direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C60, C42.3C30, C4⋊C4.6C30, C4.4(C2×C60), (C4×C20).9C6, C2.2(Q8×C30), (C4×C60).21C2, (C4×C12).9C10, (C2×Q8).7C30, C10.19(C6×Q8), C6.19(Q8×C10), C12.32(C2×C20), C60.228(C2×C4), C20.53(C2×C12), C2.5(C22×C60), (C6×Q8).10C10, (Q8×C10).14C6, (Q8×C30).20C2, C30.117(C2×Q8), C6.33(C22×C20), C30.276(C4○D4), C30.240(C22×C4), (C2×C30).454C23, (C2×C60).577C22, C10.46(C22×C12), C22.8(C22×C30), (C5×C4⋊C4).13C6, C2.3(C15×C4○D4), C6.40(C5×C4○D4), (C3×C4⋊C4).13C10, (C15×C4⋊C4).27C2, (C2×C4).12(C2×C30), C10.40(C3×C4○D4), (C2×C20).124(C2×C6), (C2×C12).80(C2×C10), (C2×C10).74(C22×C6), (C2×C6).74(C22×C10), SmallGroup(480,924)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C60
G = < a,b,c | a60=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 152 in 140 conjugacy classes, 128 normal (32 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, C20, C20, C2×C10, C2×C12, C2×C12, C3×Q8, C30, C4×Q8, C2×C20, C2×C20, C5×Q8, C4×C12, C3×C4⋊C4, C6×Q8, C60, C60, C2×C30, C4×C20, C5×C4⋊C4, Q8×C10, Q8×C12, C2×C60, C2×C60, Q8×C15, Q8×C20, C4×C60, C15×C4⋊C4, Q8×C30, Q8×C60
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, Q8, C23, C10, C12, C2×C6, C15, C22×C4, C2×Q8, C4○D4, C20, C2×C10, C2×C12, C3×Q8, C22×C6, C30, C4×Q8, C2×C20, C5×Q8, C22×C10, C22×C12, C6×Q8, C3×C4○D4, C60, C2×C30, C22×C20, Q8×C10, C5×C4○D4, Q8×C12, C2×C60, Q8×C15, C22×C30, Q8×C20, C22×C60, Q8×C30, C15×C4○D4, Q8×C60
►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)
(1 428 267 395)(2 429 268 396)(3 430 269 397)(4 431 270 398)(5 432 271 399)(6 433 272 400)(7 434 273 401)(8 435 274 402)(9 436 275 403)(10 437 276 404)(11 438 277 405)(12 439 278 406)(13 440 279 407)(14 441 280 408)(15 442 281 409)(16 443 282 410)(17 444 283 411)(18 445 284 412)(19 446 285 413)(20 447 286 414)(21 448 287 415)(22 449 288 416)(23 450 289 417)(24 451 290 418)(25 452 291 419)(26 453 292 420)(27 454 293 361)(28 455 294 362)(29 456 295 363)(30 457 296 364)(31 458 297 365)(32 459 298 366)(33 460 299 367)(34 461 300 368)(35 462 241 369)(36 463 242 370)(37 464 243 371)(38 465 244 372)(39 466 245 373)(40 467 246 374)(41 468 247 375)(42 469 248 376)(43 470 249 377)(44 471 250 378)(45 472 251 379)(46 473 252 380)(47 474 253 381)(48 475 254 382)(49 476 255 383)(50 477 256 384)(51 478 257 385)(52 479 258 386)(53 480 259 387)(54 421 260 388)(55 422 261 389)(56 423 262 390)(57 424 263 391)(58 425 264 392)(59 426 265 393)(60 427 266 394)(61 128 202 303)(62 129 203 304)(63 130 204 305)(64 131 205 306)(65 132 206 307)(66 133 207 308)(67 134 208 309)(68 135 209 310)(69 136 210 311)(70 137 211 312)(71 138 212 313)(72 139 213 314)(73 140 214 315)(74 141 215 316)(75 142 216 317)(76 143 217 318)(77 144 218 319)(78 145 219 320)(79 146 220 321)(80 147 221 322)(81 148 222 323)(82 149 223 324)(83 150 224 325)(84 151 225 326)(85 152 226 327)(86 153 227 328)(87 154 228 329)(88 155 229 330)(89 156 230 331)(90 157 231 332)(91 158 232 333)(92 159 233 334)(93 160 234 335)(94 161 235 336)(95 162 236 337)(96 163 237 338)(97 164 238 339)(98 165 239 340)(99 166 240 341)(100 167 181 342)(101 168 182 343)(102 169 183 344)(103 170 184 345)(104 171 185 346)(105 172 186 347)(106 173 187 348)(107 174 188 349)(108 175 189 350)(109 176 190 351)(110 177 191 352)(111 178 192 353)(112 179 193 354)(113 180 194 355)(114 121 195 356)(115 122 196 357)(116 123 197 358)(117 124 198 359)(118 125 199 360)(119 126 200 301)(120 127 201 302)
(1 193 267 112)(2 194 268 113)(3 195 269 114)(4 196 270 115)(5 197 271 116)(6 198 272 117)(7 199 273 118)(8 200 274 119)(9 201 275 120)(10 202 276 61)(11 203 277 62)(12 204 278 63)(13 205 279 64)(14 206 280 65)(15 207 281 66)(16 208 282 67)(17 209 283 68)(18 210 284 69)(19 211 285 70)(20 212 286 71)(21 213 287 72)(22 214 288 73)(23 215 289 74)(24 216 290 75)(25 217 291 76)(26 218 292 77)(27 219 293 78)(28 220 294 79)(29 221 295 80)(30 222 296 81)(31 223 297 82)(32 224 298 83)(33 225 299 84)(34 226 300 85)(35 227 241 86)(36 228 242 87)(37 229 243 88)(38 230 244 89)(39 231 245 90)(40 232 246 91)(41 233 247 92)(42 234 248 93)(43 235 249 94)(44 236 250 95)(45 237 251 96)(46 238 252 97)(47 239 253 98)(48 240 254 99)(49 181 255 100)(50 182 256 101)(51 183 257 102)(52 184 258 103)(53 185 259 104)(54 186 260 105)(55 187 261 106)(56 188 262 107)(57 189 263 108)(58 190 264 109)(59 191 265 110)(60 192 266 111)(121 397 356 430)(122 398 357 431)(123 399 358 432)(124 400 359 433)(125 401 360 434)(126 402 301 435)(127 403 302 436)(128 404 303 437)(129 405 304 438)(130 406 305 439)(131 407 306 440)(132 408 307 441)(133 409 308 442)(134 410 309 443)(135 411 310 444)(136 412 311 445)(137 413 312 446)(138 414 313 447)(139 415 314 448)(140 416 315 449)(141 417 316 450)(142 418 317 451)(143 419 318 452)(144 420 319 453)(145 361 320 454)(146 362 321 455)(147 363 322 456)(148 364 323 457)(149 365 324 458)(150 366 325 459)(151 367 326 460)(152 368 327 461)(153 369 328 462)(154 370 329 463)(155 371 330 464)(156 372 331 465)(157 373 332 466)(158 374 333 467)(159 375 334 468)(160 376 335 469)(161 377 336 470)(162 378 337 471)(163 379 338 472)(164 380 339 473)(165 381 340 474)(166 382 341 475)(167 383 342 476)(168 384 343 477)(169 385 344 478)(170 386 345 479)(171 387 346 480)(172 388 347 421)(173 389 348 422)(174 390 349 423)(175 391 350 424)(176 392 351 425)(177 393 352 426)(178 394 353 427)(179 395 354 428)(180 396 355 429)
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), (1,428,267,395)(2,429,268,396)(3,430,269,397)(4,431,270,398)(5,432,271,399)(6,433,272,400)(7,434,273,401)(8,435,274,402)(9,436,275,403)(10,437,276,404)(11,438,277,405)(12,439,278,406)(13,440,279,407)(14,441,280,408)(15,442,281,409)(16,443,282,410)(17,444,283,411)(18,445,284,412)(19,446,285,413)(20,447,286,414)(21,448,287,415)(22,449,288,416)(23,450,289,417)(24,451,290,418)(25,452,291,419)(26,453,292,420)(27,454,293,361)(28,455,294,362)(29,456,295,363)(30,457,296,364)(31,458,297,365)(32,459,298,366)(33,460,299,367)(34,461,300,368)(35,462,241,369)(36,463,242,370)(37,464,243,371)(38,465,244,372)(39,466,245,373)(40,467,246,374)(41,468,247,375)(42,469,248,376)(43,470,249,377)(44,471,250,378)(45,472,251,379)(46,473,252,380)(47,474,253,381)(48,475,254,382)(49,476,255,383)(50,477,256,384)(51,478,257,385)(52,479,258,386)(53,480,259,387)(54,421,260,388)(55,422,261,389)(56,423,262,390)(57,424,263,391)(58,425,264,392)(59,426,265,393)(60,427,266,394)(61,128,202,303)(62,129,203,304)(63,130,204,305)(64,131,205,306)(65,132,206,307)(66,133,207,308)(67,134,208,309)(68,135,209,310)(69,136,210,311)(70,137,211,312)(71,138,212,313)(72,139,213,314)(73,140,214,315)(74,141,215,316)(75,142,216,317)(76,143,217,318)(77,144,218,319)(78,145,219,320)(79,146,220,321)(80,147,221,322)(81,148,222,323)(82,149,223,324)(83,150,224,325)(84,151,225,326)(85,152,226,327)(86,153,227,328)(87,154,228,329)(88,155,229,330)(89,156,230,331)(90,157,231,332)(91,158,232,333)(92,159,233,334)(93,160,234,335)(94,161,235,336)(95,162,236,337)(96,163,237,338)(97,164,238,339)(98,165,239,340)(99,166,240,341)(100,167,181,342)(101,168,182,343)(102,169,183,344)(103,170,184,345)(104,171,185,346)(105,172,186,347)(106,173,187,348)(107,174,188,349)(108,175,189,350)(109,176,190,351)(110,177,191,352)(111,178,192,353)(112,179,193,354)(113,180,194,355)(114,121,195,356)(115,122,196,357)(116,123,197,358)(117,124,198,359)(118,125,199,360)(119,126,200,301)(120,127,201,302), (1,193,267,112)(2,194,268,113)(3,195,269,114)(4,196,270,115)(5,197,271,116)(6,198,272,117)(7,199,273,118)(8,200,274,119)(9,201,275,120)(10,202,276,61)(11,203,277,62)(12,204,278,63)(13,205,279,64)(14,206,280,65)(15,207,281,66)(16,208,282,67)(17,209,283,68)(18,210,284,69)(19,211,285,70)(20,212,286,71)(21,213,287,72)(22,214,288,73)(23,215,289,74)(24,216,290,75)(25,217,291,76)(26,218,292,77)(27,219,293,78)(28,220,294,79)(29,221,295,80)(30,222,296,81)(31,223,297,82)(32,224,298,83)(33,225,299,84)(34,226,300,85)(35,227,241,86)(36,228,242,87)(37,229,243,88)(38,230,244,89)(39,231,245,90)(40,232,246,91)(41,233,247,92)(42,234,248,93)(43,235,249,94)(44,236,250,95)(45,237,251,96)(46,238,252,97)(47,239,253,98)(48,240,254,99)(49,181,255,100)(50,182,256,101)(51,183,257,102)(52,184,258,103)(53,185,259,104)(54,186,260,105)(55,187,261,106)(56,188,262,107)(57,189,263,108)(58,190,264,109)(59,191,265,110)(60,192,266,111)(121,397,356,430)(122,398,357,431)(123,399,358,432)(124,400,359,433)(125,401,360,434)(126,402,301,435)(127,403,302,436)(128,404,303,437)(129,405,304,438)(130,406,305,439)(131,407,306,440)(132,408,307,441)(133,409,308,442)(134,410,309,443)(135,411,310,444)(136,412,311,445)(137,413,312,446)(138,414,313,447)(139,415,314,448)(140,416,315,449)(141,417,316,450)(142,418,317,451)(143,419,318,452)(144,420,319,453)(145,361,320,454)(146,362,321,455)(147,363,322,456)(148,364,323,457)(149,365,324,458)(150,366,325,459)(151,367,326,460)(152,368,327,461)(153,369,328,462)(154,370,329,463)(155,371,330,464)(156,372,331,465)(157,373,332,466)(158,374,333,467)(159,375,334,468)(160,376,335,469)(161,377,336,470)(162,378,337,471)(163,379,338,472)(164,380,339,473)(165,381,340,474)(166,382,341,475)(167,383,342,476)(168,384,343,477)(169,385,344,478)(170,386,345,479)(171,387,346,480)(172,388,347,421)(173,389,348,422)(174,390,349,423)(175,391,350,424)(176,392,351,425)(177,393,352,426)(178,394,353,427)(179,395,354,428)(180,396,355,429)>;
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), (1,428,267,395)(2,429,268,396)(3,430,269,397)(4,431,270,398)(5,432,271,399)(6,433,272,400)(7,434,273,401)(8,435,274,402)(9,436,275,403)(10,437,276,404)(11,438,277,405)(12,439,278,406)(13,440,279,407)(14,441,280,408)(15,442,281,409)(16,443,282,410)(17,444,283,411)(18,445,284,412)(19,446,285,413)(20,447,286,414)(21,448,287,415)(22,449,288,416)(23,450,289,417)(24,451,290,418)(25,452,291,419)(26,453,292,420)(27,454,293,361)(28,455,294,362)(29,456,295,363)(30,457,296,364)(31,458,297,365)(32,459,298,366)(33,460,299,367)(34,461,300,368)(35,462,241,369)(36,463,242,370)(37,464,243,371)(38,465,244,372)(39,466,245,373)(40,467,246,374)(41,468,247,375)(42,469,248,376)(43,470,249,377)(44,471,250,378)(45,472,251,379)(46,473,252,380)(47,474,253,381)(48,475,254,382)(49,476,255,383)(50,477,256,384)(51,478,257,385)(52,479,258,386)(53,480,259,387)(54,421,260,388)(55,422,261,389)(56,423,262,390)(57,424,263,391)(58,425,264,392)(59,426,265,393)(60,427,266,394)(61,128,202,303)(62,129,203,304)(63,130,204,305)(64,131,205,306)(65,132,206,307)(66,133,207,308)(67,134,208,309)(68,135,209,310)(69,136,210,311)(70,137,211,312)(71,138,212,313)(72,139,213,314)(73,140,214,315)(74,141,215,316)(75,142,216,317)(76,143,217,318)(77,144,218,319)(78,145,219,320)(79,146,220,321)(80,147,221,322)(81,148,222,323)(82,149,223,324)(83,150,224,325)(84,151,225,326)(85,152,226,327)(86,153,227,328)(87,154,228,329)(88,155,229,330)(89,156,230,331)(90,157,231,332)(91,158,232,333)(92,159,233,334)(93,160,234,335)(94,161,235,336)(95,162,236,337)(96,163,237,338)(97,164,238,339)(98,165,239,340)(99,166,240,341)(100,167,181,342)(101,168,182,343)(102,169,183,344)(103,170,184,345)(104,171,185,346)(105,172,186,347)(106,173,187,348)(107,174,188,349)(108,175,189,350)(109,176,190,351)(110,177,191,352)(111,178,192,353)(112,179,193,354)(113,180,194,355)(114,121,195,356)(115,122,196,357)(116,123,197,358)(117,124,198,359)(118,125,199,360)(119,126,200,301)(120,127,201,302), (1,193,267,112)(2,194,268,113)(3,195,269,114)(4,196,270,115)(5,197,271,116)(6,198,272,117)(7,199,273,118)(8,200,274,119)(9,201,275,120)(10,202,276,61)(11,203,277,62)(12,204,278,63)(13,205,279,64)(14,206,280,65)(15,207,281,66)(16,208,282,67)(17,209,283,68)(18,210,284,69)(19,211,285,70)(20,212,286,71)(21,213,287,72)(22,214,288,73)(23,215,289,74)(24,216,290,75)(25,217,291,76)(26,218,292,77)(27,219,293,78)(28,220,294,79)(29,221,295,80)(30,222,296,81)(31,223,297,82)(32,224,298,83)(33,225,299,84)(34,226,300,85)(35,227,241,86)(36,228,242,87)(37,229,243,88)(38,230,244,89)(39,231,245,90)(40,232,246,91)(41,233,247,92)(42,234,248,93)(43,235,249,94)(44,236,250,95)(45,237,251,96)(46,238,252,97)(47,239,253,98)(48,240,254,99)(49,181,255,100)(50,182,256,101)(51,183,257,102)(52,184,258,103)(53,185,259,104)(54,186,260,105)(55,187,261,106)(56,188,262,107)(57,189,263,108)(58,190,264,109)(59,191,265,110)(60,192,266,111)(121,397,356,430)(122,398,357,431)(123,399,358,432)(124,400,359,433)(125,401,360,434)(126,402,301,435)(127,403,302,436)(128,404,303,437)(129,405,304,438)(130,406,305,439)(131,407,306,440)(132,408,307,441)(133,409,308,442)(134,410,309,443)(135,411,310,444)(136,412,311,445)(137,413,312,446)(138,414,313,447)(139,415,314,448)(140,416,315,449)(141,417,316,450)(142,418,317,451)(143,419,318,452)(144,420,319,453)(145,361,320,454)(146,362,321,455)(147,363,322,456)(148,364,323,457)(149,365,324,458)(150,366,325,459)(151,367,326,460)(152,368,327,461)(153,369,328,462)(154,370,329,463)(155,371,330,464)(156,372,331,465)(157,373,332,466)(158,374,333,467)(159,375,334,468)(160,376,335,469)(161,377,336,470)(162,378,337,471)(163,379,338,472)(164,380,339,473)(165,381,340,474)(166,382,341,475)(167,383,342,476)(168,384,343,477)(169,385,344,478)(170,386,345,479)(171,387,346,480)(172,388,347,421)(173,389,348,422)(174,390,349,423)(175,391,350,424)(176,392,351,425)(177,393,352,426)(178,394,353,427)(179,395,354,428)(180,396,355,429) );
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)], [(1,428,267,395),(2,429,268,396),(3,430,269,397),(4,431,270,398),(5,432,271,399),(6,433,272,400),(7,434,273,401),(8,435,274,402),(9,436,275,403),(10,437,276,404),(11,438,277,405),(12,439,278,406),(13,440,279,407),(14,441,280,408),(15,442,281,409),(16,443,282,410),(17,444,283,411),(18,445,284,412),(19,446,285,413),(20,447,286,414),(21,448,287,415),(22,449,288,416),(23,450,289,417),(24,451,290,418),(25,452,291,419),(26,453,292,420),(27,454,293,361),(28,455,294,362),(29,456,295,363),(30,457,296,364),(31,458,297,365),(32,459,298,366),(33,460,299,367),(34,461,300,368),(35,462,241,369),(36,463,242,370),(37,464,243,371),(38,465,244,372),(39,466,245,373),(40,467,246,374),(41,468,247,375),(42,469,248,376),(43,470,249,377),(44,471,250,378),(45,472,251,379),(46,473,252,380),(47,474,253,381),(48,475,254,382),(49,476,255,383),(50,477,256,384),(51,478,257,385),(52,479,258,386),(53,480,259,387),(54,421,260,388),(55,422,261,389),(56,423,262,390),(57,424,263,391),(58,425,264,392),(59,426,265,393),(60,427,266,394),(61,128,202,303),(62,129,203,304),(63,130,204,305),(64,131,205,306),(65,132,206,307),(66,133,207,308),(67,134,208,309),(68,135,209,310),(69,136,210,311),(70,137,211,312),(71,138,212,313),(72,139,213,314),(73,140,214,315),(74,141,215,316),(75,142,216,317),(76,143,217,318),(77,144,218,319),(78,145,219,320),(79,146,220,321),(80,147,221,322),(81,148,222,323),(82,149,223,324),(83,150,224,325),(84,151,225,326),(85,152,226,327),(86,153,227,328),(87,154,228,329),(88,155,229,330),(89,156,230,331),(90,157,231,332),(91,158,232,333),(92,159,233,334),(93,160,234,335),(94,161,235,336),(95,162,236,337),(96,163,237,338),(97,164,238,339),(98,165,239,340),(99,166,240,341),(100,167,181,342),(101,168,182,343),(102,169,183,344),(103,170,184,345),(104,171,185,346),(105,172,186,347),(106,173,187,348),(107,174,188,349),(108,175,189,350),(109,176,190,351),(110,177,191,352),(111,178,192,353),(112,179,193,354),(113,180,194,355),(114,121,195,356),(115,122,196,357),(116,123,197,358),(117,124,198,359),(118,125,199,360),(119,126,200,301),(120,127,201,302)], [(1,193,267,112),(2,194,268,113),(3,195,269,114),(4,196,270,115),(5,197,271,116),(6,198,272,117),(7,199,273,118),(8,200,274,119),(9,201,275,120),(10,202,276,61),(11,203,277,62),(12,204,278,63),(13,205,279,64),(14,206,280,65),(15,207,281,66),(16,208,282,67),(17,209,283,68),(18,210,284,69),(19,211,285,70),(20,212,286,71),(21,213,287,72),(22,214,288,73),(23,215,289,74),(24,216,290,75),(25,217,291,76),(26,218,292,77),(27,219,293,78),(28,220,294,79),(29,221,295,80),(30,222,296,81),(31,223,297,82),(32,224,298,83),(33,225,299,84),(34,226,300,85),(35,227,241,86),(36,228,242,87),(37,229,243,88),(38,230,244,89),(39,231,245,90),(40,232,246,91),(41,233,247,92),(42,234,248,93),(43,235,249,94),(44,236,250,95),(45,237,251,96),(46,238,252,97),(47,239,253,98),(48,240,254,99),(49,181,255,100),(50,182,256,101),(51,183,257,102),(52,184,258,103),(53,185,259,104),(54,186,260,105),(55,187,261,106),(56,188,262,107),(57,189,263,108),(58,190,264,109),(59,191,265,110),(60,192,266,111),(121,397,356,430),(122,398,357,431),(123,399,358,432),(124,400,359,433),(125,401,360,434),(126,402,301,435),(127,403,302,436),(128,404,303,437),(129,405,304,438),(130,406,305,439),(131,407,306,440),(132,408,307,441),(133,409,308,442),(134,410,309,443),(135,411,310,444),(136,412,311,445),(137,413,312,446),(138,414,313,447),(139,415,314,448),(140,416,315,449),(141,417,316,450),(142,418,317,451),(143,419,318,452),(144,420,319,453),(145,361,320,454),(146,362,321,455),(147,363,322,456),(148,364,323,457),(149,365,324,458),(150,366,325,459),(151,367,326,460),(152,368,327,461),(153,369,328,462),(154,370,329,463),(155,371,330,464),(156,372,331,465),(157,373,332,466),(158,374,333,467),(159,375,334,468),(160,376,335,469),(161,377,336,470),(162,378,337,471),(163,379,338,472),(164,380,339,473),(165,381,340,474),(166,382,341,475),(167,383,342,476),(168,384,343,477),(169,385,344,478),(170,386,345,479),(171,387,346,480),(172,388,347,421),(173,389,348,422),(174,390,349,423),(175,391,350,424),(176,392,351,425),(177,393,352,426),(178,394,353,427),(179,395,354,428),(180,396,355,429)]])
300 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 10A | ··· | 10L | 12A | ··· | 12H | 12I | ··· | 12AF | 15A | ··· | 15H | 20A | ··· | 20P | 20Q | ··· | 20BL | 30A | ··· | 30X | 60A | ··· | 60AF | 60AG | ··· | 60DX |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
300 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C30 | C60 | Q8 | C4○D4 | C3×Q8 | C5×Q8 | C3×C4○D4 | C5×C4○D4 | Q8×C15 | C15×C4○D4 |
| kernel | Q8×C60 | C4×C60 | C15×C4⋊C4 | Q8×C30 | Q8×C20 | Q8×C15 | Q8×C12 | C4×C20 | C5×C4⋊C4 | Q8×C10 | C4×C12 | C3×C4⋊C4 | C6×Q8 | C5×Q8 | C4×Q8 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C60 | C30 | C20 | C12 | C10 | C6 | C4 | C2 |
| # reps | 1 | 3 | 3 | 1 | 2 | 8 | 4 | 6 | 6 | 2 | 12 | 12 | 4 | 16 | 8 | 32 | 24 | 24 | 8 | 64 | 2 | 2 | 4 | 8 | 4 | 8 | 16 | 16 |
Matrix representation of Q8×C60 ►in GL3(𝔽61) generated by
| 50 | 0 | 0 |
| 0 | 44 | 0 |
| 0 | 0 | 44 |
| 60 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 60 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 11 |
| 0 | 11 | 0 |
G:=sub<GL(3,GF(61))| [50,0,0,0,44,0,0,0,44],[60,0,0,0,0,60,0,1,0],[1,0,0,0,0,11,0,11,0] >;
Q8×C60 in GAP, Magma, Sage, TeX
Q_8\times C_{60} % in TeX
G:=Group("Q8xC60"); // GroupNames label
G:=SmallGroup(480,924);
// by ID
G=gap.SmallGroup(480,924);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1680,1709,848,1906]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations