metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.58D4, C30.6Q16, C12.50D20, Dic30⋊15C4, C10.4Dic12, C30.12SD16, C20.38(C4×S3), (C2×C30).22D4, (C2×C20).53D6, C12.32(C4×D5), C4⋊Dic3.1D5, C60.108(C2×C4), C15⋊8(Q8⋊C4), (C2×C10).32D12, C3⋊1(C10.Q16), C6.2(D4.D5), C6.2(C5⋊Q16), C5⋊2(C2.Dic12), C10.8(C24⋊C2), C10.22(D6⋊C4), (C2×C12).286D10, C20.14(C3⋊D4), C4.15(C3⋊D20), C4.7(D30.C2), C2.2(C5⋊Dic12), C2.8(D30⋊4C4), C6.7(D10⋊C4), C30.58(C22⋊C4), (C2×C60).130C22, (C2×Dic30).16C2, C2.2(D12.D5), C22.15(C5⋊D12), (C6×C5⋊2C8).3C2, (C2×C5⋊2C8).2S3, (C2×C4).136(S3×D5), (C5×C4⋊Dic3).1C2, (C2×C6).27(C5⋊D4), SmallGroup(480,51)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic30⋊15C4
G = < a,b,c | a60=c4=1, b2=a30, bab-1=a-1, cac-1=a11, cbc-1=a15b >
Subgroups: 412 in 84 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, Q8, C10, Dic3, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C24, Dic6, C2×Dic3, C2×C12, C30, Q8⋊C4, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C4⋊Dic3, C2×C24, C2×Dic6, C5×Dic3, Dic15, C60, C2×C30, C2×C5⋊2C8, C5×C4⋊C4, C2×Dic10, C2.Dic12, C3×C5⋊2C8, C10×Dic3, Dic30, Dic30, C2×Dic15, C2×C60, C10.Q16, C6×C5⋊2C8, C5×C4⋊Dic3, C2×Dic30, Dic30⋊15C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, SD16, Q16, D10, C4×S3, D12, C3⋊D4, Q8⋊C4, C4×D5, D20, C5⋊D4, C24⋊C2, Dic12, D6⋊C4, S3×D5, D10⋊C4, D4.D5, C5⋊Q16, C2.Dic12, D30.C2, C3⋊D20, C5⋊D12, C10.Q16, D12.D5, C5⋊Dic12, D30⋊4C4, Dic30⋊15C4
►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 92 31 62)(2 91 32 61)(3 90 33 120)(4 89 34 119)(5 88 35 118)(6 87 36 117)(7 86 37 116)(8 85 38 115)(9 84 39 114)(10 83 40 113)(11 82 41 112)(12 81 42 111)(13 80 43 110)(14 79 44 109)(15 78 45 108)(16 77 46 107)(17 76 47 106)(18 75 48 105)(19 74 49 104)(20 73 50 103)(21 72 51 102)(22 71 52 101)(23 70 53 100)(24 69 54 99)(25 68 55 98)(26 67 56 97)(27 66 57 96)(28 65 58 95)(29 64 59 94)(30 63 60 93)(121 207 151 237)(122 206 152 236)(123 205 153 235)(124 204 154 234)(125 203 155 233)(126 202 156 232)(127 201 157 231)(128 200 158 230)(129 199 159 229)(130 198 160 228)(131 197 161 227)(132 196 162 226)(133 195 163 225)(134 194 164 224)(135 193 165 223)(136 192 166 222)(137 191 167 221)(138 190 168 220)(139 189 169 219)(140 188 170 218)(141 187 171 217)(142 186 172 216)(143 185 173 215)(144 184 174 214)(145 183 175 213)(146 182 176 212)(147 181 177 211)(148 240 178 210)(149 239 179 209)(150 238 180 208)(241 368 271 398)(242 367 272 397)(243 366 273 396)(244 365 274 395)(245 364 275 394)(246 363 276 393)(247 362 277 392)(248 361 278 391)(249 420 279 390)(250 419 280 389)(251 418 281 388)(252 417 282 387)(253 416 283 386)(254 415 284 385)(255 414 285 384)(256 413 286 383)(257 412 287 382)(258 411 288 381)(259 410 289 380)(260 409 290 379)(261 408 291 378)(262 407 292 377)(263 406 293 376)(264 405 294 375)(265 404 295 374)(266 403 296 373)(267 402 297 372)(268 401 298 371)(269 400 299 370)(270 399 300 369)(301 426 331 456)(302 425 332 455)(303 424 333 454)(304 423 334 453)(305 422 335 452)(306 421 336 451)(307 480 337 450)(308 479 338 449)(309 478 339 448)(310 477 340 447)(311 476 341 446)(312 475 342 445)(313 474 343 444)(314 473 344 443)(315 472 345 442)(316 471 346 441)(317 470 347 440)(318 469 348 439)(319 468 349 438)(320 467 350 437)(321 466 351 436)(322 465 352 435)(323 464 353 434)(324 463 354 433)(325 462 355 432)(326 461 356 431)(327 460 357 430)(328 459 358 429)(329 458 359 428)(330 457 360 427)
(1 359 224 371)(2 310 225 382)(3 321 226 393)(4 332 227 404)(5 343 228 415)(6 354 229 366)(7 305 230 377)(8 316 231 388)(9 327 232 399)(10 338 233 410)(11 349 234 361)(12 360 235 372)(13 311 236 383)(14 322 237 394)(15 333 238 405)(16 344 239 416)(17 355 240 367)(18 306 181 378)(19 317 182 389)(20 328 183 400)(21 339 184 411)(22 350 185 362)(23 301 186 373)(24 312 187 384)(25 323 188 395)(26 334 189 406)(27 345 190 417)(28 356 191 368)(29 307 192 379)(30 318 193 390)(31 329 194 401)(32 340 195 412)(33 351 196 363)(34 302 197 374)(35 313 198 385)(36 324 199 396)(37 335 200 407)(38 346 201 418)(39 357 202 369)(40 308 203 380)(41 319 204 391)(42 330 205 402)(43 341 206 413)(44 352 207 364)(45 303 208 375)(46 314 209 386)(47 325 210 397)(48 336 211 408)(49 347 212 419)(50 358 213 370)(51 309 214 381)(52 320 215 392)(53 331 216 403)(54 342 217 414)(55 353 218 365)(56 304 219 376)(57 315 220 387)(58 326 221 398)(59 337 222 409)(60 348 223 420)(61 432 163 272)(62 443 164 283)(63 454 165 294)(64 465 166 245)(65 476 167 256)(66 427 168 267)(67 438 169 278)(68 449 170 289)(69 460 171 300)(70 471 172 251)(71 422 173 262)(72 433 174 273)(73 444 175 284)(74 455 176 295)(75 466 177 246)(76 477 178 257)(77 428 179 268)(78 439 180 279)(79 450 121 290)(80 461 122 241)(81 472 123 252)(82 423 124 263)(83 434 125 274)(84 445 126 285)(85 456 127 296)(86 467 128 247)(87 478 129 258)(88 429 130 269)(89 440 131 280)(90 451 132 291)(91 462 133 242)(92 473 134 253)(93 424 135 264)(94 435 136 275)(95 446 137 286)(96 457 138 297)(97 468 139 248)(98 479 140 259)(99 430 141 270)(100 441 142 281)(101 452 143 292)(102 463 144 243)(103 474 145 254)(104 425 146 265)(105 436 147 276)(106 447 148 287)(107 458 149 298)(108 469 150 249)(109 480 151 260)(110 431 152 271)(111 442 153 282)(112 453 154 293)(113 464 155 244)(114 475 156 255)(115 426 157 266)(116 437 158 277)(117 448 159 288)(118 459 160 299)(119 470 161 250)(120 421 162 261)
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), (1,92,31,62)(2,91,32,61)(3,90,33,120)(4,89,34,119)(5,88,35,118)(6,87,36,117)(7,86,37,116)(8,85,38,115)(9,84,39,114)(10,83,40,113)(11,82,41,112)(12,81,42,111)(13,80,43,110)(14,79,44,109)(15,78,45,108)(16,77,46,107)(17,76,47,106)(18,75,48,105)(19,74,49,104)(20,73,50,103)(21,72,51,102)(22,71,52,101)(23,70,53,100)(24,69,54,99)(25,68,55,98)(26,67,56,97)(27,66,57,96)(28,65,58,95)(29,64,59,94)(30,63,60,93)(121,207,151,237)(122,206,152,236)(123,205,153,235)(124,204,154,234)(125,203,155,233)(126,202,156,232)(127,201,157,231)(128,200,158,230)(129,199,159,229)(130,198,160,228)(131,197,161,227)(132,196,162,226)(133,195,163,225)(134,194,164,224)(135,193,165,223)(136,192,166,222)(137,191,167,221)(138,190,168,220)(139,189,169,219)(140,188,170,218)(141,187,171,217)(142,186,172,216)(143,185,173,215)(144,184,174,214)(145,183,175,213)(146,182,176,212)(147,181,177,211)(148,240,178,210)(149,239,179,209)(150,238,180,208)(241,368,271,398)(242,367,272,397)(243,366,273,396)(244,365,274,395)(245,364,275,394)(246,363,276,393)(247,362,277,392)(248,361,278,391)(249,420,279,390)(250,419,280,389)(251,418,281,388)(252,417,282,387)(253,416,283,386)(254,415,284,385)(255,414,285,384)(256,413,286,383)(257,412,287,382)(258,411,288,381)(259,410,289,380)(260,409,290,379)(261,408,291,378)(262,407,292,377)(263,406,293,376)(264,405,294,375)(265,404,295,374)(266,403,296,373)(267,402,297,372)(268,401,298,371)(269,400,299,370)(270,399,300,369)(301,426,331,456)(302,425,332,455)(303,424,333,454)(304,423,334,453)(305,422,335,452)(306,421,336,451)(307,480,337,450)(308,479,338,449)(309,478,339,448)(310,477,340,447)(311,476,341,446)(312,475,342,445)(313,474,343,444)(314,473,344,443)(315,472,345,442)(316,471,346,441)(317,470,347,440)(318,469,348,439)(319,468,349,438)(320,467,350,437)(321,466,351,436)(322,465,352,435)(323,464,353,434)(324,463,354,433)(325,462,355,432)(326,461,356,431)(327,460,357,430)(328,459,358,429)(329,458,359,428)(330,457,360,427), (1,359,224,371)(2,310,225,382)(3,321,226,393)(4,332,227,404)(5,343,228,415)(6,354,229,366)(7,305,230,377)(8,316,231,388)(9,327,232,399)(10,338,233,410)(11,349,234,361)(12,360,235,372)(13,311,236,383)(14,322,237,394)(15,333,238,405)(16,344,239,416)(17,355,240,367)(18,306,181,378)(19,317,182,389)(20,328,183,400)(21,339,184,411)(22,350,185,362)(23,301,186,373)(24,312,187,384)(25,323,188,395)(26,334,189,406)(27,345,190,417)(28,356,191,368)(29,307,192,379)(30,318,193,390)(31,329,194,401)(32,340,195,412)(33,351,196,363)(34,302,197,374)(35,313,198,385)(36,324,199,396)(37,335,200,407)(38,346,201,418)(39,357,202,369)(40,308,203,380)(41,319,204,391)(42,330,205,402)(43,341,206,413)(44,352,207,364)(45,303,208,375)(46,314,209,386)(47,325,210,397)(48,336,211,408)(49,347,212,419)(50,358,213,370)(51,309,214,381)(52,320,215,392)(53,331,216,403)(54,342,217,414)(55,353,218,365)(56,304,219,376)(57,315,220,387)(58,326,221,398)(59,337,222,409)(60,348,223,420)(61,432,163,272)(62,443,164,283)(63,454,165,294)(64,465,166,245)(65,476,167,256)(66,427,168,267)(67,438,169,278)(68,449,170,289)(69,460,171,300)(70,471,172,251)(71,422,173,262)(72,433,174,273)(73,444,175,284)(74,455,176,295)(75,466,177,246)(76,477,178,257)(77,428,179,268)(78,439,180,279)(79,450,121,290)(80,461,122,241)(81,472,123,252)(82,423,124,263)(83,434,125,274)(84,445,126,285)(85,456,127,296)(86,467,128,247)(87,478,129,258)(88,429,130,269)(89,440,131,280)(90,451,132,291)(91,462,133,242)(92,473,134,253)(93,424,135,264)(94,435,136,275)(95,446,137,286)(96,457,138,297)(97,468,139,248)(98,479,140,259)(99,430,141,270)(100,441,142,281)(101,452,143,292)(102,463,144,243)(103,474,145,254)(104,425,146,265)(105,436,147,276)(106,447,148,287)(107,458,149,298)(108,469,150,249)(109,480,151,260)(110,431,152,271)(111,442,153,282)(112,453,154,293)(113,464,155,244)(114,475,156,255)(115,426,157,266)(116,437,158,277)(117,448,159,288)(118,459,160,299)(119,470,161,250)(120,421,162,261)>;
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), (1,92,31,62)(2,91,32,61)(3,90,33,120)(4,89,34,119)(5,88,35,118)(6,87,36,117)(7,86,37,116)(8,85,38,115)(9,84,39,114)(10,83,40,113)(11,82,41,112)(12,81,42,111)(13,80,43,110)(14,79,44,109)(15,78,45,108)(16,77,46,107)(17,76,47,106)(18,75,48,105)(19,74,49,104)(20,73,50,103)(21,72,51,102)(22,71,52,101)(23,70,53,100)(24,69,54,99)(25,68,55,98)(26,67,56,97)(27,66,57,96)(28,65,58,95)(29,64,59,94)(30,63,60,93)(121,207,151,237)(122,206,152,236)(123,205,153,235)(124,204,154,234)(125,203,155,233)(126,202,156,232)(127,201,157,231)(128,200,158,230)(129,199,159,229)(130,198,160,228)(131,197,161,227)(132,196,162,226)(133,195,163,225)(134,194,164,224)(135,193,165,223)(136,192,166,222)(137,191,167,221)(138,190,168,220)(139,189,169,219)(140,188,170,218)(141,187,171,217)(142,186,172,216)(143,185,173,215)(144,184,174,214)(145,183,175,213)(146,182,176,212)(147,181,177,211)(148,240,178,210)(149,239,179,209)(150,238,180,208)(241,368,271,398)(242,367,272,397)(243,366,273,396)(244,365,274,395)(245,364,275,394)(246,363,276,393)(247,362,277,392)(248,361,278,391)(249,420,279,390)(250,419,280,389)(251,418,281,388)(252,417,282,387)(253,416,283,386)(254,415,284,385)(255,414,285,384)(256,413,286,383)(257,412,287,382)(258,411,288,381)(259,410,289,380)(260,409,290,379)(261,408,291,378)(262,407,292,377)(263,406,293,376)(264,405,294,375)(265,404,295,374)(266,403,296,373)(267,402,297,372)(268,401,298,371)(269,400,299,370)(270,399,300,369)(301,426,331,456)(302,425,332,455)(303,424,333,454)(304,423,334,453)(305,422,335,452)(306,421,336,451)(307,480,337,450)(308,479,338,449)(309,478,339,448)(310,477,340,447)(311,476,341,446)(312,475,342,445)(313,474,343,444)(314,473,344,443)(315,472,345,442)(316,471,346,441)(317,470,347,440)(318,469,348,439)(319,468,349,438)(320,467,350,437)(321,466,351,436)(322,465,352,435)(323,464,353,434)(324,463,354,433)(325,462,355,432)(326,461,356,431)(327,460,357,430)(328,459,358,429)(329,458,359,428)(330,457,360,427), (1,359,224,371)(2,310,225,382)(3,321,226,393)(4,332,227,404)(5,343,228,415)(6,354,229,366)(7,305,230,377)(8,316,231,388)(9,327,232,399)(10,338,233,410)(11,349,234,361)(12,360,235,372)(13,311,236,383)(14,322,237,394)(15,333,238,405)(16,344,239,416)(17,355,240,367)(18,306,181,378)(19,317,182,389)(20,328,183,400)(21,339,184,411)(22,350,185,362)(23,301,186,373)(24,312,187,384)(25,323,188,395)(26,334,189,406)(27,345,190,417)(28,356,191,368)(29,307,192,379)(30,318,193,390)(31,329,194,401)(32,340,195,412)(33,351,196,363)(34,302,197,374)(35,313,198,385)(36,324,199,396)(37,335,200,407)(38,346,201,418)(39,357,202,369)(40,308,203,380)(41,319,204,391)(42,330,205,402)(43,341,206,413)(44,352,207,364)(45,303,208,375)(46,314,209,386)(47,325,210,397)(48,336,211,408)(49,347,212,419)(50,358,213,370)(51,309,214,381)(52,320,215,392)(53,331,216,403)(54,342,217,414)(55,353,218,365)(56,304,219,376)(57,315,220,387)(58,326,221,398)(59,337,222,409)(60,348,223,420)(61,432,163,272)(62,443,164,283)(63,454,165,294)(64,465,166,245)(65,476,167,256)(66,427,168,267)(67,438,169,278)(68,449,170,289)(69,460,171,300)(70,471,172,251)(71,422,173,262)(72,433,174,273)(73,444,175,284)(74,455,176,295)(75,466,177,246)(76,477,178,257)(77,428,179,268)(78,439,180,279)(79,450,121,290)(80,461,122,241)(81,472,123,252)(82,423,124,263)(83,434,125,274)(84,445,126,285)(85,456,127,296)(86,467,128,247)(87,478,129,258)(88,429,130,269)(89,440,131,280)(90,451,132,291)(91,462,133,242)(92,473,134,253)(93,424,135,264)(94,435,136,275)(95,446,137,286)(96,457,138,297)(97,468,139,248)(98,479,140,259)(99,430,141,270)(100,441,142,281)(101,452,143,292)(102,463,144,243)(103,474,145,254)(104,425,146,265)(105,436,147,276)(106,447,148,287)(107,458,149,298)(108,469,150,249)(109,480,151,260)(110,431,152,271)(111,442,153,282)(112,453,154,293)(113,464,155,244)(114,475,156,255)(115,426,157,266)(116,437,158,277)(117,448,159,288)(118,459,160,299)(119,470,161,250)(120,421,162,261) );
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)], [(1,92,31,62),(2,91,32,61),(3,90,33,120),(4,89,34,119),(5,88,35,118),(6,87,36,117),(7,86,37,116),(8,85,38,115),(9,84,39,114),(10,83,40,113),(11,82,41,112),(12,81,42,111),(13,80,43,110),(14,79,44,109),(15,78,45,108),(16,77,46,107),(17,76,47,106),(18,75,48,105),(19,74,49,104),(20,73,50,103),(21,72,51,102),(22,71,52,101),(23,70,53,100),(24,69,54,99),(25,68,55,98),(26,67,56,97),(27,66,57,96),(28,65,58,95),(29,64,59,94),(30,63,60,93),(121,207,151,237),(122,206,152,236),(123,205,153,235),(124,204,154,234),(125,203,155,233),(126,202,156,232),(127,201,157,231),(128,200,158,230),(129,199,159,229),(130,198,160,228),(131,197,161,227),(132,196,162,226),(133,195,163,225),(134,194,164,224),(135,193,165,223),(136,192,166,222),(137,191,167,221),(138,190,168,220),(139,189,169,219),(140,188,170,218),(141,187,171,217),(142,186,172,216),(143,185,173,215),(144,184,174,214),(145,183,175,213),(146,182,176,212),(147,181,177,211),(148,240,178,210),(149,239,179,209),(150,238,180,208),(241,368,271,398),(242,367,272,397),(243,366,273,396),(244,365,274,395),(245,364,275,394),(246,363,276,393),(247,362,277,392),(248,361,278,391),(249,420,279,390),(250,419,280,389),(251,418,281,388),(252,417,282,387),(253,416,283,386),(254,415,284,385),(255,414,285,384),(256,413,286,383),(257,412,287,382),(258,411,288,381),(259,410,289,380),(260,409,290,379),(261,408,291,378),(262,407,292,377),(263,406,293,376),(264,405,294,375),(265,404,295,374),(266,403,296,373),(267,402,297,372),(268,401,298,371),(269,400,299,370),(270,399,300,369),(301,426,331,456),(302,425,332,455),(303,424,333,454),(304,423,334,453),(305,422,335,452),(306,421,336,451),(307,480,337,450),(308,479,338,449),(309,478,339,448),(310,477,340,447),(311,476,341,446),(312,475,342,445),(313,474,343,444),(314,473,344,443),(315,472,345,442),(316,471,346,441),(317,470,347,440),(318,469,348,439),(319,468,349,438),(320,467,350,437),(321,466,351,436),(322,465,352,435),(323,464,353,434),(324,463,354,433),(325,462,355,432),(326,461,356,431),(327,460,357,430),(328,459,358,429),(329,458,359,428),(330,457,360,427)], [(1,359,224,371),(2,310,225,382),(3,321,226,393),(4,332,227,404),(5,343,228,415),(6,354,229,366),(7,305,230,377),(8,316,231,388),(9,327,232,399),(10,338,233,410),(11,349,234,361),(12,360,235,372),(13,311,236,383),(14,322,237,394),(15,333,238,405),(16,344,239,416),(17,355,240,367),(18,306,181,378),(19,317,182,389),(20,328,183,400),(21,339,184,411),(22,350,185,362),(23,301,186,373),(24,312,187,384),(25,323,188,395),(26,334,189,406),(27,345,190,417),(28,356,191,368),(29,307,192,379),(30,318,193,390),(31,329,194,401),(32,340,195,412),(33,351,196,363),(34,302,197,374),(35,313,198,385),(36,324,199,396),(37,335,200,407),(38,346,201,418),(39,357,202,369),(40,308,203,380),(41,319,204,391),(42,330,205,402),(43,341,206,413),(44,352,207,364),(45,303,208,375),(46,314,209,386),(47,325,210,397),(48,336,211,408),(49,347,212,419),(50,358,213,370),(51,309,214,381),(52,320,215,392),(53,331,216,403),(54,342,217,414),(55,353,218,365),(56,304,219,376),(57,315,220,387),(58,326,221,398),(59,337,222,409),(60,348,223,420),(61,432,163,272),(62,443,164,283),(63,454,165,294),(64,465,166,245),(65,476,167,256),(66,427,168,267),(67,438,169,278),(68,449,170,289),(69,460,171,300),(70,471,172,251),(71,422,173,262),(72,433,174,273),(73,444,175,284),(74,455,176,295),(75,466,177,246),(76,477,178,257),(77,428,179,268),(78,439,180,279),(79,450,121,290),(80,461,122,241),(81,472,123,252),(82,423,124,263),(83,434,125,274),(84,445,126,285),(85,456,127,296),(86,467,128,247),(87,478,129,258),(88,429,130,269),(89,440,131,280),(90,451,132,291),(91,462,133,242),(92,473,134,253),(93,424,135,264),(94,435,136,275),(95,446,137,286),(96,457,138,297),(97,468,139,248),(98,479,140,259),(99,430,141,270),(100,441,142,281),(101,452,143,292),(102,463,144,243),(103,474,145,254),(104,425,146,265),(105,436,147,276),(106,447,148,287),(107,458,149,298),(108,469,150,249),(109,480,151,260),(110,431,152,271),(111,442,153,282),(112,453,154,293),(113,464,155,244),(114,475,156,255),(115,426,157,266),(116,437,158,277),(117,448,159,288),(118,459,160,299),(119,470,161,250),(120,421,162,261)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | - | + | + | + | - | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D5 | D6 | SD16 | Q16 | D10 | C4×S3 | C3⋊D4 | D12 | C4×D5 | D20 | C5⋊D4 | C24⋊C2 | Dic12 | S3×D5 | D4.D5 | C5⋊Q16 | D30.C2 | C3⋊D20 | C5⋊D12 | D12.D5 | C5⋊Dic12 |
| kernel | Dic30⋊15C4 | C6×C5⋊2C8 | C5×C4⋊Dic3 | C2×Dic30 | Dic30 | C2×C5⋊2C8 | C60 | C2×C30 | C4⋊Dic3 | C2×C20 | C30 | C30 | C2×C12 | C20 | C20 | C2×C10 | C12 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of Dic30⋊15C4 ►in GL4(𝔽241) generated by
| 142 | 142 | 0 | 0 |
| 99 | 43 | 0 | 0 |
| 0 | 0 | 1 | 240 |
| 0 | 0 | 191 | 51 |
| 20 | 110 | 0 | 0 |
| 130 | 221 | 0 | 0 |
| 0 | 0 | 108 | 56 |
| 0 | 0 | 235 | 133 |
| 55 | 18 | 0 | 0 |
| 73 | 186 | 0 | 0 |
| 0 | 0 | 44 | 156 |
| 0 | 0 | 88 | 197 |
G:=sub<GL(4,GF(241))| [142,99,0,0,142,43,0,0,0,0,1,191,0,0,240,51],[20,130,0,0,110,221,0,0,0,0,108,235,0,0,56,133],[55,73,0,0,18,186,0,0,0,0,44,88,0,0,156,197] >;
Dic30⋊15C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{30}\rtimes_{15}C_4 % in TeX
G:=Group("Dic30:15C4"); // GroupNames label
G:=SmallGroup(480,51);
// by ID
G=gap.SmallGroup(480,51);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,197,176,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=c^4=1,b^2=a^30,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations