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## G = C60.Q8order 480 = 25·3·5

### 4th non-split extension by C60 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C60.Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×C60 — C3×C4⋊Dic5 — C60.Q8
 Lower central C15 — C30 — C60 — C60.Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for C60.Q8
G = < a,b,c | a60=1, b4=a30, c2=a15b2, bab-1=a41, cac-1=a19, cbc-1=b3 >

Subgroups: 332 in 72 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C30, C4.Q8, C40, C2×Dic5, C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C3×Dic5, Dic15, C60, C2×C30, C4⋊Dic5, C4⋊Dic5, C2×C40, C12.Q8, C5×C3⋊C8, C6×Dic5, C2×Dic15, C2×C60, C406C4, C3×C4⋊Dic5, C10×C3⋊C8, C605C4, C60.Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, SD16, Dic5, D10, Dic6, C4×S3, C3⋊D4, C4.Q8, Dic10, D20, C2×Dic5, Dic3⋊C4, D4.S3, Q82S3, S3×D5, C40⋊C2, C4⋊Dic5, C12.Q8, S3×Dic5, C3⋊D20, C15⋊Q8, C406C4, C6.D20, C15⋊SD16, C6.Dic10, C60.Q8

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A ··· 20H 30A ··· 30F 40A ··· 40P 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 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 2 2 20 20 60 60 2 2 2 2 2 6 6 6 6 2 ··· 2 4 4 20 20 20 20 4 4 2 ··· 2 4 ··· 4 6 ··· 6 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + - + + + - + - - + - + + - - + - + image C1 C2 C2 C2 C4 S3 Q8 D4 D5 D6 SD16 Dic5 D10 Dic6 C4×S3 C3⋊D4 Dic10 D20 C40⋊C2 D4.S3 Q8⋊2S3 S3×D5 S3×Dic5 C15⋊Q8 C3⋊D20 C6.D20 C15⋊SD16 kernel C60.Q8 C3×C4⋊Dic5 C10×C3⋊C8 C60⋊5C4 C5×C3⋊C8 C4⋊Dic5 C60 C2×C30 C2×C3⋊C8 C2×C20 C30 C3⋊C8 C2×C12 C20 C20 C2×C10 C12 C2×C6 C6 C10 C10 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 2 1 4 4 2 2 2 2 4 4 16 1 1 2 2 2 2 4 4

Matrix representation of C60.Q8 in GL6(𝔽241)

 51 52 0 0 0 0 190 0 0 0 0 0 0 0 122 119 0 0 0 0 122 200 0 0 0 0 0 0 1 54 0 0 0 0 174 239
,
 197 156 0 0 0 0 88 44 0 0 0 0 0 0 75 169 0 0 0 0 72 204 0 0 0 0 0 0 77 172 0 0 0 0 44 164
,
 77 210 0 0 0 0 98 164 0 0 0 0 0 0 167 79 0 0 0 0 71 74 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(241))| [51,190,0,0,0,0,52,0,0,0,0,0,0,0,122,122,0,0,0,0,119,200,0,0,0,0,0,0,1,174,0,0,0,0,54,239],[197,88,0,0,0,0,156,44,0,0,0,0,0,0,75,72,0,0,0,0,169,204,0,0,0,0,0,0,77,44,0,0,0,0,172,164],[77,98,0,0,0,0,210,164,0,0,0,0,0,0,167,71,0,0,0,0,79,74,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C60.Q8 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(480,63);`
`// by ID`

`G=gap.SmallGroup(480,63);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,675,80,1356,18822]);`
`// 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^19,c*b*c^-1=b^3>;`
`// generators/relations`

׿
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