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## G = C60.24Q8order 480 = 25·3·5

### 3rd non-split extension by C60 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.24Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×Dic15 — C30.4Q8 — C60.24Q8
 Lower central C15 — C2×C30 — C60.24Q8
 Upper central C1 — C22 — C42

Generators and relations for C60.24Q8
G = < a,b,c | a60=b4=1, c2=a30b2, ab=ba, cac-1=a-1, cbc-1=a30b-1 >

Subgroups: 516 in 112 conjugacy classes, 55 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, C10, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, Dic5, C20, C20, C2×C10, C2×Dic3, C2×C12, C2×C12, C30, C30, C42.C2, C2×Dic5, C2×C20, C2×C20, Dic3⋊C4, C4⋊Dic3, C4×C12, Dic15, C60, C60, C2×C30, C10.D4, C4⋊Dic5, C4×C20, C12.6Q8, C2×Dic15, C2×C60, C2×C60, C20.6Q8, C30.4Q8, C605C4, C4×C60, C60.24Q8
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, Dic6, C22×S3, D15, C42.C2, Dic10, C22×D5, C2×Dic6, C4○D12, D30, C2×Dic10, C4○D20, C12.6Q8, Dic30, C22×D15, C20.6Q8, C2×Dic30, D6011C2, C60.24Q8

Smallest permutation representation of C60.24Q8
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)
(1 306 74 191)(2 307 75 192)(3 308 76 193)(4 309 77 194)(5 310 78 195)(6 311 79 196)(7 312 80 197)(8 313 81 198)(9 314 82 199)(10 315 83 200)(11 316 84 201)(12 317 85 202)(13 318 86 203)(14 319 87 204)(15 320 88 205)(16 321 89 206)(17 322 90 207)(18 323 91 208)(19 324 92 209)(20 325 93 210)(21 326 94 211)(22 327 95 212)(23 328 96 213)(24 329 97 214)(25 330 98 215)(26 331 99 216)(27 332 100 217)(28 333 101 218)(29 334 102 219)(30 335 103 220)(31 336 104 221)(32 337 105 222)(33 338 106 223)(34 339 107 224)(35 340 108 225)(36 341 109 226)(37 342 110 227)(38 343 111 228)(39 344 112 229)(40 345 113 230)(41 346 114 231)(42 347 115 232)(43 348 116 233)(44 349 117 234)(45 350 118 235)(46 351 119 236)(47 352 120 237)(48 353 61 238)(49 354 62 239)(50 355 63 240)(51 356 64 181)(52 357 65 182)(53 358 66 183)(54 359 67 184)(55 360 68 185)(56 301 69 186)(57 302 70 187)(58 303 71 188)(59 304 72 189)(60 305 73 190)(121 385 456 275)(122 386 457 276)(123 387 458 277)(124 388 459 278)(125 389 460 279)(126 390 461 280)(127 391 462 281)(128 392 463 282)(129 393 464 283)(130 394 465 284)(131 395 466 285)(132 396 467 286)(133 397 468 287)(134 398 469 288)(135 399 470 289)(136 400 471 290)(137 401 472 291)(138 402 473 292)(139 403 474 293)(140 404 475 294)(141 405 476 295)(142 406 477 296)(143 407 478 297)(144 408 479 298)(145 409 480 299)(146 410 421 300)(147 411 422 241)(148 412 423 242)(149 413 424 243)(150 414 425 244)(151 415 426 245)(152 416 427 246)(153 417 428 247)(154 418 429 248)(155 419 430 249)(156 420 431 250)(157 361 432 251)(158 362 433 252)(159 363 434 253)(160 364 435 254)(161 365 436 255)(162 366 437 256)(163 367 438 257)(164 368 439 258)(165 369 440 259)(166 370 441 260)(167 371 442 261)(168 372 443 262)(169 373 444 263)(170 374 445 264)(171 375 446 265)(172 376 447 266)(173 377 448 267)(174 378 449 268)(175 379 450 269)(176 380 451 270)(177 381 452 271)(178 382 453 272)(179 383 454 273)(180 384 455 274)
(1 268 104 408)(2 267 105 407)(3 266 106 406)(4 265 107 405)(5 264 108 404)(6 263 109 403)(7 262 110 402)(8 261 111 401)(9 260 112 400)(10 259 113 399)(11 258 114 398)(12 257 115 397)(13 256 116 396)(14 255 117 395)(15 254 118 394)(16 253 119 393)(17 252 120 392)(18 251 61 391)(19 250 62 390)(20 249 63 389)(21 248 64 388)(22 247 65 387)(23 246 66 386)(24 245 67 385)(25 244 68 384)(26 243 69 383)(27 242 70 382)(28 241 71 381)(29 300 72 380)(30 299 73 379)(31 298 74 378)(32 297 75 377)(33 296 76 376)(34 295 77 375)(35 294 78 374)(36 293 79 373)(37 292 80 372)(38 291 81 371)(39 290 82 370)(40 289 83 369)(41 288 84 368)(42 287 85 367)(43 286 86 366)(44 285 87 365)(45 284 88 364)(46 283 89 363)(47 282 90 362)(48 281 91 361)(49 280 92 420)(50 279 93 419)(51 278 94 418)(52 277 95 417)(53 276 96 416)(54 275 97 415)(55 274 98 414)(56 273 99 413)(57 272 100 412)(58 271 101 411)(59 270 102 410)(60 269 103 409)(121 359 426 214)(122 358 427 213)(123 357 428 212)(124 356 429 211)(125 355 430 210)(126 354 431 209)(127 353 432 208)(128 352 433 207)(129 351 434 206)(130 350 435 205)(131 349 436 204)(132 348 437 203)(133 347 438 202)(134 346 439 201)(135 345 440 200)(136 344 441 199)(137 343 442 198)(138 342 443 197)(139 341 444 196)(140 340 445 195)(141 339 446 194)(142 338 447 193)(143 337 448 192)(144 336 449 191)(145 335 450 190)(146 334 451 189)(147 333 452 188)(148 332 453 187)(149 331 454 186)(150 330 455 185)(151 329 456 184)(152 328 457 183)(153 327 458 182)(154 326 459 181)(155 325 460 240)(156 324 461 239)(157 323 462 238)(158 322 463 237)(159 321 464 236)(160 320 465 235)(161 319 466 234)(162 318 467 233)(163 317 468 232)(164 316 469 231)(165 315 470 230)(166 314 471 229)(167 313 472 228)(168 312 473 227)(169 311 474 226)(170 310 475 225)(171 309 476 224)(172 308 477 223)(173 307 478 222)(174 306 479 221)(175 305 480 220)(176 304 421 219)(177 303 422 218)(178 302 423 217)(179 301 424 216)(180 360 425 215)```

`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), (1,306,74,191)(2,307,75,192)(3,308,76,193)(4,309,77,194)(5,310,78,195)(6,311,79,196)(7,312,80,197)(8,313,81,198)(9,314,82,199)(10,315,83,200)(11,316,84,201)(12,317,85,202)(13,318,86,203)(14,319,87,204)(15,320,88,205)(16,321,89,206)(17,322,90,207)(18,323,91,208)(19,324,92,209)(20,325,93,210)(21,326,94,211)(22,327,95,212)(23,328,96,213)(24,329,97,214)(25,330,98,215)(26,331,99,216)(27,332,100,217)(28,333,101,218)(29,334,102,219)(30,335,103,220)(31,336,104,221)(32,337,105,222)(33,338,106,223)(34,339,107,224)(35,340,108,225)(36,341,109,226)(37,342,110,227)(38,343,111,228)(39,344,112,229)(40,345,113,230)(41,346,114,231)(42,347,115,232)(43,348,116,233)(44,349,117,234)(45,350,118,235)(46,351,119,236)(47,352,120,237)(48,353,61,238)(49,354,62,239)(50,355,63,240)(51,356,64,181)(52,357,65,182)(53,358,66,183)(54,359,67,184)(55,360,68,185)(56,301,69,186)(57,302,70,187)(58,303,71,188)(59,304,72,189)(60,305,73,190)(121,385,456,275)(122,386,457,276)(123,387,458,277)(124,388,459,278)(125,389,460,279)(126,390,461,280)(127,391,462,281)(128,392,463,282)(129,393,464,283)(130,394,465,284)(131,395,466,285)(132,396,467,286)(133,397,468,287)(134,398,469,288)(135,399,470,289)(136,400,471,290)(137,401,472,291)(138,402,473,292)(139,403,474,293)(140,404,475,294)(141,405,476,295)(142,406,477,296)(143,407,478,297)(144,408,479,298)(145,409,480,299)(146,410,421,300)(147,411,422,241)(148,412,423,242)(149,413,424,243)(150,414,425,244)(151,415,426,245)(152,416,427,246)(153,417,428,247)(154,418,429,248)(155,419,430,249)(156,420,431,250)(157,361,432,251)(158,362,433,252)(159,363,434,253)(160,364,435,254)(161,365,436,255)(162,366,437,256)(163,367,438,257)(164,368,439,258)(165,369,440,259)(166,370,441,260)(167,371,442,261)(168,372,443,262)(169,373,444,263)(170,374,445,264)(171,375,446,265)(172,376,447,266)(173,377,448,267)(174,378,449,268)(175,379,450,269)(176,380,451,270)(177,381,452,271)(178,382,453,272)(179,383,454,273)(180,384,455,274), (1,268,104,408)(2,267,105,407)(3,266,106,406)(4,265,107,405)(5,264,108,404)(6,263,109,403)(7,262,110,402)(8,261,111,401)(9,260,112,400)(10,259,113,399)(11,258,114,398)(12,257,115,397)(13,256,116,396)(14,255,117,395)(15,254,118,394)(16,253,119,393)(17,252,120,392)(18,251,61,391)(19,250,62,390)(20,249,63,389)(21,248,64,388)(22,247,65,387)(23,246,66,386)(24,245,67,385)(25,244,68,384)(26,243,69,383)(27,242,70,382)(28,241,71,381)(29,300,72,380)(30,299,73,379)(31,298,74,378)(32,297,75,377)(33,296,76,376)(34,295,77,375)(35,294,78,374)(36,293,79,373)(37,292,80,372)(38,291,81,371)(39,290,82,370)(40,289,83,369)(41,288,84,368)(42,287,85,367)(43,286,86,366)(44,285,87,365)(45,284,88,364)(46,283,89,363)(47,282,90,362)(48,281,91,361)(49,280,92,420)(50,279,93,419)(51,278,94,418)(52,277,95,417)(53,276,96,416)(54,275,97,415)(55,274,98,414)(56,273,99,413)(57,272,100,412)(58,271,101,411)(59,270,102,410)(60,269,103,409)(121,359,426,214)(122,358,427,213)(123,357,428,212)(124,356,429,211)(125,355,430,210)(126,354,431,209)(127,353,432,208)(128,352,433,207)(129,351,434,206)(130,350,435,205)(131,349,436,204)(132,348,437,203)(133,347,438,202)(134,346,439,201)(135,345,440,200)(136,344,441,199)(137,343,442,198)(138,342,443,197)(139,341,444,196)(140,340,445,195)(141,339,446,194)(142,338,447,193)(143,337,448,192)(144,336,449,191)(145,335,450,190)(146,334,451,189)(147,333,452,188)(148,332,453,187)(149,331,454,186)(150,330,455,185)(151,329,456,184)(152,328,457,183)(153,327,458,182)(154,326,459,181)(155,325,460,240)(156,324,461,239)(157,323,462,238)(158,322,463,237)(159,321,464,236)(160,320,465,235)(161,319,466,234)(162,318,467,233)(163,317,468,232)(164,316,469,231)(165,315,470,230)(166,314,471,229)(167,313,472,228)(168,312,473,227)(169,311,474,226)(170,310,475,225)(171,309,476,224)(172,308,477,223)(173,307,478,222)(174,306,479,221)(175,305,480,220)(176,304,421,219)(177,303,422,218)(178,302,423,217)(179,301,424,216)(180,360,425,215)>;`

`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), (1,306,74,191)(2,307,75,192)(3,308,76,193)(4,309,77,194)(5,310,78,195)(6,311,79,196)(7,312,80,197)(8,313,81,198)(9,314,82,199)(10,315,83,200)(11,316,84,201)(12,317,85,202)(13,318,86,203)(14,319,87,204)(15,320,88,205)(16,321,89,206)(17,322,90,207)(18,323,91,208)(19,324,92,209)(20,325,93,210)(21,326,94,211)(22,327,95,212)(23,328,96,213)(24,329,97,214)(25,330,98,215)(26,331,99,216)(27,332,100,217)(28,333,101,218)(29,334,102,219)(30,335,103,220)(31,336,104,221)(32,337,105,222)(33,338,106,223)(34,339,107,224)(35,340,108,225)(36,341,109,226)(37,342,110,227)(38,343,111,228)(39,344,112,229)(40,345,113,230)(41,346,114,231)(42,347,115,232)(43,348,116,233)(44,349,117,234)(45,350,118,235)(46,351,119,236)(47,352,120,237)(48,353,61,238)(49,354,62,239)(50,355,63,240)(51,356,64,181)(52,357,65,182)(53,358,66,183)(54,359,67,184)(55,360,68,185)(56,301,69,186)(57,302,70,187)(58,303,71,188)(59,304,72,189)(60,305,73,190)(121,385,456,275)(122,386,457,276)(123,387,458,277)(124,388,459,278)(125,389,460,279)(126,390,461,280)(127,391,462,281)(128,392,463,282)(129,393,464,283)(130,394,465,284)(131,395,466,285)(132,396,467,286)(133,397,468,287)(134,398,469,288)(135,399,470,289)(136,400,471,290)(137,401,472,291)(138,402,473,292)(139,403,474,293)(140,404,475,294)(141,405,476,295)(142,406,477,296)(143,407,478,297)(144,408,479,298)(145,409,480,299)(146,410,421,300)(147,411,422,241)(148,412,423,242)(149,413,424,243)(150,414,425,244)(151,415,426,245)(152,416,427,246)(153,417,428,247)(154,418,429,248)(155,419,430,249)(156,420,431,250)(157,361,432,251)(158,362,433,252)(159,363,434,253)(160,364,435,254)(161,365,436,255)(162,366,437,256)(163,367,438,257)(164,368,439,258)(165,369,440,259)(166,370,441,260)(167,371,442,261)(168,372,443,262)(169,373,444,263)(170,374,445,264)(171,375,446,265)(172,376,447,266)(173,377,448,267)(174,378,449,268)(175,379,450,269)(176,380,451,270)(177,381,452,271)(178,382,453,272)(179,383,454,273)(180,384,455,274), (1,268,104,408)(2,267,105,407)(3,266,106,406)(4,265,107,405)(5,264,108,404)(6,263,109,403)(7,262,110,402)(8,261,111,401)(9,260,112,400)(10,259,113,399)(11,258,114,398)(12,257,115,397)(13,256,116,396)(14,255,117,395)(15,254,118,394)(16,253,119,393)(17,252,120,392)(18,251,61,391)(19,250,62,390)(20,249,63,389)(21,248,64,388)(22,247,65,387)(23,246,66,386)(24,245,67,385)(25,244,68,384)(26,243,69,383)(27,242,70,382)(28,241,71,381)(29,300,72,380)(30,299,73,379)(31,298,74,378)(32,297,75,377)(33,296,76,376)(34,295,77,375)(35,294,78,374)(36,293,79,373)(37,292,80,372)(38,291,81,371)(39,290,82,370)(40,289,83,369)(41,288,84,368)(42,287,85,367)(43,286,86,366)(44,285,87,365)(45,284,88,364)(46,283,89,363)(47,282,90,362)(48,281,91,361)(49,280,92,420)(50,279,93,419)(51,278,94,418)(52,277,95,417)(53,276,96,416)(54,275,97,415)(55,274,98,414)(56,273,99,413)(57,272,100,412)(58,271,101,411)(59,270,102,410)(60,269,103,409)(121,359,426,214)(122,358,427,213)(123,357,428,212)(124,356,429,211)(125,355,430,210)(126,354,431,209)(127,353,432,208)(128,352,433,207)(129,351,434,206)(130,350,435,205)(131,349,436,204)(132,348,437,203)(133,347,438,202)(134,346,439,201)(135,345,440,200)(136,344,441,199)(137,343,442,198)(138,342,443,197)(139,341,444,196)(140,340,445,195)(141,339,446,194)(142,338,447,193)(143,337,448,192)(144,336,449,191)(145,335,450,190)(146,334,451,189)(147,333,452,188)(148,332,453,187)(149,331,454,186)(150,330,455,185)(151,329,456,184)(152,328,457,183)(153,327,458,182)(154,326,459,181)(155,325,460,240)(156,324,461,239)(157,323,462,238)(158,322,463,237)(159,321,464,236)(160,320,465,235)(161,319,466,234)(162,318,467,233)(163,317,468,232)(164,316,469,231)(165,315,470,230)(166,314,471,229)(167,313,472,228)(168,312,473,227)(169,311,474,226)(170,310,475,225)(171,309,476,224)(172,308,477,223)(173,307,478,222)(174,306,479,221)(175,305,480,220)(176,304,421,219)(177,303,422,218)(178,302,423,217)(179,301,424,216)(180,360,425,215) );`

`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)], [(1,306,74,191),(2,307,75,192),(3,308,76,193),(4,309,77,194),(5,310,78,195),(6,311,79,196),(7,312,80,197),(8,313,81,198),(9,314,82,199),(10,315,83,200),(11,316,84,201),(12,317,85,202),(13,318,86,203),(14,319,87,204),(15,320,88,205),(16,321,89,206),(17,322,90,207),(18,323,91,208),(19,324,92,209),(20,325,93,210),(21,326,94,211),(22,327,95,212),(23,328,96,213),(24,329,97,214),(25,330,98,215),(26,331,99,216),(27,332,100,217),(28,333,101,218),(29,334,102,219),(30,335,103,220),(31,336,104,221),(32,337,105,222),(33,338,106,223),(34,339,107,224),(35,340,108,225),(36,341,109,226),(37,342,110,227),(38,343,111,228),(39,344,112,229),(40,345,113,230),(41,346,114,231),(42,347,115,232),(43,348,116,233),(44,349,117,234),(45,350,118,235),(46,351,119,236),(47,352,120,237),(48,353,61,238),(49,354,62,239),(50,355,63,240),(51,356,64,181),(52,357,65,182),(53,358,66,183),(54,359,67,184),(55,360,68,185),(56,301,69,186),(57,302,70,187),(58,303,71,188),(59,304,72,189),(60,305,73,190),(121,385,456,275),(122,386,457,276),(123,387,458,277),(124,388,459,278),(125,389,460,279),(126,390,461,280),(127,391,462,281),(128,392,463,282),(129,393,464,283),(130,394,465,284),(131,395,466,285),(132,396,467,286),(133,397,468,287),(134,398,469,288),(135,399,470,289),(136,400,471,290),(137,401,472,291),(138,402,473,292),(139,403,474,293),(140,404,475,294),(141,405,476,295),(142,406,477,296),(143,407,478,297),(144,408,479,298),(145,409,480,299),(146,410,421,300),(147,411,422,241),(148,412,423,242),(149,413,424,243),(150,414,425,244),(151,415,426,245),(152,416,427,246),(153,417,428,247),(154,418,429,248),(155,419,430,249),(156,420,431,250),(157,361,432,251),(158,362,433,252),(159,363,434,253),(160,364,435,254),(161,365,436,255),(162,366,437,256),(163,367,438,257),(164,368,439,258),(165,369,440,259),(166,370,441,260),(167,371,442,261),(168,372,443,262),(169,373,444,263),(170,374,445,264),(171,375,446,265),(172,376,447,266),(173,377,448,267),(174,378,449,268),(175,379,450,269),(176,380,451,270),(177,381,452,271),(178,382,453,272),(179,383,454,273),(180,384,455,274)], [(1,268,104,408),(2,267,105,407),(3,266,106,406),(4,265,107,405),(5,264,108,404),(6,263,109,403),(7,262,110,402),(8,261,111,401),(9,260,112,400),(10,259,113,399),(11,258,114,398),(12,257,115,397),(13,256,116,396),(14,255,117,395),(15,254,118,394),(16,253,119,393),(17,252,120,392),(18,251,61,391),(19,250,62,390),(20,249,63,389),(21,248,64,388),(22,247,65,387),(23,246,66,386),(24,245,67,385),(25,244,68,384),(26,243,69,383),(27,242,70,382),(28,241,71,381),(29,300,72,380),(30,299,73,379),(31,298,74,378),(32,297,75,377),(33,296,76,376),(34,295,77,375),(35,294,78,374),(36,293,79,373),(37,292,80,372),(38,291,81,371),(39,290,82,370),(40,289,83,369),(41,288,84,368),(42,287,85,367),(43,286,86,366),(44,285,87,365),(45,284,88,364),(46,283,89,363),(47,282,90,362),(48,281,91,361),(49,280,92,420),(50,279,93,419),(51,278,94,418),(52,277,95,417),(53,276,96,416),(54,275,97,415),(55,274,98,414),(56,273,99,413),(57,272,100,412),(58,271,101,411),(59,270,102,410),(60,269,103,409),(121,359,426,214),(122,358,427,213),(123,357,428,212),(124,356,429,211),(125,355,430,210),(126,354,431,209),(127,353,432,208),(128,352,433,207),(129,351,434,206),(130,350,435,205),(131,349,436,204),(132,348,437,203),(133,347,438,202),(134,346,439,201),(135,345,440,200),(136,344,441,199),(137,343,442,198),(138,342,443,197),(139,341,444,196),(140,340,445,195),(141,339,446,194),(142,338,447,193),(143,337,448,192),(144,336,449,191),(145,335,450,190),(146,334,451,189),(147,333,452,188),(148,332,453,187),(149,331,454,186),(150,330,455,185),(151,329,456,184),(152,328,457,183),(153,327,458,182),(154,326,459,181),(155,325,460,240),(156,324,461,239),(157,323,462,238),(158,322,463,237),(159,321,464,236),(160,320,465,235),(161,319,466,234),(162,318,467,233),(163,317,468,232),(164,316,469,231),(165,315,470,230),(166,314,471,229),(167,313,472,228),(168,312,473,227),(169,311,474,226),(170,310,475,225),(171,309,476,224),(172,308,477,223),(173,307,478,222),(174,306,479,221),(175,305,480,220),(176,304,421,219),(177,303,422,218),(178,302,423,217),(179,301,424,216),(180,360,425,215)]])`

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 Q8 D5 D6 C4○D4 D10 Dic6 D15 Dic10 C4○D12 D30 C4○D20 Dic30 D60⋊11C2 kernel C60.24Q8 C30.4Q8 C60⋊5C4 C4×C60 C4×C20 C60 C4×C12 C2×C20 C30 C2×C12 C20 C42 C12 C10 C2×C4 C6 C4 C2 # reps 1 4 2 1 1 2 2 3 4 6 4 4 8 8 12 16 16 32

Matrix representation of C60.24Q8 in GL4(𝔽61) generated by

 58 0 0 0 0 20 0 0 0 0 56 8 0 0 11 31
,
 50 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 0 1 0 0 60 0 0 0 0 0 12 36 0 0 35 49
`G:=sub<GL(4,GF(61))| [58,0,0,0,0,20,0,0,0,0,56,11,0,0,8,31],[50,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[0,60,0,0,1,0,0,0,0,0,12,35,0,0,36,49] >;`

C60.24Q8 in GAP, Magma, Sage, TeX

`C_{60}._{24}Q_8`
`% in TeX`

`G:=Group("C60.24Q8");`
`// GroupNames label`

`G:=SmallGroup(480,835);`
`// by ID`

`G=gap.SmallGroup(480,835);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,64,254,100,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^4=1,c^2=a^30*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^30*b^-1>;`
`// generators/relations`

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