metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.14Q8, Dic15⋊5C8, C60.103D4, C20.14Dic6, C12.14Dic10, C30.18M4(2), C15⋊8(C4⋊C8), C6.5(C8×D5), C10.14(S3×C8), C30.32(C2×C8), C5⋊3(Dic3⋊C8), C4.11(C15⋊Q8), C30.23(C4⋊C4), (C2×C20).325D6, C6.3(C8⋊D5), C3⋊1(C20.8Q8), C10.8(C8⋊S3), (C2×C12).329D10, C20.88(C3⋊D4), C12.88(C5⋊D4), C4.30(C15⋊D4), C2.5(D15⋊2C8), C10.8(Dic3⋊C4), (C2×C60).227C22, (C4×Dic15).19C2, (C2×Dic15).20C4, C6.2(C10.D4), C2.1(Dic15⋊5C4), C2.3(D30.5C4), C22.11(D30.C2), (C2×C3⋊C8).10D5, (C10×C3⋊C8).12C2, (C2×C5⋊2C8).9S3, (C2×C6).19(C4×D5), (C6×C5⋊2C8).11C2, (C2×C10).42(C4×S3), (C2×C30).87(C2×C4), (C2×C4).230(S3×D5), SmallGroup(480,59)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.14Q8
G = < a,b,c | a60=1, b4=a30, c2=b2, bab-1=a41, cac-1=a49, cbc-1=a15b3 >
Subgroups: 284 in 76 conjugacy classes, 42 normal (40 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C4⋊C8, C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, Dic15, Dic15, C60, C2×C30, C2×C5⋊2C8, C4×Dic5, C2×C40, Dic3⋊C8, C5×C3⋊C8, C3×C5⋊2C8, C2×Dic15, C2×C60, C20.8Q8, C6×C5⋊2C8, C10×C3⋊C8, C4×Dic15, C60.14Q8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, D5, D6, C4⋊C4, C2×C8, M4(2), D10, Dic6, C4×S3, C3⋊D4, C4⋊C8, Dic10, C4×D5, C5⋊D4, S3×C8, C8⋊S3, Dic3⋊C4, S3×D5, C8×D5, C8⋊D5, C10.D4, Dic3⋊C8, D30.C2, C15⋊D4, C15⋊Q8, C20.8Q8, D15⋊2C8, D30.5C4, Dic15⋊5C4, C60.14Q8
►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)
(1 226 140 258 31 196 170 288)(2 207 141 299 32 237 171 269)(3 188 142 280 33 218 172 250)(4 229 143 261 34 199 173 291)(5 210 144 242 35 240 174 272)(6 191 145 283 36 221 175 253)(7 232 146 264 37 202 176 294)(8 213 147 245 38 183 177 275)(9 194 148 286 39 224 178 256)(10 235 149 267 40 205 179 297)(11 216 150 248 41 186 180 278)(12 197 151 289 42 227 121 259)(13 238 152 270 43 208 122 300)(14 219 153 251 44 189 123 281)(15 200 154 292 45 230 124 262)(16 181 155 273 46 211 125 243)(17 222 156 254 47 192 126 284)(18 203 157 295 48 233 127 265)(19 184 158 276 49 214 128 246)(20 225 159 257 50 195 129 287)(21 206 160 298 51 236 130 268)(22 187 161 279 52 217 131 249)(23 228 162 260 53 198 132 290)(24 209 163 241 54 239 133 271)(25 190 164 282 55 220 134 252)(26 231 165 263 56 201 135 293)(27 212 166 244 57 182 136 274)(28 193 167 285 58 223 137 255)(29 234 168 266 59 204 138 296)(30 215 169 247 60 185 139 277)(61 423 373 334 91 453 403 304)(62 464 374 315 92 434 404 345)(63 445 375 356 93 475 405 326)(64 426 376 337 94 456 406 307)(65 467 377 318 95 437 407 348)(66 448 378 359 96 478 408 329)(67 429 379 340 97 459 409 310)(68 470 380 321 98 440 410 351)(69 451 381 302 99 421 411 332)(70 432 382 343 100 462 412 313)(71 473 383 324 101 443 413 354)(72 454 384 305 102 424 414 335)(73 435 385 346 103 465 415 316)(74 476 386 327 104 446 416 357)(75 457 387 308 105 427 417 338)(76 438 388 349 106 468 418 319)(77 479 389 330 107 449 419 360)(78 460 390 311 108 430 420 341)(79 441 391 352 109 471 361 322)(80 422 392 333 110 452 362 303)(81 463 393 314 111 433 363 344)(82 444 394 355 112 474 364 325)(83 425 395 336 113 455 365 306)(84 466 396 317 114 436 366 347)(85 447 397 358 115 477 367 328)(86 428 398 339 116 458 368 309)(87 469 399 320 117 439 369 350)(88 450 400 301 118 480 370 331)(89 431 401 342 119 461 371 312)(90 472 402 323 120 442 372 353)
(1 96 140 408 31 66 170 378)(2 85 141 397 32 115 171 367)(3 74 142 386 33 104 172 416)(4 63 143 375 34 93 173 405)(5 112 144 364 35 82 174 394)(6 101 145 413 36 71 175 383)(7 90 146 402 37 120 176 372)(8 79 147 391 38 109 177 361)(9 68 148 380 39 98 178 410)(10 117 149 369 40 87 179 399)(11 106 150 418 41 76 180 388)(12 95 151 407 42 65 121 377)(13 84 152 396 43 114 122 366)(14 73 153 385 44 103 123 415)(15 62 154 374 45 92 124 404)(16 111 155 363 46 81 125 393)(17 100 156 412 47 70 126 382)(18 89 157 401 48 119 127 371)(19 78 158 390 49 108 128 420)(20 67 159 379 50 97 129 409)(21 116 160 368 51 86 130 398)(22 105 161 417 52 75 131 387)(23 94 162 406 53 64 132 376)(24 83 163 395 54 113 133 365)(25 72 164 384 55 102 134 414)(26 61 165 373 56 91 135 403)(27 110 166 362 57 80 136 392)(28 99 167 411 58 69 137 381)(29 88 168 400 59 118 138 370)(30 77 169 389 60 107 139 419)(181 359 273 478 211 329 243 448)(182 348 274 467 212 318 244 437)(183 337 275 456 213 307 245 426)(184 326 276 445 214 356 246 475)(185 315 277 434 215 345 247 464)(186 304 278 423 216 334 248 453)(187 353 279 472 217 323 249 442)(188 342 280 461 218 312 250 431)(189 331 281 450 219 301 251 480)(190 320 282 439 220 350 252 469)(191 309 283 428 221 339 253 458)(192 358 284 477 222 328 254 447)(193 347 285 466 223 317 255 436)(194 336 286 455 224 306 256 425)(195 325 287 444 225 355 257 474)(196 314 288 433 226 344 258 463)(197 303 289 422 227 333 259 452)(198 352 290 471 228 322 260 441)(199 341 291 460 229 311 261 430)(200 330 292 449 230 360 262 479)(201 319 293 438 231 349 263 468)(202 308 294 427 232 338 264 457)(203 357 295 476 233 327 265 446)(204 346 296 465 234 316 266 435)(205 335 297 454 235 305 267 424)(206 324 298 443 236 354 268 473)(207 313 299 432 237 343 269 462)(208 302 300 421 238 332 270 451)(209 351 241 470 239 321 271 440)(210 340 242 459 240 310 272 429)
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), (1,226,140,258,31,196,170,288)(2,207,141,299,32,237,171,269)(3,188,142,280,33,218,172,250)(4,229,143,261,34,199,173,291)(5,210,144,242,35,240,174,272)(6,191,145,283,36,221,175,253)(7,232,146,264,37,202,176,294)(8,213,147,245,38,183,177,275)(9,194,148,286,39,224,178,256)(10,235,149,267,40,205,179,297)(11,216,150,248,41,186,180,278)(12,197,151,289,42,227,121,259)(13,238,152,270,43,208,122,300)(14,219,153,251,44,189,123,281)(15,200,154,292,45,230,124,262)(16,181,155,273,46,211,125,243)(17,222,156,254,47,192,126,284)(18,203,157,295,48,233,127,265)(19,184,158,276,49,214,128,246)(20,225,159,257,50,195,129,287)(21,206,160,298,51,236,130,268)(22,187,161,279,52,217,131,249)(23,228,162,260,53,198,132,290)(24,209,163,241,54,239,133,271)(25,190,164,282,55,220,134,252)(26,231,165,263,56,201,135,293)(27,212,166,244,57,182,136,274)(28,193,167,285,58,223,137,255)(29,234,168,266,59,204,138,296)(30,215,169,247,60,185,139,277)(61,423,373,334,91,453,403,304)(62,464,374,315,92,434,404,345)(63,445,375,356,93,475,405,326)(64,426,376,337,94,456,406,307)(65,467,377,318,95,437,407,348)(66,448,378,359,96,478,408,329)(67,429,379,340,97,459,409,310)(68,470,380,321,98,440,410,351)(69,451,381,302,99,421,411,332)(70,432,382,343,100,462,412,313)(71,473,383,324,101,443,413,354)(72,454,384,305,102,424,414,335)(73,435,385,346,103,465,415,316)(74,476,386,327,104,446,416,357)(75,457,387,308,105,427,417,338)(76,438,388,349,106,468,418,319)(77,479,389,330,107,449,419,360)(78,460,390,311,108,430,420,341)(79,441,391,352,109,471,361,322)(80,422,392,333,110,452,362,303)(81,463,393,314,111,433,363,344)(82,444,394,355,112,474,364,325)(83,425,395,336,113,455,365,306)(84,466,396,317,114,436,366,347)(85,447,397,358,115,477,367,328)(86,428,398,339,116,458,368,309)(87,469,399,320,117,439,369,350)(88,450,400,301,118,480,370,331)(89,431,401,342,119,461,371,312)(90,472,402,323,120,442,372,353), (1,96,140,408,31,66,170,378)(2,85,141,397,32,115,171,367)(3,74,142,386,33,104,172,416)(4,63,143,375,34,93,173,405)(5,112,144,364,35,82,174,394)(6,101,145,413,36,71,175,383)(7,90,146,402,37,120,176,372)(8,79,147,391,38,109,177,361)(9,68,148,380,39,98,178,410)(10,117,149,369,40,87,179,399)(11,106,150,418,41,76,180,388)(12,95,151,407,42,65,121,377)(13,84,152,396,43,114,122,366)(14,73,153,385,44,103,123,415)(15,62,154,374,45,92,124,404)(16,111,155,363,46,81,125,393)(17,100,156,412,47,70,126,382)(18,89,157,401,48,119,127,371)(19,78,158,390,49,108,128,420)(20,67,159,379,50,97,129,409)(21,116,160,368,51,86,130,398)(22,105,161,417,52,75,131,387)(23,94,162,406,53,64,132,376)(24,83,163,395,54,113,133,365)(25,72,164,384,55,102,134,414)(26,61,165,373,56,91,135,403)(27,110,166,362,57,80,136,392)(28,99,167,411,58,69,137,381)(29,88,168,400,59,118,138,370)(30,77,169,389,60,107,139,419)(181,359,273,478,211,329,243,448)(182,348,274,467,212,318,244,437)(183,337,275,456,213,307,245,426)(184,326,276,445,214,356,246,475)(185,315,277,434,215,345,247,464)(186,304,278,423,216,334,248,453)(187,353,279,472,217,323,249,442)(188,342,280,461,218,312,250,431)(189,331,281,450,219,301,251,480)(190,320,282,439,220,350,252,469)(191,309,283,428,221,339,253,458)(192,358,284,477,222,328,254,447)(193,347,285,466,223,317,255,436)(194,336,286,455,224,306,256,425)(195,325,287,444,225,355,257,474)(196,314,288,433,226,344,258,463)(197,303,289,422,227,333,259,452)(198,352,290,471,228,322,260,441)(199,341,291,460,229,311,261,430)(200,330,292,449,230,360,262,479)(201,319,293,438,231,349,263,468)(202,308,294,427,232,338,264,457)(203,357,295,476,233,327,265,446)(204,346,296,465,234,316,266,435)(205,335,297,454,235,305,267,424)(206,324,298,443,236,354,268,473)(207,313,299,432,237,343,269,462)(208,302,300,421,238,332,270,451)(209,351,241,470,239,321,271,440)(210,340,242,459,240,310,272,429)>;
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), (1,226,140,258,31,196,170,288)(2,207,141,299,32,237,171,269)(3,188,142,280,33,218,172,250)(4,229,143,261,34,199,173,291)(5,210,144,242,35,240,174,272)(6,191,145,283,36,221,175,253)(7,232,146,264,37,202,176,294)(8,213,147,245,38,183,177,275)(9,194,148,286,39,224,178,256)(10,235,149,267,40,205,179,297)(11,216,150,248,41,186,180,278)(12,197,151,289,42,227,121,259)(13,238,152,270,43,208,122,300)(14,219,153,251,44,189,123,281)(15,200,154,292,45,230,124,262)(16,181,155,273,46,211,125,243)(17,222,156,254,47,192,126,284)(18,203,157,295,48,233,127,265)(19,184,158,276,49,214,128,246)(20,225,159,257,50,195,129,287)(21,206,160,298,51,236,130,268)(22,187,161,279,52,217,131,249)(23,228,162,260,53,198,132,290)(24,209,163,241,54,239,133,271)(25,190,164,282,55,220,134,252)(26,231,165,263,56,201,135,293)(27,212,166,244,57,182,136,274)(28,193,167,285,58,223,137,255)(29,234,168,266,59,204,138,296)(30,215,169,247,60,185,139,277)(61,423,373,334,91,453,403,304)(62,464,374,315,92,434,404,345)(63,445,375,356,93,475,405,326)(64,426,376,337,94,456,406,307)(65,467,377,318,95,437,407,348)(66,448,378,359,96,478,408,329)(67,429,379,340,97,459,409,310)(68,470,380,321,98,440,410,351)(69,451,381,302,99,421,411,332)(70,432,382,343,100,462,412,313)(71,473,383,324,101,443,413,354)(72,454,384,305,102,424,414,335)(73,435,385,346,103,465,415,316)(74,476,386,327,104,446,416,357)(75,457,387,308,105,427,417,338)(76,438,388,349,106,468,418,319)(77,479,389,330,107,449,419,360)(78,460,390,311,108,430,420,341)(79,441,391,352,109,471,361,322)(80,422,392,333,110,452,362,303)(81,463,393,314,111,433,363,344)(82,444,394,355,112,474,364,325)(83,425,395,336,113,455,365,306)(84,466,396,317,114,436,366,347)(85,447,397,358,115,477,367,328)(86,428,398,339,116,458,368,309)(87,469,399,320,117,439,369,350)(88,450,400,301,118,480,370,331)(89,431,401,342,119,461,371,312)(90,472,402,323,120,442,372,353), (1,96,140,408,31,66,170,378)(2,85,141,397,32,115,171,367)(3,74,142,386,33,104,172,416)(4,63,143,375,34,93,173,405)(5,112,144,364,35,82,174,394)(6,101,145,413,36,71,175,383)(7,90,146,402,37,120,176,372)(8,79,147,391,38,109,177,361)(9,68,148,380,39,98,178,410)(10,117,149,369,40,87,179,399)(11,106,150,418,41,76,180,388)(12,95,151,407,42,65,121,377)(13,84,152,396,43,114,122,366)(14,73,153,385,44,103,123,415)(15,62,154,374,45,92,124,404)(16,111,155,363,46,81,125,393)(17,100,156,412,47,70,126,382)(18,89,157,401,48,119,127,371)(19,78,158,390,49,108,128,420)(20,67,159,379,50,97,129,409)(21,116,160,368,51,86,130,398)(22,105,161,417,52,75,131,387)(23,94,162,406,53,64,132,376)(24,83,163,395,54,113,133,365)(25,72,164,384,55,102,134,414)(26,61,165,373,56,91,135,403)(27,110,166,362,57,80,136,392)(28,99,167,411,58,69,137,381)(29,88,168,400,59,118,138,370)(30,77,169,389,60,107,139,419)(181,359,273,478,211,329,243,448)(182,348,274,467,212,318,244,437)(183,337,275,456,213,307,245,426)(184,326,276,445,214,356,246,475)(185,315,277,434,215,345,247,464)(186,304,278,423,216,334,248,453)(187,353,279,472,217,323,249,442)(188,342,280,461,218,312,250,431)(189,331,281,450,219,301,251,480)(190,320,282,439,220,350,252,469)(191,309,283,428,221,339,253,458)(192,358,284,477,222,328,254,447)(193,347,285,466,223,317,255,436)(194,336,286,455,224,306,256,425)(195,325,287,444,225,355,257,474)(196,314,288,433,226,344,258,463)(197,303,289,422,227,333,259,452)(198,352,290,471,228,322,260,441)(199,341,291,460,229,311,261,430)(200,330,292,449,230,360,262,479)(201,319,293,438,231,349,263,468)(202,308,294,427,232,338,264,457)(203,357,295,476,233,327,265,446)(204,346,296,465,234,316,266,435)(205,335,297,454,235,305,267,424)(206,324,298,443,236,354,268,473)(207,313,299,432,237,343,269,462)(208,302,300,421,238,332,270,451)(209,351,241,470,239,321,271,440)(210,340,242,459,240,310,272,429) );
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)], [(1,226,140,258,31,196,170,288),(2,207,141,299,32,237,171,269),(3,188,142,280,33,218,172,250),(4,229,143,261,34,199,173,291),(5,210,144,242,35,240,174,272),(6,191,145,283,36,221,175,253),(7,232,146,264,37,202,176,294),(8,213,147,245,38,183,177,275),(9,194,148,286,39,224,178,256),(10,235,149,267,40,205,179,297),(11,216,150,248,41,186,180,278),(12,197,151,289,42,227,121,259),(13,238,152,270,43,208,122,300),(14,219,153,251,44,189,123,281),(15,200,154,292,45,230,124,262),(16,181,155,273,46,211,125,243),(17,222,156,254,47,192,126,284),(18,203,157,295,48,233,127,265),(19,184,158,276,49,214,128,246),(20,225,159,257,50,195,129,287),(21,206,160,298,51,236,130,268),(22,187,161,279,52,217,131,249),(23,228,162,260,53,198,132,290),(24,209,163,241,54,239,133,271),(25,190,164,282,55,220,134,252),(26,231,165,263,56,201,135,293),(27,212,166,244,57,182,136,274),(28,193,167,285,58,223,137,255),(29,234,168,266,59,204,138,296),(30,215,169,247,60,185,139,277),(61,423,373,334,91,453,403,304),(62,464,374,315,92,434,404,345),(63,445,375,356,93,475,405,326),(64,426,376,337,94,456,406,307),(65,467,377,318,95,437,407,348),(66,448,378,359,96,478,408,329),(67,429,379,340,97,459,409,310),(68,470,380,321,98,440,410,351),(69,451,381,302,99,421,411,332),(70,432,382,343,100,462,412,313),(71,473,383,324,101,443,413,354),(72,454,384,305,102,424,414,335),(73,435,385,346,103,465,415,316),(74,476,386,327,104,446,416,357),(75,457,387,308,105,427,417,338),(76,438,388,349,106,468,418,319),(77,479,389,330,107,449,419,360),(78,460,390,311,108,430,420,341),(79,441,391,352,109,471,361,322),(80,422,392,333,110,452,362,303),(81,463,393,314,111,433,363,344),(82,444,394,355,112,474,364,325),(83,425,395,336,113,455,365,306),(84,466,396,317,114,436,366,347),(85,447,397,358,115,477,367,328),(86,428,398,339,116,458,368,309),(87,469,399,320,117,439,369,350),(88,450,400,301,118,480,370,331),(89,431,401,342,119,461,371,312),(90,472,402,323,120,442,372,353)], [(1,96,140,408,31,66,170,378),(2,85,141,397,32,115,171,367),(3,74,142,386,33,104,172,416),(4,63,143,375,34,93,173,405),(5,112,144,364,35,82,174,394),(6,101,145,413,36,71,175,383),(7,90,146,402,37,120,176,372),(8,79,147,391,38,109,177,361),(9,68,148,380,39,98,178,410),(10,117,149,369,40,87,179,399),(11,106,150,418,41,76,180,388),(12,95,151,407,42,65,121,377),(13,84,152,396,43,114,122,366),(14,73,153,385,44,103,123,415),(15,62,154,374,45,92,124,404),(16,111,155,363,46,81,125,393),(17,100,156,412,47,70,126,382),(18,89,157,401,48,119,127,371),(19,78,158,390,49,108,128,420),(20,67,159,379,50,97,129,409),(21,116,160,368,51,86,130,398),(22,105,161,417,52,75,131,387),(23,94,162,406,53,64,132,376),(24,83,163,395,54,113,133,365),(25,72,164,384,55,102,134,414),(26,61,165,373,56,91,135,403),(27,110,166,362,57,80,136,392),(28,99,167,411,58,69,137,381),(29,88,168,400,59,118,138,370),(30,77,169,389,60,107,139,419),(181,359,273,478,211,329,243,448),(182,348,274,467,212,318,244,437),(183,337,275,456,213,307,245,426),(184,326,276,445,214,356,246,475),(185,315,277,434,215,345,247,464),(186,304,278,423,216,334,248,453),(187,353,279,472,217,323,249,442),(188,342,280,461,218,312,250,431),(189,331,281,450,219,301,251,480),(190,320,282,439,220,350,252,469),(191,309,283,428,221,339,253,458),(192,358,284,477,222,328,254,447),(193,347,285,466,223,317,255,436),(194,336,286,455,224,306,256,425),(195,325,287,444,225,355,257,474),(196,314,288,433,226,344,258,463),(197,303,289,422,227,333,259,452),(198,352,290,471,228,322,260,441),(199,341,291,460,229,311,261,430),(200,330,292,449,230,360,262,479),(201,319,293,438,231,349,263,468),(202,308,294,427,232,338,264,457),(203,357,295,476,233,327,265,446),(204,346,296,465,234,316,266,435),(205,335,297,454,235,305,267,424),(206,324,298,443,236,354,268,473),(207,313,299,432,237,343,269,462),(208,302,300,421,238,332,270,451),(209,351,241,470,239,321,271,440),(210,340,242,459,240,310,272,429)]])
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 | 24A | ··· | 24H | 30A | ··· | 30F | 40A | ··· | 40P | 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 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 6 | ··· | 6 | 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) | D10 | Dic6 | C3⋊D4 | C4×S3 | Dic10 | C5⋊D4 | C4×D5 | S3×C8 | C8⋊S3 | C8×D5 | C8⋊D5 | S3×D5 | C15⋊D4 | C15⋊Q8 | D30.C2 | D15⋊2C8 | D30.5C4 |
| kernel | C60.14Q8 | C6×C5⋊2C8 | C10×C3⋊C8 | C4×Dic15 | C2×Dic15 | Dic15 | C2×C5⋊2C8 | C60 | C60 | C2×C3⋊C8 | C2×C20 | C30 | C2×C12 | C20 | C20 | C2×C10 | C12 | C12 | C2×C6 | C10 | C10 | C6 | C6 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C60.14Q8 ►in GL4(𝔽241) generated by
| 52 | 52 | 0 | 0 |
| 189 | 240 | 0 | 0 |
| 0 | 0 | 177 | 64 |
| 0 | 0 | 177 | 0 |
| 41 | 85 | 0 | 0 |
| 156 | 200 | 0 | 0 |
| 0 | 0 | 128 | 41 |
| 0 | 0 | 169 | 113 |
| 11 | 124 | 0 | 0 |
| 34 | 230 | 0 | 0 |
| 0 | 0 | 172 | 138 |
| 0 | 0 | 103 | 69 |
G:=sub<GL(4,GF(241))| [52,189,0,0,52,240,0,0,0,0,177,177,0,0,64,0],[41,156,0,0,85,200,0,0,0,0,128,169,0,0,41,113],[11,34,0,0,124,230,0,0,0,0,172,103,0,0,138,69] >;
C60.14Q8 in GAP, Magma, Sage, TeX
C_{60}._{14}Q_8 % in TeX
G:=Group("C60.14Q8"); // GroupNames label
G:=SmallGroup(480,59);
// by ID
G=gap.SmallGroup(480,59);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,36,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=b^2,b*a*b^-1=a^41,c*a*c^-1=a^49,c*b*c^-1=a^15*b^3>;
// generators/relations