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G = C60.Q8order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.4Q8, C30.15SD16, C20.10Dic6, C12.2Dic10, C3⋊C81Dic5, C31(C406C4), C156(C4.Q8), C4.1(C15⋊Q8), C60.96(C2×C4), (C2×C30).31D4, (C2×C6).34D20, C4⋊Dic5.2S3, C30.27(C4⋊C4), C20.100(C4×S3), (C2×C12).60D10, (C2×C20).283D6, C6.8(C40⋊C2), C6.2(C4⋊Dic5), C605C4.16C2, C4.11(S3×Dic5), C12.4(C2×Dic5), C52(C12.Q8), C10.3(D4.S3), (C2×C60).102C22, C10.3(Q82S3), C2.3(C15⋊SD16), C2.3(C6.D20), C2.3(C6.Dic10), C10.11(Dic3⋊C4), C22.16(C3⋊D20), (C5×C3⋊C8)⋊7C4, (C2×C3⋊C8).3D5, (C10×C3⋊C8).4C2, (C2×C4).88(S3×D5), (C3×C4⋊Dic5).2C2, (C2×C10).27(C3⋊D4), SmallGroup(480,63)

Series: Derived Chief Lower central Upper central

C1C60 — C60.Q8
C1C5C15C30C2×C30C2×C60C3×C4⋊Dic5 — C60.Q8
C15C30C60 — C60.Q8
C1C22C2×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 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3, C12 [×2], C12, C2×C6, C15, C4⋊C4 [×2], C2×C8, Dic5 [×2], C20 [×2], C2×C10, C3⋊C8 [×2], C2×Dic3, C2×C12, C2×C12, C30 [×3], C4.Q8, C40 [×2], C2×Dic5 [×2], C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C3×Dic5, Dic15, C60 [×2], C2×C30, C4⋊Dic5, C4⋊Dic5, C2×C40, C12.Q8, C5×C3⋊C8 [×2], C6×Dic5, C2×Dic15, C2×C60, C406C4, C3×C4⋊Dic5, C10×C3⋊C8, C605C4, C60.Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, SD16 [×2], Dic5 [×2], D10, Dic6, C4×S3, C3⋊D4, C4.Q8, Dic10, D20, C2×Dic5, Dic3⋊C4, D4.S3, Q82S3, S3×D5, C40⋊C2 [×2], 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)
(1 350 275 387 31 320 245 417)(2 331 276 368 32 301 246 398)(3 312 277 409 33 342 247 379)(4 353 278 390 34 323 248 420)(5 334 279 371 35 304 249 401)(6 315 280 412 36 345 250 382)(7 356 281 393 37 326 251 363)(8 337 282 374 38 307 252 404)(9 318 283 415 39 348 253 385)(10 359 284 396 40 329 254 366)(11 340 285 377 41 310 255 407)(12 321 286 418 42 351 256 388)(13 302 287 399 43 332 257 369)(14 343 288 380 44 313 258 410)(15 324 289 361 45 354 259 391)(16 305 290 402 46 335 260 372)(17 346 291 383 47 316 261 413)(18 327 292 364 48 357 262 394)(19 308 293 405 49 338 263 375)(20 349 294 386 50 319 264 416)(21 330 295 367 51 360 265 397)(22 311 296 408 52 341 266 378)(23 352 297 389 53 322 267 419)(24 333 298 370 54 303 268 400)(25 314 299 411 55 344 269 381)(26 355 300 392 56 325 270 362)(27 336 241 373 57 306 271 403)(28 317 242 414 58 347 272 384)(29 358 243 395 59 328 273 365)(30 339 244 376 60 309 274 406)(61 225 435 157 91 195 465 127)(62 206 436 138 92 236 466 168)(63 187 437 179 93 217 467 149)(64 228 438 160 94 198 468 130)(65 209 439 141 95 239 469 171)(66 190 440 122 96 220 470 152)(67 231 441 163 97 201 471 133)(68 212 442 144 98 182 472 174)(69 193 443 125 99 223 473 155)(70 234 444 166 100 204 474 136)(71 215 445 147 101 185 475 177)(72 196 446 128 102 226 476 158)(73 237 447 169 103 207 477 139)(74 218 448 150 104 188 478 180)(75 199 449 131 105 229 479 161)(76 240 450 172 106 210 480 142)(77 221 451 153 107 191 421 123)(78 202 452 134 108 232 422 164)(79 183 453 175 109 213 423 145)(80 224 454 156 110 194 424 126)(81 205 455 137 111 235 425 167)(82 186 456 178 112 216 426 148)(83 227 457 159 113 197 427 129)(84 208 458 140 114 238 428 170)(85 189 459 121 115 219 429 151)(86 230 460 162 116 200 430 132)(87 211 461 143 117 181 431 173)(88 192 462 124 118 222 432 154)(89 233 463 165 119 203 433 135)(90 214 464 146 120 184 434 176)
(1 146 290 199)(2 165 291 218)(3 124 292 237)(4 143 293 196)(5 162 294 215)(6 121 295 234)(7 140 296 193)(8 159 297 212)(9 178 298 231)(10 137 299 190)(11 156 300 209)(12 175 241 228)(13 134 242 187)(14 153 243 206)(15 172 244 225)(16 131 245 184)(17 150 246 203)(18 169 247 222)(19 128 248 181)(20 147 249 200)(21 166 250 219)(22 125 251 238)(23 144 252 197)(24 163 253 216)(25 122 254 235)(26 141 255 194)(27 160 256 213)(28 179 257 232)(29 138 258 191)(30 157 259 210)(31 176 260 229)(32 135 261 188)(33 154 262 207)(34 173 263 226)(35 132 264 185)(36 151 265 204)(37 170 266 223)(38 129 267 182)(39 148 268 201)(40 167 269 220)(41 126 270 239)(42 145 271 198)(43 164 272 217)(44 123 273 236)(45 142 274 195)(46 161 275 214)(47 180 276 233)(48 139 277 192)(49 158 278 211)(50 177 279 230)(51 136 280 189)(52 155 281 208)(53 174 282 227)(54 133 283 186)(55 152 284 205)(56 171 285 224)(57 130 286 183)(58 149 287 202)(59 168 288 221)(60 127 289 240)(61 354 450 406)(62 313 451 365)(63 332 452 384)(64 351 453 403)(65 310 454 362)(66 329 455 381)(67 348 456 400)(68 307 457 419)(69 326 458 378)(70 345 459 397)(71 304 460 416)(72 323 461 375)(73 342 462 394)(74 301 463 413)(75 320 464 372)(76 339 465 391)(77 358 466 410)(78 317 467 369)(79 336 468 388)(80 355 469 407)(81 314 470 366)(82 333 471 385)(83 352 472 404)(84 311 473 363)(85 330 474 382)(86 349 475 401)(87 308 476 420)(88 327 477 379)(89 346 478 398)(90 305 479 417)(91 324 480 376)(92 343 421 395)(93 302 422 414)(94 321 423 373)(95 340 424 392)(96 359 425 411)(97 318 426 370)(98 337 427 389)(99 356 428 408)(100 315 429 367)(101 334 430 386)(102 353 431 405)(103 312 432 364)(104 331 433 383)(105 350 434 402)(106 309 435 361)(107 328 436 380)(108 347 437 399)(109 306 438 418)(110 325 439 377)(111 344 440 396)(112 303 441 415)(113 322 442 374)(114 341 443 393)(115 360 444 412)(116 319 445 371)(117 338 446 390)(118 357 447 409)(119 316 448 368)(120 335 449 387)

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

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

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

72 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D12E12F15A15B20A···20H30A···30F40A···40P60A···60H
order1222344444455666888810···10121212121212151520···2030···3040···4060···60
size1111222202060602222266662···24420202020442···24···46···64···4

72 irreducible representations

dim111112222222222222244444444
type+++++-+++-+--+-++--+-+
imageC1C2C2C2C4S3Q8D4D5D6SD16Dic5D10Dic6C4×S3C3⋊D4Dic10D20C40⋊C2D4.S3Q82S3S3×D5S3×Dic5C15⋊Q8C3⋊D20C6.D20C15⋊SD16
kernelC60.Q8C3×C4⋊Dic5C10×C3⋊C8C605C4C5×C3⋊C8C4⋊Dic5C60C2×C30C2×C3⋊C8C2×C20C30C3⋊C8C2×C12C20C20C2×C10C12C2×C6C6C10C10C2×C4C4C4C22C2C2
# reps1111411121442222441611222244

Matrix representation of C60.Q8 in GL6(𝔽241)

51520000
19000000
0012211900
0012220000
0000154
0000174239
,
1971560000
88440000
007516900
007220400
000077172
000044164
,
772100000
981640000
001677900
00717400
000010
000001

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

׿
×
𝔽