metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.13Q8, C20.58D12, C60.102D4, C20.13Dic6, C12.13Dic10, C30.17M4(2), C15⋊7(C4⋊C8), C6.15(C8×D5), C5⋊3(C12⋊C8), C30.31(C2×C8), Dic5⋊1(C3⋊C8), (C3×Dic5)⋊2C8, C4.10(C15⋊Q8), C30.22(C4⋊C4), (C2×C20).324D6, C6.9(C8⋊D5), (C4×Dic5).7S3, (C6×Dic5).8C4, C3⋊2(C20.8Q8), (C2×C12).328D10, C12.66(C5⋊D4), C4.30(C5⋊D12), C10.8(C4⋊Dic3), (C12×Dic5).8C2, (C2×C60).226C22, (C2×Dic5).5Dic3, C6.1(C10.D4), C10.8(C4.Dic3), C2.1(C30.Q8), C22.11(D5×Dic3), C2.3(C20.32D6), C2.5(D5×C3⋊C8), (C2×C3⋊C8).9D5, C10.14(C2×C3⋊C8), (C10×C3⋊C8).11C2, (C2×C6).47(C4×D5), (C2×C30).86(C2×C4), (C2×C4).229(S3×D5), (C2×C15⋊3C8).21C2, (C2×C10).32(C2×Dic3), SmallGroup(480,58)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.13Q8
G = < a,b,c | a60=1, b4=a30, c2=a15b2, bab-1=a41, cac-1=a49, cbc-1=a15b3 >
Subgroups: 236 in 76 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, Dic5, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C30, C4⋊C8, C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C3⋊C8, C2×C3⋊C8, C4×C12, C3×Dic5, C3×Dic5, C60, C2×C30, C2×C5⋊2C8, C4×Dic5, C2×C40, C12⋊C8, C5×C3⋊C8, C15⋊3C8, C6×Dic5, C2×C60, C20.8Q8, C12×Dic5, C10×C3⋊C8, C2×C15⋊3C8, C60.13Q8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, D5, Dic3, D6, C4⋊C4, C2×C8, M4(2), D10, C3⋊C8, Dic6, D12, C2×Dic3, C4⋊C8, Dic10, C4×D5, C5⋊D4, C2×C3⋊C8, C4.Dic3, C4⋊Dic3, S3×D5, C8×D5, C8⋊D5, C10.D4, C12⋊C8, D5×Dic3, C5⋊D12, C15⋊Q8, C20.8Q8, D5×C3⋊C8, C20.32D6, C30.Q8, C60.13Q8
►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)
(1 258 107 161 31 288 77 131)(2 299 108 142 32 269 78 172)(3 280 109 123 33 250 79 153)(4 261 110 164 34 291 80 134)(5 242 111 145 35 272 81 175)(6 283 112 126 36 253 82 156)(7 264 113 167 37 294 83 137)(8 245 114 148 38 275 84 178)(9 286 115 129 39 256 85 159)(10 267 116 170 40 297 86 140)(11 248 117 151 41 278 87 121)(12 289 118 132 42 259 88 162)(13 270 119 173 43 300 89 143)(14 251 120 154 44 281 90 124)(15 292 61 135 45 262 91 165)(16 273 62 176 46 243 92 146)(17 254 63 157 47 284 93 127)(18 295 64 138 48 265 94 168)(19 276 65 179 49 246 95 149)(20 257 66 160 50 287 96 130)(21 298 67 141 51 268 97 171)(22 279 68 122 52 249 98 152)(23 260 69 163 53 290 99 133)(24 241 70 144 54 271 100 174)(25 282 71 125 55 252 101 155)(26 263 72 166 56 293 102 136)(27 244 73 147 57 274 103 177)(28 285 74 128 58 255 104 158)(29 266 75 169 59 296 105 139)(30 247 76 150 60 277 106 180)(181 330 388 446 211 360 418 476)(182 311 389 427 212 341 419 457)(183 352 390 468 213 322 420 438)(184 333 391 449 214 303 361 479)(185 314 392 430 215 344 362 460)(186 355 393 471 216 325 363 441)(187 336 394 452 217 306 364 422)(188 317 395 433 218 347 365 463)(189 358 396 474 219 328 366 444)(190 339 397 455 220 309 367 425)(191 320 398 436 221 350 368 466)(192 301 399 477 222 331 369 447)(193 342 400 458 223 312 370 428)(194 323 401 439 224 353 371 469)(195 304 402 480 225 334 372 450)(196 345 403 461 226 315 373 431)(197 326 404 442 227 356 374 472)(198 307 405 423 228 337 375 453)(199 348 406 464 229 318 376 434)(200 329 407 445 230 359 377 475)(201 310 408 426 231 340 378 456)(202 351 409 467 232 321 379 437)(203 332 410 448 233 302 380 478)(204 313 411 429 234 343 381 459)(205 354 412 470 235 324 382 440)(206 335 413 451 236 305 383 421)(207 316 414 432 237 346 384 462)(208 357 415 473 238 327 385 443)(209 338 416 454 239 308 386 424)(210 319 417 435 240 349 387 465)
(1 393 62 231)(2 382 63 220)(3 371 64 209)(4 420 65 198)(5 409 66 187)(6 398 67 236)(7 387 68 225)(8 376 69 214)(9 365 70 203)(10 414 71 192)(11 403 72 181)(12 392 73 230)(13 381 74 219)(14 370 75 208)(15 419 76 197)(16 408 77 186)(17 397 78 235)(18 386 79 224)(19 375 80 213)(20 364 81 202)(21 413 82 191)(22 402 83 240)(23 391 84 229)(24 380 85 218)(25 369 86 207)(26 418 87 196)(27 407 88 185)(28 396 89 234)(29 385 90 223)(30 374 91 212)(31 363 92 201)(32 412 93 190)(33 401 94 239)(34 390 95 228)(35 379 96 217)(36 368 97 206)(37 417 98 195)(38 406 99 184)(39 395 100 233)(40 384 101 222)(41 373 102 211)(42 362 103 200)(43 411 104 189)(44 400 105 238)(45 389 106 227)(46 378 107 216)(47 367 108 205)(48 416 109 194)(49 405 110 183)(50 394 111 232)(51 383 112 221)(52 372 113 210)(53 361 114 199)(54 410 115 188)(55 399 116 237)(56 388 117 226)(57 377 118 215)(58 366 119 204)(59 415 120 193)(60 404 61 182)(121 476 263 345)(122 465 264 334)(123 454 265 323)(124 443 266 312)(125 432 267 301)(126 421 268 350)(127 470 269 339)(128 459 270 328)(129 448 271 317)(130 437 272 306)(131 426 273 355)(132 475 274 344)(133 464 275 333)(134 453 276 322)(135 442 277 311)(136 431 278 360)(137 480 279 349)(138 469 280 338)(139 458 281 327)(140 447 282 316)(141 436 283 305)(142 425 284 354)(143 474 285 343)(144 463 286 332)(145 452 287 321)(146 441 288 310)(147 430 289 359)(148 479 290 348)(149 468 291 337)(150 457 292 326)(151 446 293 315)(152 435 294 304)(153 424 295 353)(154 473 296 342)(155 462 297 331)(156 451 298 320)(157 440 299 309)(158 429 300 358)(159 478 241 347)(160 467 242 336)(161 456 243 325)(162 445 244 314)(163 434 245 303)(164 423 246 352)(165 472 247 341)(166 461 248 330)(167 450 249 319)(168 439 250 308)(169 428 251 357)(170 477 252 346)(171 466 253 335)(172 455 254 324)(173 444 255 313)(174 433 256 302)(175 422 257 351)(176 471 258 340)(177 460 259 329)(178 449 260 318)(179 438 261 307)(180 427 262 356)
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), (1,258,107,161,31,288,77,131)(2,299,108,142,32,269,78,172)(3,280,109,123,33,250,79,153)(4,261,110,164,34,291,80,134)(5,242,111,145,35,272,81,175)(6,283,112,126,36,253,82,156)(7,264,113,167,37,294,83,137)(8,245,114,148,38,275,84,178)(9,286,115,129,39,256,85,159)(10,267,116,170,40,297,86,140)(11,248,117,151,41,278,87,121)(12,289,118,132,42,259,88,162)(13,270,119,173,43,300,89,143)(14,251,120,154,44,281,90,124)(15,292,61,135,45,262,91,165)(16,273,62,176,46,243,92,146)(17,254,63,157,47,284,93,127)(18,295,64,138,48,265,94,168)(19,276,65,179,49,246,95,149)(20,257,66,160,50,287,96,130)(21,298,67,141,51,268,97,171)(22,279,68,122,52,249,98,152)(23,260,69,163,53,290,99,133)(24,241,70,144,54,271,100,174)(25,282,71,125,55,252,101,155)(26,263,72,166,56,293,102,136)(27,244,73,147,57,274,103,177)(28,285,74,128,58,255,104,158)(29,266,75,169,59,296,105,139)(30,247,76,150,60,277,106,180)(181,330,388,446,211,360,418,476)(182,311,389,427,212,341,419,457)(183,352,390,468,213,322,420,438)(184,333,391,449,214,303,361,479)(185,314,392,430,215,344,362,460)(186,355,393,471,216,325,363,441)(187,336,394,452,217,306,364,422)(188,317,395,433,218,347,365,463)(189,358,396,474,219,328,366,444)(190,339,397,455,220,309,367,425)(191,320,398,436,221,350,368,466)(192,301,399,477,222,331,369,447)(193,342,400,458,223,312,370,428)(194,323,401,439,224,353,371,469)(195,304,402,480,225,334,372,450)(196,345,403,461,226,315,373,431)(197,326,404,442,227,356,374,472)(198,307,405,423,228,337,375,453)(199,348,406,464,229,318,376,434)(200,329,407,445,230,359,377,475)(201,310,408,426,231,340,378,456)(202,351,409,467,232,321,379,437)(203,332,410,448,233,302,380,478)(204,313,411,429,234,343,381,459)(205,354,412,470,235,324,382,440)(206,335,413,451,236,305,383,421)(207,316,414,432,237,346,384,462)(208,357,415,473,238,327,385,443)(209,338,416,454,239,308,386,424)(210,319,417,435,240,349,387,465), (1,393,62,231)(2,382,63,220)(3,371,64,209)(4,420,65,198)(5,409,66,187)(6,398,67,236)(7,387,68,225)(8,376,69,214)(9,365,70,203)(10,414,71,192)(11,403,72,181)(12,392,73,230)(13,381,74,219)(14,370,75,208)(15,419,76,197)(16,408,77,186)(17,397,78,235)(18,386,79,224)(19,375,80,213)(20,364,81,202)(21,413,82,191)(22,402,83,240)(23,391,84,229)(24,380,85,218)(25,369,86,207)(26,418,87,196)(27,407,88,185)(28,396,89,234)(29,385,90,223)(30,374,91,212)(31,363,92,201)(32,412,93,190)(33,401,94,239)(34,390,95,228)(35,379,96,217)(36,368,97,206)(37,417,98,195)(38,406,99,184)(39,395,100,233)(40,384,101,222)(41,373,102,211)(42,362,103,200)(43,411,104,189)(44,400,105,238)(45,389,106,227)(46,378,107,216)(47,367,108,205)(48,416,109,194)(49,405,110,183)(50,394,111,232)(51,383,112,221)(52,372,113,210)(53,361,114,199)(54,410,115,188)(55,399,116,237)(56,388,117,226)(57,377,118,215)(58,366,119,204)(59,415,120,193)(60,404,61,182)(121,476,263,345)(122,465,264,334)(123,454,265,323)(124,443,266,312)(125,432,267,301)(126,421,268,350)(127,470,269,339)(128,459,270,328)(129,448,271,317)(130,437,272,306)(131,426,273,355)(132,475,274,344)(133,464,275,333)(134,453,276,322)(135,442,277,311)(136,431,278,360)(137,480,279,349)(138,469,280,338)(139,458,281,327)(140,447,282,316)(141,436,283,305)(142,425,284,354)(143,474,285,343)(144,463,286,332)(145,452,287,321)(146,441,288,310)(147,430,289,359)(148,479,290,348)(149,468,291,337)(150,457,292,326)(151,446,293,315)(152,435,294,304)(153,424,295,353)(154,473,296,342)(155,462,297,331)(156,451,298,320)(157,440,299,309)(158,429,300,358)(159,478,241,347)(160,467,242,336)(161,456,243,325)(162,445,244,314)(163,434,245,303)(164,423,246,352)(165,472,247,341)(166,461,248,330)(167,450,249,319)(168,439,250,308)(169,428,251,357)(170,477,252,346)(171,466,253,335)(172,455,254,324)(173,444,255,313)(174,433,256,302)(175,422,257,351)(176,471,258,340)(177,460,259,329)(178,449,260,318)(179,438,261,307)(180,427,262,356)>;
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), (1,258,107,161,31,288,77,131)(2,299,108,142,32,269,78,172)(3,280,109,123,33,250,79,153)(4,261,110,164,34,291,80,134)(5,242,111,145,35,272,81,175)(6,283,112,126,36,253,82,156)(7,264,113,167,37,294,83,137)(8,245,114,148,38,275,84,178)(9,286,115,129,39,256,85,159)(10,267,116,170,40,297,86,140)(11,248,117,151,41,278,87,121)(12,289,118,132,42,259,88,162)(13,270,119,173,43,300,89,143)(14,251,120,154,44,281,90,124)(15,292,61,135,45,262,91,165)(16,273,62,176,46,243,92,146)(17,254,63,157,47,284,93,127)(18,295,64,138,48,265,94,168)(19,276,65,179,49,246,95,149)(20,257,66,160,50,287,96,130)(21,298,67,141,51,268,97,171)(22,279,68,122,52,249,98,152)(23,260,69,163,53,290,99,133)(24,241,70,144,54,271,100,174)(25,282,71,125,55,252,101,155)(26,263,72,166,56,293,102,136)(27,244,73,147,57,274,103,177)(28,285,74,128,58,255,104,158)(29,266,75,169,59,296,105,139)(30,247,76,150,60,277,106,180)(181,330,388,446,211,360,418,476)(182,311,389,427,212,341,419,457)(183,352,390,468,213,322,420,438)(184,333,391,449,214,303,361,479)(185,314,392,430,215,344,362,460)(186,355,393,471,216,325,363,441)(187,336,394,452,217,306,364,422)(188,317,395,433,218,347,365,463)(189,358,396,474,219,328,366,444)(190,339,397,455,220,309,367,425)(191,320,398,436,221,350,368,466)(192,301,399,477,222,331,369,447)(193,342,400,458,223,312,370,428)(194,323,401,439,224,353,371,469)(195,304,402,480,225,334,372,450)(196,345,403,461,226,315,373,431)(197,326,404,442,227,356,374,472)(198,307,405,423,228,337,375,453)(199,348,406,464,229,318,376,434)(200,329,407,445,230,359,377,475)(201,310,408,426,231,340,378,456)(202,351,409,467,232,321,379,437)(203,332,410,448,233,302,380,478)(204,313,411,429,234,343,381,459)(205,354,412,470,235,324,382,440)(206,335,413,451,236,305,383,421)(207,316,414,432,237,346,384,462)(208,357,415,473,238,327,385,443)(209,338,416,454,239,308,386,424)(210,319,417,435,240,349,387,465), (1,393,62,231)(2,382,63,220)(3,371,64,209)(4,420,65,198)(5,409,66,187)(6,398,67,236)(7,387,68,225)(8,376,69,214)(9,365,70,203)(10,414,71,192)(11,403,72,181)(12,392,73,230)(13,381,74,219)(14,370,75,208)(15,419,76,197)(16,408,77,186)(17,397,78,235)(18,386,79,224)(19,375,80,213)(20,364,81,202)(21,413,82,191)(22,402,83,240)(23,391,84,229)(24,380,85,218)(25,369,86,207)(26,418,87,196)(27,407,88,185)(28,396,89,234)(29,385,90,223)(30,374,91,212)(31,363,92,201)(32,412,93,190)(33,401,94,239)(34,390,95,228)(35,379,96,217)(36,368,97,206)(37,417,98,195)(38,406,99,184)(39,395,100,233)(40,384,101,222)(41,373,102,211)(42,362,103,200)(43,411,104,189)(44,400,105,238)(45,389,106,227)(46,378,107,216)(47,367,108,205)(48,416,109,194)(49,405,110,183)(50,394,111,232)(51,383,112,221)(52,372,113,210)(53,361,114,199)(54,410,115,188)(55,399,116,237)(56,388,117,226)(57,377,118,215)(58,366,119,204)(59,415,120,193)(60,404,61,182)(121,476,263,345)(122,465,264,334)(123,454,265,323)(124,443,266,312)(125,432,267,301)(126,421,268,350)(127,470,269,339)(128,459,270,328)(129,448,271,317)(130,437,272,306)(131,426,273,355)(132,475,274,344)(133,464,275,333)(134,453,276,322)(135,442,277,311)(136,431,278,360)(137,480,279,349)(138,469,280,338)(139,458,281,327)(140,447,282,316)(141,436,283,305)(142,425,284,354)(143,474,285,343)(144,463,286,332)(145,452,287,321)(146,441,288,310)(147,430,289,359)(148,479,290,348)(149,468,291,337)(150,457,292,326)(151,446,293,315)(152,435,294,304)(153,424,295,353)(154,473,296,342)(155,462,297,331)(156,451,298,320)(157,440,299,309)(158,429,300,358)(159,478,241,347)(160,467,242,336)(161,456,243,325)(162,445,244,314)(163,434,245,303)(164,423,246,352)(165,472,247,341)(166,461,248,330)(167,450,249,319)(168,439,250,308)(169,428,251,357)(170,477,252,346)(171,466,253,335)(172,455,254,324)(173,444,255,313)(174,433,256,302)(175,422,257,351)(176,471,258,340)(177,460,259,329)(178,449,260,318)(179,438,261,307)(180,427,262,356) );
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)], [(1,258,107,161,31,288,77,131),(2,299,108,142,32,269,78,172),(3,280,109,123,33,250,79,153),(4,261,110,164,34,291,80,134),(5,242,111,145,35,272,81,175),(6,283,112,126,36,253,82,156),(7,264,113,167,37,294,83,137),(8,245,114,148,38,275,84,178),(9,286,115,129,39,256,85,159),(10,267,116,170,40,297,86,140),(11,248,117,151,41,278,87,121),(12,289,118,132,42,259,88,162),(13,270,119,173,43,300,89,143),(14,251,120,154,44,281,90,124),(15,292,61,135,45,262,91,165),(16,273,62,176,46,243,92,146),(17,254,63,157,47,284,93,127),(18,295,64,138,48,265,94,168),(19,276,65,179,49,246,95,149),(20,257,66,160,50,287,96,130),(21,298,67,141,51,268,97,171),(22,279,68,122,52,249,98,152),(23,260,69,163,53,290,99,133),(24,241,70,144,54,271,100,174),(25,282,71,125,55,252,101,155),(26,263,72,166,56,293,102,136),(27,244,73,147,57,274,103,177),(28,285,74,128,58,255,104,158),(29,266,75,169,59,296,105,139),(30,247,76,150,60,277,106,180),(181,330,388,446,211,360,418,476),(182,311,389,427,212,341,419,457),(183,352,390,468,213,322,420,438),(184,333,391,449,214,303,361,479),(185,314,392,430,215,344,362,460),(186,355,393,471,216,325,363,441),(187,336,394,452,217,306,364,422),(188,317,395,433,218,347,365,463),(189,358,396,474,219,328,366,444),(190,339,397,455,220,309,367,425),(191,320,398,436,221,350,368,466),(192,301,399,477,222,331,369,447),(193,342,400,458,223,312,370,428),(194,323,401,439,224,353,371,469),(195,304,402,480,225,334,372,450),(196,345,403,461,226,315,373,431),(197,326,404,442,227,356,374,472),(198,307,405,423,228,337,375,453),(199,348,406,464,229,318,376,434),(200,329,407,445,230,359,377,475),(201,310,408,426,231,340,378,456),(202,351,409,467,232,321,379,437),(203,332,410,448,233,302,380,478),(204,313,411,429,234,343,381,459),(205,354,412,470,235,324,382,440),(206,335,413,451,236,305,383,421),(207,316,414,432,237,346,384,462),(208,357,415,473,238,327,385,443),(209,338,416,454,239,308,386,424),(210,319,417,435,240,349,387,465)], [(1,393,62,231),(2,382,63,220),(3,371,64,209),(4,420,65,198),(5,409,66,187),(6,398,67,236),(7,387,68,225),(8,376,69,214),(9,365,70,203),(10,414,71,192),(11,403,72,181),(12,392,73,230),(13,381,74,219),(14,370,75,208),(15,419,76,197),(16,408,77,186),(17,397,78,235),(18,386,79,224),(19,375,80,213),(20,364,81,202),(21,413,82,191),(22,402,83,240),(23,391,84,229),(24,380,85,218),(25,369,86,207),(26,418,87,196),(27,407,88,185),(28,396,89,234),(29,385,90,223),(30,374,91,212),(31,363,92,201),(32,412,93,190),(33,401,94,239),(34,390,95,228),(35,379,96,217),(36,368,97,206),(37,417,98,195),(38,406,99,184),(39,395,100,233),(40,384,101,222),(41,373,102,211),(42,362,103,200),(43,411,104,189),(44,400,105,238),(45,389,106,227),(46,378,107,216),(47,367,108,205),(48,416,109,194),(49,405,110,183),(50,394,111,232),(51,383,112,221),(52,372,113,210),(53,361,114,199),(54,410,115,188),(55,399,116,237),(56,388,117,226),(57,377,118,215),(58,366,119,204),(59,415,120,193),(60,404,61,182),(121,476,263,345),(122,465,264,334),(123,454,265,323),(124,443,266,312),(125,432,267,301),(126,421,268,350),(127,470,269,339),(128,459,270,328),(129,448,271,317),(130,437,272,306),(131,426,273,355),(132,475,274,344),(133,464,275,333),(134,453,276,322),(135,442,277,311),(136,431,278,360),(137,480,279,349),(138,469,280,338),(139,458,281,327),(140,447,282,316),(141,436,283,305),(142,425,284,354),(143,474,285,343),(144,463,286,332),(145,452,287,321),(146,441,288,310),(147,430,289,359),(148,479,290,348),(149,468,291,337),(150,457,292,326),(151,446,293,315),(152,435,294,304),(153,424,295,353),(154,473,296,342),(155,462,297,331),(156,451,298,320),(157,440,299,309),(158,429,300,358),(159,478,241,347),(160,467,242,336),(161,456,243,325),(162,445,244,314),(163,434,245,303),(164,423,246,352),(165,472,247,341),(166,461,248,330),(167,450,249,319),(168,439,250,308),(169,428,251,357),(170,477,252,346),(171,466,253,335),(172,455,254,324),(173,444,255,313),(174,433,256,302),(175,422,257,351),(176,471,258,340),(177,460,259,329),(178,449,260,318),(179,438,261,307),(180,427,262,356)]])
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 | 12E | ··· | 12L | 15A | 15B | 20A | ··· | 20H | 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 | 12 | ··· | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 2 | ··· | 2 | 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 | Dic3 | D6 | M4(2) | D10 | C3⋊C8 | Dic6 | D12 | Dic10 | C5⋊D4 | C4×D5 | C4.Dic3 | C8×D5 | C8⋊D5 | S3×D5 | C5⋊D12 | C15⋊Q8 | D5×Dic3 | D5×C3⋊C8 | C20.32D6 |
| kernel | C60.13Q8 | C12×Dic5 | C10×C3⋊C8 | C2×C15⋊3C8 | C6×Dic5 | C3×Dic5 | C4×Dic5 | C60 | C60 | C2×C3⋊C8 | C2×Dic5 | C2×C20 | C30 | C2×C12 | Dic5 | C20 | C20 | C12 | C12 | C2×C6 | C10 | C6 | C6 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C60.13Q8 ►in GL4(𝔽241) generated by
| 64 | 46 | 0 | 0 |
| 195 | 46 | 0 | 0 |
| 0 | 0 | 1 | 240 |
| 0 | 0 | 1 | 0 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 26 | 69 |
| 0 | 0 | 95 | 215 |
| 194 | 20 | 0 | 0 |
| 227 | 47 | 0 | 0 |
| 0 | 0 | 99 | 43 |
| 0 | 0 | 198 | 142 |
G:=sub<GL(4,GF(241))| [64,195,0,0,46,46,0,0,0,0,1,1,0,0,240,0],[30,0,0,0,0,30,0,0,0,0,26,95,0,0,69,215],[194,227,0,0,20,47,0,0,0,0,99,198,0,0,43,142] >;
C60.13Q8 in GAP, Magma, Sage, TeX
C_{60}._{13}Q_8 % in TeX
G:=Group("C60.13Q8"); // GroupNames label
G:=SmallGroup(480,58);
// by ID
G=gap.SmallGroup(480,58);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,36,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^15*b^2,b*a*b^-1=a^41,c*a*c^-1=a^49,c*b*c^-1=a^15*b^3>;
// generators/relations