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G = C60.15Q8order 480 = 25·3·5

15th non-split extension by C60 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.15Q8, C12.58D20, C60.104D4, C20.15Dic6, C12.15Dic10, C30.19M4(2), C159(C4⋊C8), C31(C203C8), C10.24(S3×C8), C30.33(C2×C8), Dic3⋊(C52C8), C54(Dic3⋊C8), (C5×Dic3)⋊4C8, C4.12(C15⋊Q8), C30.24(C4⋊C4), (C2×C20).326D6, C6.1(C4⋊Dic5), (C4×Dic3).7D5, (C2×C12).330D10, C20.63(C3⋊D4), C4.30(C3⋊D20), (Dic3×C20).8C2, C10.14(C8⋊S3), C6.3(C4.Dic5), C10.9(Dic3⋊C4), (C2×C60).228C22, (C10×Dic3).15C4, (C2×Dic3).4Dic5, C2.3(D6.Dic5), C2.1(C6.Dic10), C22.11(S3×Dic5), C6.5(C2×C52C8), C2.5(S3×C52C8), (C6×C52C8).12C2, (C2×C52C8).10S3, (C2×C30).88(C2×C4), (C2×C10).72(C4×S3), (C2×C4).231(S3×D5), (C2×C153C8).22C2, (C2×C6).12(C2×Dic5), SmallGroup(480,60)

Series: Derived Chief Lower central Upper central

C1C30 — C60.15Q8
C1C5C15C30C60C2×C60C6×C52C8 — C60.15Q8
C15C30 — C60.15Q8
C1C2×C4

Generators and relations for C60.15Q8
 G = < a,b,c | a60=b4=1, c2=a45b2, bab-1=a41, cac-1=a49, cbc-1=b-1 >

Subgroups: 220 in 76 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3 [×2], Dic3, C12 [×2], C2×C6, C15, C42, C2×C8 [×2], C20 [×2], C20 [×3], C2×C10, C3⋊C8, C24, C2×Dic3 [×2], C2×C12, C30 [×3], C4⋊C8, C52C8 [×2], C2×C20, C2×C20 [×2], C2×C3⋊C8, C4×Dic3, C2×C24, C5×Dic3 [×2], C5×Dic3, C60 [×2], C2×C30, C2×C52C8, C2×C52C8, C4×C20, Dic3⋊C8, C3×C52C8, C153C8, C10×Dic3 [×2], C2×C60, C203C8, C6×C52C8, Dic3×C20, C2×C153C8, C60.15Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4, Q8, D5, D6, C4⋊C4, C2×C8, M4(2), Dic5 [×2], D10, Dic6, C4×S3, C3⋊D4, C4⋊C8, C52C8 [×2], Dic10, D20, C2×Dic5, S3×C8, C8⋊S3, Dic3⋊C4, S3×D5, C2×C52C8, C4.Dic5, C4⋊Dic5, Dic3⋊C8, S3×Dic5, C3⋊D20, C15⋊Q8, C203C8, S3×C52C8, D6.Dic5, C6.Dic10, C60.15Q8

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

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B6C8A8B8C8D8E8F8G8H10A···10F12A12B12C12D15A15B20A···20H20I···20X24A···24H30A···30F60A···60H
order1222344444444556668888888810···1012121212151520···2020···2024···2430···3060···60
size11112111166662222210101010303030302···22222442···26···610···104···44···4

84 irreducible representations

dim11111122222222222222222444444
type++++++-++-+--+++--
imageC1C2C2C2C4C8S3D4Q8D5D6M4(2)Dic5D10Dic6C3⋊D4C4×S3C52C8Dic10D20S3×C8C8⋊S3C4.Dic5S3×D5C3⋊D20C15⋊Q8S3×Dic5S3×C52C8D6.Dic5
kernelC60.15Q8C6×C52C8Dic3×C20C2×C153C8C10×Dic3C5×Dic3C2×C52C8C60C60C4×Dic3C2×C20C30C2×Dic3C2×C12C20C20C2×C10Dic3C12C12C10C10C6C2×C4C4C4C22C2C2
# reps11114811121242222844448222244

Matrix representation of C60.15Q8 in GL4(𝔽241) generated by

13113100
1106400
001240
0010
,
44300
23819700
00519
0024236
,
1013000
12014000
00171140
0010170
G:=sub<GL(4,GF(241))| [131,110,0,0,131,64,0,0,0,0,1,1,0,0,240,0],[44,238,0,0,3,197,0,0,0,0,5,24,0,0,19,236],[101,120,0,0,30,140,0,0,0,0,171,101,0,0,140,70] >;

C60.15Q8 in GAP, Magma, Sage, TeX

C_{60}._{15}Q_8
% in TeX

G:=Group("C60.15Q8");
// GroupNames label

G:=SmallGroup(480,60);
// by ID

G=gap.SmallGroup(480,60);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=1,c^2=a^45*b^2,b*a*b^-1=a^41,c*a*c^-1=a^49,c*b*c^-1=b^-1>;
// generators/relations

׿
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