metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.7Q8, C20.1Dic6, C30.13SD16, C12.10Dic10, C5⋊2(C8⋊Dic3), C15⋊4(C4.Q8), C4.4(C15⋊Q8), C12.68(C4×D5), (C2×C20).58D6, (C2×C30).29D4, C5⋊2C8⋊1Dic3, C6.3(Q8⋊D5), C4⋊Dic3.2D5, C30.25(C4⋊C4), C60.111(C2×C4), (C2×C10).34D12, C6.3(D4.D5), C60⋊5C4.22C2, C4.11(D5×Dic3), C3⋊1(C20.Q8), C10.6(C24⋊C2), (C2×C12).287D10, C10.9(C4⋊Dic3), C20.25(C2×Dic3), (C2×C60).131C22, C2.3(Dic6⋊D5), C2.3(D12.D5), C6.3(C10.D4), C2.3(C30.Q8), C22.16(C5⋊D12), (C3×C5⋊2C8)⋊1C4, (C6×C5⋊2C8).4C2, (C2×C5⋊2C8).3S3, (C2×C4).139(S3×D5), (C5×C4⋊Dic3).2C2, (C2×C6).28(C5⋊D4), SmallGroup(480,61)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.7Q8
G = < a,b,c | a60=b4=1, c2=a45b2, bab-1=a11, cac-1=a49, cbc-1=a15b-1 >
Subgroups: 316 in 72 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, C20, C2×C10, C24, C2×Dic3, C2×C12, C30, C4.Q8, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C4⋊Dic3, C4⋊Dic3, C2×C24, C5×Dic3, Dic15, C60, C2×C30, C2×C5⋊2C8, C4⋊Dic5, C5×C4⋊C4, C8⋊Dic3, C3×C5⋊2C8, C10×Dic3, C2×Dic15, C2×C60, C20.Q8, C6×C5⋊2C8, C5×C4⋊Dic3, C60⋊5C4, C60.7Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, Dic3, D6, C4⋊C4, SD16, D10, Dic6, D12, C2×Dic3, C4.Q8, Dic10, C4×D5, C5⋊D4, C24⋊C2, C4⋊Dic3, S3×D5, C10.D4, D4.D5, Q8⋊D5, C8⋊Dic3, D5×Dic3, C5⋊D12, C15⋊Q8, C20.Q8, D12.D5, Dic6⋊D5, C30.Q8, C60.7Q8
►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 91 296 180)(2 102 297 131)(3 113 298 142)(4 64 299 153)(5 75 300 164)(6 86 241 175)(7 97 242 126)(8 108 243 137)(9 119 244 148)(10 70 245 159)(11 81 246 170)(12 92 247 121)(13 103 248 132)(14 114 249 143)(15 65 250 154)(16 76 251 165)(17 87 252 176)(18 98 253 127)(19 109 254 138)(20 120 255 149)(21 71 256 160)(22 82 257 171)(23 93 258 122)(24 104 259 133)(25 115 260 144)(26 66 261 155)(27 77 262 166)(28 88 263 177)(29 99 264 128)(30 110 265 139)(31 61 266 150)(32 72 267 161)(33 83 268 172)(34 94 269 123)(35 105 270 134)(36 116 271 145)(37 67 272 156)(38 78 273 167)(39 89 274 178)(40 100 275 129)(41 111 276 140)(42 62 277 151)(43 73 278 162)(44 84 279 173)(45 95 280 124)(46 106 281 135)(47 117 282 146)(48 68 283 157)(49 79 284 168)(50 90 285 179)(51 101 286 130)(52 112 287 141)(53 63 288 152)(54 74 289 163)(55 85 290 174)(56 96 291 125)(57 107 292 136)(58 118 293 147)(59 69 294 158)(60 80 295 169)(181 323 446 365)(182 334 447 376)(183 345 448 387)(184 356 449 398)(185 307 450 409)(186 318 451 420)(187 329 452 371)(188 340 453 382)(189 351 454 393)(190 302 455 404)(191 313 456 415)(192 324 457 366)(193 335 458 377)(194 346 459 388)(195 357 460 399)(196 308 461 410)(197 319 462 361)(198 330 463 372)(199 341 464 383)(200 352 465 394)(201 303 466 405)(202 314 467 416)(203 325 468 367)(204 336 469 378)(205 347 470 389)(206 358 471 400)(207 309 472 411)(208 320 473 362)(209 331 474 373)(210 342 475 384)(211 353 476 395)(212 304 477 406)(213 315 478 417)(214 326 479 368)(215 337 480 379)(216 348 421 390)(217 359 422 401)(218 310 423 412)(219 321 424 363)(220 332 425 374)(221 343 426 385)(222 354 427 396)(223 305 428 407)(224 316 429 418)(225 327 430 369)(226 338 431 380)(227 349 432 391)(228 360 433 402)(229 311 434 413)(230 322 435 364)(231 333 436 375)(232 344 437 386)(233 355 438 397)(234 306 439 408)(235 317 440 419)(236 328 441 370)(237 339 442 381)(238 350 443 392)(239 301 444 403)(240 312 445 414)
(1 352 281 379 31 322 251 409)(2 341 282 368 32 311 252 398)(3 330 283 417 33 360 253 387)(4 319 284 406 34 349 254 376)(5 308 285 395 35 338 255 365)(6 357 286 384 36 327 256 414)(7 346 287 373 37 316 257 403)(8 335 288 362 38 305 258 392)(9 324 289 411 39 354 259 381)(10 313 290 400 40 343 260 370)(11 302 291 389 41 332 261 419)(12 351 292 378 42 321 262 408)(13 340 293 367 43 310 263 397)(14 329 294 416 44 359 264 386)(15 318 295 405 45 348 265 375)(16 307 296 394 46 337 266 364)(17 356 297 383 47 326 267 413)(18 345 298 372 48 315 268 402)(19 334 299 361 49 304 269 391)(20 323 300 410 50 353 270 380)(21 312 241 399 51 342 271 369)(22 301 242 388 52 331 272 418)(23 350 243 377 53 320 273 407)(24 339 244 366 54 309 274 396)(25 328 245 415 55 358 275 385)(26 317 246 404 56 347 276 374)(27 306 247 393 57 336 277 363)(28 355 248 382 58 325 278 412)(29 344 249 371 59 314 279 401)(30 333 250 420 60 303 280 390)(61 185 135 435 91 215 165 465)(62 234 136 424 92 204 166 454)(63 223 137 473 93 193 167 443)(64 212 138 462 94 182 168 432)(65 201 139 451 95 231 169 421)(66 190 140 440 96 220 170 470)(67 239 141 429 97 209 171 459)(68 228 142 478 98 198 172 448)(69 217 143 467 99 187 173 437)(70 206 144 456 100 236 174 426)(71 195 145 445 101 225 175 475)(72 184 146 434 102 214 176 464)(73 233 147 423 103 203 177 453)(74 222 148 472 104 192 178 442)(75 211 149 461 105 181 179 431)(76 200 150 450 106 230 180 480)(77 189 151 439 107 219 121 469)(78 238 152 428 108 208 122 458)(79 227 153 477 109 197 123 447)(80 216 154 466 110 186 124 436)(81 205 155 455 111 235 125 425)(82 194 156 444 112 224 126 474)(83 183 157 433 113 213 127 463)(84 232 158 422 114 202 128 452)(85 221 159 471 115 191 129 441)(86 210 160 460 116 240 130 430)(87 199 161 449 117 229 131 479)(88 188 162 438 118 218 132 468)(89 237 163 427 119 207 133 457)(90 226 164 476 120 196 134 446)
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), (1,91,296,180)(2,102,297,131)(3,113,298,142)(4,64,299,153)(5,75,300,164)(6,86,241,175)(7,97,242,126)(8,108,243,137)(9,119,244,148)(10,70,245,159)(11,81,246,170)(12,92,247,121)(13,103,248,132)(14,114,249,143)(15,65,250,154)(16,76,251,165)(17,87,252,176)(18,98,253,127)(19,109,254,138)(20,120,255,149)(21,71,256,160)(22,82,257,171)(23,93,258,122)(24,104,259,133)(25,115,260,144)(26,66,261,155)(27,77,262,166)(28,88,263,177)(29,99,264,128)(30,110,265,139)(31,61,266,150)(32,72,267,161)(33,83,268,172)(34,94,269,123)(35,105,270,134)(36,116,271,145)(37,67,272,156)(38,78,273,167)(39,89,274,178)(40,100,275,129)(41,111,276,140)(42,62,277,151)(43,73,278,162)(44,84,279,173)(45,95,280,124)(46,106,281,135)(47,117,282,146)(48,68,283,157)(49,79,284,168)(50,90,285,179)(51,101,286,130)(52,112,287,141)(53,63,288,152)(54,74,289,163)(55,85,290,174)(56,96,291,125)(57,107,292,136)(58,118,293,147)(59,69,294,158)(60,80,295,169)(181,323,446,365)(182,334,447,376)(183,345,448,387)(184,356,449,398)(185,307,450,409)(186,318,451,420)(187,329,452,371)(188,340,453,382)(189,351,454,393)(190,302,455,404)(191,313,456,415)(192,324,457,366)(193,335,458,377)(194,346,459,388)(195,357,460,399)(196,308,461,410)(197,319,462,361)(198,330,463,372)(199,341,464,383)(200,352,465,394)(201,303,466,405)(202,314,467,416)(203,325,468,367)(204,336,469,378)(205,347,470,389)(206,358,471,400)(207,309,472,411)(208,320,473,362)(209,331,474,373)(210,342,475,384)(211,353,476,395)(212,304,477,406)(213,315,478,417)(214,326,479,368)(215,337,480,379)(216,348,421,390)(217,359,422,401)(218,310,423,412)(219,321,424,363)(220,332,425,374)(221,343,426,385)(222,354,427,396)(223,305,428,407)(224,316,429,418)(225,327,430,369)(226,338,431,380)(227,349,432,391)(228,360,433,402)(229,311,434,413)(230,322,435,364)(231,333,436,375)(232,344,437,386)(233,355,438,397)(234,306,439,408)(235,317,440,419)(236,328,441,370)(237,339,442,381)(238,350,443,392)(239,301,444,403)(240,312,445,414), (1,352,281,379,31,322,251,409)(2,341,282,368,32,311,252,398)(3,330,283,417,33,360,253,387)(4,319,284,406,34,349,254,376)(5,308,285,395,35,338,255,365)(6,357,286,384,36,327,256,414)(7,346,287,373,37,316,257,403)(8,335,288,362,38,305,258,392)(9,324,289,411,39,354,259,381)(10,313,290,400,40,343,260,370)(11,302,291,389,41,332,261,419)(12,351,292,378,42,321,262,408)(13,340,293,367,43,310,263,397)(14,329,294,416,44,359,264,386)(15,318,295,405,45,348,265,375)(16,307,296,394,46,337,266,364)(17,356,297,383,47,326,267,413)(18,345,298,372,48,315,268,402)(19,334,299,361,49,304,269,391)(20,323,300,410,50,353,270,380)(21,312,241,399,51,342,271,369)(22,301,242,388,52,331,272,418)(23,350,243,377,53,320,273,407)(24,339,244,366,54,309,274,396)(25,328,245,415,55,358,275,385)(26,317,246,404,56,347,276,374)(27,306,247,393,57,336,277,363)(28,355,248,382,58,325,278,412)(29,344,249,371,59,314,279,401)(30,333,250,420,60,303,280,390)(61,185,135,435,91,215,165,465)(62,234,136,424,92,204,166,454)(63,223,137,473,93,193,167,443)(64,212,138,462,94,182,168,432)(65,201,139,451,95,231,169,421)(66,190,140,440,96,220,170,470)(67,239,141,429,97,209,171,459)(68,228,142,478,98,198,172,448)(69,217,143,467,99,187,173,437)(70,206,144,456,100,236,174,426)(71,195,145,445,101,225,175,475)(72,184,146,434,102,214,176,464)(73,233,147,423,103,203,177,453)(74,222,148,472,104,192,178,442)(75,211,149,461,105,181,179,431)(76,200,150,450,106,230,180,480)(77,189,151,439,107,219,121,469)(78,238,152,428,108,208,122,458)(79,227,153,477,109,197,123,447)(80,216,154,466,110,186,124,436)(81,205,155,455,111,235,125,425)(82,194,156,444,112,224,126,474)(83,183,157,433,113,213,127,463)(84,232,158,422,114,202,128,452)(85,221,159,471,115,191,129,441)(86,210,160,460,116,240,130,430)(87,199,161,449,117,229,131,479)(88,188,162,438,118,218,132,468)(89,237,163,427,119,207,133,457)(90,226,164,476,120,196,134,446)>;
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), (1,91,296,180)(2,102,297,131)(3,113,298,142)(4,64,299,153)(5,75,300,164)(6,86,241,175)(7,97,242,126)(8,108,243,137)(9,119,244,148)(10,70,245,159)(11,81,246,170)(12,92,247,121)(13,103,248,132)(14,114,249,143)(15,65,250,154)(16,76,251,165)(17,87,252,176)(18,98,253,127)(19,109,254,138)(20,120,255,149)(21,71,256,160)(22,82,257,171)(23,93,258,122)(24,104,259,133)(25,115,260,144)(26,66,261,155)(27,77,262,166)(28,88,263,177)(29,99,264,128)(30,110,265,139)(31,61,266,150)(32,72,267,161)(33,83,268,172)(34,94,269,123)(35,105,270,134)(36,116,271,145)(37,67,272,156)(38,78,273,167)(39,89,274,178)(40,100,275,129)(41,111,276,140)(42,62,277,151)(43,73,278,162)(44,84,279,173)(45,95,280,124)(46,106,281,135)(47,117,282,146)(48,68,283,157)(49,79,284,168)(50,90,285,179)(51,101,286,130)(52,112,287,141)(53,63,288,152)(54,74,289,163)(55,85,290,174)(56,96,291,125)(57,107,292,136)(58,118,293,147)(59,69,294,158)(60,80,295,169)(181,323,446,365)(182,334,447,376)(183,345,448,387)(184,356,449,398)(185,307,450,409)(186,318,451,420)(187,329,452,371)(188,340,453,382)(189,351,454,393)(190,302,455,404)(191,313,456,415)(192,324,457,366)(193,335,458,377)(194,346,459,388)(195,357,460,399)(196,308,461,410)(197,319,462,361)(198,330,463,372)(199,341,464,383)(200,352,465,394)(201,303,466,405)(202,314,467,416)(203,325,468,367)(204,336,469,378)(205,347,470,389)(206,358,471,400)(207,309,472,411)(208,320,473,362)(209,331,474,373)(210,342,475,384)(211,353,476,395)(212,304,477,406)(213,315,478,417)(214,326,479,368)(215,337,480,379)(216,348,421,390)(217,359,422,401)(218,310,423,412)(219,321,424,363)(220,332,425,374)(221,343,426,385)(222,354,427,396)(223,305,428,407)(224,316,429,418)(225,327,430,369)(226,338,431,380)(227,349,432,391)(228,360,433,402)(229,311,434,413)(230,322,435,364)(231,333,436,375)(232,344,437,386)(233,355,438,397)(234,306,439,408)(235,317,440,419)(236,328,441,370)(237,339,442,381)(238,350,443,392)(239,301,444,403)(240,312,445,414), (1,352,281,379,31,322,251,409)(2,341,282,368,32,311,252,398)(3,330,283,417,33,360,253,387)(4,319,284,406,34,349,254,376)(5,308,285,395,35,338,255,365)(6,357,286,384,36,327,256,414)(7,346,287,373,37,316,257,403)(8,335,288,362,38,305,258,392)(9,324,289,411,39,354,259,381)(10,313,290,400,40,343,260,370)(11,302,291,389,41,332,261,419)(12,351,292,378,42,321,262,408)(13,340,293,367,43,310,263,397)(14,329,294,416,44,359,264,386)(15,318,295,405,45,348,265,375)(16,307,296,394,46,337,266,364)(17,356,297,383,47,326,267,413)(18,345,298,372,48,315,268,402)(19,334,299,361,49,304,269,391)(20,323,300,410,50,353,270,380)(21,312,241,399,51,342,271,369)(22,301,242,388,52,331,272,418)(23,350,243,377,53,320,273,407)(24,339,244,366,54,309,274,396)(25,328,245,415,55,358,275,385)(26,317,246,404,56,347,276,374)(27,306,247,393,57,336,277,363)(28,355,248,382,58,325,278,412)(29,344,249,371,59,314,279,401)(30,333,250,420,60,303,280,390)(61,185,135,435,91,215,165,465)(62,234,136,424,92,204,166,454)(63,223,137,473,93,193,167,443)(64,212,138,462,94,182,168,432)(65,201,139,451,95,231,169,421)(66,190,140,440,96,220,170,470)(67,239,141,429,97,209,171,459)(68,228,142,478,98,198,172,448)(69,217,143,467,99,187,173,437)(70,206,144,456,100,236,174,426)(71,195,145,445,101,225,175,475)(72,184,146,434,102,214,176,464)(73,233,147,423,103,203,177,453)(74,222,148,472,104,192,178,442)(75,211,149,461,105,181,179,431)(76,200,150,450,106,230,180,480)(77,189,151,439,107,219,121,469)(78,238,152,428,108,208,122,458)(79,227,153,477,109,197,123,447)(80,216,154,466,110,186,124,436)(81,205,155,455,111,235,125,425)(82,194,156,444,112,224,126,474)(83,183,157,433,113,213,127,463)(84,232,158,422,114,202,128,452)(85,221,159,471,115,191,129,441)(86,210,160,460,116,240,130,430)(87,199,161,449,117,229,131,479)(88,188,162,438,118,218,132,468)(89,237,163,427,119,207,133,457)(90,226,164,476,120,196,134,446) );
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)], [(1,91,296,180),(2,102,297,131),(3,113,298,142),(4,64,299,153),(5,75,300,164),(6,86,241,175),(7,97,242,126),(8,108,243,137),(9,119,244,148),(10,70,245,159),(11,81,246,170),(12,92,247,121),(13,103,248,132),(14,114,249,143),(15,65,250,154),(16,76,251,165),(17,87,252,176),(18,98,253,127),(19,109,254,138),(20,120,255,149),(21,71,256,160),(22,82,257,171),(23,93,258,122),(24,104,259,133),(25,115,260,144),(26,66,261,155),(27,77,262,166),(28,88,263,177),(29,99,264,128),(30,110,265,139),(31,61,266,150),(32,72,267,161),(33,83,268,172),(34,94,269,123),(35,105,270,134),(36,116,271,145),(37,67,272,156),(38,78,273,167),(39,89,274,178),(40,100,275,129),(41,111,276,140),(42,62,277,151),(43,73,278,162),(44,84,279,173),(45,95,280,124),(46,106,281,135),(47,117,282,146),(48,68,283,157),(49,79,284,168),(50,90,285,179),(51,101,286,130),(52,112,287,141),(53,63,288,152),(54,74,289,163),(55,85,290,174),(56,96,291,125),(57,107,292,136),(58,118,293,147),(59,69,294,158),(60,80,295,169),(181,323,446,365),(182,334,447,376),(183,345,448,387),(184,356,449,398),(185,307,450,409),(186,318,451,420),(187,329,452,371),(188,340,453,382),(189,351,454,393),(190,302,455,404),(191,313,456,415),(192,324,457,366),(193,335,458,377),(194,346,459,388),(195,357,460,399),(196,308,461,410),(197,319,462,361),(198,330,463,372),(199,341,464,383),(200,352,465,394),(201,303,466,405),(202,314,467,416),(203,325,468,367),(204,336,469,378),(205,347,470,389),(206,358,471,400),(207,309,472,411),(208,320,473,362),(209,331,474,373),(210,342,475,384),(211,353,476,395),(212,304,477,406),(213,315,478,417),(214,326,479,368),(215,337,480,379),(216,348,421,390),(217,359,422,401),(218,310,423,412),(219,321,424,363),(220,332,425,374),(221,343,426,385),(222,354,427,396),(223,305,428,407),(224,316,429,418),(225,327,430,369),(226,338,431,380),(227,349,432,391),(228,360,433,402),(229,311,434,413),(230,322,435,364),(231,333,436,375),(232,344,437,386),(233,355,438,397),(234,306,439,408),(235,317,440,419),(236,328,441,370),(237,339,442,381),(238,350,443,392),(239,301,444,403),(240,312,445,414)], [(1,352,281,379,31,322,251,409),(2,341,282,368,32,311,252,398),(3,330,283,417,33,360,253,387),(4,319,284,406,34,349,254,376),(5,308,285,395,35,338,255,365),(6,357,286,384,36,327,256,414),(7,346,287,373,37,316,257,403),(8,335,288,362,38,305,258,392),(9,324,289,411,39,354,259,381),(10,313,290,400,40,343,260,370),(11,302,291,389,41,332,261,419),(12,351,292,378,42,321,262,408),(13,340,293,367,43,310,263,397),(14,329,294,416,44,359,264,386),(15,318,295,405,45,348,265,375),(16,307,296,394,46,337,266,364),(17,356,297,383,47,326,267,413),(18,345,298,372,48,315,268,402),(19,334,299,361,49,304,269,391),(20,323,300,410,50,353,270,380),(21,312,241,399,51,342,271,369),(22,301,242,388,52,331,272,418),(23,350,243,377,53,320,273,407),(24,339,244,366,54,309,274,396),(25,328,245,415,55,358,275,385),(26,317,246,404,56,347,276,374),(27,306,247,393,57,336,277,363),(28,355,248,382,58,325,278,412),(29,344,249,371,59,314,279,401),(30,333,250,420,60,303,280,390),(61,185,135,435,91,215,165,465),(62,234,136,424,92,204,166,454),(63,223,137,473,93,193,167,443),(64,212,138,462,94,182,168,432),(65,201,139,451,95,231,169,421),(66,190,140,440,96,220,170,470),(67,239,141,429,97,209,171,459),(68,228,142,478,98,198,172,448),(69,217,143,467,99,187,173,437),(70,206,144,456,100,236,174,426),(71,195,145,445,101,225,175,475),(72,184,146,434,102,214,176,464),(73,233,147,423,103,203,177,453),(74,222,148,472,104,192,178,442),(75,211,149,461,105,181,179,431),(76,200,150,450,106,230,180,480),(77,189,151,439,107,219,121,469),(78,238,152,428,108,208,122,458),(79,227,153,477,109,197,123,447),(80,216,154,466,110,186,124,436),(81,205,155,455,111,235,125,425),(82,194,156,444,112,224,126,474),(83,183,157,433,113,213,127,463),(84,232,158,422,114,202,128,452),(85,221,159,471,115,191,129,441),(86,210,160,460,116,240,130,430),(87,199,161,449,117,229,131,479),(88,188,162,438,118,218,132,468),(89,237,163,427,119,207,133,457),(90,226,164,476,120,196,134,446)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | + | + | - | + | - | + | - | + | - | - | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D5 | Dic3 | D6 | SD16 | D10 | Dic6 | D12 | Dic10 | C4×D5 | C5⋊D4 | C24⋊C2 | S3×D5 | D4.D5 | Q8⋊D5 | D5×Dic3 | C15⋊Q8 | C5⋊D12 | D12.D5 | Dic6⋊D5 |
| kernel | C60.7Q8 | C6×C5⋊2C8 | C5×C4⋊Dic3 | C60⋊5C4 | C3×C5⋊2C8 | C2×C5⋊2C8 | C60 | C2×C30 | C4⋊Dic3 | C5⋊2C8 | C2×C20 | C30 | C2×C12 | C20 | C2×C10 | C12 | C12 | C2×C6 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C60.7Q8 ►in GL4(𝔽241) generated by
| 198 | 99 | 0 | 0 |
| 142 | 99 | 0 | 0 |
| 0 | 0 | 240 | 190 |
| 0 | 0 | 51 | 190 |
| 23 | 78 | 0 | 0 |
| 55 | 218 | 0 | 0 |
| 0 | 0 | 197 | 3 |
| 0 | 0 | 238 | 44 |
| 66 | 94 | 0 | 0 |
| 147 | 213 | 0 | 0 |
| 0 | 0 | 183 | 128 |
| 0 | 0 | 194 | 58 |
G:=sub<GL(4,GF(241))| [198,142,0,0,99,99,0,0,0,0,240,51,0,0,190,190],[23,55,0,0,78,218,0,0,0,0,197,238,0,0,3,44],[66,147,0,0,94,213,0,0,0,0,183,194,0,0,128,58] >;
C60.7Q8 in GAP, Magma, Sage, TeX
C_{60}._7Q_8 % in TeX
G:=Group("C60.7Q8"); // GroupNames label
G:=SmallGroup(480,61);
// by ID
G=gap.SmallGroup(480,61);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,36,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^45*b^2,b*a*b^-1=a^11,c*a*c^-1=a^49,c*b*c^-1=a^15*b^-1>;
// generators/relations