metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.26Q8, Dic15⋊3C8, C60.208D4, C4.8Dic30, C20.23Dic6, C12.23Dic10, C30.25M4(2), C15⋊13(C4⋊C8), C6.9(C8×D5), (C2×C40).3S3, C2.4(C8×D15), (C2×C24).3D5, (C2×C8).1D15, C10.18(S3×C8), (C2×C120).5C2, C30.46(C2×C8), C5⋊5(Dic3⋊C8), (C2×C4).92D30, C30.38(C4⋊C4), (C2×C20).406D6, C6.4(C8⋊D5), C3⋊3(C20.8Q8), C22.9(C4×D15), C2.1(C40⋊S3), C10.9(C8⋊S3), (C2×C12).410D10, C4.26(C15⋊7D4), (C4×Dic15).5C2, C20.105(C3⋊D4), C12.105(C5⋊D4), (C2×C60).492C22, (C2×Dic15).12C4, C2.1(C30.4Q8), C10.19(Dic3⋊C4), C6.12(C10.D4), (C2×C6).27(C4×D5), (C2×C15⋊3C8).9C2, (C2×C10).52(C4×S3), (C2×C30).134(C2×C4), SmallGroup(480,174)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.26Q8
G = < a,b,c | a60=1, b4=a30, c2=a15b2, ab=ba, cac-1=a29, cbc-1=a15b3 >
Subgroups: 308 in 76 conjugacy classes, 43 normal (41 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C2×C6, C15, C42, C2×C8, C2×C8, Dic5, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C4⋊C8, C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, Dic15, Dic15, C60, C2×C30, C2×C5⋊2C8, C4×Dic5, C2×C40, Dic3⋊C8, C15⋊3C8, C120, C2×Dic15, C2×C60, C20.8Q8, C2×C15⋊3C8, C4×Dic15, C2×C120, C60.26Q8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, D5, D6, C4⋊C4, C2×C8, M4(2), D10, Dic6, C4×S3, C3⋊D4, D15, C4⋊C8, Dic10, C4×D5, C5⋊D4, S3×C8, C8⋊S3, Dic3⋊C4, D30, C8×D5, C8⋊D5, C10.D4, Dic3⋊C8, Dic30, C4×D15, C15⋊7D4, C20.8Q8, C8×D15, C40⋊S3, C30.4Q8, C60.26Q8
►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)
(1 311 469 371 31 341 439 401)(2 312 470 372 32 342 440 402)(3 313 471 373 33 343 441 403)(4 314 472 374 34 344 442 404)(5 315 473 375 35 345 443 405)(6 316 474 376 36 346 444 406)(7 317 475 377 37 347 445 407)(8 318 476 378 38 348 446 408)(9 319 477 379 39 349 447 409)(10 320 478 380 40 350 448 410)(11 321 479 381 41 351 449 411)(12 322 480 382 42 352 450 412)(13 323 421 383 43 353 451 413)(14 324 422 384 44 354 452 414)(15 325 423 385 45 355 453 415)(16 326 424 386 46 356 454 416)(17 327 425 387 47 357 455 417)(18 328 426 388 48 358 456 418)(19 329 427 389 49 359 457 419)(20 330 428 390 50 360 458 420)(21 331 429 391 51 301 459 361)(22 332 430 392 52 302 460 362)(23 333 431 393 53 303 461 363)(24 334 432 394 54 304 462 364)(25 335 433 395 55 305 463 365)(26 336 434 396 56 306 464 366)(27 337 435 397 57 307 465 367)(28 338 436 398 58 308 466 368)(29 339 437 399 59 309 467 369)(30 340 438 400 60 310 468 370)(61 287 163 191 91 257 133 221)(62 288 164 192 92 258 134 222)(63 289 165 193 93 259 135 223)(64 290 166 194 94 260 136 224)(65 291 167 195 95 261 137 225)(66 292 168 196 96 262 138 226)(67 293 169 197 97 263 139 227)(68 294 170 198 98 264 140 228)(69 295 171 199 99 265 141 229)(70 296 172 200 100 266 142 230)(71 297 173 201 101 267 143 231)(72 298 174 202 102 268 144 232)(73 299 175 203 103 269 145 233)(74 300 176 204 104 270 146 234)(75 241 177 205 105 271 147 235)(76 242 178 206 106 272 148 236)(77 243 179 207 107 273 149 237)(78 244 180 208 108 274 150 238)(79 245 121 209 109 275 151 239)(80 246 122 210 110 276 152 240)(81 247 123 211 111 277 153 181)(82 248 124 212 112 278 154 182)(83 249 125 213 113 279 155 183)(84 250 126 214 114 280 156 184)(85 251 127 215 115 281 157 185)(86 252 128 216 116 282 158 186)(87 253 129 217 117 283 159 187)(88 254 130 218 118 284 160 188)(89 255 131 219 119 285 161 189)(90 256 132 220 120 286 162 190)
(1 280 424 199)(2 249 425 228)(3 278 426 197)(4 247 427 226)(5 276 428 195)(6 245 429 224)(7 274 430 193)(8 243 431 222)(9 272 432 191)(10 241 433 220)(11 270 434 189)(12 299 435 218)(13 268 436 187)(14 297 437 216)(15 266 438 185)(16 295 439 214)(17 264 440 183)(18 293 441 212)(19 262 442 181)(20 291 443 210)(21 260 444 239)(22 289 445 208)(23 258 446 237)(24 287 447 206)(25 256 448 235)(26 285 449 204)(27 254 450 233)(28 283 451 202)(29 252 452 231)(30 281 453 200)(31 250 454 229)(32 279 455 198)(33 248 456 227)(34 277 457 196)(35 246 458 225)(36 275 459 194)(37 244 460 223)(38 273 461 192)(39 242 462 221)(40 271 463 190)(41 300 464 219)(42 269 465 188)(43 298 466 217)(44 267 467 186)(45 296 468 215)(46 265 469 184)(47 294 470 213)(48 263 471 182)(49 292 472 211)(50 261 473 240)(51 290 474 209)(52 259 475 238)(53 288 476 207)(54 257 477 236)(55 286 478 205)(56 255 479 234)(57 284 480 203)(58 253 421 232)(59 282 422 201)(60 251 423 230)(61 364 178 349)(62 393 179 318)(63 362 180 347)(64 391 121 316)(65 420 122 345)(66 389 123 314)(67 418 124 343)(68 387 125 312)(69 416 126 341)(70 385 127 310)(71 414 128 339)(72 383 129 308)(73 412 130 337)(74 381 131 306)(75 410 132 335)(76 379 133 304)(77 408 134 333)(78 377 135 302)(79 406 136 331)(80 375 137 360)(81 404 138 329)(82 373 139 358)(83 402 140 327)(84 371 141 356)(85 400 142 325)(86 369 143 354)(87 398 144 323)(88 367 145 352)(89 396 146 321)(90 365 147 350)(91 394 148 319)(92 363 149 348)(93 392 150 317)(94 361 151 346)(95 390 152 315)(96 419 153 344)(97 388 154 313)(98 417 155 342)(99 386 156 311)(100 415 157 340)(101 384 158 309)(102 413 159 338)(103 382 160 307)(104 411 161 336)(105 380 162 305)(106 409 163 334)(107 378 164 303)(108 407 165 332)(109 376 166 301)(110 405 167 330)(111 374 168 359)(112 403 169 328)(113 372 170 357)(114 401 171 326)(115 370 172 355)(116 399 173 324)(117 368 174 353)(118 397 175 322)(119 366 176 351)(120 395 177 320)
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), (1,311,469,371,31,341,439,401)(2,312,470,372,32,342,440,402)(3,313,471,373,33,343,441,403)(4,314,472,374,34,344,442,404)(5,315,473,375,35,345,443,405)(6,316,474,376,36,346,444,406)(7,317,475,377,37,347,445,407)(8,318,476,378,38,348,446,408)(9,319,477,379,39,349,447,409)(10,320,478,380,40,350,448,410)(11,321,479,381,41,351,449,411)(12,322,480,382,42,352,450,412)(13,323,421,383,43,353,451,413)(14,324,422,384,44,354,452,414)(15,325,423,385,45,355,453,415)(16,326,424,386,46,356,454,416)(17,327,425,387,47,357,455,417)(18,328,426,388,48,358,456,418)(19,329,427,389,49,359,457,419)(20,330,428,390,50,360,458,420)(21,331,429,391,51,301,459,361)(22,332,430,392,52,302,460,362)(23,333,431,393,53,303,461,363)(24,334,432,394,54,304,462,364)(25,335,433,395,55,305,463,365)(26,336,434,396,56,306,464,366)(27,337,435,397,57,307,465,367)(28,338,436,398,58,308,466,368)(29,339,437,399,59,309,467,369)(30,340,438,400,60,310,468,370)(61,287,163,191,91,257,133,221)(62,288,164,192,92,258,134,222)(63,289,165,193,93,259,135,223)(64,290,166,194,94,260,136,224)(65,291,167,195,95,261,137,225)(66,292,168,196,96,262,138,226)(67,293,169,197,97,263,139,227)(68,294,170,198,98,264,140,228)(69,295,171,199,99,265,141,229)(70,296,172,200,100,266,142,230)(71,297,173,201,101,267,143,231)(72,298,174,202,102,268,144,232)(73,299,175,203,103,269,145,233)(74,300,176,204,104,270,146,234)(75,241,177,205,105,271,147,235)(76,242,178,206,106,272,148,236)(77,243,179,207,107,273,149,237)(78,244,180,208,108,274,150,238)(79,245,121,209,109,275,151,239)(80,246,122,210,110,276,152,240)(81,247,123,211,111,277,153,181)(82,248,124,212,112,278,154,182)(83,249,125,213,113,279,155,183)(84,250,126,214,114,280,156,184)(85,251,127,215,115,281,157,185)(86,252,128,216,116,282,158,186)(87,253,129,217,117,283,159,187)(88,254,130,218,118,284,160,188)(89,255,131,219,119,285,161,189)(90,256,132,220,120,286,162,190), (1,280,424,199)(2,249,425,228)(3,278,426,197)(4,247,427,226)(5,276,428,195)(6,245,429,224)(7,274,430,193)(8,243,431,222)(9,272,432,191)(10,241,433,220)(11,270,434,189)(12,299,435,218)(13,268,436,187)(14,297,437,216)(15,266,438,185)(16,295,439,214)(17,264,440,183)(18,293,441,212)(19,262,442,181)(20,291,443,210)(21,260,444,239)(22,289,445,208)(23,258,446,237)(24,287,447,206)(25,256,448,235)(26,285,449,204)(27,254,450,233)(28,283,451,202)(29,252,452,231)(30,281,453,200)(31,250,454,229)(32,279,455,198)(33,248,456,227)(34,277,457,196)(35,246,458,225)(36,275,459,194)(37,244,460,223)(38,273,461,192)(39,242,462,221)(40,271,463,190)(41,300,464,219)(42,269,465,188)(43,298,466,217)(44,267,467,186)(45,296,468,215)(46,265,469,184)(47,294,470,213)(48,263,471,182)(49,292,472,211)(50,261,473,240)(51,290,474,209)(52,259,475,238)(53,288,476,207)(54,257,477,236)(55,286,478,205)(56,255,479,234)(57,284,480,203)(58,253,421,232)(59,282,422,201)(60,251,423,230)(61,364,178,349)(62,393,179,318)(63,362,180,347)(64,391,121,316)(65,420,122,345)(66,389,123,314)(67,418,124,343)(68,387,125,312)(69,416,126,341)(70,385,127,310)(71,414,128,339)(72,383,129,308)(73,412,130,337)(74,381,131,306)(75,410,132,335)(76,379,133,304)(77,408,134,333)(78,377,135,302)(79,406,136,331)(80,375,137,360)(81,404,138,329)(82,373,139,358)(83,402,140,327)(84,371,141,356)(85,400,142,325)(86,369,143,354)(87,398,144,323)(88,367,145,352)(89,396,146,321)(90,365,147,350)(91,394,148,319)(92,363,149,348)(93,392,150,317)(94,361,151,346)(95,390,152,315)(96,419,153,344)(97,388,154,313)(98,417,155,342)(99,386,156,311)(100,415,157,340)(101,384,158,309)(102,413,159,338)(103,382,160,307)(104,411,161,336)(105,380,162,305)(106,409,163,334)(107,378,164,303)(108,407,165,332)(109,376,166,301)(110,405,167,330)(111,374,168,359)(112,403,169,328)(113,372,170,357)(114,401,171,326)(115,370,172,355)(116,399,173,324)(117,368,174,353)(118,397,175,322)(119,366,176,351)(120,395,177,320)>;
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), (1,311,469,371,31,341,439,401)(2,312,470,372,32,342,440,402)(3,313,471,373,33,343,441,403)(4,314,472,374,34,344,442,404)(5,315,473,375,35,345,443,405)(6,316,474,376,36,346,444,406)(7,317,475,377,37,347,445,407)(8,318,476,378,38,348,446,408)(9,319,477,379,39,349,447,409)(10,320,478,380,40,350,448,410)(11,321,479,381,41,351,449,411)(12,322,480,382,42,352,450,412)(13,323,421,383,43,353,451,413)(14,324,422,384,44,354,452,414)(15,325,423,385,45,355,453,415)(16,326,424,386,46,356,454,416)(17,327,425,387,47,357,455,417)(18,328,426,388,48,358,456,418)(19,329,427,389,49,359,457,419)(20,330,428,390,50,360,458,420)(21,331,429,391,51,301,459,361)(22,332,430,392,52,302,460,362)(23,333,431,393,53,303,461,363)(24,334,432,394,54,304,462,364)(25,335,433,395,55,305,463,365)(26,336,434,396,56,306,464,366)(27,337,435,397,57,307,465,367)(28,338,436,398,58,308,466,368)(29,339,437,399,59,309,467,369)(30,340,438,400,60,310,468,370)(61,287,163,191,91,257,133,221)(62,288,164,192,92,258,134,222)(63,289,165,193,93,259,135,223)(64,290,166,194,94,260,136,224)(65,291,167,195,95,261,137,225)(66,292,168,196,96,262,138,226)(67,293,169,197,97,263,139,227)(68,294,170,198,98,264,140,228)(69,295,171,199,99,265,141,229)(70,296,172,200,100,266,142,230)(71,297,173,201,101,267,143,231)(72,298,174,202,102,268,144,232)(73,299,175,203,103,269,145,233)(74,300,176,204,104,270,146,234)(75,241,177,205,105,271,147,235)(76,242,178,206,106,272,148,236)(77,243,179,207,107,273,149,237)(78,244,180,208,108,274,150,238)(79,245,121,209,109,275,151,239)(80,246,122,210,110,276,152,240)(81,247,123,211,111,277,153,181)(82,248,124,212,112,278,154,182)(83,249,125,213,113,279,155,183)(84,250,126,214,114,280,156,184)(85,251,127,215,115,281,157,185)(86,252,128,216,116,282,158,186)(87,253,129,217,117,283,159,187)(88,254,130,218,118,284,160,188)(89,255,131,219,119,285,161,189)(90,256,132,220,120,286,162,190), (1,280,424,199)(2,249,425,228)(3,278,426,197)(4,247,427,226)(5,276,428,195)(6,245,429,224)(7,274,430,193)(8,243,431,222)(9,272,432,191)(10,241,433,220)(11,270,434,189)(12,299,435,218)(13,268,436,187)(14,297,437,216)(15,266,438,185)(16,295,439,214)(17,264,440,183)(18,293,441,212)(19,262,442,181)(20,291,443,210)(21,260,444,239)(22,289,445,208)(23,258,446,237)(24,287,447,206)(25,256,448,235)(26,285,449,204)(27,254,450,233)(28,283,451,202)(29,252,452,231)(30,281,453,200)(31,250,454,229)(32,279,455,198)(33,248,456,227)(34,277,457,196)(35,246,458,225)(36,275,459,194)(37,244,460,223)(38,273,461,192)(39,242,462,221)(40,271,463,190)(41,300,464,219)(42,269,465,188)(43,298,466,217)(44,267,467,186)(45,296,468,215)(46,265,469,184)(47,294,470,213)(48,263,471,182)(49,292,472,211)(50,261,473,240)(51,290,474,209)(52,259,475,238)(53,288,476,207)(54,257,477,236)(55,286,478,205)(56,255,479,234)(57,284,480,203)(58,253,421,232)(59,282,422,201)(60,251,423,230)(61,364,178,349)(62,393,179,318)(63,362,180,347)(64,391,121,316)(65,420,122,345)(66,389,123,314)(67,418,124,343)(68,387,125,312)(69,416,126,341)(70,385,127,310)(71,414,128,339)(72,383,129,308)(73,412,130,337)(74,381,131,306)(75,410,132,335)(76,379,133,304)(77,408,134,333)(78,377,135,302)(79,406,136,331)(80,375,137,360)(81,404,138,329)(82,373,139,358)(83,402,140,327)(84,371,141,356)(85,400,142,325)(86,369,143,354)(87,398,144,323)(88,367,145,352)(89,396,146,321)(90,365,147,350)(91,394,148,319)(92,363,149,348)(93,392,150,317)(94,361,151,346)(95,390,152,315)(96,419,153,344)(97,388,154,313)(98,417,155,342)(99,386,156,311)(100,415,157,340)(101,384,158,309)(102,413,159,338)(103,382,160,307)(104,411,161,336)(105,380,162,305)(106,409,163,334)(107,378,164,303)(108,407,165,332)(109,376,166,301)(110,405,167,330)(111,374,168,359)(112,403,169,328)(113,372,170,357)(114,401,171,326)(115,370,172,355)(116,399,173,324)(117,368,174,353)(118,397,175,322)(119,366,176,351)(120,395,177,320) );
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)], [(1,311,469,371,31,341,439,401),(2,312,470,372,32,342,440,402),(3,313,471,373,33,343,441,403),(4,314,472,374,34,344,442,404),(5,315,473,375,35,345,443,405),(6,316,474,376,36,346,444,406),(7,317,475,377,37,347,445,407),(8,318,476,378,38,348,446,408),(9,319,477,379,39,349,447,409),(10,320,478,380,40,350,448,410),(11,321,479,381,41,351,449,411),(12,322,480,382,42,352,450,412),(13,323,421,383,43,353,451,413),(14,324,422,384,44,354,452,414),(15,325,423,385,45,355,453,415),(16,326,424,386,46,356,454,416),(17,327,425,387,47,357,455,417),(18,328,426,388,48,358,456,418),(19,329,427,389,49,359,457,419),(20,330,428,390,50,360,458,420),(21,331,429,391,51,301,459,361),(22,332,430,392,52,302,460,362),(23,333,431,393,53,303,461,363),(24,334,432,394,54,304,462,364),(25,335,433,395,55,305,463,365),(26,336,434,396,56,306,464,366),(27,337,435,397,57,307,465,367),(28,338,436,398,58,308,466,368),(29,339,437,399,59,309,467,369),(30,340,438,400,60,310,468,370),(61,287,163,191,91,257,133,221),(62,288,164,192,92,258,134,222),(63,289,165,193,93,259,135,223),(64,290,166,194,94,260,136,224),(65,291,167,195,95,261,137,225),(66,292,168,196,96,262,138,226),(67,293,169,197,97,263,139,227),(68,294,170,198,98,264,140,228),(69,295,171,199,99,265,141,229),(70,296,172,200,100,266,142,230),(71,297,173,201,101,267,143,231),(72,298,174,202,102,268,144,232),(73,299,175,203,103,269,145,233),(74,300,176,204,104,270,146,234),(75,241,177,205,105,271,147,235),(76,242,178,206,106,272,148,236),(77,243,179,207,107,273,149,237),(78,244,180,208,108,274,150,238),(79,245,121,209,109,275,151,239),(80,246,122,210,110,276,152,240),(81,247,123,211,111,277,153,181),(82,248,124,212,112,278,154,182),(83,249,125,213,113,279,155,183),(84,250,126,214,114,280,156,184),(85,251,127,215,115,281,157,185),(86,252,128,216,116,282,158,186),(87,253,129,217,117,283,159,187),(88,254,130,218,118,284,160,188),(89,255,131,219,119,285,161,189),(90,256,132,220,120,286,162,190)], [(1,280,424,199),(2,249,425,228),(3,278,426,197),(4,247,427,226),(5,276,428,195),(6,245,429,224),(7,274,430,193),(8,243,431,222),(9,272,432,191),(10,241,433,220),(11,270,434,189),(12,299,435,218),(13,268,436,187),(14,297,437,216),(15,266,438,185),(16,295,439,214),(17,264,440,183),(18,293,441,212),(19,262,442,181),(20,291,443,210),(21,260,444,239),(22,289,445,208),(23,258,446,237),(24,287,447,206),(25,256,448,235),(26,285,449,204),(27,254,450,233),(28,283,451,202),(29,252,452,231),(30,281,453,200),(31,250,454,229),(32,279,455,198),(33,248,456,227),(34,277,457,196),(35,246,458,225),(36,275,459,194),(37,244,460,223),(38,273,461,192),(39,242,462,221),(40,271,463,190),(41,300,464,219),(42,269,465,188),(43,298,466,217),(44,267,467,186),(45,296,468,215),(46,265,469,184),(47,294,470,213),(48,263,471,182),(49,292,472,211),(50,261,473,240),(51,290,474,209),(52,259,475,238),(53,288,476,207),(54,257,477,236),(55,286,478,205),(56,255,479,234),(57,284,480,203),(58,253,421,232),(59,282,422,201),(60,251,423,230),(61,364,178,349),(62,393,179,318),(63,362,180,347),(64,391,121,316),(65,420,122,345),(66,389,123,314),(67,418,124,343),(68,387,125,312),(69,416,126,341),(70,385,127,310),(71,414,128,339),(72,383,129,308),(73,412,130,337),(74,381,131,306),(75,410,132,335),(76,379,133,304),(77,408,134,333),(78,377,135,302),(79,406,136,331),(80,375,137,360),(81,404,138,329),(82,373,139,358),(83,402,140,327),(84,371,141,356),(85,400,142,325),(86,369,143,354),(87,398,144,323),(88,367,145,352),(89,396,146,321),(90,365,147,350),(91,394,148,319),(92,363,149,348),(93,392,150,317),(94,361,151,346),(95,390,152,315),(96,419,153,344),(97,388,154,313),(98,417,155,342),(99,386,156,311),(100,415,157,340),(101,384,158,309),(102,413,159,338),(103,382,160,307),(104,411,161,336),(105,380,162,305),(106,409,163,334),(107,378,164,303),(108,407,165,332),(109,376,166,301),(110,405,167,330),(111,374,168,359),(112,403,169,328),(113,372,170,357),(114,401,171,326),(115,370,172,355),(116,399,173,324),(117,368,174,353),(118,397,175,322),(119,366,176,351),(120,395,177,320)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | S3 | D4 | Q8 | D5 | D6 | M4(2) | D10 | Dic6 | C3⋊D4 | C4×S3 | D15 | Dic10 | C5⋊D4 | C4×D5 | S3×C8 | C8⋊S3 | D30 | C8×D5 | C8⋊D5 | Dic30 | C15⋊7D4 | C4×D15 | C8×D15 | C40⋊S3 |
| kernel | C60.26Q8 | C2×C15⋊3C8 | C4×Dic15 | C2×C120 | C2×Dic15 | Dic15 | C2×C40 | C60 | C60 | C2×C24 | C2×C20 | C30 | C2×C12 | C20 | C20 | C2×C10 | C2×C8 | C12 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of C60.26Q8 ►in GL3(𝔽241) generated by
| 64 | 0 | 0 |
| 0 | 14 | 181 |
| 0 | 60 | 1 |
| 233 | 0 | 0 |
| 0 | 25 | 140 |
| 0 | 101 | 216 |
| 1 | 0 | 0 |
| 0 | 35 | 2 |
| 0 | 110 | 206 |
G:=sub<GL(3,GF(241))| [64,0,0,0,14,60,0,181,1],[233,0,0,0,25,101,0,140,216],[1,0,0,0,35,110,0,2,206] >;
C60.26Q8 in GAP, Magma, Sage, TeX
C_{60}._{26}Q_8 % in TeX
G:=Group("C60.26Q8"); // GroupNames label
G:=SmallGroup(480,174);
// by ID
G=gap.SmallGroup(480,174);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,36,100,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^15*b^2,a*b=b*a,c*a*c^-1=a^29,c*b*c^-1=a^15*b^3>;
// generators/relations