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G = C60.1Q8order 480 = 25·3·5

1st non-split extension by C60 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.1Q8, C30.37D8, C30.16Q16, C20.5Dic6, C4.1Dic30, C12.5Dic10, C153C87C4, C4⋊C4.1D15, C157(C2.D8), C20.45(C4×S3), C60.75(C2×C4), C12.13(C4×D5), (C2×C4).32D30, (C2×C20).64D6, C4.11(C4×D15), C54(C6.Q16), C30.36(C4⋊C4), C6.15(D4⋊D5), C2.1(D4⋊D15), (C2×C12).65D10, (C2×C30).135D4, C33(C10.D8), C605C4.10C2, C6.7(C5⋊Q16), C10.15(D4⋊S3), (C2×C60).50C22, C2.1(C157Q16), C10.7(C3⋊Q16), C2.3(C30.4Q8), C10.17(Dic3⋊C4), C6.10(C10.D4), C22.12(C157D4), (C5×C4⋊C4).1S3, (C3×C4⋊C4).1D5, (C15×C4⋊C4).1C2, (C2×C153C8).1C2, (C2×C6).67(C5⋊D4), (C2×C10).67(C3⋊D4), SmallGroup(480,167)

Series: Derived Chief Lower central Upper central

C1C60 — C60.1Q8
C1C5C15C30C2×C30C2×C60C2×C153C8 — C60.1Q8
C15C30C60 — C60.1Q8
C1C22C2×C4C4⋊C4

Generators and relations for C60.1Q8
 G = < a,b,c | a60=b4=1, c2=a45b2, bab-1=a31, cac-1=a29, cbc-1=a45b-1 >

Subgroups: 324 in 72 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3, C12 [×2], C12, C2×C6, C15, C4⋊C4, C4⋊C4, C2×C8, Dic5, C20 [×2], C20, C2×C10, C3⋊C8 [×2], C2×Dic3, C2×C12, C2×C12, C30 [×3], C2.D8, C52C8 [×2], C2×Dic5, C2×C20, C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, Dic15, C60 [×2], C60, C2×C30, C2×C52C8, C4⋊Dic5, C5×C4⋊C4, C6.Q16, C153C8 [×2], C2×Dic15, C2×C60, C2×C60, C10.D8, C2×C153C8, C605C4, C15×C4⋊C4, C60.1Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, D8, Q16, D10, Dic6, C4×S3, C3⋊D4, D15, C2.D8, Dic10, C4×D5, C5⋊D4, Dic3⋊C4, D4⋊S3, C3⋊Q16, D30, C10.D4, D4⋊D5, C5⋊Q16, C6.Q16, Dic30, C4×D15, C157D4, C10.D8, C30.4Q8, D4⋊D15, C157Q16, C60.1Q8

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

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222344444455666888810···1012···121515151520···2030···3060···60
size111122244606022222303030302···24···422224···42···24···4

84 irreducible representations

dim111112222222222222222222444444
type+++++-++++-+-+-+-+-+-+-
imageC1C2C2C2C4S3Q8D4D5D6D8Q16D10Dic6C4×S3C3⋊D4D15Dic10C4×D5C5⋊D4D30Dic30C4×D15C157D4D4⋊S3C3⋊Q16D4⋊D5C5⋊Q16D4⋊D15C157Q16
kernelC60.1Q8C2×C153C8C605C4C15×C4⋊C4C153C8C5×C4⋊C4C60C2×C30C3×C4⋊C4C2×C20C30C30C2×C12C20C20C2×C10C4⋊C4C12C12C2×C6C2×C4C4C4C22C10C10C6C6C2C2
# reps111141112122222244444888112244

Matrix representation of C60.1Q8 in GL4(𝔽241) generated by

8011000
13114700
002402
002401
,
64000
06400
0076138
00145165
,
2012000
1494000
0022219
00110
G:=sub<GL(4,GF(241))| [80,131,0,0,110,147,0,0,0,0,240,240,0,0,2,1],[64,0,0,0,0,64,0,0,0,0,76,145,0,0,138,165],[201,149,0,0,20,40,0,0,0,0,22,11,0,0,219,0] >;

C60.1Q8 in GAP, Magma, Sage, TeX

C_{60}._1Q_8
% in TeX

G:=Group("C60.1Q8");
// GroupNames label

G:=SmallGroup(480,167);
// by ID

G=gap.SmallGroup(480,167);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,36,346,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=1,c^2=a^45*b^2,b*a*b^-1=a^31,c*a*c^-1=a^29,c*b*c^-1=a^45*b^-1>;
// generators/relations

׿
×
𝔽