Copied to
clipboard

## G = C60.8Q8order 480 = 25·3·5

### 8th non-split extension by C60 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C60.8Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×C60 — C6×C5⋊2C8 — C60.8Q8
 Lower central C15 — C30 — C60 — C60.8Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for C60.8Q8
G = < a,b,c | a60=b4=1, c2=a45b2, bab-1=a11, cac-1=a49, cbc-1=a45b-1 >

Subgroups: 316 in 72 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, C20, C2×C10, C24, C2×Dic3, C2×C12, C30, C2.D8, C52C8, C2×Dic5, C2×C20, C2×C20, C4⋊Dic3, C4⋊Dic3, C2×C24, C5×Dic3, Dic15, C60, C2×C30, C2×C52C8, C4⋊Dic5, C5×C4⋊C4, C241C4, C3×C52C8, C10×Dic3, C2×Dic15, C2×C60, C10.D8, C6×C52C8, C5×C4⋊Dic3, C605C4, C60.8Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, Dic3, D6, C4⋊C4, D8, Q16, D10, Dic6, D12, C2×Dic3, C2.D8, Dic10, C4×D5, C5⋊D4, D24, Dic12, C4⋊Dic3, S3×D5, C10.D4, D4⋊D5, C5⋊Q16, C241C4, D5×Dic3, C5⋊D12, C15⋊Q8, C10.D8, C5⋊D24, C5⋊Dic12, C30.Q8, C60.8Q8

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

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

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

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

66 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 15A 15B 20A 20B 20C 20D 20E ··· 20L 24A ··· 24H 30A ··· 30F 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 15 15 20 20 20 20 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 12 12 60 60 2 2 2 2 2 10 10 10 10 2 ··· 2 2 2 2 2 4 4 4 4 4 4 12 ··· 12 10 ··· 10 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 2 2 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 Dic3 D6 D8 Q16 D10 Dic6 D12 Dic10 C4×D5 C5⋊D4 D24 Dic12 S3×D5 D4⋊D5 C5⋊Q16 D5×Dic3 C15⋊Q8 C5⋊D12 C5⋊D24 C5⋊Dic12 kernel C60.8Q8 C6×C5⋊2C8 C5×C4⋊Dic3 C60⋊5C4 C3×C5⋊2C8 C2×C5⋊2C8 C60 C2×C30 C4⋊Dic3 C5⋊2C8 C2×C20 C30 C30 C2×C12 C20 C2×C10 C12 C12 C2×C6 C10 C10 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 2 2 1 2 2 2 2 2 4 4 4 4 4 2 2 2 2 2 2 4 4

Matrix representation of C60.8Q8 in GL6(𝔽241)

 0 1 0 0 0 0 240 240 0 0 0 0 0 0 240 118 0 0 0 0 49 1 0 0 0 0 0 0 0 190 0 0 0 0 52 189
,
 81 21 0 0 0 0 181 160 0 0 0 0 0 0 23 11 0 0 0 0 149 218 0 0 0 0 0 0 41 3 0 0 0 0 82 200
,
 99 198 0 0 0 0 43 142 0 0 0 0 0 0 219 93 0 0 0 0 57 0 0 0 0 0 0 0 123 154 0 0 0 0 221 118

`G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,240,0,0,0,0,0,0,240,49,0,0,0,0,118,1,0,0,0,0,0,0,0,52,0,0,0,0,190,189],[81,181,0,0,0,0,21,160,0,0,0,0,0,0,23,149,0,0,0,0,11,218,0,0,0,0,0,0,41,82,0,0,0,0,3,200],[99,43,0,0,0,0,198,142,0,0,0,0,0,0,219,57,0,0,0,0,93,0,0,0,0,0,0,0,123,221,0,0,0,0,154,118] >;`

C60.8Q8 in GAP, Magma, Sage, TeX

`C_{60}._8Q_8`
`% in TeX`

`G:=Group("C60.8Q8");`
`// GroupNames label`

`G:=SmallGroup(480,64);`
`// by ID`

`G=gap.SmallGroup(480,64);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,36,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^4=1,c^2=a^45*b^2,b*a*b^-1=a^11,c*a*c^-1=a^49,c*b*c^-1=a^45*b^-1>;`
`// generators/relations`

׿
×
𝔽