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G = C60.14Q8order 480 = 25·3·5

14th non-split extension by C60 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.14Q8, Dic155C8, C60.103D4, C20.14Dic6, C12.14Dic10, C30.18M4(2), C158(C4⋊C8), C6.5(C8×D5), C10.14(S3×C8), C30.32(C2×C8), C53(Dic3⋊C8), C4.11(C15⋊Q8), C30.23(C4⋊C4), (C2×C20).325D6, C6.3(C8⋊D5), C31(C20.8Q8), C10.8(C8⋊S3), (C2×C12).329D10, C20.88(C3⋊D4), C12.88(C5⋊D4), C4.30(C15⋊D4), C2.5(D152C8), C10.8(Dic3⋊C4), (C2×C60).227C22, (C4×Dic15).19C2, (C2×Dic15).20C4, C6.2(C10.D4), C2.1(Dic155C4), C2.3(D30.5C4), C22.11(D30.C2), (C2×C3⋊C8).10D5, (C10×C3⋊C8).12C2, (C2×C52C8).9S3, (C2×C6).19(C4×D5), (C6×C52C8).11C2, (C2×C10).42(C4×S3), (C2×C30).87(C2×C4), (C2×C4).230(S3×D5), SmallGroup(480,59)

Series: Derived Chief Lower central Upper central

C1C30 — C60.14Q8
C1C5C15C30C60C2×C60C6×C52C8 — C60.14Q8
C15C30 — C60.14Q8
C1C2×C4

Generators and relations for C60.14Q8
 G = < a,b,c | a60=1, b4=a30, c2=b2, bab-1=a41, cac-1=a49, cbc-1=a15b3 >

Subgroups: 284 in 76 conjugacy classes, 42 normal (40 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3 [×3], C12 [×2], C2×C6, C15, C42, C2×C8 [×2], Dic5 [×3], C20 [×2], C2×C10, C3⋊C8, C24, C2×Dic3 [×2], C2×C12, C30 [×3], C4⋊C8, C52C8, C40, C2×Dic5 [×2], C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, Dic15 [×2], Dic15, C60 [×2], C2×C30, C2×C52C8, C4×Dic5, C2×C40, Dic3⋊C8, C5×C3⋊C8, C3×C52C8, C2×Dic15 [×2], C2×C60, C20.8Q8, C6×C52C8, C10×C3⋊C8, C4×Dic15, C60.14Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4, Q8, D5, D6, C4⋊C4, C2×C8, M4(2), D10, Dic6, C4×S3, C3⋊D4, C4⋊C8, Dic10, C4×D5, C5⋊D4, S3×C8, C8⋊S3, Dic3⋊C4, S3×D5, C8×D5, C8⋊D5, C10.D4, Dic3⋊C8, D30.C2, C15⋊D4, C15⋊Q8, C20.8Q8, D152C8, D30.5C4, Dic155C4, C60.14Q8

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

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B6C8A8B8C8D8E8F8G8H10A···10F12A12B12C12D15A15B20A···20H24A···24H30A···30F40A···40P60A···60H
order1222344444444556668888888810···1012121212151520···2024···2430···3040···4060···60
size11112111130303030222226666101010102···22222442···210···104···46···64···4

84 irreducible representations

dim11111122222222222222222444444
type++++++-+++--+--+
imageC1C2C2C2C4C8S3D4Q8D5D6M4(2)D10Dic6C3⋊D4C4×S3Dic10C5⋊D4C4×D5S3×C8C8⋊S3C8×D5C8⋊D5S3×D5C15⋊D4C15⋊Q8D30.C2D152C8D30.5C4
kernelC60.14Q8C6×C52C8C10×C3⋊C8C4×Dic15C2×Dic15Dic15C2×C52C8C60C60C2×C3⋊C8C2×C20C30C2×C12C20C20C2×C10C12C12C2×C6C10C10C6C6C2×C4C4C4C22C2C2
# reps11114811121222224444488222244

Matrix representation of C60.14Q8 in GL4(𝔽241) generated by

525200
18924000
0017764
001770
,
418500
15620000
0012841
00169113
,
1112400
3423000
00172138
0010369
G:=sub<GL(4,GF(241))| [52,189,0,0,52,240,0,0,0,0,177,177,0,0,64,0],[41,156,0,0,85,200,0,0,0,0,128,169,0,0,41,113],[11,34,0,0,124,230,0,0,0,0,172,103,0,0,138,69] >;

C60.14Q8 in GAP, Magma, Sage, TeX

C_{60}._{14}Q_8
% in TeX

G:=Group("C60.14Q8");
// GroupNames label

G:=SmallGroup(480,59);
// by ID

G=gap.SmallGroup(480,59);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,36,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=b^2,b*a*b^-1=a^41,c*a*c^-1=a^49,c*b*c^-1=a^15*b^3>;
// generators/relations

׿
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