 261 104 191 465 291)(75 190 436 290 105 220 466 260)(76 219 437 259 106 189 467 289)(77 188 438 288 107 218 468 258)(78 217 439 257 108 187 469 287)(79 186 440 286 109 216 470 256)(80 215 441 255 110 185 471 285)(81 184 442 284 111 214 472 254)(82 213 443 253 112 183 473 283)(83 182 444 282 113 212 474 252)(84 211 445 251 114 181 475 281)(85 240 446 280 115 210 476 250)(86 209 447 249 116 239 477 279)(87 238 448 278 117 208 478 248)(88 207 449 247 118 237 479 277)(89 236 450 276 119 206 480 246)(90 205 451 245 120 235 421 275)
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), (1,287,367,232)(2,258,368,203)(3,289,369,234)(4,260,370,205)(5,291,371,236)(6,262,372,207)(7,293,373,238)(8,264,374,209)(9,295,375,240)(10,266,376,211)(11,297,377,182)(12,268,378,213)(13,299,379,184)(14,270,380,215)(15,241,381,186)(16,272,382,217)(17,243,383,188)(18,274,384,219)(19,245,385,190)(20,276,386,221)(21,247,387,192)(22,278,388,223)(23,249,389,194)(24,280,390,225)(25,251,391,196)(26,282,392,227)(27,253,393,198)(28,284,394,229)(29,255,395,200)(30,286,396,231)(31,257,397,202)(32,288,398,233)(33,259,399,204)(34,290,400,235)(35,261,401,206)(36,292,402,237)(37,263,403,208)(38,294,404,239)(39,265,405,210)(40,296,406,181)(41,267,407,212)(42,298,408,183)(43,269,409,214)(44,300,410,185)(45,271,411,216)(46,242,412,187)(47,273,413,218)(48,244,414,189)(49,275,415,220)(50,246,416,191)(51,277,417,222)(52,248,418,193)(53,279,419,224)(54,250,420,195)(55,281,361,226)(56,252,362,197)(57,283,363,228)(58,254,364,199)(59,285,365,230)(60,256,366,201)(61,322,437,164)(62,353,438,135)(63,324,439,166)(64,355,440,137)(65,326,441,168)(66,357,442,139)(67,328,443,170)(68,359,444,141)(69,330,445,172)(70,301,446,143)(71,332,447,174)(72,303,448,145)(73,334,449,176)(74,305,450,147)(75,336,451,178)(76,307,452,149)(77,338,453,180)(78,309,454,151)(79,340,455,122)(80,311,456,153)(81,342,457,124)(82,313,458,155)(83,344,459,126)(84,315,460,157)(85,346,461,128)(86,317,462,159)(87,348,463,130)(88,319,464,161)(89,350,465,132)(90,321,466,163)(91,352,467,134)(92,323,468,165)(93,354,469,136)(94,325,470,167)(95,356,471,138)(96,327,472,169)(97,358,473,140)(98,329,474,171)(99,360,475,142)(100,331,476,173)(101,302,477,144)(102,333,478,175)(103,304,479,146)(104,335,480,177)(105,306,421,148)(106,337,422,179)(107,308,423,150)(108,339,424,121)(109,310,425,152)(110,341,426,123)(111,312,427,154)(112,343,428,125)(113,314,429,156)(114,345,430,127)(115,316,431,158)(116,347,432,129)(117,318,433,160)(118,349,434,131)(119,320,435,162)(120,351,436,133), (1,324,412,151,31,354,382,121)(2,353,413,180,32,323,383,150)(3,322,414,149,33,352,384,179)(4,351,415,178,34,321,385,148)(5,320,416,147,35,350,386,177)(6,349,417,176,36,319,387,146)(7,318,418,145,37,348,388,175)(8,347,419,174,38,317,389,144)(9,316,420,143,39,346,390,173)(10,345,361,172,40,315,391,142)(11,314,362,141,41,344,392,171)(12,343,363,170,42,313,393,140)(13,312,364,139,43,342,394,169)(14,341,365,168,44,311,395,138)(15,310,366,137,45,340,396,167)(16,339,367,166,46,309,397,136)(17,308,368,135,47,338,398,165)(18,337,369,164,48,307,399,134)(19,306,370,133,49,336,400,163)(20,335,371,162,50,305,401,132)(21,304,372,131,51,334,402,161)(22,333,373,160,52,303,403,130)(23,302,374,129,53,332,404,159)(24,331,375,158,54,301,405,128)(25,360,376,127,55,330,406,157)(26,329,377,156,56,359,407,126)(27,358,378,125,57,328,408,155)(28,327,379,154,58,357,409,124)(29,356,380,123,59,326,410,153)(30,325,381,152,60,355,411,122)(61,204,422,244,91,234,452,274)(62,233,423,273,92,203,453,243)(63,202,424,242,93,232,454,272)(64,231,425,271,94,201,455,241)(65,200,426,300,95,230,456,270)(66,229,427,269,96,199,457,299)(67,198,428,298,97,228,458,268)(68,227,429,267,98,197,459,297)(69,196,430,296,99,226,460,266)(70,225,431,265,100,195,461,295)(71,194,432,294,101,224,462,264)(72,223,433,263,102,193,463,293)(73,192,434,292,103,222,464,262)(74,221,435,261,104,191,465,291)(75,190,436,290,105,220,466,260)(76,219,437,259,106,189,467,289)(77,188,438,288,107,218,468,258)(78,217,439,257,108,187,469,287)(79,186,440,286,109,216,470,256)(80,215,441,255,110,185,471,285)(81,184,442,284,111,214,472,254)(82,213,443,253,112,183,473,283)(83,182,444,282,113,212,474,252)(84,211,445,251,114,181,475,281)(85,240,446,280,115,210,476,250)(86,209,447,249,116,239,477,279)(87,238,448,278,117,208,478,248)(88,207,449,247,118,237,479,277)(89,236,450,276,119,206,480,246)(90,205,451,245,120,235,421,275)>;
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), (1,287,367,232)(2,258,368,203)(3,289,369,234)(4,260,370,205)(5,291,371,236)(6,262,372,207)(7,293,373,238)(8,264,374,209)(9,295,375,240)(10,266,376,211)(11,297,377,182)(12,268,378,213)(13,299,379,184)(14,270,380,215)(15,241,381,186)(16,272,382,217)(17,243,383,188)(18,274,384,219)(19,245,385,190)(20,276,386,221)(21,247,387,192)(22,278,388,223)(23,249,389,194)(24,280,390,225)(25,251,391,196)(26,282,392,227)(27,253,393,198)(28,284,394,229)(29,255,395,200)(30,286,396,231)(31,257,397,202)(32,288,398,233)(33,259,399,204)(34,290,400,235)(35,261,401,206)(36,292,402,237)(37,263,403,208)(38,294,404,239)(39,265,405,210)(40,296,406,181)(41,267,407,212)(42,298,408,183)(43,269,409,214)(44,300,410,185)(45,271,411,216)(46,242,412,187)(47,273,413,218)(48,244,414,189)(49,275,415,220)(50,246,416,191)(51,277,417,222)(52,248,418,193)(53,279,419,224)(54,250,420,195)(55,281,361,226)(56,252,362,197)(57,283,363,228)(58,254,364,199)(59,285,365,230)(60,256,366,201)(61,322,437,164)(62,353,438,135)(63,324,439,166)(64,355,440,137)(65,326,441,168)(66,357,442,139)(67,328,443,170)(68,359,444,141)(69,330,445,172)(70,301,446,143)(71,332,447,174)(72,303,448,145)(73,334,449,176)(74,305,450,147)(75,336,451,178)(76,307,452,149)(77,338,453,180)(78,309,454,151)(79,340,455,122)(80,311,456,153)(81,342,457,124)(82,313,458,155)(83,344,459,126)(84,315,460,157)(85,346,461,128)(86,317,462,159)(87,348,463,130)(88,319,464,161)(89,350,465,132)(90,321,466,163)(91,352,467,134)(92,323,468,165)(93,354,469,136)(94,325,470,167)(95,356,471,138)(96,327,472,169)(97,358,473,140)(98,329,474,171)(99,360,475,142)(100,331,476,173)(101,302,477,144)(102,333,478,175)(103,304,479,146)(104,335,480,177)(105,306,421,148)(106,337,422,179)(107,308,423,150)(108,339,424,121)(109,310,425,152)(110,341,426,123)(111,312,427,154)(112,343,428,125)(113,314,429,156)(114,345,430,127)(115,316,431,158)(116,347,432,129)(117,318,433,160)(118,349,434,131)(119,320,435,162)(120,351,436,133), (1,324,412,151,31,354,382,121)(2,353,413,180,32,323,383,150)(3,322,414,149,33,352,384,179)(4,351,415,178,34,321,385,148)(5,320,416,147,35,350,386,177)(6,349,417,176,36,319,387,146)(7,318,418,145,37,348,388,175)(8,347,419,174,38,317,389,144)(9,316,420,143,39,346,390,173)(10,345,361,172,40,315,391,142)(11,314,362,141,41,344,392,171)(12,343,363,170,42,313,393,140)(13,312,364,139,43,342,394,169)(14,341,365,168,44,311,395,138)(15,310,366,137,45,340,396,167)(16,339,367,166,46,309,397,136)(17,308,368,135,47,338,398,165)(18,337,369,164,48,307,399,134)(19,306,370,133,49,336,400,163)(20,335,371,162,50,305,401,132)(21,304,372,131,51,334,402,161)(22,333,373,160,52,303,403,130)(23,302,374,129,53,332,404,159)(24,331,375,158,54,301,405,128)(25,360,376,127,55,330,406,157)(26,329,377,156,56,359,407,126)(27,358,378,125,57,328,408,155)(28,327,379,154,58,357,409,124)(29,356,380,123,59,326,410,153)(30,325,381,152,60,355,411,122)(61,204,422,244,91,234,452,274)(62,233,423,273,92,203,453,243)(63,202,424,242,93,232,454,272)(64,231,425,271,94,201,455,241)(65,200,426,300,95,230,456,270)(66,229,427,269,96,199,457,299)(67,198,428,298,97,228,458,268)(68,227,429,267,98,197,459,297)(69,196,430,296,99,226,460,266)(70,225,431,265,100,195,461,295)(71,194,432,294,101,224,462,264)(72,223,433,263,102,193,463,293)(73,192,434,292,103,222,464,262)(74,221,435,261,104,191,465,291)(75,190,436,290,105,220,466,260)(76,219,437,259,106,189,467,289)(77,188,438,288,107,218,468,258)(78,217,439,257,108,187,469,287)(79,186,440,286,109,216,470,256)(80,215,441,255,110,185,471,285)(81,184,442,284,111,214,472,254)(82,213,443,253,112,183,473,283)(83,182,444,282,113,212,474,252)(84,211,445,251,114,181,475,281)(85,240,446,280,115,210,476,250)(86,209,447,249,116,239,477,279)(87,238,448,278,117,208,478,248)(88,207,449,247,118,237,479,277)(89,236,450,276,119,206,480,246)(90,205,451,245,120,235,421,275) );
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)], [(1,287,367,232),(2,258,368,203),(3,289,369,234),(4,260,370,205),(5,291,371,236),(6,262,372,207),(7,293,373,238),(8,264,374,209),(9,295,375,240),(10,266,376,211),(11,297,377,182),(12,268,378,213),(13,299,379,184),(14,270,380,215),(15,241,381,186),(16,272,382,217),(17,243,383,188),(18,274,384,219),(19,245,385,190),(20,276,386,221),(21,247,387,192),(22,278,388,223),(23,249,389,194),(24,280,390,225),(25,251,391,196),(26,282,392,227),(27,253,393,198),(28,284,394,229),(29,255,395,200),(30,286,396,231),(31,257,397,202),(32,288,398,233),(33,259,399,204),(34,290,400,235),(35,261,401,206),(36,292,402,237),(37,263,403,208),(38,294,404,239),(39,265,405,210),(40,296,406,181),(41,267,407,212),(42,298,408,183),(43,269,409,214),(44,300,410,185),(45,271,411,216),(46,242,412,187),(47,273,413,218),(48,244,414,189),(49,275,415,220),(50,246,416,191),(51,277,417,222),(52,248,418,193),(53,279,419,224),(54,250,420,195),(55,281,361,226),(56,252,362,197),(57,283,363,228),(58,254,364,199),(59,285,365,230),(60,256,366,201),(61,322,437,164),(62,353,438,135),(63,324,439,166),(64,355,440,137),(65,326,441,168),(66,357,442,139),(67,328,443,170),(68,359,444,141),(69,330,445,172),(70,301,446,143),(71,332,447,174),(72,303,448,145),(73,334,449,176),(74,305,450,147),(75,336,451,178),(76,307,452,149),(77,338,453,180),(78,309,454,151),(79,340,455,122),(80,311,456,153),(81,342,457,124),(82,313,458,155),(83,344,459,126),(84,315,460,157),(85,346,461,128),(86,317,462,159),(87,348,463,130),(88,319,464,161),(89,350,465,132),(90,321,466,163),(91,352,467,134),(92,323,468,165),(93,354,469,136),(94,325,470,167),(95,356,471,138),(96,327,472,169),(97,358,473,140),(98,329,474,171),(99,360,475,142),(100,331,476,173),(101,302,477,144),(102,333,478,175),(103,304,479,146),(104,335,480,177),(105,306,421,148),(106,337,422,179),(107,308,423,150),(108,339,424,121),(109,310,425,152),(110,341,426,123),(111,312,427,154),(112,343,428,125),(113,314,429,156),(114,345,430,127),(115,316,431,158),(116,347,432,129),(117,318,433,160),(118,349,434,131),(119,320,435,162),(120,351,436,133)], [(1,324,412,151,31,354,382,121),(2,353,413,180,32,323,383,150),(3,322,414,149,33,352,384,179),(4,351,415,178,34,321,385,148),(5,320,416,147,35,350,386,177),(6,349,417,176,36,319,387,146),(7,318,418,145,37,348,388,175),(8,347,419,174,38,317,389,144),(9,316,420,143,39,346,390,173),(10,345,361,172,40,315,391,142),(11,314,362,141,41,344,392,171),(12,343,363,170,42,313,393,140),(13,312,364,139,43,342,394,169),(14,341,365,168,44,311,395,138),(15,310,366,137,45,340,396,167),(16,339,367,166,46,309,397,136),(17,308,368,135,47,338,398,165),(18,337,369,164,48,307,399,134),(19,306,370,133,49,336,400,163),(20,335,371,162,50,305,401,132),(21,304,372,131,51,334,402,161),(22,333,373,160,52,303,403,130),(23,302,374,129,53,332,404,159),(24,331,375,158,54,301,405,128),(25,360,376,127,55,330,406,157),(26,329,377,156,56,359,407,126),(27,358,378,125,57,328,408,155),(28,327,379,154,58,357,409,124),(29,356,380,123,59,326,410,153),(30,325,381,152,60,355,411,122),(61,204,422,244,91,234,452,274),(62,233,423,273,92,203,453,243),(63,202,424,242,93,232,454,272),(64,231,425,271,94,201,455,241),(65,200,426,300,95,230,456,270),(66,229,427,269,96,199,457,299),(67,198,428,298,97,228,458,268),(68,227,429,267,98,197,459,297),(69,196,430,296,99,226,460,266),(70,225,431,265,100,195,461,295),(71,194,432,294,101,224,462,264),(72,223,433,263,102,193,463,293),(73,192,434,292,103,222,464,262),(74,221,435,261,104,191,465,291),(75,190,436,290,105,220,466,260),(76,219,437,259,106,189,467,289),(77,188,438,288,107,218,468,258),(78,217,439,257,108,187,469,287),(79,186,440,286,109,216,470,256),(80,215,441,255,110,185,471,285),(81,184,442,284,111,214,472,254),(82,213,443,253,112,183,473,283),(83,182,444,282,113,212,474,252),(84,211,445,251,114,181,475,281),(85,240,446,280,115,210,476,250),(86,209,447,249,116,239,477,279),(87,238,448,278,117,208,478,248),(88,207,449,247,118,237,479,277),(89,236,450,276,119,206,480,246),(90,205,451,245,120,235,421,275)]])
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 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | + | - | + | - | - | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D5 | D6 | SD16 | D10 | Dic6 | C4×S3 | C3⋊D4 | D15 | Dic10 | C4×D5 | C5⋊D4 | D30 | Dic30 | C4×D15 | C15⋊7D4 | D4.S3 | Q8⋊2S3 | D4.D5 | Q8⋊D5 | D4.D15 | Q8⋊2D15 |
| kernel | C60.2Q8 | C2×C15⋊3C8 | C60⋊5C4 | C15×C4⋊C4 | C15⋊3C8 | C5×C4⋊C4 | C60 | C2×C30 | C3×C4⋊C4 | C2×C20 | 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 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C60.2Q8 ►in GL4(𝔽241) generated by
| 16 | 173 | 0 | 0 |
| 68 | 178 | 0 | 0 |
| 0 | 0 | 1 | 54 |
| 0 | 0 | 116 | 240 |
| 41 | 85 | 0 | 0 |
| 156 | 200 | 0 | 0 |
| 0 | 0 | 212 | 13 |
| 0 | 0 | 232 | 29 |
| 63 | 163 | 0 | 0 |
| 20 | 178 | 0 | 0 |
| 0 | 0 | 0 | 62 |
| 0 | 0 | 35 | 203 |
G:=sub<GL(4,GF(241))| [16,68,0,0,173,178,0,0,0,0,1,116,0,0,54,240],[41,156,0,0,85,200,0,0,0,0,212,232,0,0,13,29],[63,20,0,0,163,178,0,0,0,0,0,35,0,0,62,203] >;
C60.2Q8 in GAP, Magma, Sage, TeX
C_{60}._2Q_8 % in TeX
G:=Group("C60.2Q8"); // GroupNames label
G:=SmallGroup(480,168);
// by ID
G=gap.SmallGroup(480,168);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,36,346,80,2693,18822]);
// 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^15*b^-1>;
// generators/relations