metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊3Q8, C20⋊1Dic6, C12⋊4Dic10, Dic5⋊4Dic6, Dic5.9D12, C15⋊7(C4⋊Q8), C3⋊1(C20⋊Q8), C4⋊2(C15⋊Q8), C6.25(D4×D5), C6.37(Q8×D5), (C3×Dic5)⋊7Q8, C2.27(D5×D12), C5⋊1(C12⋊2Q8), C30.65(C2×D4), C4⋊Dic3.7D5, C30.56(C2×Q8), C10.26(C2×D12), (C2×C20).135D6, (C4×Dic5).5S3, C60⋊5C4.26C2, 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
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, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×Dic10, C12⋊2Q8, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C20⋊Q8, C30.Q8, C12×Dic5, C5×C4⋊Dic3, C60⋊5C4, C2×C15⋊Q8, C60⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, D10, Dic6, D12, C22×S3, C4⋊Q8, Dic10, C22×D5, C2×Dic6, C2×D12, S3×D5, C2×Dic10, D4×D5, Q8×D5, C12⋊2Q8, C15⋊Q8, C2×S3×D5, C20⋊Q8, D5×Dic6, D5×D12, C2×C15⋊Q8, 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)
(1 336 196 437)(2 347 197 448)(3 358 198 459)(4 309 199 470)(5 320 200 421)(6 331 201 432)(7 342 202 443)(8 353 203 454)(9 304 204 465)(10 315 205 476)(11 326 206 427)(12 337 207 438)(13 348 208 449)(14 359 209 460)(15 310 210 471)(16 321 211 422)(17 332 212 433)(18 343 213 444)(19 354 214 455)(20 305 215 466)(21 316 216 477)(22 327 217 428)(23 338 218 439)(24 349 219 450)(25 360 220 461)(26 311 221 472)(27 322 222 423)(28 333 223 434)(29 344 224 445)(30 355 225 456)(31 306 226 467)(32 317 227 478)(33 328 228 429)(34 339 229 440)(35 350 230 451)(36 301 231 462)(37 312 232 473)(38 323 233 424)(39 334 234 435)(40 345 235 446)(41 356 236 457)(42 307 237 468)(43 318 238 479)(44 329 239 430)(45 340 240 441)(46 351 181 452)(47 302 182 463)(48 313 183 474)(49 324 184 425)(50 335 185 436)(51 346 186 447)(52 357 187 458)(53 308 188 469)(54 319 189 480)(55 330 190 431)(56 341 191 442)(57 352 192 453)(58 303 193 464)(59 314 194 475)(60 325 195 426)(61 300 173 364)(62 251 174 375)(63 262 175 386)(64 273 176 397)(65 284 177 408)(66 295 178 419)(67 246 179 370)(68 257 180 381)(69 268 121 392)(70 279 122 403)(71 290 123 414)(72 241 124 365)(73 252 125 376)(74 263 126 387)(75 274 127 398)(76 285 128 409)(77 296 129 420)(78 247 130 371)(79 258 131 382)(80 269 132 393)(81 280 133 404)(82 291 134 415)(83 242 135 366)(84 253 136 377)(85 264 137 388)(86 275 138 399)(87 286 139 410)(88 297 140 361)(89 248 141 372)(90 259 142 383)(91 270 143 394)(92 281 144 405)(93 292 145 416)(94 243 146 367)(95 254 147 378)(96 265 148 389)(97 276 149 400)(98 287 150 411)(99 298 151 362)(100 249 152 373)(101 260 153 384)(102 271 154 395)(103 282 155 406)(104 293 156 417)(105 244 157 368)(106 255 158 379)(107 266 159 390)(108 277 160 401)(109 288 161 412)(110 299 162 363)(111 250 163 374)(112 261 164 385)(113 272 165 396)(114 283 166 407)(115 294 167 418)(116 245 168 369)(117 256 169 380)(118 267 170 391)(119 278 171 402)(120 289 172 413)
(1 161 196 109)(2 150 197 98)(3 139 198 87)(4 128 199 76)(5 177 200 65)(6 166 201 114)(7 155 202 103)(8 144 203 92)(9 133 204 81)(10 122 205 70)(11 171 206 119)(12 160 207 108)(13 149 208 97)(14 138 209 86)(15 127 210 75)(16 176 211 64)(17 165 212 113)(18 154 213 102)(19 143 214 91)(20 132 215 80)(21 121 216 69)(22 170 217 118)(23 159 218 107)(24 148 219 96)(25 137 220 85)(26 126 221 74)(27 175 222 63)(28 164 223 112)(29 153 224 101)(30 142 225 90)(31 131 226 79)(32 180 227 68)(33 169 228 117)(34 158 229 106)(35 147 230 95)(36 136 231 84)(37 125 232 73)(38 174 233 62)(39 163 234 111)(40 152 235 100)(41 141 236 89)(42 130 237 78)(43 179 238 67)(44 168 239 116)(45 157 240 105)(46 146 181 94)(47 135 182 83)(48 124 183 72)(49 173 184 61)(50 162 185 110)(51 151 186 99)(52 140 187 88)(53 129 188 77)(54 178 189 66)(55 167 190 115)(56 156 191 104)(57 145 192 93)(58 134 193 82)(59 123 194 71)(60 172 195 120)(241 474 365 313)(242 463 366 302)(243 452 367 351)(244 441 368 340)(245 430 369 329)(246 479 370 318)(247 468 371 307)(248 457 372 356)(249 446 373 345)(250 435 374 334)(251 424 375 323)(252 473 376 312)(253 462 377 301)(254 451 378 350)(255 440 379 339)(256 429 380 328)(257 478 381 317)(258 467 382 306)(259 456 383 355)(260 445 384 344)(261 434 385 333)(262 423 386 322)(263 472 387 311)(264 461 388 360)(265 450 389 349)(266 439 390 338)(267 428 391 327)(268 477 392 316)(269 466 393 305)(270 455 394 354)(271 444 395 343)(272 433 396 332)(273 422 397 321)(274 471 398 310)(275 460 399 359)(276 449 400 348)(277 438 401 337)(278 427 402 326)(279 476 403 315)(280 465 404 304)(281 454 405 353)(282 443 406 342)(283 432 407 331)(284 421 408 320)(285 470 409 309)(286 459 410 358)(287 448 411 347)(288 437 412 336)(289 426 413 325)(290 475 414 314)(291 464 415 303)(292 453 416 352)(293 442 417 341)(294 431 418 330)(295 480 419 319)(296 469 420 308)(297 458 361 357)(298 447 362 346)(299 436 363 335)(300 425 364 324)
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), (1,336,196,437)(2,347,197,448)(3,358,198,459)(4,309,199,470)(5,320,200,421)(6,331,201,432)(7,342,202,443)(8,353,203,454)(9,304,204,465)(10,315,205,476)(11,326,206,427)(12,337,207,438)(13,348,208,449)(14,359,209,460)(15,310,210,471)(16,321,211,422)(17,332,212,433)(18,343,213,444)(19,354,214,455)(20,305,215,466)(21,316,216,477)(22,327,217,428)(23,338,218,439)(24,349,219,450)(25,360,220,461)(26,311,221,472)(27,322,222,423)(28,333,223,434)(29,344,224,445)(30,355,225,456)(31,306,226,467)(32,317,227,478)(33,328,228,429)(34,339,229,440)(35,350,230,451)(36,301,231,462)(37,312,232,473)(38,323,233,424)(39,334,234,435)(40,345,235,446)(41,356,236,457)(42,307,237,468)(43,318,238,479)(44,329,239,430)(45,340,240,441)(46,351,181,452)(47,302,182,463)(48,313,183,474)(49,324,184,425)(50,335,185,436)(51,346,186,447)(52,357,187,458)(53,308,188,469)(54,319,189,480)(55,330,190,431)(56,341,191,442)(57,352,192,453)(58,303,193,464)(59,314,194,475)(60,325,195,426)(61,300,173,364)(62,251,174,375)(63,262,175,386)(64,273,176,397)(65,284,177,408)(66,295,178,419)(67,246,179,370)(68,257,180,381)(69,268,121,392)(70,279,122,403)(71,290,123,414)(72,241,124,365)(73,252,125,376)(74,263,126,387)(75,274,127,398)(76,285,128,409)(77,296,129,420)(78,247,130,371)(79,258,131,382)(80,269,132,393)(81,280,133,404)(82,291,134,415)(83,242,135,366)(84,253,136,377)(85,264,137,388)(86,275,138,399)(87,286,139,410)(88,297,140,361)(89,248,141,372)(90,259,142,383)(91,270,143,394)(92,281,144,405)(93,292,145,416)(94,243,146,367)(95,254,147,378)(96,265,148,389)(97,276,149,400)(98,287,150,411)(99,298,151,362)(100,249,152,373)(101,260,153,384)(102,271,154,395)(103,282,155,406)(104,293,156,417)(105,244,157,368)(106,255,158,379)(107,266,159,390)(108,277,160,401)(109,288,161,412)(110,299,162,363)(111,250,163,374)(112,261,164,385)(113,272,165,396)(114,283,166,407)(115,294,167,418)(116,245,168,369)(117,256,169,380)(118,267,170,391)(119,278,171,402)(120,289,172,413), (1,161,196,109)(2,150,197,98)(3,139,198,87)(4,128,199,76)(5,177,200,65)(6,166,201,114)(7,155,202,103)(8,144,203,92)(9,133,204,81)(10,122,205,70)(11,171,206,119)(12,160,207,108)(13,149,208,97)(14,138,209,86)(15,127,210,75)(16,176,211,64)(17,165,212,113)(18,154,213,102)(19,143,214,91)(20,132,215,80)(21,121,216,69)(22,170,217,118)(23,159,218,107)(24,148,219,96)(25,137,220,85)(26,126,221,74)(27,175,222,63)(28,164,223,112)(29,153,224,101)(30,142,225,90)(31,131,226,79)(32,180,227,68)(33,169,228,117)(34,158,229,106)(35,147,230,95)(36,136,231,84)(37,125,232,73)(38,174,233,62)(39,163,234,111)(40,152,235,100)(41,141,236,89)(42,130,237,78)(43,179,238,67)(44,168,239,116)(45,157,240,105)(46,146,181,94)(47,135,182,83)(48,124,183,72)(49,173,184,61)(50,162,185,110)(51,151,186,99)(52,140,187,88)(53,129,188,77)(54,178,189,66)(55,167,190,115)(56,156,191,104)(57,145,192,93)(58,134,193,82)(59,123,194,71)(60,172,195,120)(241,474,365,313)(242,463,366,302)(243,452,367,351)(244,441,368,340)(245,430,369,329)(246,479,370,318)(247,468,371,307)(248,457,372,356)(249,446,373,345)(250,435,374,334)(251,424,375,323)(252,473,376,312)(253,462,377,301)(254,451,378,350)(255,440,379,339)(256,429,380,328)(257,478,381,317)(258,467,382,306)(259,456,383,355)(260,445,384,344)(261,434,385,333)(262,423,386,322)(263,472,387,311)(264,461,388,360)(265,450,389,349)(266,439,390,338)(267,428,391,327)(268,477,392,316)(269,466,393,305)(270,455,394,354)(271,444,395,343)(272,433,396,332)(273,422,397,321)(274,471,398,310)(275,460,399,359)(276,449,400,348)(277,438,401,337)(278,427,402,326)(279,476,403,315)(280,465,404,304)(281,454,405,353)(282,443,406,342)(283,432,407,331)(284,421,408,320)(285,470,409,309)(286,459,410,358)(287,448,411,347)(288,437,412,336)(289,426,413,325)(290,475,414,314)(291,464,415,303)(292,453,416,352)(293,442,417,341)(294,431,418,330)(295,480,419,319)(296,469,420,308)(297,458,361,357)(298,447,362,346)(299,436,363,335)(300,425,364,324)>;
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), (1,336,196,437)(2,347,197,448)(3,358,198,459)(4,309,199,470)(5,320,200,421)(6,331,201,432)(7,342,202,443)(8,353,203,454)(9,304,204,465)(10,315,205,476)(11,326,206,427)(12,337,207,438)(13,348,208,449)(14,359,209,460)(15,310,210,471)(16,321,211,422)(17,332,212,433)(18,343,213,444)(19,354,214,455)(20,305,215,466)(21,316,216,477)(22,327,217,428)(23,338,218,439)(24,349,219,450)(25,360,220,461)(26,311,221,472)(27,322,222,423)(28,333,223,434)(29,344,224,445)(30,355,225,456)(31,306,226,467)(32,317,227,478)(33,328,228,429)(34,339,229,440)(35,350,230,451)(36,301,231,462)(37,312,232,473)(38,323,233,424)(39,334,234,435)(40,345,235,446)(41,356,236,457)(42,307,237,468)(43,318,238,479)(44,329,239,430)(45,340,240,441)(46,351,181,452)(47,302,182,463)(48,313,183,474)(49,324,184,425)(50,335,185,436)(51,346,186,447)(52,357,187,458)(53,308,188,469)(54,319,189,480)(55,330,190,431)(56,341,191,442)(57,352,192,453)(58,303,193,464)(59,314,194,475)(60,325,195,426)(61,300,173,364)(62,251,174,375)(63,262,175,386)(64,273,176,397)(65,284,177,408)(66,295,178,419)(67,246,179,370)(68,257,180,381)(69,268,121,392)(70,279,122,403)(71,290,123,414)(72,241,124,365)(73,252,125,376)(74,263,126,387)(75,274,127,398)(76,285,128,409)(77,296,129,420)(78,247,130,371)(79,258,131,382)(80,269,132,393)(81,280,133,404)(82,291,134,415)(83,242,135,366)(84,253,136,377)(85,264,137,388)(86,275,138,399)(87,286,139,410)(88,297,140,361)(89,248,141,372)(90,259,142,383)(91,270,143,394)(92,281,144,405)(93,292,145,416)(94,243,146,367)(95,254,147,378)(96,265,148,389)(97,276,149,400)(98,287,150,411)(99,298,151,362)(100,249,152,373)(101,260,153,384)(102,271,154,395)(103,282,155,406)(104,293,156,417)(105,244,157,368)(106,255,158,379)(107,266,159,390)(108,277,160,401)(109,288,161,412)(110,299,162,363)(111,250,163,374)(112,261,164,385)(113,272,165,396)(114,283,166,407)(115,294,167,418)(116,245,168,369)(117,256,169,380)(118,267,170,391)(119,278,171,402)(120,289,172,413), (1,161,196,109)(2,150,197,98)(3,139,198,87)(4,128,199,76)(5,177,200,65)(6,166,201,114)(7,155,202,103)(8,144,203,92)(9,133,204,81)(10,122,205,70)(11,171,206,119)(12,160,207,108)(13,149,208,97)(14,138,209,86)(15,127,210,75)(16,176,211,64)(17,165,212,113)(18,154,213,102)(19,143,214,91)(20,132,215,80)(21,121,216,69)(22,170,217,118)(23,159,218,107)(24,148,219,96)(25,137,220,85)(26,126,221,74)(27,175,222,63)(28,164,223,112)(29,153,224,101)(30,142,225,90)(31,131,226,79)(32,180,227,68)(33,169,228,117)(34,158,229,106)(35,147,230,95)(36,136,231,84)(37,125,232,73)(38,174,233,62)(39,163,234,111)(40,152,235,100)(41,141,236,89)(42,130,237,78)(43,179,238,67)(44,168,239,116)(45,157,240,105)(46,146,181,94)(47,135,182,83)(48,124,183,72)(49,173,184,61)(50,162,185,110)(51,151,186,99)(52,140,187,88)(53,129,188,77)(54,178,189,66)(55,167,190,115)(56,156,191,104)(57,145,192,93)(58,134,193,82)(59,123,194,71)(60,172,195,120)(241,474,365,313)(242,463,366,302)(243,452,367,351)(244,441,368,340)(245,430,369,329)(246,479,370,318)(247,468,371,307)(248,457,372,356)(249,446,373,345)(250,435,374,334)(251,424,375,323)(252,473,376,312)(253,462,377,301)(254,451,378,350)(255,440,379,339)(256,429,380,328)(257,478,381,317)(258,467,382,306)(259,456,383,355)(260,445,384,344)(261,434,385,333)(262,423,386,322)(263,472,387,311)(264,461,388,360)(265,450,389,349)(266,439,390,338)(267,428,391,327)(268,477,392,316)(269,466,393,305)(270,455,394,354)(271,444,395,343)(272,433,396,332)(273,422,397,321)(274,471,398,310)(275,460,399,359)(276,449,400,348)(277,438,401,337)(278,427,402,326)(279,476,403,315)(280,465,404,304)(281,454,405,353)(282,443,406,342)(283,432,407,331)(284,421,408,320)(285,470,409,309)(286,459,410,358)(287,448,411,347)(288,437,412,336)(289,426,413,325)(290,475,414,314)(291,464,415,303)(292,453,416,352)(293,442,417,341)(294,431,418,330)(295,480,419,319)(296,469,420,308)(297,458,361,357)(298,447,362,346)(299,436,363,335)(300,425,364,324) );
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)], [(1,336,196,437),(2,347,197,448),(3,358,198,459),(4,309,199,470),(5,320,200,421),(6,331,201,432),(7,342,202,443),(8,353,203,454),(9,304,204,465),(10,315,205,476),(11,326,206,427),(12,337,207,438),(13,348,208,449),(14,359,209,460),(15,310,210,471),(16,321,211,422),(17,332,212,433),(18,343,213,444),(19,354,214,455),(20,305,215,466),(21,316,216,477),(22,327,217,428),(23,338,218,439),(24,349,219,450),(25,360,220,461),(26,311,221,472),(27,322,222,423),(28,333,223,434),(29,344,224,445),(30,355,225,456),(31,306,226,467),(32,317,227,478),(33,328,228,429),(34,339,229,440),(35,350,230,451),(36,301,231,462),(37,312,232,473),(38,323,233,424),(39,334,234,435),(40,345,235,446),(41,356,236,457),(42,307,237,468),(43,318,238,479),(44,329,239,430),(45,340,240,441),(46,351,181,452),(47,302,182,463),(48,313,183,474),(49,324,184,425),(50,335,185,436),(51,346,186,447),(52,357,187,458),(53,308,188,469),(54,319,189,480),(55,330,190,431),(56,341,191,442),(57,352,192,453),(58,303,193,464),(59,314,194,475),(60,325,195,426),(61,300,173,364),(62,251,174,375),(63,262,175,386),(64,273,176,397),(65,284,177,408),(66,295,178,419),(67,246,179,370),(68,257,180,381),(69,268,121,392),(70,279,122,403),(71,290,123,414),(72,241,124,365),(73,252,125,376),(74,263,126,387),(75,274,127,398),(76,285,128,409),(77,296,129,420),(78,247,130,371),(79,258,131,382),(80,269,132,393),(81,280,133,404),(82,291,134,415),(83,242,135,366),(84,253,136,377),(85,264,137,388),(86,275,138,399),(87,286,139,410),(88,297,140,361),(89,248,141,372),(90,259,142,383),(91,270,143,394),(92,281,144,405),(93,292,145,416),(94,243,146,367),(95,254,147,378),(96,265,148,389),(97,276,149,400),(98,287,150,411),(99,298,151,362),(100,249,152,373),(101,260,153,384),(102,271,154,395),(103,282,155,406),(104,293,156,417),(105,244,157,368),(106,255,158,379),(107,266,159,390),(108,277,160,401),(109,288,161,412),(110,299,162,363),(111,250,163,374),(112,261,164,385),(113,272,165,396),(114,283,166,407),(115,294,167,418),(116,245,168,369),(117,256,169,380),(118,267,170,391),(119,278,171,402),(120,289,172,413)], [(1,161,196,109),(2,150,197,98),(3,139,198,87),(4,128,199,76),(5,177,200,65),(6,166,201,114),(7,155,202,103),(8,144,203,92),(9,133,204,81),(10,122,205,70),(11,171,206,119),(12,160,207,108),(13,149,208,97),(14,138,209,86),(15,127,210,75),(16,176,211,64),(17,165,212,113),(18,154,213,102),(19,143,214,91),(20,132,215,80),(21,121,216,69),(22,170,217,118),(23,159,218,107),(24,148,219,96),(25,137,220,85),(26,126,221,74),(27,175,222,63),(28,164,223,112),(29,153,224,101),(30,142,225,90),(31,131,226,79),(32,180,227,68),(33,169,228,117),(34,158,229,106),(35,147,230,95),(36,136,231,84),(37,125,232,73),(38,174,233,62),(39,163,234,111),(40,152,235,100),(41,141,236,89),(42,130,237,78),(43,179,238,67),(44,168,239,116),(45,157,240,105),(46,146,181,94),(47,135,182,83),(48,124,183,72),(49,173,184,61),(50,162,185,110),(51,151,186,99),(52,140,187,88),(53,129,188,77),(54,178,189,66),(55,167,190,115),(56,156,191,104),(57,145,192,93),(58,134,193,82),(59,123,194,71),(60,172,195,120),(241,474,365,313),(242,463,366,302),(243,452,367,351),(244,441,368,340),(245,430,369,329),(246,479,370,318),(247,468,371,307),(248,457,372,356),(249,446,373,345),(250,435,374,334),(251,424,375,323),(252,473,376,312),(253,462,377,301),(254,451,378,350),(255,440,379,339),(256,429,380,328),(257,478,381,317),(258,467,382,306),(259,456,383,355),(260,445,384,344),(261,434,385,333),(262,423,386,322),(263,472,387,311),(264,461,388,360),(265,450,389,349),(266,439,390,338),(267,428,391,327),(268,477,392,316),(269,466,393,305),(270,455,394,354),(271,444,395,343),(272,433,396,332),(273,422,397,321),(274,471,398,310),(275,460,399,359),(276,449,400,348),(277,438,401,337),(278,427,402,326),(279,476,403,315),(280,465,404,304),(281,454,405,353),(282,443,406,342),(283,432,407,331),(284,421,408,320),(285,470,409,309),(286,459,410,358),(287,448,411,347),(288,437,412,336),(289,426,413,325),(290,475,414,314),(291,464,415,303),(292,453,416,352),(293,442,417,341),(294,431,418,330),(295,480,419,319),(296,469,420,308),(297,458,361,357),(298,447,362,346),(299,436,363,335),(300,425,364,324)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 12 | 12 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | + | + | - | + | - | - | + | + | - | - | + | - | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | Q8 | D5 | D6 | D6 | D10 | D10 | Dic6 | D12 | Dic6 | Dic10 | S3×D5 | D4×D5 | Q8×D5 | C15⋊Q8 | C2×S3×D5 | D5×Dic6 | D5×D12 |
| kernel | C60⋊Q8 | C30.Q8 | C12×Dic5 | C5×C4⋊Dic3 | C60⋊5C4 | C2×C15⋊Q8 | C4×Dic5 | C3×Dic5 | C3×Dic5 | C60 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C2×Dic3 | C2×C12 | Dic5 | Dic5 | C20 | C12 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C60⋊Q8 ►in GL4(𝔽61) generated by
| 15 | 38 | 0 | 0 |
| 23 | 38 | 0 | 0 |
| 0 | 0 | 0 | 43 |
| 0 | 0 | 17 | 44 |
| 31 | 4 | 0 | 0 |
| 34 | 30 | 0 | 0 |
| 0 | 0 | 29 | 57 |
| 0 | 0 | 58 | 32 |
| 38 | 15 | 0 | 0 |
| 46 | 23 | 0 | 0 |
| 0 | 0 | 58 | 9 |
| 0 | 0 | 26 | 3 |
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