metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊8Q8, C4.4D60, C4⋊2Dic30, C20⋊7Dic6, C12⋊7Dic10, C20.29D12, C12.29D20, C60.160D4, C42.4D15, (C4×C60).2C2, (C4×C20).2S3, C15⋊10(C4⋊Q8), C2.4(C2×D60), (C4×C12).2D5, C5⋊2(C12⋊2Q8), (C2×C4).76D30, C3⋊2(C20⋊2Q8), C6.29(C2×D20), C30.66(C2×Q8), C60⋊5C4.6C2, (C2×C20).388D6, C30.258(C2×D4), C10.30(C2×D12), C2.4(C2×Dic30), (C2×C12).391D10, (C2×Dic30).3C2, C6.33(C2×Dic10), C10.33(C2×Dic6), (C2×C60).475C22, (C2×C30).266C23, (C2×Dic15).1C22, C22.34(C22×D15), (C2×C6).262(C22×D5), (C2×C10).261(C22×S3), SmallGroup(480,834)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60⋊8Q8
G = < a,b,c | a60=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 708 in 136 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, C20, C2×C10, Dic6, C2×Dic3, C2×C12, C2×C12, C30, C30, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C4⋊Dic3, C4×C12, C2×Dic6, Dic15, C60, C2×C30, C4⋊Dic5, C4×C20, C2×Dic10, C12⋊2Q8, Dic30, C2×Dic15, C2×C60, C2×C60, C20⋊2Q8, C60⋊5C4, C4×C60, C2×Dic30, C60⋊8Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, D10, Dic6, D12, C22×S3, D15, C4⋊Q8, Dic10, D20, C22×D5, C2×Dic6, C2×D12, D30, C2×Dic10, C2×D20, C12⋊2Q8, Dic30, D60, C22×D15, C20⋊2Q8, C2×Dic30, C2×D60, C60⋊8Q8
►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)
(1 474 253 82)(2 475 254 83)(3 476 255 84)(4 477 256 85)(5 478 257 86)(6 479 258 87)(7 480 259 88)(8 421 260 89)(9 422 261 90)(10 423 262 91)(11 424 263 92)(12 425 264 93)(13 426 265 94)(14 427 266 95)(15 428 267 96)(16 429 268 97)(17 430 269 98)(18 431 270 99)(19 432 271 100)(20 433 272 101)(21 434 273 102)(22 435 274 103)(23 436 275 104)(24 437 276 105)(25 438 277 106)(26 439 278 107)(27 440 279 108)(28 441 280 109)(29 442 281 110)(30 443 282 111)(31 444 283 112)(32 445 284 113)(33 446 285 114)(34 447 286 115)(35 448 287 116)(36 449 288 117)(37 450 289 118)(38 451 290 119)(39 452 291 120)(40 453 292 61)(41 454 293 62)(42 455 294 63)(43 456 295 64)(44 457 296 65)(45 458 297 66)(46 459 298 67)(47 460 299 68)(48 461 300 69)(49 462 241 70)(50 463 242 71)(51 464 243 72)(52 465 244 73)(53 466 245 74)(54 467 246 75)(55 468 247 76)(56 469 248 77)(57 470 249 78)(58 471 250 79)(59 472 251 80)(60 473 252 81)(121 416 320 199)(122 417 321 200)(123 418 322 201)(124 419 323 202)(125 420 324 203)(126 361 325 204)(127 362 326 205)(128 363 327 206)(129 364 328 207)(130 365 329 208)(131 366 330 209)(132 367 331 210)(133 368 332 211)(134 369 333 212)(135 370 334 213)(136 371 335 214)(137 372 336 215)(138 373 337 216)(139 374 338 217)(140 375 339 218)(141 376 340 219)(142 377 341 220)(143 378 342 221)(144 379 343 222)(145 380 344 223)(146 381 345 224)(147 382 346 225)(148 383 347 226)(149 384 348 227)(150 385 349 228)(151 386 350 229)(152 387 351 230)(153 388 352 231)(154 389 353 232)(155 390 354 233)(156 391 355 234)(157 392 356 235)(158 393 357 236)(159 394 358 237)(160 395 359 238)(161 396 360 239)(162 397 301 240)(163 398 302 181)(164 399 303 182)(165 400 304 183)(166 401 305 184)(167 402 306 185)(168 403 307 186)(169 404 308 187)(170 405 309 188)(171 406 310 189)(172 407 311 190)(173 408 312 191)(174 409 313 192)(175 410 314 193)(176 411 315 194)(177 412 316 195)(178 413 317 196)(179 414 318 197)(180 415 319 198)
(1 180 253 319)(2 179 254 318)(3 178 255 317)(4 177 256 316)(5 176 257 315)(6 175 258 314)(7 174 259 313)(8 173 260 312)(9 172 261 311)(10 171 262 310)(11 170 263 309)(12 169 264 308)(13 168 265 307)(14 167 266 306)(15 166 267 305)(16 165 268 304)(17 164 269 303)(18 163 270 302)(19 162 271 301)(20 161 272 360)(21 160 273 359)(22 159 274 358)(23 158 275 357)(24 157 276 356)(25 156 277 355)(26 155 278 354)(27 154 279 353)(28 153 280 352)(29 152 281 351)(30 151 282 350)(31 150 283 349)(32 149 284 348)(33 148 285 347)(34 147 286 346)(35 146 287 345)(36 145 288 344)(37 144 289 343)(38 143 290 342)(39 142 291 341)(40 141 292 340)(41 140 293 339)(42 139 294 338)(43 138 295 337)(44 137 296 336)(45 136 297 335)(46 135 298 334)(47 134 299 333)(48 133 300 332)(49 132 241 331)(50 131 242 330)(51 130 243 329)(52 129 244 328)(53 128 245 327)(54 127 246 326)(55 126 247 325)(56 125 248 324)(57 124 249 323)(58 123 250 322)(59 122 251 321)(60 121 252 320)(61 376 453 219)(62 375 454 218)(63 374 455 217)(64 373 456 216)(65 372 457 215)(66 371 458 214)(67 370 459 213)(68 369 460 212)(69 368 461 211)(70 367 462 210)(71 366 463 209)(72 365 464 208)(73 364 465 207)(74 363 466 206)(75 362 467 205)(76 361 468 204)(77 420 469 203)(78 419 470 202)(79 418 471 201)(80 417 472 200)(81 416 473 199)(82 415 474 198)(83 414 475 197)(84 413 476 196)(85 412 477 195)(86 411 478 194)(87 410 479 193)(88 409 480 192)(89 408 421 191)(90 407 422 190)(91 406 423 189)(92 405 424 188)(93 404 425 187)(94 403 426 186)(95 402 427 185)(96 401 428 184)(97 400 429 183)(98 399 430 182)(99 398 431 181)(100 397 432 240)(101 396 433 239)(102 395 434 238)(103 394 435 237)(104 393 436 236)(105 392 437 235)(106 391 438 234)(107 390 439 233)(108 389 440 232)(109 388 441 231)(110 387 442 230)(111 386 443 229)(112 385 444 228)(113 384 445 227)(114 383 446 226)(115 382 447 225)(116 381 448 224)(117 380 449 223)(118 379 450 222)(119 378 451 221)(120 377 452 220)
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), (1,474,253,82)(2,475,254,83)(3,476,255,84)(4,477,256,85)(5,478,257,86)(6,479,258,87)(7,480,259,88)(8,421,260,89)(9,422,261,90)(10,423,262,91)(11,424,263,92)(12,425,264,93)(13,426,265,94)(14,427,266,95)(15,428,267,96)(16,429,268,97)(17,430,269,98)(18,431,270,99)(19,432,271,100)(20,433,272,101)(21,434,273,102)(22,435,274,103)(23,436,275,104)(24,437,276,105)(25,438,277,106)(26,439,278,107)(27,440,279,108)(28,441,280,109)(29,442,281,110)(30,443,282,111)(31,444,283,112)(32,445,284,113)(33,446,285,114)(34,447,286,115)(35,448,287,116)(36,449,288,117)(37,450,289,118)(38,451,290,119)(39,452,291,120)(40,453,292,61)(41,454,293,62)(42,455,294,63)(43,456,295,64)(44,457,296,65)(45,458,297,66)(46,459,298,67)(47,460,299,68)(48,461,300,69)(49,462,241,70)(50,463,242,71)(51,464,243,72)(52,465,244,73)(53,466,245,74)(54,467,246,75)(55,468,247,76)(56,469,248,77)(57,470,249,78)(58,471,250,79)(59,472,251,80)(60,473,252,81)(121,416,320,199)(122,417,321,200)(123,418,322,201)(124,419,323,202)(125,420,324,203)(126,361,325,204)(127,362,326,205)(128,363,327,206)(129,364,328,207)(130,365,329,208)(131,366,330,209)(132,367,331,210)(133,368,332,211)(134,369,333,212)(135,370,334,213)(136,371,335,214)(137,372,336,215)(138,373,337,216)(139,374,338,217)(140,375,339,218)(141,376,340,219)(142,377,341,220)(143,378,342,221)(144,379,343,222)(145,380,344,223)(146,381,345,224)(147,382,346,225)(148,383,347,226)(149,384,348,227)(150,385,349,228)(151,386,350,229)(152,387,351,230)(153,388,352,231)(154,389,353,232)(155,390,354,233)(156,391,355,234)(157,392,356,235)(158,393,357,236)(159,394,358,237)(160,395,359,238)(161,396,360,239)(162,397,301,240)(163,398,302,181)(164,399,303,182)(165,400,304,183)(166,401,305,184)(167,402,306,185)(168,403,307,186)(169,404,308,187)(170,405,309,188)(171,406,310,189)(172,407,311,190)(173,408,312,191)(174,409,313,192)(175,410,314,193)(176,411,315,194)(177,412,316,195)(178,413,317,196)(179,414,318,197)(180,415,319,198), (1,180,253,319)(2,179,254,318)(3,178,255,317)(4,177,256,316)(5,176,257,315)(6,175,258,314)(7,174,259,313)(8,173,260,312)(9,172,261,311)(10,171,262,310)(11,170,263,309)(12,169,264,308)(13,168,265,307)(14,167,266,306)(15,166,267,305)(16,165,268,304)(17,164,269,303)(18,163,270,302)(19,162,271,301)(20,161,272,360)(21,160,273,359)(22,159,274,358)(23,158,275,357)(24,157,276,356)(25,156,277,355)(26,155,278,354)(27,154,279,353)(28,153,280,352)(29,152,281,351)(30,151,282,350)(31,150,283,349)(32,149,284,348)(33,148,285,347)(34,147,286,346)(35,146,287,345)(36,145,288,344)(37,144,289,343)(38,143,290,342)(39,142,291,341)(40,141,292,340)(41,140,293,339)(42,139,294,338)(43,138,295,337)(44,137,296,336)(45,136,297,335)(46,135,298,334)(47,134,299,333)(48,133,300,332)(49,132,241,331)(50,131,242,330)(51,130,243,329)(52,129,244,328)(53,128,245,327)(54,127,246,326)(55,126,247,325)(56,125,248,324)(57,124,249,323)(58,123,250,322)(59,122,251,321)(60,121,252,320)(61,376,453,219)(62,375,454,218)(63,374,455,217)(64,373,456,216)(65,372,457,215)(66,371,458,214)(67,370,459,213)(68,369,460,212)(69,368,461,211)(70,367,462,210)(71,366,463,209)(72,365,464,208)(73,364,465,207)(74,363,466,206)(75,362,467,205)(76,361,468,204)(77,420,469,203)(78,419,470,202)(79,418,471,201)(80,417,472,200)(81,416,473,199)(82,415,474,198)(83,414,475,197)(84,413,476,196)(85,412,477,195)(86,411,478,194)(87,410,479,193)(88,409,480,192)(89,408,421,191)(90,407,422,190)(91,406,423,189)(92,405,424,188)(93,404,425,187)(94,403,426,186)(95,402,427,185)(96,401,428,184)(97,400,429,183)(98,399,430,182)(99,398,431,181)(100,397,432,240)(101,396,433,239)(102,395,434,238)(103,394,435,237)(104,393,436,236)(105,392,437,235)(106,391,438,234)(107,390,439,233)(108,389,440,232)(109,388,441,231)(110,387,442,230)(111,386,443,229)(112,385,444,228)(113,384,445,227)(114,383,446,226)(115,382,447,225)(116,381,448,224)(117,380,449,223)(118,379,450,222)(119,378,451,221)(120,377,452,220)>;
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), (1,474,253,82)(2,475,254,83)(3,476,255,84)(4,477,256,85)(5,478,257,86)(6,479,258,87)(7,480,259,88)(8,421,260,89)(9,422,261,90)(10,423,262,91)(11,424,263,92)(12,425,264,93)(13,426,265,94)(14,427,266,95)(15,428,267,96)(16,429,268,97)(17,430,269,98)(18,431,270,99)(19,432,271,100)(20,433,272,101)(21,434,273,102)(22,435,274,103)(23,436,275,104)(24,437,276,105)(25,438,277,106)(26,439,278,107)(27,440,279,108)(28,441,280,109)(29,442,281,110)(30,443,282,111)(31,444,283,112)(32,445,284,113)(33,446,285,114)(34,447,286,115)(35,448,287,116)(36,449,288,117)(37,450,289,118)(38,451,290,119)(39,452,291,120)(40,453,292,61)(41,454,293,62)(42,455,294,63)(43,456,295,64)(44,457,296,65)(45,458,297,66)(46,459,298,67)(47,460,299,68)(48,461,300,69)(49,462,241,70)(50,463,242,71)(51,464,243,72)(52,465,244,73)(53,466,245,74)(54,467,246,75)(55,468,247,76)(56,469,248,77)(57,470,249,78)(58,471,250,79)(59,472,251,80)(60,473,252,81)(121,416,320,199)(122,417,321,200)(123,418,322,201)(124,419,323,202)(125,420,324,203)(126,361,325,204)(127,362,326,205)(128,363,327,206)(129,364,328,207)(130,365,329,208)(131,366,330,209)(132,367,331,210)(133,368,332,211)(134,369,333,212)(135,370,334,213)(136,371,335,214)(137,372,336,215)(138,373,337,216)(139,374,338,217)(140,375,339,218)(141,376,340,219)(142,377,341,220)(143,378,342,221)(144,379,343,222)(145,380,344,223)(146,381,345,224)(147,382,346,225)(148,383,347,226)(149,384,348,227)(150,385,349,228)(151,386,350,229)(152,387,351,230)(153,388,352,231)(154,389,353,232)(155,390,354,233)(156,391,355,234)(157,392,356,235)(158,393,357,236)(159,394,358,237)(160,395,359,238)(161,396,360,239)(162,397,301,240)(163,398,302,181)(164,399,303,182)(165,400,304,183)(166,401,305,184)(167,402,306,185)(168,403,307,186)(169,404,308,187)(170,405,309,188)(171,406,310,189)(172,407,311,190)(173,408,312,191)(174,409,313,192)(175,410,314,193)(176,411,315,194)(177,412,316,195)(178,413,317,196)(179,414,318,197)(180,415,319,198), (1,180,253,319)(2,179,254,318)(3,178,255,317)(4,177,256,316)(5,176,257,315)(6,175,258,314)(7,174,259,313)(8,173,260,312)(9,172,261,311)(10,171,262,310)(11,170,263,309)(12,169,264,308)(13,168,265,307)(14,167,266,306)(15,166,267,305)(16,165,268,304)(17,164,269,303)(18,163,270,302)(19,162,271,301)(20,161,272,360)(21,160,273,359)(22,159,274,358)(23,158,275,357)(24,157,276,356)(25,156,277,355)(26,155,278,354)(27,154,279,353)(28,153,280,352)(29,152,281,351)(30,151,282,350)(31,150,283,349)(32,149,284,348)(33,148,285,347)(34,147,286,346)(35,146,287,345)(36,145,288,344)(37,144,289,343)(38,143,290,342)(39,142,291,341)(40,141,292,340)(41,140,293,339)(42,139,294,338)(43,138,295,337)(44,137,296,336)(45,136,297,335)(46,135,298,334)(47,134,299,333)(48,133,300,332)(49,132,241,331)(50,131,242,330)(51,130,243,329)(52,129,244,328)(53,128,245,327)(54,127,246,326)(55,126,247,325)(56,125,248,324)(57,124,249,323)(58,123,250,322)(59,122,251,321)(60,121,252,320)(61,376,453,219)(62,375,454,218)(63,374,455,217)(64,373,456,216)(65,372,457,215)(66,371,458,214)(67,370,459,213)(68,369,460,212)(69,368,461,211)(70,367,462,210)(71,366,463,209)(72,365,464,208)(73,364,465,207)(74,363,466,206)(75,362,467,205)(76,361,468,204)(77,420,469,203)(78,419,470,202)(79,418,471,201)(80,417,472,200)(81,416,473,199)(82,415,474,198)(83,414,475,197)(84,413,476,196)(85,412,477,195)(86,411,478,194)(87,410,479,193)(88,409,480,192)(89,408,421,191)(90,407,422,190)(91,406,423,189)(92,405,424,188)(93,404,425,187)(94,403,426,186)(95,402,427,185)(96,401,428,184)(97,400,429,183)(98,399,430,182)(99,398,431,181)(100,397,432,240)(101,396,433,239)(102,395,434,238)(103,394,435,237)(104,393,436,236)(105,392,437,235)(106,391,438,234)(107,390,439,233)(108,389,440,232)(109,388,441,231)(110,387,442,230)(111,386,443,229)(112,385,444,228)(113,384,445,227)(114,383,446,226)(115,382,447,225)(116,381,448,224)(117,380,449,223)(118,379,450,222)(119,378,451,221)(120,377,452,220) );
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)], [(1,474,253,82),(2,475,254,83),(3,476,255,84),(4,477,256,85),(5,478,257,86),(6,479,258,87),(7,480,259,88),(8,421,260,89),(9,422,261,90),(10,423,262,91),(11,424,263,92),(12,425,264,93),(13,426,265,94),(14,427,266,95),(15,428,267,96),(16,429,268,97),(17,430,269,98),(18,431,270,99),(19,432,271,100),(20,433,272,101),(21,434,273,102),(22,435,274,103),(23,436,275,104),(24,437,276,105),(25,438,277,106),(26,439,278,107),(27,440,279,108),(28,441,280,109),(29,442,281,110),(30,443,282,111),(31,444,283,112),(32,445,284,113),(33,446,285,114),(34,447,286,115),(35,448,287,116),(36,449,288,117),(37,450,289,118),(38,451,290,119),(39,452,291,120),(40,453,292,61),(41,454,293,62),(42,455,294,63),(43,456,295,64),(44,457,296,65),(45,458,297,66),(46,459,298,67),(47,460,299,68),(48,461,300,69),(49,462,241,70),(50,463,242,71),(51,464,243,72),(52,465,244,73),(53,466,245,74),(54,467,246,75),(55,468,247,76),(56,469,248,77),(57,470,249,78),(58,471,250,79),(59,472,251,80),(60,473,252,81),(121,416,320,199),(122,417,321,200),(123,418,322,201),(124,419,323,202),(125,420,324,203),(126,361,325,204),(127,362,326,205),(128,363,327,206),(129,364,328,207),(130,365,329,208),(131,366,330,209),(132,367,331,210),(133,368,332,211),(134,369,333,212),(135,370,334,213),(136,371,335,214),(137,372,336,215),(138,373,337,216),(139,374,338,217),(140,375,339,218),(141,376,340,219),(142,377,341,220),(143,378,342,221),(144,379,343,222),(145,380,344,223),(146,381,345,224),(147,382,346,225),(148,383,347,226),(149,384,348,227),(150,385,349,228),(151,386,350,229),(152,387,351,230),(153,388,352,231),(154,389,353,232),(155,390,354,233),(156,391,355,234),(157,392,356,235),(158,393,357,236),(159,394,358,237),(160,395,359,238),(161,396,360,239),(162,397,301,240),(163,398,302,181),(164,399,303,182),(165,400,304,183),(166,401,305,184),(167,402,306,185),(168,403,307,186),(169,404,308,187),(170,405,309,188),(171,406,310,189),(172,407,311,190),(173,408,312,191),(174,409,313,192),(175,410,314,193),(176,411,315,194),(177,412,316,195),(178,413,317,196),(179,414,318,197),(180,415,319,198)], [(1,180,253,319),(2,179,254,318),(3,178,255,317),(4,177,256,316),(5,176,257,315),(6,175,258,314),(7,174,259,313),(8,173,260,312),(9,172,261,311),(10,171,262,310),(11,170,263,309),(12,169,264,308),(13,168,265,307),(14,167,266,306),(15,166,267,305),(16,165,268,304),(17,164,269,303),(18,163,270,302),(19,162,271,301),(20,161,272,360),(21,160,273,359),(22,159,274,358),(23,158,275,357),(24,157,276,356),(25,156,277,355),(26,155,278,354),(27,154,279,353),(28,153,280,352),(29,152,281,351),(30,151,282,350),(31,150,283,349),(32,149,284,348),(33,148,285,347),(34,147,286,346),(35,146,287,345),(36,145,288,344),(37,144,289,343),(38,143,290,342),(39,142,291,341),(40,141,292,340),(41,140,293,339),(42,139,294,338),(43,138,295,337),(44,137,296,336),(45,136,297,335),(46,135,298,334),(47,134,299,333),(48,133,300,332),(49,132,241,331),(50,131,242,330),(51,130,243,329),(52,129,244,328),(53,128,245,327),(54,127,246,326),(55,126,247,325),(56,125,248,324),(57,124,249,323),(58,123,250,322),(59,122,251,321),(60,121,252,320),(61,376,453,219),(62,375,454,218),(63,374,455,217),(64,373,456,216),(65,372,457,215),(66,371,458,214),(67,370,459,213),(68,369,460,212),(69,368,461,211),(70,367,462,210),(71,366,463,209),(72,365,464,208),(73,364,465,207),(74,363,466,206),(75,362,467,205),(76,361,468,204),(77,420,469,203),(78,419,470,202),(79,418,471,201),(80,417,472,200),(81,416,473,199),(82,415,474,198),(83,414,475,197),(84,413,476,196),(85,412,477,195),(86,411,478,194),(87,410,479,193),(88,409,480,192),(89,408,421,191),(90,407,422,190),(91,406,423,189),(92,405,424,188),(93,404,425,187),(94,403,426,186),(95,402,427,185),(96,401,428,184),(97,400,429,183),(98,399,430,182),(99,398,431,181),(100,397,432,240),(101,396,433,239),(102,395,434,238),(103,394,435,237),(104,393,436,236),(105,392,437,235),(106,391,438,234),(107,390,439,233),(108,389,440,232),(109,388,441,231),(110,387,442,230),(111,386,443,229),(112,385,444,228),(113,384,445,227),(114,383,446,226),(115,382,447,225),(116,381,448,224),(117,380,449,223),(118,379,450,222),(119,378,451,221),(120,377,452,220)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 60 | 60 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | + | - | + | + | - | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | D10 | Dic6 | D12 | D15 | Dic10 | D20 | D30 | Dic30 | D60 |
| kernel | C60⋊8Q8 | C60⋊5C4 | C4×C60 | C2×Dic30 | C4×C20 | C60 | C60 | C4×C12 | C2×C20 | C2×C12 | C20 | C20 | C42 | C12 | C12 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 1 | 2 | 4 | 2 | 3 | 6 | 8 | 4 | 4 | 16 | 8 | 12 | 32 | 16 |
Matrix representation of C60⋊8Q8 ►in GL4(𝔽61) generated by
| 30 | 14 | 0 | 0 |
| 47 | 24 | 0 | 0 |
| 0 | 0 | 55 | 53 |
| 0 | 0 | 8 | 41 |
| 32 | 7 | 0 | 0 |
| 54 | 29 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 24 | 53 | 0 | 0 |
| 34 | 37 | 0 | 0 |
| 0 | 0 | 49 | 55 |
| 0 | 0 | 34 | 12 |
G:=sub<GL(4,GF(61))| [30,47,0,0,14,24,0,0,0,0,55,8,0,0,53,41],[32,54,0,0,7,29,0,0,0,0,1,0,0,0,0,1],[24,34,0,0,53,37,0,0,0,0,49,34,0,0,55,12] >;
C60⋊8Q8 in GAP, Magma, Sage, TeX
C_{60}\rtimes_8Q_8 % in TeX
G:=Group("C60:8Q8"); // GroupNames label
G:=SmallGroup(480,834);
// by ID
G=gap.SmallGroup(480,834);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,120,254,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations