metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.48D4, C12.21D20, Dic3⋊3Dic10, C15⋊6(C4⋊Q8), (C5×Dic3)⋊6Q8, C3⋊3(C20⋊2Q8), C6.58(C2×D20), C10.30(S3×Q8), C30.32(C2×Q8), C30.122(C2×D4), (C2×C20).297D6, (C4×Dic3).4D5, (C2×C12).122D10, C5⋊1(Dic3⋊Q8), C20.59(C3⋊D4), C4.11(C3⋊D20), (C2×C30).79C23, (C2×Dic5).27D6, (C2×Dic10).5S3, (Dic3×C20).4C2, (C6×Dic10).5C2, C6.12(C2×Dic10), C2.14(S3×Dic10), (C2×C60).116C22, (C2×Dic30).15C2, C6.Dic10.15C2, (C2×Dic3).148D10, (C6×Dic5).46C22, (C2×Dic15).68C22, (C10×Dic3).172C22, (C2×C4).107(S3×D5), C10.13(C2×C3⋊D4), C2.17(C2×C3⋊D20), C22.164(C2×S3×D5), (C2×C6).91(C22×D5), (C2×C10).91(C22×S3), SmallGroup(480,465)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.48D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=a41, cac-1=a-1, cbc-1=b-1 >
Subgroups: 604 in 136 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C10, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C30, C30, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C2×Dic6, C6×Q8, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4⋊Dic5, C4×C20, C2×Dic10, C2×Dic10, Dic3⋊Q8, C3×Dic10, C6×Dic5, C10×Dic3, Dic30, C2×Dic15, C2×C60, C20⋊2Q8, C6.Dic10, Dic3×C20, C6×Dic10, C2×Dic30, C60.48D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, D10, C3⋊D4, C22×S3, C4⋊Q8, Dic10, D20, C22×D5, S3×Q8, C2×C3⋊D4, S3×D5, C2×Dic10, C2×D20, Dic3⋊Q8, C3⋊D20, C2×S3×D5, C20⋊2Q8, S3×Dic10, C2×C3⋊D20, C60.48D4
►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)
(1 110 395 297)(2 91 396 278)(3 72 397 259)(4 113 398 300)(5 94 399 281)(6 75 400 262)(7 116 401 243)(8 97 402 284)(9 78 403 265)(10 119 404 246)(11 100 405 287)(12 81 406 268)(13 62 407 249)(14 103 408 290)(15 84 409 271)(16 65 410 252)(17 106 411 293)(18 87 412 274)(19 68 413 255)(20 109 414 296)(21 90 415 277)(22 71 416 258)(23 112 417 299)(24 93 418 280)(25 74 419 261)(26 115 420 242)(27 96 361 283)(28 77 362 264)(29 118 363 245)(30 99 364 286)(31 80 365 267)(32 61 366 248)(33 102 367 289)(34 83 368 270)(35 64 369 251)(36 105 370 292)(37 86 371 273)(38 67 372 254)(39 108 373 295)(40 89 374 276)(41 70 375 257)(42 111 376 298)(43 92 377 279)(44 73 378 260)(45 114 379 241)(46 95 380 282)(47 76 381 263)(48 117 382 244)(49 98 383 285)(50 79 384 266)(51 120 385 247)(52 101 386 288)(53 82 387 269)(54 63 388 250)(55 104 389 291)(56 85 390 272)(57 66 391 253)(58 107 392 294)(59 88 393 275)(60 69 394 256)(121 204 323 474)(122 185 324 455)(123 226 325 436)(124 207 326 477)(125 188 327 458)(126 229 328 439)(127 210 329 480)(128 191 330 461)(129 232 331 442)(130 213 332 423)(131 194 333 464)(132 235 334 445)(133 216 335 426)(134 197 336 467)(135 238 337 448)(136 219 338 429)(137 200 339 470)(138 181 340 451)(139 222 341 432)(140 203 342 473)(141 184 343 454)(142 225 344 435)(143 206 345 476)(144 187 346 457)(145 228 347 438)(146 209 348 479)(147 190 349 460)(148 231 350 441)(149 212 351 422)(150 193 352 463)(151 234 353 444)(152 215 354 425)(153 196 355 466)(154 237 356 447)(155 218 357 428)(156 199 358 469)(157 240 359 450)(158 221 360 431)(159 202 301 472)(160 183 302 453)(161 224 303 434)(162 205 304 475)(163 186 305 456)(164 227 306 437)(165 208 307 478)(166 189 308 459)(167 230 309 440)(168 211 310 421)(169 192 311 462)(170 233 312 443)(171 214 313 424)(172 195 314 465)(173 236 315 446)(174 217 316 427)(175 198 317 468)(176 239 318 449)(177 220 319 430)(178 201 320 471)(179 182 321 452)(180 223 322 433)
(1 155 31 125)(2 154 32 124)(3 153 33 123)(4 152 34 122)(5 151 35 121)(6 150 36 180)(7 149 37 179)(8 148 38 178)(9 147 39 177)(10 146 40 176)(11 145 41 175)(12 144 42 174)(13 143 43 173)(14 142 44 172)(15 141 45 171)(16 140 46 170)(17 139 47 169)(18 138 48 168)(19 137 49 167)(20 136 50 166)(21 135 51 165)(22 134 52 164)(23 133 53 163)(24 132 54 162)(25 131 55 161)(26 130 56 160)(27 129 57 159)(28 128 58 158)(29 127 59 157)(30 126 60 156)(61 477 91 447)(62 476 92 446)(63 475 93 445)(64 474 94 444)(65 473 95 443)(66 472 96 442)(67 471 97 441)(68 470 98 440)(69 469 99 439)(70 468 100 438)(71 467 101 437)(72 466 102 436)(73 465 103 435)(74 464 104 434)(75 463 105 433)(76 462 106 432)(77 461 107 431)(78 460 108 430)(79 459 109 429)(80 458 110 428)(81 457 111 427)(82 456 112 426)(83 455 113 425)(84 454 114 424)(85 453 115 423)(86 452 116 422)(87 451 117 421)(88 450 118 480)(89 449 119 479)(90 448 120 478)(181 244 211 274)(182 243 212 273)(183 242 213 272)(184 241 214 271)(185 300 215 270)(186 299 216 269)(187 298 217 268)(188 297 218 267)(189 296 219 266)(190 295 220 265)(191 294 221 264)(192 293 222 263)(193 292 223 262)(194 291 224 261)(195 290 225 260)(196 289 226 259)(197 288 227 258)(198 287 228 257)(199 286 229 256)(200 285 230 255)(201 284 231 254)(202 283 232 253)(203 282 233 252)(204 281 234 251)(205 280 235 250)(206 279 236 249)(207 278 237 248)(208 277 238 247)(209 276 239 246)(210 275 240 245)(301 361 331 391)(302 420 332 390)(303 419 333 389)(304 418 334 388)(305 417 335 387)(306 416 336 386)(307 415 337 385)(308 414 338 384)(309 413 339 383)(310 412 340 382)(311 411 341 381)(312 410 342 380)(313 409 343 379)(314 408 344 378)(315 407 345 377)(316 406 346 376)(317 405 347 375)(318 404 348 374)(319 403 349 373)(320 402 350 372)(321 401 351 371)(322 400 352 370)(323 399 353 369)(324 398 354 368)(325 397 355 367)(326 396 356 366)(327 395 357 365)(328 394 358 364)(329 393 359 363)(330 392 360 362)
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), (1,110,395,297)(2,91,396,278)(3,72,397,259)(4,113,398,300)(5,94,399,281)(6,75,400,262)(7,116,401,243)(8,97,402,284)(9,78,403,265)(10,119,404,246)(11,100,405,287)(12,81,406,268)(13,62,407,249)(14,103,408,290)(15,84,409,271)(16,65,410,252)(17,106,411,293)(18,87,412,274)(19,68,413,255)(20,109,414,296)(21,90,415,277)(22,71,416,258)(23,112,417,299)(24,93,418,280)(25,74,419,261)(26,115,420,242)(27,96,361,283)(28,77,362,264)(29,118,363,245)(30,99,364,286)(31,80,365,267)(32,61,366,248)(33,102,367,289)(34,83,368,270)(35,64,369,251)(36,105,370,292)(37,86,371,273)(38,67,372,254)(39,108,373,295)(40,89,374,276)(41,70,375,257)(42,111,376,298)(43,92,377,279)(44,73,378,260)(45,114,379,241)(46,95,380,282)(47,76,381,263)(48,117,382,244)(49,98,383,285)(50,79,384,266)(51,120,385,247)(52,101,386,288)(53,82,387,269)(54,63,388,250)(55,104,389,291)(56,85,390,272)(57,66,391,253)(58,107,392,294)(59,88,393,275)(60,69,394,256)(121,204,323,474)(122,185,324,455)(123,226,325,436)(124,207,326,477)(125,188,327,458)(126,229,328,439)(127,210,329,480)(128,191,330,461)(129,232,331,442)(130,213,332,423)(131,194,333,464)(132,235,334,445)(133,216,335,426)(134,197,336,467)(135,238,337,448)(136,219,338,429)(137,200,339,470)(138,181,340,451)(139,222,341,432)(140,203,342,473)(141,184,343,454)(142,225,344,435)(143,206,345,476)(144,187,346,457)(145,228,347,438)(146,209,348,479)(147,190,349,460)(148,231,350,441)(149,212,351,422)(150,193,352,463)(151,234,353,444)(152,215,354,425)(153,196,355,466)(154,237,356,447)(155,218,357,428)(156,199,358,469)(157,240,359,450)(158,221,360,431)(159,202,301,472)(160,183,302,453)(161,224,303,434)(162,205,304,475)(163,186,305,456)(164,227,306,437)(165,208,307,478)(166,189,308,459)(167,230,309,440)(168,211,310,421)(169,192,311,462)(170,233,312,443)(171,214,313,424)(172,195,314,465)(173,236,315,446)(174,217,316,427)(175,198,317,468)(176,239,318,449)(177,220,319,430)(178,201,320,471)(179,182,321,452)(180,223,322,433), (1,155,31,125)(2,154,32,124)(3,153,33,123)(4,152,34,122)(5,151,35,121)(6,150,36,180)(7,149,37,179)(8,148,38,178)(9,147,39,177)(10,146,40,176)(11,145,41,175)(12,144,42,174)(13,143,43,173)(14,142,44,172)(15,141,45,171)(16,140,46,170)(17,139,47,169)(18,138,48,168)(19,137,49,167)(20,136,50,166)(21,135,51,165)(22,134,52,164)(23,133,53,163)(24,132,54,162)(25,131,55,161)(26,130,56,160)(27,129,57,159)(28,128,58,158)(29,127,59,157)(30,126,60,156)(61,477,91,447)(62,476,92,446)(63,475,93,445)(64,474,94,444)(65,473,95,443)(66,472,96,442)(67,471,97,441)(68,470,98,440)(69,469,99,439)(70,468,100,438)(71,467,101,437)(72,466,102,436)(73,465,103,435)(74,464,104,434)(75,463,105,433)(76,462,106,432)(77,461,107,431)(78,460,108,430)(79,459,109,429)(80,458,110,428)(81,457,111,427)(82,456,112,426)(83,455,113,425)(84,454,114,424)(85,453,115,423)(86,452,116,422)(87,451,117,421)(88,450,118,480)(89,449,119,479)(90,448,120,478)(181,244,211,274)(182,243,212,273)(183,242,213,272)(184,241,214,271)(185,300,215,270)(186,299,216,269)(187,298,217,268)(188,297,218,267)(189,296,219,266)(190,295,220,265)(191,294,221,264)(192,293,222,263)(193,292,223,262)(194,291,224,261)(195,290,225,260)(196,289,226,259)(197,288,227,258)(198,287,228,257)(199,286,229,256)(200,285,230,255)(201,284,231,254)(202,283,232,253)(203,282,233,252)(204,281,234,251)(205,280,235,250)(206,279,236,249)(207,278,237,248)(208,277,238,247)(209,276,239,246)(210,275,240,245)(301,361,331,391)(302,420,332,390)(303,419,333,389)(304,418,334,388)(305,417,335,387)(306,416,336,386)(307,415,337,385)(308,414,338,384)(309,413,339,383)(310,412,340,382)(311,411,341,381)(312,410,342,380)(313,409,343,379)(314,408,344,378)(315,407,345,377)(316,406,346,376)(317,405,347,375)(318,404,348,374)(319,403,349,373)(320,402,350,372)(321,401,351,371)(322,400,352,370)(323,399,353,369)(324,398,354,368)(325,397,355,367)(326,396,356,366)(327,395,357,365)(328,394,358,364)(329,393,359,363)(330,392,360,362)>;
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), (1,110,395,297)(2,91,396,278)(3,72,397,259)(4,113,398,300)(5,94,399,281)(6,75,400,262)(7,116,401,243)(8,97,402,284)(9,78,403,265)(10,119,404,246)(11,100,405,287)(12,81,406,268)(13,62,407,249)(14,103,408,290)(15,84,409,271)(16,65,410,252)(17,106,411,293)(18,87,412,274)(19,68,413,255)(20,109,414,296)(21,90,415,277)(22,71,416,258)(23,112,417,299)(24,93,418,280)(25,74,419,261)(26,115,420,242)(27,96,361,283)(28,77,362,264)(29,118,363,245)(30,99,364,286)(31,80,365,267)(32,61,366,248)(33,102,367,289)(34,83,368,270)(35,64,369,251)(36,105,370,292)(37,86,371,273)(38,67,372,254)(39,108,373,295)(40,89,374,276)(41,70,375,257)(42,111,376,298)(43,92,377,279)(44,73,378,260)(45,114,379,241)(46,95,380,282)(47,76,381,263)(48,117,382,244)(49,98,383,285)(50,79,384,266)(51,120,385,247)(52,101,386,288)(53,82,387,269)(54,63,388,250)(55,104,389,291)(56,85,390,272)(57,66,391,253)(58,107,392,294)(59,88,393,275)(60,69,394,256)(121,204,323,474)(122,185,324,455)(123,226,325,436)(124,207,326,477)(125,188,327,458)(126,229,328,439)(127,210,329,480)(128,191,330,461)(129,232,331,442)(130,213,332,423)(131,194,333,464)(132,235,334,445)(133,216,335,426)(134,197,336,467)(135,238,337,448)(136,219,338,429)(137,200,339,470)(138,181,340,451)(139,222,341,432)(140,203,342,473)(141,184,343,454)(142,225,344,435)(143,206,345,476)(144,187,346,457)(145,228,347,438)(146,209,348,479)(147,190,349,460)(148,231,350,441)(149,212,351,422)(150,193,352,463)(151,234,353,444)(152,215,354,425)(153,196,355,466)(154,237,356,447)(155,218,357,428)(156,199,358,469)(157,240,359,450)(158,221,360,431)(159,202,301,472)(160,183,302,453)(161,224,303,434)(162,205,304,475)(163,186,305,456)(164,227,306,437)(165,208,307,478)(166,189,308,459)(167,230,309,440)(168,211,310,421)(169,192,311,462)(170,233,312,443)(171,214,313,424)(172,195,314,465)(173,236,315,446)(174,217,316,427)(175,198,317,468)(176,239,318,449)(177,220,319,430)(178,201,320,471)(179,182,321,452)(180,223,322,433), (1,155,31,125)(2,154,32,124)(3,153,33,123)(4,152,34,122)(5,151,35,121)(6,150,36,180)(7,149,37,179)(8,148,38,178)(9,147,39,177)(10,146,40,176)(11,145,41,175)(12,144,42,174)(13,143,43,173)(14,142,44,172)(15,141,45,171)(16,140,46,170)(17,139,47,169)(18,138,48,168)(19,137,49,167)(20,136,50,166)(21,135,51,165)(22,134,52,164)(23,133,53,163)(24,132,54,162)(25,131,55,161)(26,130,56,160)(27,129,57,159)(28,128,58,158)(29,127,59,157)(30,126,60,156)(61,477,91,447)(62,476,92,446)(63,475,93,445)(64,474,94,444)(65,473,95,443)(66,472,96,442)(67,471,97,441)(68,470,98,440)(69,469,99,439)(70,468,100,438)(71,467,101,437)(72,466,102,436)(73,465,103,435)(74,464,104,434)(75,463,105,433)(76,462,106,432)(77,461,107,431)(78,460,108,430)(79,459,109,429)(80,458,110,428)(81,457,111,427)(82,456,112,426)(83,455,113,425)(84,454,114,424)(85,453,115,423)(86,452,116,422)(87,451,117,421)(88,450,118,480)(89,449,119,479)(90,448,120,478)(181,244,211,274)(182,243,212,273)(183,242,213,272)(184,241,214,271)(185,300,215,270)(186,299,216,269)(187,298,217,268)(188,297,218,267)(189,296,219,266)(190,295,220,265)(191,294,221,264)(192,293,222,263)(193,292,223,262)(194,291,224,261)(195,290,225,260)(196,289,226,259)(197,288,227,258)(198,287,228,257)(199,286,229,256)(200,285,230,255)(201,284,231,254)(202,283,232,253)(203,282,233,252)(204,281,234,251)(205,280,235,250)(206,279,236,249)(207,278,237,248)(208,277,238,247)(209,276,239,246)(210,275,240,245)(301,361,331,391)(302,420,332,390)(303,419,333,389)(304,418,334,388)(305,417,335,387)(306,416,336,386)(307,415,337,385)(308,414,338,384)(309,413,339,383)(310,412,340,382)(311,411,341,381)(312,410,342,380)(313,409,343,379)(314,408,344,378)(315,407,345,377)(316,406,346,376)(317,405,347,375)(318,404,348,374)(319,403,349,373)(320,402,350,372)(321,401,351,371)(322,400,352,370)(323,399,353,369)(324,398,354,368)(325,397,355,367)(326,396,356,366)(327,395,357,365)(328,394,358,364)(329,393,359,363)(330,392,360,362) );
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)], [(1,110,395,297),(2,91,396,278),(3,72,397,259),(4,113,398,300),(5,94,399,281),(6,75,400,262),(7,116,401,243),(8,97,402,284),(9,78,403,265),(10,119,404,246),(11,100,405,287),(12,81,406,268),(13,62,407,249),(14,103,408,290),(15,84,409,271),(16,65,410,252),(17,106,411,293),(18,87,412,274),(19,68,413,255),(20,109,414,296),(21,90,415,277),(22,71,416,258),(23,112,417,299),(24,93,418,280),(25,74,419,261),(26,115,420,242),(27,96,361,283),(28,77,362,264),(29,118,363,245),(30,99,364,286),(31,80,365,267),(32,61,366,248),(33,102,367,289),(34,83,368,270),(35,64,369,251),(36,105,370,292),(37,86,371,273),(38,67,372,254),(39,108,373,295),(40,89,374,276),(41,70,375,257),(42,111,376,298),(43,92,377,279),(44,73,378,260),(45,114,379,241),(46,95,380,282),(47,76,381,263),(48,117,382,244),(49,98,383,285),(50,79,384,266),(51,120,385,247),(52,101,386,288),(53,82,387,269),(54,63,388,250),(55,104,389,291),(56,85,390,272),(57,66,391,253),(58,107,392,294),(59,88,393,275),(60,69,394,256),(121,204,323,474),(122,185,324,455),(123,226,325,436),(124,207,326,477),(125,188,327,458),(126,229,328,439),(127,210,329,480),(128,191,330,461),(129,232,331,442),(130,213,332,423),(131,194,333,464),(132,235,334,445),(133,216,335,426),(134,197,336,467),(135,238,337,448),(136,219,338,429),(137,200,339,470),(138,181,340,451),(139,222,341,432),(140,203,342,473),(141,184,343,454),(142,225,344,435),(143,206,345,476),(144,187,346,457),(145,228,347,438),(146,209,348,479),(147,190,349,460),(148,231,350,441),(149,212,351,422),(150,193,352,463),(151,234,353,444),(152,215,354,425),(153,196,355,466),(154,237,356,447),(155,218,357,428),(156,199,358,469),(157,240,359,450),(158,221,360,431),(159,202,301,472),(160,183,302,453),(161,224,303,434),(162,205,304,475),(163,186,305,456),(164,227,306,437),(165,208,307,478),(166,189,308,459),(167,230,309,440),(168,211,310,421),(169,192,311,462),(170,233,312,443),(171,214,313,424),(172,195,314,465),(173,236,315,446),(174,217,316,427),(175,198,317,468),(176,239,318,449),(177,220,319,430),(178,201,320,471),(179,182,321,452),(180,223,322,433)], [(1,155,31,125),(2,154,32,124),(3,153,33,123),(4,152,34,122),(5,151,35,121),(6,150,36,180),(7,149,37,179),(8,148,38,178),(9,147,39,177),(10,146,40,176),(11,145,41,175),(12,144,42,174),(13,143,43,173),(14,142,44,172),(15,141,45,171),(16,140,46,170),(17,139,47,169),(18,138,48,168),(19,137,49,167),(20,136,50,166),(21,135,51,165),(22,134,52,164),(23,133,53,163),(24,132,54,162),(25,131,55,161),(26,130,56,160),(27,129,57,159),(28,128,58,158),(29,127,59,157),(30,126,60,156),(61,477,91,447),(62,476,92,446),(63,475,93,445),(64,474,94,444),(65,473,95,443),(66,472,96,442),(67,471,97,441),(68,470,98,440),(69,469,99,439),(70,468,100,438),(71,467,101,437),(72,466,102,436),(73,465,103,435),(74,464,104,434),(75,463,105,433),(76,462,106,432),(77,461,107,431),(78,460,108,430),(79,459,109,429),(80,458,110,428),(81,457,111,427),(82,456,112,426),(83,455,113,425),(84,454,114,424),(85,453,115,423),(86,452,116,422),(87,451,117,421),(88,450,118,480),(89,449,119,479),(90,448,120,478),(181,244,211,274),(182,243,212,273),(183,242,213,272),(184,241,214,271),(185,300,215,270),(186,299,216,269),(187,298,217,268),(188,297,218,267),(189,296,219,266),(190,295,220,265),(191,294,221,264),(192,293,222,263),(193,292,223,262),(194,291,224,261),(195,290,225,260),(196,289,226,259),(197,288,227,258),(198,287,228,257),(199,286,229,256),(200,285,230,255),(201,284,231,254),(202,283,232,253),(203,282,233,252),(204,281,234,251),(205,280,235,250),(206,279,236,249),(207,278,237,248),(208,277,238,247),(209,276,239,246),(210,275,240,245),(301,361,331,391),(302,420,332,390),(303,419,333,389),(304,418,334,388),(305,417,335,387),(306,416,336,386),(307,415,337,385),(308,414,338,384),(309,413,339,383),(310,412,340,382),(311,411,341,381),(312,410,342,380),(313,409,343,379),(314,408,344,378),(315,407,345,377),(316,406,346,376),(317,405,347,375),(318,404,348,374),(319,403,349,373),(320,402,350,372),(321,401,351,371),(322,400,352,370),(323,399,353,369),(324,398,354,368),(325,397,355,367),(326,396,356,366),(327,395,357,365),(328,394,358,364),(329,393,359,363),(330,392,360,362)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 20 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | - | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D5 | D6 | D6 | D10 | D10 | C3⋊D4 | Dic10 | D20 | S3×Q8 | S3×D5 | C3⋊D20 | C2×S3×D5 | S3×Dic10 |
| kernel | C60.48D4 | C6.Dic10 | Dic3×C20 | C6×Dic10 | C2×Dic30 | C2×Dic10 | C5×Dic3 | C60 | C4×Dic3 | C2×Dic5 | C2×C20 | C2×Dic3 | C2×C12 | C20 | Dic3 | C12 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 16 | 8 | 2 | 2 | 4 | 2 | 8 |
Matrix representation of C60.48D4 ►in GL8(𝔽61)
| 18 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 43 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 30 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 42 | 31 |
| 2 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 53 | 59 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 57 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 30 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 42 | 31 |
G:=sub<GL(8,GF(61))| [18,43,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,59,60,0,0,0,0,0,0,0,0,30,42,0,0,0,0,0,0,12,31],[2,53,0,0,0,0,0,0,8,59,0,0,0,0,0,0,0,0,29,40,0,0,0,0,0,0,40,32,0,0,0,0,0,0,0,0,21,19,0,0,0,0,0,0,57,40,0,0,0,0,0,0,0,0,30,42,0,0,0,0,0,0,12,31] >;
C60.48D4 in GAP, Magma, Sage, TeX
C_{60}._{48}D_4 % in TeX
G:=Group("C60.48D4"); // GroupNames label
G:=SmallGroup(480,465);
// by ID
G=gap.SmallGroup(480,465);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,64,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^41,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations