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G = C60.13Q8order 480 = 25·3·5

13rd non-split extension by C60 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.13Q8, C20.58D12, C60.102D4, C20.13Dic6, C12.13Dic10, C30.17M4(2), C157(C4⋊C8), C6.15(C8×D5), C53(C12⋊C8), C30.31(C2×C8), Dic51(C3⋊C8), (C3×Dic5)⋊2C8, C4.10(C15⋊Q8), C30.22(C4⋊C4), (C2×C20).324D6, C6.9(C8⋊D5), (C4×Dic5).7S3, (C6×Dic5).8C4, C32(C20.8Q8), (C2×C12).328D10, C12.66(C5⋊D4), C4.30(C5⋊D12), C10.8(C4⋊Dic3), (C12×Dic5).8C2, (C2×C60).226C22, (C2×Dic5).5Dic3, C6.1(C10.D4), C10.8(C4.Dic3), C2.1(C30.Q8), C22.11(D5×Dic3), C2.3(C20.32D6), C2.5(D5×C3⋊C8), (C2×C3⋊C8).9D5, C10.14(C2×C3⋊C8), (C10×C3⋊C8).11C2, (C2×C6).47(C4×D5), (C2×C30).86(C2×C4), (C2×C4).229(S3×D5), (C2×C153C8).21C2, (C2×C10).32(C2×Dic3), SmallGroup(480,58)

Series: Derived Chief Lower central Upper central

C1C30 — C60.13Q8
C1C5C15C30C60C2×C60C12×Dic5 — C60.13Q8
C15C30 — C60.13Q8
C1C2×C4

Generators and relations for C60.13Q8
 G = < a,b,c | a60=1, b4=a30, c2=a15b2, bab-1=a41, cac-1=a49, cbc-1=a15b3 >

Subgroups: 236 in 76 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], C12 [×2], C12 [×3], C2×C6, C15, C42, C2×C8 [×2], Dic5 [×2], Dic5, C20 [×2], C2×C10, C3⋊C8 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4⋊C8, C52C8, C40, C2×Dic5 [×2], C2×C20, C2×C3⋊C8, C2×C3⋊C8, C4×C12, C3×Dic5 [×2], C3×Dic5, C60 [×2], C2×C30, C2×C52C8, C4×Dic5, C2×C40, C12⋊C8, C5×C3⋊C8, C153C8, C6×Dic5 [×2], C2×C60, C20.8Q8, C12×Dic5, C10×C3⋊C8, C2×C153C8, C60.13Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4, Q8, D5, Dic3 [×2], D6, C4⋊C4, C2×C8, M4(2), D10, C3⋊C8 [×2], Dic6, D12, C2×Dic3, C4⋊C8, Dic10, C4×D5, C5⋊D4, C2×C3⋊C8, C4.Dic3, C4⋊Dic3, S3×D5, C8×D5, C8⋊D5, C10.D4, C12⋊C8, D5×Dic3, C5⋊D12, C15⋊Q8, C20.8Q8, D5×C3⋊C8, C20.32D6, C30.Q8, C60.13Q8

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

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B6C8A8B8C8D8E8F8G8H10A···10F12A12B12C12D12E···12L15A15B20A···20H30A···30F40A···40P60A···60H
order1222344444444556668888888810···101212121212···12151520···2030···3040···4060···60
size11112111110101010222226666303030302···2222210···10442···24···46···64···4

84 irreducible representations

dim11111122222222222222222444444
type++++++-+-++-+-++--
imageC1C2C2C2C4C8S3D4Q8D5Dic3D6M4(2)D10C3⋊C8Dic6D12Dic10C5⋊D4C4×D5C4.Dic3C8×D5C8⋊D5S3×D5C5⋊D12C15⋊Q8D5×Dic3D5×C3⋊C8C20.32D6
kernelC60.13Q8C12×Dic5C10×C3⋊C8C2×C153C8C6×Dic5C3×Dic5C4×Dic5C60C60C2×C3⋊C8C2×Dic5C2×C20C30C2×C12Dic5C20C20C12C12C2×C6C10C6C6C2×C4C4C4C22C2C2
# reps11114811122122422444488222244

Matrix representation of C60.13Q8 in GL4(𝔽241) generated by

644600
1954600
001240
0010
,
30000
03000
002669
0095215
,
1942000
2274700
009943
00198142
G:=sub<GL(4,GF(241))| [64,195,0,0,46,46,0,0,0,0,1,1,0,0,240,0],[30,0,0,0,0,30,0,0,0,0,26,95,0,0,69,215],[194,227,0,0,20,47,0,0,0,0,99,198,0,0,43,142] >;

C60.13Q8 in GAP, Magma, Sage, TeX

C_{60}._{13}Q_8
% in TeX

G:=Group("C60.13Q8");
// GroupNames label

G:=SmallGroup(480,58);
// by ID

G=gap.SmallGroup(480,58);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,36,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^15*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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