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G = C60.8Q8order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.8Q8, C30.19D8, C30.7Q16, C10.12D24, C20.3Dic6, C10.5Dic12, C12.11Dic10, C52(C241C4), C154(C2.D8), C4.5(C15⋊Q8), C6.7(D4⋊D5), (C2×C30).32D4, (C2×C20).60D6, C12.69(C4×D5), C52C82Dic3, C4⋊Dic3.3D5, C30.28(C4⋊C4), C60.112(C2×C4), (C2×C10).35D12, C2.3(C5⋊D24), C31(C10.D8), C605C4.23C2, C4.12(D5×Dic3), C6.3(C5⋊Q16), (C2×C12).288D10, C20.26(C2×Dic3), C2.3(C5⋊Dic12), (C2×C60).132C22, C10.10(C4⋊Dic3), C6.5(C10.D4), C2.4(C30.Q8), C22.17(C5⋊D12), (C3×C52C8)⋊2C4, (C6×C52C8).5C2, (C2×C52C8).4S3, (C2×C4).140(S3×D5), (C5×C4⋊Dic3).3C2, (C2×C6).29(C5⋊D4), SmallGroup(480,64)

Series: Derived Chief Lower central Upper central

C1C60 — C60.8Q8
C1C5C15C30C2×C30C2×C60C6×C52C8 — C60.8Q8
C15C30C60 — C60.8Q8
C1C22C2×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 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3 [×2], C12 [×2], C2×C6, C15, C4⋊C4 [×2], C2×C8, Dic5, C20 [×2], C20, C2×C10, C24 [×2], C2×Dic3 [×2], C2×C12, C30 [×3], C2.D8, C52C8 [×2], C2×Dic5, C2×C20, C2×C20, C4⋊Dic3, C4⋊Dic3, C2×C24, C5×Dic3, Dic15, C60 [×2], C2×C30, C2×C52C8, C4⋊Dic5, C5×C4⋊C4, C241C4, C3×C52C8 [×2], C10×Dic3, C2×Dic15, C2×C60, C10.D8, C6×C52C8, C5×C4⋊Dic3, C605C4, C60.8Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, Dic3 [×2], 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)
(1 401 234 98)(2 412 235 109)(3 363 236 120)(4 374 237 71)(5 385 238 82)(6 396 239 93)(7 407 240 104)(8 418 181 115)(9 369 182 66)(10 380 183 77)(11 391 184 88)(12 402 185 99)(13 413 186 110)(14 364 187 61)(15 375 188 72)(16 386 189 83)(17 397 190 94)(18 408 191 105)(19 419 192 116)(20 370 193 67)(21 381 194 78)(22 392 195 89)(23 403 196 100)(24 414 197 111)(25 365 198 62)(26 376 199 73)(27 387 200 84)(28 398 201 95)(29 409 202 106)(30 420 203 117)(31 371 204 68)(32 382 205 79)(33 393 206 90)(34 404 207 101)(35 415 208 112)(36 366 209 63)(37 377 210 74)(38 388 211 85)(39 399 212 96)(40 410 213 107)(41 361 214 118)(42 372 215 69)(43 383 216 80)(44 394 217 91)(45 405 218 102)(46 416 219 113)(47 367 220 64)(48 378 221 75)(49 389 222 86)(50 400 223 97)(51 411 224 108)(52 362 225 119)(53 373 226 70)(54 384 227 81)(55 395 228 92)(56 406 229 103)(57 417 230 114)(58 368 231 65)(59 379 232 76)(60 390 233 87)(121 356 443 264)(122 307 444 275)(123 318 445 286)(124 329 446 297)(125 340 447 248)(126 351 448 259)(127 302 449 270)(128 313 450 281)(129 324 451 292)(130 335 452 243)(131 346 453 254)(132 357 454 265)(133 308 455 276)(134 319 456 287)(135 330 457 298)(136 341 458 249)(137 352 459 260)(138 303 460 271)(139 314 461 282)(140 325 462 293)(141 336 463 244)(142 347 464 255)(143 358 465 266)(144 309 466 277)(145 320 467 288)(146 331 468 299)(147 342 469 250)(148 353 470 261)(149 304 471 272)(150 315 472 283)(151 326 473 294)(152 337 474 245)(153 348 475 256)(154 359 476 267)(155 310 477 278)(156 321 478 289)(157 332 479 300)(158 343 480 251)(159 354 421 262)(160 305 422 273)(161 316 423 284)(162 327 424 295)(163 338 425 246)(164 349 426 257)(165 360 427 268)(166 311 428 279)(167 322 429 290)(168 333 430 241)(169 344 431 252)(170 355 432 263)(171 306 433 274)(172 317 434 285)(173 328 435 296)(174 339 436 247)(175 350 437 258)(176 301 438 269)(177 312 439 280)(178 323 440 291)(179 334 441 242)(180 345 442 253)
(1 438 219 161 31 468 189 131)(2 427 220 150 32 457 190 180)(3 476 221 139 33 446 191 169)(4 465 222 128 34 435 192 158)(5 454 223 177 35 424 193 147)(6 443 224 166 36 473 194 136)(7 432 225 155 37 462 195 125)(8 421 226 144 38 451 196 174)(9 470 227 133 39 440 197 163)(10 459 228 122 40 429 198 152)(11 448 229 171 41 478 199 141)(12 437 230 160 42 467 200 130)(13 426 231 149 43 456 201 179)(14 475 232 138 44 445 202 168)(15 464 233 127 45 434 203 157)(16 453 234 176 46 423 204 146)(17 442 235 165 47 472 205 135)(18 431 236 154 48 461 206 124)(19 480 237 143 49 450 207 173)(20 469 238 132 50 439 208 162)(21 458 239 121 51 428 209 151)(22 447 240 170 52 477 210 140)(23 436 181 159 53 466 211 129)(24 425 182 148 54 455 212 178)(25 474 183 137 55 444 213 167)(26 463 184 126 56 433 214 156)(27 452 185 175 57 422 215 145)(28 441 186 164 58 471 216 134)(29 430 187 153 59 460 217 123)(30 479 188 142 60 449 218 172)(61 241 409 318 91 271 379 348)(62 290 410 307 92 260 380 337)(63 279 411 356 93 249 381 326)(64 268 412 345 94 298 382 315)(65 257 413 334 95 287 383 304)(66 246 414 323 96 276 384 353)(67 295 415 312 97 265 385 342)(68 284 416 301 98 254 386 331)(69 273 417 350 99 243 387 320)(70 262 418 339 100 292 388 309)(71 251 419 328 101 281 389 358)(72 300 420 317 102 270 390 347)(73 289 361 306 103 259 391 336)(74 278 362 355 104 248 392 325)(75 267 363 344 105 297 393 314)(76 256 364 333 106 286 394 303)(77 245 365 322 107 275 395 352)(78 294 366 311 108 264 396 341)(79 283 367 360 109 253 397 330)(80 272 368 349 110 242 398 319)(81 261 369 338 111 291 399 308)(82 250 370 327 112 280 400 357)(83 299 371 316 113 269 401 346)(84 288 372 305 114 258 402 335)(85 277 373 354 115 247 403 324)(86 266 374 343 116 296 404 313)(87 255 375 332 117 285 405 302)(88 244 376 321 118 274 406 351)(89 293 377 310 119 263 407 340)(90 282 378 359 120 252 408 329)

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

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

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

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L24A···24H30A···30F60A···60H
order1222344444455666888810···101212121215152020202020···2024···2430···3060···60
size11112221212606022222101010102···2222244444412···1210···104···44···4

66 irreducible representations

dim11111222222222222222244444444
type+++++-++-++-+-+-+-++---++-
imageC1C2C2C2C4S3Q8D4D5Dic3D6D8Q16D10Dic6D12Dic10C4×D5C5⋊D4D24Dic12S3×D5D4⋊D5C5⋊Q16D5×Dic3C15⋊Q8C5⋊D12C5⋊D24C5⋊Dic12
kernelC60.8Q8C6×C52C8C5×C4⋊Dic3C605C4C3×C52C8C2×C52C8C60C2×C30C4⋊Dic3C52C8C2×C20C30C30C2×C12C20C2×C10C12C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps11114111221222224444422222244

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

010000
2402400000
0024011800
0049100
00000190
000052189
,
81210000
1811600000
00231100
0014921800
0000413
000082200
,
991980000
431420000
002199300
0057000
0000123154
0000221118

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