Copied to
clipboard

## G = C60.48D4order 480 = 25·3·5

### 48th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.48D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C6.Dic10 — C60.48D4
 Lower central C15 — C2×C30 — C60.48D4
 Upper central C1 — C22 — C2×C4

Generators and relations for C60.48D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=a41, cac-1=a-1, cbc-1=b-1 >

Subgroups: 604 in 136 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×8], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×6], Q8 [×4], C10, C10 [×2], Dic3 [×4], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×4], C20 [×2], C20 [×4], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C30, C30 [×2], C4⋊Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3, Dic3⋊C4 [×4], C2×Dic6, C6×Q8, C5×Dic3 [×4], C3×Dic5 [×2], Dic15 [×2], C60 [×2], C2×C30, C4⋊Dic5 [×4], C4×C20, C2×Dic10, C2×Dic10, Dic3⋊Q8, C3×Dic10 [×2], C6×Dic5 [×2], C10×Dic3 [×2], Dic30 [×2], C2×Dic15 [×2], C2×C60, C202Q8, C6.Dic10 [×4], Dic3×C20, C6×Dic10, C2×Dic30, C60.48D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D5, D6 [×3], C2×D4, C2×Q8 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, C4⋊Q8, Dic10 [×4], D20 [×2], C22×D5, S3×Q8 [×2], C2×C3⋊D4, S3×D5, C2×Dic10 [×2], C2×D20, Dic3⋊Q8, C3⋊D20 [×2], C2×S3×D5, C202Q8, S3×Dic10 [×2], C2×C3⋊D20, C60.48D4

Smallest permutation representation of C60.48D4
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)
(1 171 204 105)(2 152 205 86)(3 133 206 67)(4 174 207 108)(5 155 208 89)(6 136 209 70)(7 177 210 111)(8 158 211 92)(9 139 212 73)(10 180 213 114)(11 161 214 95)(12 142 215 76)(13 123 216 117)(14 164 217 98)(15 145 218 79)(16 126 219 120)(17 167 220 101)(18 148 221 82)(19 129 222 63)(20 170 223 104)(21 151 224 85)(22 132 225 66)(23 173 226 107)(24 154 227 88)(25 135 228 69)(26 176 229 110)(27 157 230 91)(28 138 231 72)(29 179 232 113)(30 160 233 94)(31 141 234 75)(32 122 235 116)(33 163 236 97)(34 144 237 78)(35 125 238 119)(36 166 239 100)(37 147 240 81)(38 128 181 62)(39 169 182 103)(40 150 183 84)(41 131 184 65)(42 172 185 106)(43 153 186 87)(44 134 187 68)(45 175 188 109)(46 156 189 90)(47 137 190 71)(48 178 191 112)(49 159 192 93)(50 140 193 74)(51 121 194 115)(52 162 195 96)(53 143 196 77)(54 124 197 118)(55 165 198 99)(56 146 199 80)(57 127 200 61)(58 168 201 102)(59 149 202 83)(60 130 203 64)(241 395 323 436)(242 376 324 477)(243 417 325 458)(244 398 326 439)(245 379 327 480)(246 420 328 461)(247 401 329 442)(248 382 330 423)(249 363 331 464)(250 404 332 445)(251 385 333 426)(252 366 334 467)(253 407 335 448)(254 388 336 429)(255 369 337 470)(256 410 338 451)(257 391 339 432)(258 372 340 473)(259 413 341 454)(260 394 342 435)(261 375 343 476)(262 416 344 457)(263 397 345 438)(264 378 346 479)(265 419 347 460)(266 400 348 441)(267 381 349 422)(268 362 350 463)(269 403 351 444)(270 384 352 425)(271 365 353 466)(272 406 354 447)(273 387 355 428)(274 368 356 469)(275 409 357 450)(276 390 358 431)(277 371 359 472)(278 412 360 453)(279 393 301 434)(280 374 302 475)(281 415 303 456)(282 396 304 437)(283 377 305 478)(284 418 306 459)(285 399 307 440)(286 380 308 421)(287 361 309 462)(288 402 310 443)(289 383 311 424)(290 364 312 465)(291 405 313 446)(292 386 314 427)(293 367 315 468)(294 408 316 449)(295 389 317 430)(296 370 318 471)(297 411 319 452)(298 392 320 433)(299 373 321 474)(300 414 322 455)
(1 449 31 479)(2 448 32 478)(3 447 33 477)(4 446 34 476)(5 445 35 475)(6 444 36 474)(7 443 37 473)(8 442 38 472)(9 441 39 471)(10 440 40 470)(11 439 41 469)(12 438 42 468)(13 437 43 467)(14 436 44 466)(15 435 45 465)(16 434 46 464)(17 433 47 463)(18 432 48 462)(19 431 49 461)(20 430 50 460)(21 429 51 459)(22 428 52 458)(23 427 53 457)(24 426 54 456)(25 425 55 455)(26 424 56 454)(27 423 57 453)(28 422 58 452)(29 421 59 451)(30 480 60 450)(61 278 91 248)(62 277 92 247)(63 276 93 246)(64 275 94 245)(65 274 95 244)(66 273 96 243)(67 272 97 242)(68 271 98 241)(69 270 99 300)(70 269 100 299)(71 268 101 298)(72 267 102 297)(73 266 103 296)(74 265 104 295)(75 264 105 294)(76 263 106 293)(77 262 107 292)(78 261 108 291)(79 260 109 290)(80 259 110 289)(81 258 111 288)(82 257 112 287)(83 256 113 286)(84 255 114 285)(85 254 115 284)(86 253 116 283)(87 252 117 282)(88 251 118 281)(89 250 119 280)(90 249 120 279)(121 306 151 336)(122 305 152 335)(123 304 153 334)(124 303 154 333)(125 302 155 332)(126 301 156 331)(127 360 157 330)(128 359 158 329)(129 358 159 328)(130 357 160 327)(131 356 161 326)(132 355 162 325)(133 354 163 324)(134 353 164 323)(135 352 165 322)(136 351 166 321)(137 350 167 320)(138 349 168 319)(139 348 169 318)(140 347 170 317)(141 346 171 316)(142 345 172 315)(143 344 173 314)(144 343 174 313)(145 342 175 312)(146 341 176 311)(147 340 177 310)(148 339 178 309)(149 338 179 308)(150 337 180 307)(181 371 211 401)(182 370 212 400)(183 369 213 399)(184 368 214 398)(185 367 215 397)(186 366 216 396)(187 365 217 395)(188 364 218 394)(189 363 219 393)(190 362 220 392)(191 361 221 391)(192 420 222 390)(193 419 223 389)(194 418 224 388)(195 417 225 387)(196 416 226 386)(197 415 227 385)(198 414 228 384)(199 413 229 383)(200 412 230 382)(201 411 231 381)(202 410 232 380)(203 409 233 379)(204 408 234 378)(205 407 235 377)(206 406 236 376)(207 405 237 375)(208 404 238 374)(209 403 239 373)(210 402 240 372)```

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A ··· 20H 20I ··· 20X 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 12 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 6 6 6 6 20 20 60 60 2 2 2 2 2 2 ··· 2 4 4 20 20 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + - + + + + + + - + - + + + - image C1 C2 C2 C2 C2 S3 Q8 D4 D5 D6 D6 D10 D10 C3⋊D4 Dic10 D20 S3×Q8 S3×D5 C3⋊D20 C2×S3×D5 S3×Dic10 kernel C60.48D4 C6.Dic10 Dic3×C20 C6×Dic10 C2×Dic30 C2×Dic10 C5×Dic3 C60 C4×Dic3 C2×Dic5 C2×C20 C2×Dic3 C2×C12 C20 Dic3 C12 C10 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 1 4 2 2 2 1 4 2 4 16 8 2 2 4 2 8

Matrix representation of C60.48D4 in GL8(𝔽61)

 18 17 0 0 0 0 0 0 43 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 2 0 0 0 0 0 0 60 1 0 0 0 0 0 0 0 0 60 1 0 0 0 0 0 0 60 0
,
 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 59 0 0 0 0 0 0 1 60 0 0 0 0 0 0 0 0 30 12 0 0 0 0 0 0 42 31
,
 2 8 0 0 0 0 0 0 53 59 0 0 0 0 0 0 0 0 29 40 0 0 0 0 0 0 40 32 0 0 0 0 0 0 0 0 21 57 0 0 0 0 0 0 19 40 0 0 0 0 0 0 0 0 30 12 0 0 0 0 0 0 42 31

`G:=sub<GL(8,GF(61))| [18,43,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,59,60,0,0,0,0,0,0,0,0,30,42,0,0,0,0,0,0,12,31],[2,53,0,0,0,0,0,0,8,59,0,0,0,0,0,0,0,0,29,40,0,0,0,0,0,0,40,32,0,0,0,0,0,0,0,0,21,19,0,0,0,0,0,0,57,40,0,0,0,0,0,0,0,0,30,42,0,0,0,0,0,0,12,31] >;`

C60.48D4 in GAP, Magma, Sage, TeX

`C_{60}._{48}D_4`
`% in TeX`

`G:=Group("C60.48D4");`
`// GroupNames label`

`G:=SmallGroup(480,465);`
`// by ID`

`G=gap.SmallGroup(480,465);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,64,422,100,1356,18822]);`
`// Polycyclic`

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

׿
×
𝔽