metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.15Q8, C12.58D20, C60.104D4, C20.15Dic6, C12.15Dic10, C30.19M4(2), C15⋊9(C4⋊C8), C3⋊1(C20⋊3C8), C10.24(S3×C8), C30.33(C2×C8), Dic3⋊(C5⋊2C8), C5⋊4(Dic3⋊C8), (C5×Dic3)⋊4C8, C4.12(C15⋊Q8), C30.24(C4⋊C4), (C2×C20).326D6, C6.1(C4⋊Dic5), (C4×Dic3).7D5, (C2×C12).330D10, C20.63(C3⋊D4), C4.30(C3⋊D20), (Dic3×C20).8C2, C10.14(C8⋊S3), C6.3(C4.Dic5), C10.9(Dic3⋊C4), (C2×C60).228C22, (C10×Dic3).15C4, (C2×Dic3).4Dic5, C2.3(D6.Dic5), C2.1(C6.Dic10), C22.11(S3×Dic5), C6.5(C2×C5⋊2C8), C2.5(S3×C5⋊2C8), (C6×C5⋊2C8).12C2, (C2×C5⋊2C8).10S3, (C2×C30).88(C2×C4), (C2×C10).72(C4×S3), (C2×C4).231(S3×D5), (C2×C15⋊3C8).22C2, (C2×C6).12(C2×Dic5), SmallGroup(480,60)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.15Q8
G = < a,b,c | a60=b4=1, c2=a45b2, bab-1=a41, cac-1=a49, cbc-1=b-1 >
Subgroups: 220 in 76 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C2×C8, C20, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C4⋊C8, C5⋊2C8, C2×C20, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, C5×Dic3, C5×Dic3, C60, C2×C30, C2×C5⋊2C8, C2×C5⋊2C8, C4×C20, Dic3⋊C8, C3×C5⋊2C8, C15⋊3C8, C10×Dic3, C2×C60, C20⋊3C8, C6×C5⋊2C8, Dic3×C20, C2×C15⋊3C8, C60.15Q8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, D5, D6, C4⋊C4, C2×C8, M4(2), Dic5, D10, Dic6, C4×S3, C3⋊D4, C4⋊C8, C5⋊2C8, Dic10, D20, C2×Dic5, S3×C8, C8⋊S3, Dic3⋊C4, S3×D5, C2×C5⋊2C8, C4.Dic5, C4⋊Dic5, Dic3⋊C8, S3×Dic5, C3⋊D20, C15⋊Q8, C20⋊3C8, S3×C5⋊2C8, D6.Dic5, C6.Dic10, C60.15Q8
►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)
(1 182 363 144)(2 223 364 125)(3 204 365 166)(4 185 366 147)(5 226 367 128)(6 207 368 169)(7 188 369 150)(8 229 370 131)(9 210 371 172)(10 191 372 153)(11 232 373 134)(12 213 374 175)(13 194 375 156)(14 235 376 137)(15 216 377 178)(16 197 378 159)(17 238 379 140)(18 219 380 121)(19 200 381 162)(20 181 382 143)(21 222 383 124)(22 203 384 165)(23 184 385 146)(24 225 386 127)(25 206 387 168)(26 187 388 149)(27 228 389 130)(28 209 390 171)(29 190 391 152)(30 231 392 133)(31 212 393 174)(32 193 394 155)(33 234 395 136)(34 215 396 177)(35 196 397 158)(36 237 398 139)(37 218 399 180)(38 199 400 161)(39 240 401 142)(40 221 402 123)(41 202 403 164)(42 183 404 145)(43 224 405 126)(44 205 406 167)(45 186 407 148)(46 227 408 129)(47 208 409 170)(48 189 410 151)(49 230 411 132)(50 211 412 173)(51 192 413 154)(52 233 414 135)(53 214 415 176)(54 195 416 157)(55 236 417 138)(56 217 418 179)(57 198 419 160)(58 239 420 141)(59 220 361 122)(60 201 362 163)(61 268 348 442)(62 249 349 423)(63 290 350 464)(64 271 351 445)(65 252 352 426)(66 293 353 467)(67 274 354 448)(68 255 355 429)(69 296 356 470)(70 277 357 451)(71 258 358 432)(72 299 359 473)(73 280 360 454)(74 261 301 435)(75 242 302 476)(76 283 303 457)(77 264 304 438)(78 245 305 479)(79 286 306 460)(80 267 307 441)(81 248 308 422)(82 289 309 463)(83 270 310 444)(84 251 311 425)(85 292 312 466)(86 273 313 447)(87 254 314 428)(88 295 315 469)(89 276 316 450)(90 257 317 431)(91 298 318 472)(92 279 319 453)(93 260 320 434)(94 241 321 475)(95 282 322 456)(96 263 323 437)(97 244 324 478)(98 285 325 459)(99 266 326 440)(100 247 327 421)(101 288 328 462)(102 269 329 443)(103 250 330 424)(104 291 331 465)(105 272 332 446)(106 253 333 427)(107 294 334 468)(108 275 335 449)(109 256 336 430)(110 297 337 471)(111 278 338 452)(112 259 339 433)(113 300 340 474)(114 281 341 455)(115 262 342 436)(116 243 343 477)(117 284 344 458)(118 265 345 439)(119 246 346 480)(120 287 347 461)
(1 277 408 436 31 247 378 466)(2 266 409 425 32 296 379 455)(3 255 410 474 33 285 380 444)(4 244 411 463 34 274 381 433)(5 293 412 452 35 263 382 422)(6 282 413 441 36 252 383 471)(7 271 414 430 37 241 384 460)(8 260 415 479 38 290 385 449)(9 249 416 468 39 279 386 438)(10 298 417 457 40 268 387 427)(11 287 418 446 41 257 388 476)(12 276 419 435 42 246 389 465)(13 265 420 424 43 295 390 454)(14 254 361 473 44 284 391 443)(15 243 362 462 45 273 392 432)(16 292 363 451 46 262 393 421)(17 281 364 440 47 251 394 470)(18 270 365 429 48 300 395 459)(19 259 366 478 49 289 396 448)(20 248 367 467 50 278 397 437)(21 297 368 456 51 267 398 426)(22 286 369 445 52 256 399 475)(23 275 370 434 53 245 400 464)(24 264 371 423 54 294 401 453)(25 253 372 472 55 283 402 442)(26 242 373 461 56 272 403 431)(27 291 374 450 57 261 404 480)(28 280 375 439 58 250 405 469)(29 269 376 428 59 299 406 458)(30 258 377 477 60 288 407 447)(61 168 333 191 91 138 303 221)(62 157 334 240 92 127 304 210)(63 146 335 229 93 176 305 199)(64 135 336 218 94 165 306 188)(65 124 337 207 95 154 307 237)(66 173 338 196 96 143 308 226)(67 162 339 185 97 132 309 215)(68 151 340 234 98 121 310 204)(69 140 341 223 99 170 311 193)(70 129 342 212 100 159 312 182)(71 178 343 201 101 148 313 231)(72 167 344 190 102 137 314 220)(73 156 345 239 103 126 315 209)(74 145 346 228 104 175 316 198)(75 134 347 217 105 164 317 187)(76 123 348 206 106 153 318 236)(77 172 349 195 107 142 319 225)(78 161 350 184 108 131 320 214)(79 150 351 233 109 180 321 203)(80 139 352 222 110 169 322 192)(81 128 353 211 111 158 323 181)(82 177 354 200 112 147 324 230)(83 166 355 189 113 136 325 219)(84 155 356 238 114 125 326 208)(85 144 357 227 115 174 327 197)(86 133 358 216 116 163 328 186)(87 122 359 205 117 152 329 235)(88 171 360 194 118 141 330 224)(89 160 301 183 119 130 331 213)(90 149 302 232 120 179 332 202)
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), (1,182,363,144)(2,223,364,125)(3,204,365,166)(4,185,366,147)(5,226,367,128)(6,207,368,169)(7,188,369,150)(8,229,370,131)(9,210,371,172)(10,191,372,153)(11,232,373,134)(12,213,374,175)(13,194,375,156)(14,235,376,137)(15,216,377,178)(16,197,378,159)(17,238,379,140)(18,219,380,121)(19,200,381,162)(20,181,382,143)(21,222,383,124)(22,203,384,165)(23,184,385,146)(24,225,386,127)(25,206,387,168)(26,187,388,149)(27,228,389,130)(28,209,390,171)(29,190,391,152)(30,231,392,133)(31,212,393,174)(32,193,394,155)(33,234,395,136)(34,215,396,177)(35,196,397,158)(36,237,398,139)(37,218,399,180)(38,199,400,161)(39,240,401,142)(40,221,402,123)(41,202,403,164)(42,183,404,145)(43,224,405,126)(44,205,406,167)(45,186,407,148)(46,227,408,129)(47,208,409,170)(48,189,410,151)(49,230,411,132)(50,211,412,173)(51,192,413,154)(52,233,414,135)(53,214,415,176)(54,195,416,157)(55,236,417,138)(56,217,418,179)(57,198,419,160)(58,239,420,141)(59,220,361,122)(60,201,362,163)(61,268,348,442)(62,249,349,423)(63,290,350,464)(64,271,351,445)(65,252,352,426)(66,293,353,467)(67,274,354,448)(68,255,355,429)(69,296,356,470)(70,277,357,451)(71,258,358,432)(72,299,359,473)(73,280,360,454)(74,261,301,435)(75,242,302,476)(76,283,303,457)(77,264,304,438)(78,245,305,479)(79,286,306,460)(80,267,307,441)(81,248,308,422)(82,289,309,463)(83,270,310,444)(84,251,311,425)(85,292,312,466)(86,273,313,447)(87,254,314,428)(88,295,315,469)(89,276,316,450)(90,257,317,431)(91,298,318,472)(92,279,319,453)(93,260,320,434)(94,241,321,475)(95,282,322,456)(96,263,323,437)(97,244,324,478)(98,285,325,459)(99,266,326,440)(100,247,327,421)(101,288,328,462)(102,269,329,443)(103,250,330,424)(104,291,331,465)(105,272,332,446)(106,253,333,427)(107,294,334,468)(108,275,335,449)(109,256,336,430)(110,297,337,471)(111,278,338,452)(112,259,339,433)(113,300,340,474)(114,281,341,455)(115,262,342,436)(116,243,343,477)(117,284,344,458)(118,265,345,439)(119,246,346,480)(120,287,347,461), (1,277,408,436,31,247,378,466)(2,266,409,425,32,296,379,455)(3,255,410,474,33,285,380,444)(4,244,411,463,34,274,381,433)(5,293,412,452,35,263,382,422)(6,282,413,441,36,252,383,471)(7,271,414,430,37,241,384,460)(8,260,415,479,38,290,385,449)(9,249,416,468,39,279,386,438)(10,298,417,457,40,268,387,427)(11,287,418,446,41,257,388,476)(12,276,419,435,42,246,389,465)(13,265,420,424,43,295,390,454)(14,254,361,473,44,284,391,443)(15,243,362,462,45,273,392,432)(16,292,363,451,46,262,393,421)(17,281,364,440,47,251,394,470)(18,270,365,429,48,300,395,459)(19,259,366,478,49,289,396,448)(20,248,367,467,50,278,397,437)(21,297,368,456,51,267,398,426)(22,286,369,445,52,256,399,475)(23,275,370,434,53,245,400,464)(24,264,371,423,54,294,401,453)(25,253,372,472,55,283,402,442)(26,242,373,461,56,272,403,431)(27,291,374,450,57,261,404,480)(28,280,375,439,58,250,405,469)(29,269,376,428,59,299,406,458)(30,258,377,477,60,288,407,447)(61,168,333,191,91,138,303,221)(62,157,334,240,92,127,304,210)(63,146,335,229,93,176,305,199)(64,135,336,218,94,165,306,188)(65,124,337,207,95,154,307,237)(66,173,338,196,96,143,308,226)(67,162,339,185,97,132,309,215)(68,151,340,234,98,121,310,204)(69,140,341,223,99,170,311,193)(70,129,342,212,100,159,312,182)(71,178,343,201,101,148,313,231)(72,167,344,190,102,137,314,220)(73,156,345,239,103,126,315,209)(74,145,346,228,104,175,316,198)(75,134,347,217,105,164,317,187)(76,123,348,206,106,153,318,236)(77,172,349,195,107,142,319,225)(78,161,350,184,108,131,320,214)(79,150,351,233,109,180,321,203)(80,139,352,222,110,169,322,192)(81,128,353,211,111,158,323,181)(82,177,354,200,112,147,324,230)(83,166,355,189,113,136,325,219)(84,155,356,238,114,125,326,208)(85,144,357,227,115,174,327,197)(86,133,358,216,116,163,328,186)(87,122,359,205,117,152,329,235)(88,171,360,194,118,141,330,224)(89,160,301,183,119,130,331,213)(90,149,302,232,120,179,332,202)>;
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), (1,182,363,144)(2,223,364,125)(3,204,365,166)(4,185,366,147)(5,226,367,128)(6,207,368,169)(7,188,369,150)(8,229,370,131)(9,210,371,172)(10,191,372,153)(11,232,373,134)(12,213,374,175)(13,194,375,156)(14,235,376,137)(15,216,377,178)(16,197,378,159)(17,238,379,140)(18,219,380,121)(19,200,381,162)(20,181,382,143)(21,222,383,124)(22,203,384,165)(23,184,385,146)(24,225,386,127)(25,206,387,168)(26,187,388,149)(27,228,389,130)(28,209,390,171)(29,190,391,152)(30,231,392,133)(31,212,393,174)(32,193,394,155)(33,234,395,136)(34,215,396,177)(35,196,397,158)(36,237,398,139)(37,218,399,180)(38,199,400,161)(39,240,401,142)(40,221,402,123)(41,202,403,164)(42,183,404,145)(43,224,405,126)(44,205,406,167)(45,186,407,148)(46,227,408,129)(47,208,409,170)(48,189,410,151)(49,230,411,132)(50,211,412,173)(51,192,413,154)(52,233,414,135)(53,214,415,176)(54,195,416,157)(55,236,417,138)(56,217,418,179)(57,198,419,160)(58,239,420,141)(59,220,361,122)(60,201,362,163)(61,268,348,442)(62,249,349,423)(63,290,350,464)(64,271,351,445)(65,252,352,426)(66,293,353,467)(67,274,354,448)(68,255,355,429)(69,296,356,470)(70,277,357,451)(71,258,358,432)(72,299,359,473)(73,280,360,454)(74,261,301,435)(75,242,302,476)(76,283,303,457)(77,264,304,438)(78,245,305,479)(79,286,306,460)(80,267,307,441)(81,248,308,422)(82,289,309,463)(83,270,310,444)(84,251,311,425)(85,292,312,466)(86,273,313,447)(87,254,314,428)(88,295,315,469)(89,276,316,450)(90,257,317,431)(91,298,318,472)(92,279,319,453)(93,260,320,434)(94,241,321,475)(95,282,322,456)(96,263,323,437)(97,244,324,478)(98,285,325,459)(99,266,326,440)(100,247,327,421)(101,288,328,462)(102,269,329,443)(103,250,330,424)(104,291,331,465)(105,272,332,446)(106,253,333,427)(107,294,334,468)(108,275,335,449)(109,256,336,430)(110,297,337,471)(111,278,338,452)(112,259,339,433)(113,300,340,474)(114,281,341,455)(115,262,342,436)(116,243,343,477)(117,284,344,458)(118,265,345,439)(119,246,346,480)(120,287,347,461), (1,277,408,436,31,247,378,466)(2,266,409,425,32,296,379,455)(3,255,410,474,33,285,380,444)(4,244,411,463,34,274,381,433)(5,293,412,452,35,263,382,422)(6,282,413,441,36,252,383,471)(7,271,414,430,37,241,384,460)(8,260,415,479,38,290,385,449)(9,249,416,468,39,279,386,438)(10,298,417,457,40,268,387,427)(11,287,418,446,41,257,388,476)(12,276,419,435,42,246,389,465)(13,265,420,424,43,295,390,454)(14,254,361,473,44,284,391,443)(15,243,362,462,45,273,392,432)(16,292,363,451,46,262,393,421)(17,281,364,440,47,251,394,470)(18,270,365,429,48,300,395,459)(19,259,366,478,49,289,396,448)(20,248,367,467,50,278,397,437)(21,297,368,456,51,267,398,426)(22,286,369,445,52,256,399,475)(23,275,370,434,53,245,400,464)(24,264,371,423,54,294,401,453)(25,253,372,472,55,283,402,442)(26,242,373,461,56,272,403,431)(27,291,374,450,57,261,404,480)(28,280,375,439,58,250,405,469)(29,269,376,428,59,299,406,458)(30,258,377,477,60,288,407,447)(61,168,333,191,91,138,303,221)(62,157,334,240,92,127,304,210)(63,146,335,229,93,176,305,199)(64,135,336,218,94,165,306,188)(65,124,337,207,95,154,307,237)(66,173,338,196,96,143,308,226)(67,162,339,185,97,132,309,215)(68,151,340,234,98,121,310,204)(69,140,341,223,99,170,311,193)(70,129,342,212,100,159,312,182)(71,178,343,201,101,148,313,231)(72,167,344,190,102,137,314,220)(73,156,345,239,103,126,315,209)(74,145,346,228,104,175,316,198)(75,134,347,217,105,164,317,187)(76,123,348,206,106,153,318,236)(77,172,349,195,107,142,319,225)(78,161,350,184,108,131,320,214)(79,150,351,233,109,180,321,203)(80,139,352,222,110,169,322,192)(81,128,353,211,111,158,323,181)(82,177,354,200,112,147,324,230)(83,166,355,189,113,136,325,219)(84,155,356,238,114,125,326,208)(85,144,357,227,115,174,327,197)(86,133,358,216,116,163,328,186)(87,122,359,205,117,152,329,235)(88,171,360,194,118,141,330,224)(89,160,301,183,119,130,331,213)(90,149,302,232,120,179,332,202) );
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)], [(1,182,363,144),(2,223,364,125),(3,204,365,166),(4,185,366,147),(5,226,367,128),(6,207,368,169),(7,188,369,150),(8,229,370,131),(9,210,371,172),(10,191,372,153),(11,232,373,134),(12,213,374,175),(13,194,375,156),(14,235,376,137),(15,216,377,178),(16,197,378,159),(17,238,379,140),(18,219,380,121),(19,200,381,162),(20,181,382,143),(21,222,383,124),(22,203,384,165),(23,184,385,146),(24,225,386,127),(25,206,387,168),(26,187,388,149),(27,228,389,130),(28,209,390,171),(29,190,391,152),(30,231,392,133),(31,212,393,174),(32,193,394,155),(33,234,395,136),(34,215,396,177),(35,196,397,158),(36,237,398,139),(37,218,399,180),(38,199,400,161),(39,240,401,142),(40,221,402,123),(41,202,403,164),(42,183,404,145),(43,224,405,126),(44,205,406,167),(45,186,407,148),(46,227,408,129),(47,208,409,170),(48,189,410,151),(49,230,411,132),(50,211,412,173),(51,192,413,154),(52,233,414,135),(53,214,415,176),(54,195,416,157),(55,236,417,138),(56,217,418,179),(57,198,419,160),(58,239,420,141),(59,220,361,122),(60,201,362,163),(61,268,348,442),(62,249,349,423),(63,290,350,464),(64,271,351,445),(65,252,352,426),(66,293,353,467),(67,274,354,448),(68,255,355,429),(69,296,356,470),(70,277,357,451),(71,258,358,432),(72,299,359,473),(73,280,360,454),(74,261,301,435),(75,242,302,476),(76,283,303,457),(77,264,304,438),(78,245,305,479),(79,286,306,460),(80,267,307,441),(81,248,308,422),(82,289,309,463),(83,270,310,444),(84,251,311,425),(85,292,312,466),(86,273,313,447),(87,254,314,428),(88,295,315,469),(89,276,316,450),(90,257,317,431),(91,298,318,472),(92,279,319,453),(93,260,320,434),(94,241,321,475),(95,282,322,456),(96,263,323,437),(97,244,324,478),(98,285,325,459),(99,266,326,440),(100,247,327,421),(101,288,328,462),(102,269,329,443),(103,250,330,424),(104,291,331,465),(105,272,332,446),(106,253,333,427),(107,294,334,468),(108,275,335,449),(109,256,336,430),(110,297,337,471),(111,278,338,452),(112,259,339,433),(113,300,340,474),(114,281,341,455),(115,262,342,436),(116,243,343,477),(117,284,344,458),(118,265,345,439),(119,246,346,480),(120,287,347,461)], [(1,277,408,436,31,247,378,466),(2,266,409,425,32,296,379,455),(3,255,410,474,33,285,380,444),(4,244,411,463,34,274,381,433),(5,293,412,452,35,263,382,422),(6,282,413,441,36,252,383,471),(7,271,414,430,37,241,384,460),(8,260,415,479,38,290,385,449),(9,249,416,468,39,279,386,438),(10,298,417,457,40,268,387,427),(11,287,418,446,41,257,388,476),(12,276,419,435,42,246,389,465),(13,265,420,424,43,295,390,454),(14,254,361,473,44,284,391,443),(15,243,362,462,45,273,392,432),(16,292,363,451,46,262,393,421),(17,281,364,440,47,251,394,470),(18,270,365,429,48,300,395,459),(19,259,366,478,49,289,396,448),(20,248,367,467,50,278,397,437),(21,297,368,456,51,267,398,426),(22,286,369,445,52,256,399,475),(23,275,370,434,53,245,400,464),(24,264,371,423,54,294,401,453),(25,253,372,472,55,283,402,442),(26,242,373,461,56,272,403,431),(27,291,374,450,57,261,404,480),(28,280,375,439,58,250,405,469),(29,269,376,428,59,299,406,458),(30,258,377,477,60,288,407,447),(61,168,333,191,91,138,303,221),(62,157,334,240,92,127,304,210),(63,146,335,229,93,176,305,199),(64,135,336,218,94,165,306,188),(65,124,337,207,95,154,307,237),(66,173,338,196,96,143,308,226),(67,162,339,185,97,132,309,215),(68,151,340,234,98,121,310,204),(69,140,341,223,99,170,311,193),(70,129,342,212,100,159,312,182),(71,178,343,201,101,148,313,231),(72,167,344,190,102,137,314,220),(73,156,345,239,103,126,315,209),(74,145,346,228,104,175,316,198),(75,134,347,217,105,164,317,187),(76,123,348,206,106,153,318,236),(77,172,349,195,107,142,319,225),(78,161,350,184,108,131,320,214),(79,150,351,233,109,180,321,203),(80,139,352,222,110,169,322,192),(81,128,353,211,111,158,323,181),(82,177,354,200,112,147,324,230),(83,166,355,189,113,136,325,219),(84,155,356,238,114,125,326,208),(85,144,357,227,115,174,327,197),(86,133,358,216,116,163,328,186),(87,122,359,205,117,152,329,235),(88,171,360,194,118,141,330,224),(89,160,301,183,119,130,331,213),(90,149,302,232,120,179,332,202)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C8 | S3 | D4 | Q8 | D5 | D6 | M4(2) | Dic5 | D10 | Dic6 | C3⋊D4 | C4×S3 | C5⋊2C8 | Dic10 | D20 | S3×C8 | C8⋊S3 | C4.Dic5 | S3×D5 | C3⋊D20 | C15⋊Q8 | S3×Dic5 | S3×C5⋊2C8 | D6.Dic5 |
| kernel | C60.15Q8 | C6×C5⋊2C8 | Dic3×C20 | C2×C15⋊3C8 | C10×Dic3 | C5×Dic3 | C2×C5⋊2C8 | C60 | C60 | C4×Dic3 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C20 | C20 | C2×C10 | Dic3 | C12 | C12 | C10 | C10 | C6 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C60.15Q8 ►in GL4(𝔽241) generated by
| 131 | 131 | 0 | 0 |
| 110 | 64 | 0 | 0 |
| 0 | 0 | 1 | 240 |
| 0 | 0 | 1 | 0 |
| 44 | 3 | 0 | 0 |
| 238 | 197 | 0 | 0 |
| 0 | 0 | 5 | 19 |
| 0 | 0 | 24 | 236 |
| 101 | 30 | 0 | 0 |
| 120 | 140 | 0 | 0 |
| 0 | 0 | 171 | 140 |
| 0 | 0 | 101 | 70 |
G:=sub<GL(4,GF(241))| [131,110,0,0,131,64,0,0,0,0,1,1,0,0,240,0],[44,238,0,0,3,197,0,0,0,0,5,24,0,0,19,236],[101,120,0,0,30,140,0,0,0,0,171,101,0,0,140,70] >;
C60.15Q8 in GAP, Magma, Sage, TeX
C_{60}._{15}Q_8 % in TeX
G:=Group("C60.15Q8"); // GroupNames label
G:=SmallGroup(480,60);
// by ID
G=gap.SmallGroup(480,60);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^45*b^2,b*a*b^-1=a^41,c*a*c^-1=a^49,c*b*c^-1=b^-1>;
// generators/relations