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G = C60⋊Q8order 480 = 25·3·5

3rd semidirect product of C60 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C603Q8, C201Dic6, C124Dic10, Dic54Dic6, Dic5.9D12, C157(C4⋊Q8), C31(C20⋊Q8), C42(C15⋊Q8), C6.25(D4×D5), C6.37(Q8×D5), (C3×Dic5)⋊7Q8, C2.27(D5×D12), C51(C122Q8), C30.65(C2×D4), C4⋊Dic3.7D5, C30.56(C2×Q8), C10.26(C2×D12), (C2×C20).135D6, (C4×Dic5).5S3, C605C4.26C2, C2.19(D5×Dic6), (C2×C12).308D10, (C12×Dic5).5C2, (C3×Dic5).51D4, C6.23(C2×Dic10), C10.19(C2×Dic6), (C2×C60).152C22, (C2×C30).158C23, (C2×Dic3).50D10, (C2×Dic5).187D6, C30.Q8.16C2, (C6×Dic5).214C22, (C10×Dic3).95C22, (C2×Dic15).116C22, C2.8(C2×C15⋊Q8), (C2×C15⋊Q8).7C2, (C2×C4).165(S3×D5), (C5×C4⋊Dic3).6C2, C22.209(C2×S3×D5), (C2×C6).170(C22×D5), (C2×C10).170(C22×S3), SmallGroup(480,544)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60⋊Q8
C1C5C15C30C2×C30C6×Dic5C2×C15⋊Q8 — C60⋊Q8
C15C2×C30 — C60⋊Q8
C1C22C2×C4

Generators and relations for C60⋊Q8
 G = < a,b,c | a60=b4=1, c2=b2, bab-1=a11, cac-1=a49, cbc-1=b-1 >

Subgroups: 652 in 136 conjugacy classes, 60 normal (34 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×8], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×4], C12 [×2], C12 [×4], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4⋊Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4⋊Dic3, C4⋊Dic3 [×3], C4×C12, C2×Dic6 [×2], C5×Dic3 [×2], C3×Dic5 [×4], Dic15 [×2], C60 [×2], C2×C30, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4, C2×Dic10 [×2], C122Q8, C15⋊Q8 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, C20⋊Q8, C30.Q8 [×2], C12×Dic5, C5×C4⋊Dic3, C605C4, C2×C15⋊Q8 [×2], C60⋊Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D5, D6 [×3], C2×D4, C2×Q8 [×2], D10 [×3], Dic6 [×4], D12 [×2], C22×S3, C4⋊Q8, Dic10 [×2], C22×D5, C2×Dic6 [×2], C2×D12, S3×D5, C2×Dic10, D4×D5, Q8×D5, C122Q8, C15⋊Q8 [×2], C2×S3×D5, C20⋊Q8, D5×Dic6, D5×D12, C2×C15⋊Q8, 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)
(1 231 261 358)(2 182 262 309)(3 193 263 320)(4 204 264 331)(5 215 265 342)(6 226 266 353)(7 237 267 304)(8 188 268 315)(9 199 269 326)(10 210 270 337)(11 221 271 348)(12 232 272 359)(13 183 273 310)(14 194 274 321)(15 205 275 332)(16 216 276 343)(17 227 277 354)(18 238 278 305)(19 189 279 316)(20 200 280 327)(21 211 281 338)(22 222 282 349)(23 233 283 360)(24 184 284 311)(25 195 285 322)(26 206 286 333)(27 217 287 344)(28 228 288 355)(29 239 289 306)(30 190 290 317)(31 201 291 328)(32 212 292 339)(33 223 293 350)(34 234 294 301)(35 185 295 312)(36 196 296 323)(37 207 297 334)(38 218 298 345)(39 229 299 356)(40 240 300 307)(41 191 241 318)(42 202 242 329)(43 213 243 340)(44 224 244 351)(45 235 245 302)(46 186 246 313)(47 197 247 324)(48 208 248 335)(49 219 249 346)(50 230 250 357)(51 181 251 308)(52 192 252 319)(53 203 253 330)(54 214 254 341)(55 225 255 352)(56 236 256 303)(57 187 257 314)(58 198 258 325)(59 209 259 336)(60 220 260 347)(61 389 467 129)(62 400 468 140)(63 411 469 151)(64 362 470 162)(65 373 471 173)(66 384 472 124)(67 395 473 135)(68 406 474 146)(69 417 475 157)(70 368 476 168)(71 379 477 179)(72 390 478 130)(73 401 479 141)(74 412 480 152)(75 363 421 163)(76 374 422 174)(77 385 423 125)(78 396 424 136)(79 407 425 147)(80 418 426 158)(81 369 427 169)(82 380 428 180)(83 391 429 131)(84 402 430 142)(85 413 431 153)(86 364 432 164)(87 375 433 175)(88 386 434 126)(89 397 435 137)(90 408 436 148)(91 419 437 159)(92 370 438 170)(93 381 439 121)(94 392 440 132)(95 403 441 143)(96 414 442 154)(97 365 443 165)(98 376 444 176)(99 387 445 127)(100 398 446 138)(101 409 447 149)(102 420 448 160)(103 371 449 171)(104 382 450 122)(105 393 451 133)(106 404 452 144)(107 415 453 155)(108 366 454 166)(109 377 455 177)(110 388 456 128)(111 399 457 139)(112 410 458 150)(113 361 459 161)(114 372 460 172)(115 383 461 123)(116 394 462 134)(117 405 463 145)(118 416 464 156)(119 367 465 167)(120 378 466 178)
(1 443 261 97)(2 432 262 86)(3 421 263 75)(4 470 264 64)(5 459 265 113)(6 448 266 102)(7 437 267 91)(8 426 268 80)(9 475 269 69)(10 464 270 118)(11 453 271 107)(12 442 272 96)(13 431 273 85)(14 480 274 74)(15 469 275 63)(16 458 276 112)(17 447 277 101)(18 436 278 90)(19 425 279 79)(20 474 280 68)(21 463 281 117)(22 452 282 106)(23 441 283 95)(24 430 284 84)(25 479 285 73)(26 468 286 62)(27 457 287 111)(28 446 288 100)(29 435 289 89)(30 424 290 78)(31 473 291 67)(32 462 292 116)(33 451 293 105)(34 440 294 94)(35 429 295 83)(36 478 296 72)(37 467 297 61)(38 456 298 110)(39 445 299 99)(40 434 300 88)(41 423 241 77)(42 472 242 66)(43 461 243 115)(44 450 244 104)(45 439 245 93)(46 428 246 82)(47 477 247 71)(48 466 248 120)(49 455 249 109)(50 444 250 98)(51 433 251 87)(52 422 252 76)(53 471 253 65)(54 460 254 114)(55 449 255 103)(56 438 256 92)(57 427 257 81)(58 476 258 70)(59 465 259 119)(60 454 260 108)(121 235 381 302)(122 224 382 351)(123 213 383 340)(124 202 384 329)(125 191 385 318)(126 240 386 307)(127 229 387 356)(128 218 388 345)(129 207 389 334)(130 196 390 323)(131 185 391 312)(132 234 392 301)(133 223 393 350)(134 212 394 339)(135 201 395 328)(136 190 396 317)(137 239 397 306)(138 228 398 355)(139 217 399 344)(140 206 400 333)(141 195 401 322)(142 184 402 311)(143 233 403 360)(144 222 404 349)(145 211 405 338)(146 200 406 327)(147 189 407 316)(148 238 408 305)(149 227 409 354)(150 216 410 343)(151 205 411 332)(152 194 412 321)(153 183 413 310)(154 232 414 359)(155 221 415 348)(156 210 416 337)(157 199 417 326)(158 188 418 315)(159 237 419 304)(160 226 420 353)(161 215 361 342)(162 204 362 331)(163 193 363 320)(164 182 364 309)(165 231 365 358)(166 220 366 347)(167 209 367 336)(168 198 368 325)(169 187 369 314)(170 236 370 303)(171 225 371 352)(172 214 372 341)(173 203 373 330)(174 192 374 319)(175 181 375 308)(176 230 376 357)(177 219 377 346)(178 208 378 335)(179 197 379 324)(180 186 380 313)

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

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

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

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A···10F12A12B12C12D12E···12L15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444445566610···101212121212···1215152020202020···2030···3060···60
size11112221010101012126060222222···2222210···1044444412···124···44···4

66 irreducible representations

dim11111122222222222224444444
type++++++++--+++++-+--++--+-+
imageC1C2C2C2C2C2S3D4Q8Q8D5D6D6D10D10Dic6D12Dic6Dic10S3×D5D4×D5Q8×D5C15⋊Q8C2×S3×D5D5×Dic6D5×D12
kernelC60⋊Q8C30.Q8C12×Dic5C5×C4⋊Dic3C605C4C2×C15⋊Q8C4×Dic5C3×Dic5C3×Dic5C60C4⋊Dic3C2×Dic5C2×C20C2×Dic3C2×C12Dic5Dic5C20C12C2×C4C6C6C4C22C2C2
# reps12111212222214244482224244

Matrix representation of C60⋊Q8 in GL4(𝔽61) generated by

153800
233800
00043
001744
,
31400
343000
002957
005832
,
381500
462300
00589
00263
G:=sub<GL(4,GF(61))| [15,23,0,0,38,38,0,0,0,0,0,17,0,0,43,44],[31,34,0,0,4,30,0,0,0,0,29,58,0,0,57,32],[38,46,0,0,15,23,0,0,0,0,58,26,0,0,9,3] >;

C60⋊Q8 in GAP, Magma, Sage, TeX

C_{60}\rtimes Q_8
% in TeX

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

G:=SmallGroup(480,544);
// by ID

G=gap.SmallGroup(480,544);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,64,219,100,1356,18822]);
// Polycyclic

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

׿
×
𝔽