metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.35D4, C30.5Q16, C20.50D12, C6.4Dic20, Dic30⋊12C4, C30.11SD16, C12.6(C4×D5), C20.64(C4×S3), C60.93(C2×C4), (C2×C6).32D20, (C2×C30).21D4, C4⋊Dic5.1S3, C15⋊7(Q8⋊C4), (C2×C12).54D10, (C2×C20).282D6, C6.7(C40⋊C2), C5⋊2(C6.SD16), C10.21(D6⋊C4), C12.16(C5⋊D4), C4.15(C5⋊D12), C3⋊1(C20.44D4), C4.2(D30.C2), C10.2(D4.S3), C2.2(C3⋊Dic20), C10.2(C3⋊Q16), C2.7(D30⋊4C4), C6.6(D10⋊C4), C30.57(C22⋊C4), (C2×C60).101C22, (C2×Dic30).12C2, C2.2(C6.D20), C22.15(C3⋊D20), (C2×C3⋊C8).2D5, (C10×C3⋊C8).3C2, (C2×C4).85(S3×D5), (C3×C4⋊Dic5).1C2, (C2×C10).26(C3⋊D4), SmallGroup(480,50)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic30⋊12C4
G = < a,b,c | a60=c4=1, b2=a30, bab-1=a-1, cac-1=a19, cbc-1=a15b >
Subgroups: 428 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, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C2×C10, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C30, Q8⋊C4, C40, Dic10, C2×Dic5, C2×C20, C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C3×Dic5, Dic15, C60, C2×C30, C4⋊Dic5, C2×C40, C2×Dic10, C6.SD16, C5×C3⋊C8, C6×Dic5, Dic30, Dic30, C2×Dic15, C2×C60, C20.44D4, C3×C4⋊Dic5, C10×C3⋊C8, C2×Dic30, Dic30⋊12C4
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, D6⋊C4, D4.S3, C3⋊Q16, S3×D5, C40⋊C2, Dic20, D10⋊C4, C6.SD16, D30.C2, C3⋊D20, C5⋊D12, C20.44D4, C6.D20, C3⋊Dic20, D30⋊4C4, Dic30⋊12C4
►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)
(1 366 31 396)(2 365 32 395)(3 364 33 394)(4 363 34 393)(5 362 35 392)(6 361 36 391)(7 420 37 390)(8 419 38 389)(9 418 39 388)(10 417 40 387)(11 416 41 386)(12 415 42 385)(13 414 43 384)(14 413 44 383)(15 412 45 382)(16 411 46 381)(17 410 47 380)(18 409 48 379)(19 408 49 378)(20 407 50 377)(21 406 51 376)(22 405 52 375)(23 404 53 374)(24 403 54 373)(25 402 55 372)(26 401 56 371)(27 400 57 370)(28 399 58 369)(29 398 59 368)(30 397 60 367)(61 427 91 457)(62 426 92 456)(63 425 93 455)(64 424 94 454)(65 423 95 453)(66 422 96 452)(67 421 97 451)(68 480 98 450)(69 479 99 449)(70 478 100 448)(71 477 101 447)(72 476 102 446)(73 475 103 445)(74 474 104 444)(75 473 105 443)(76 472 106 442)(77 471 107 441)(78 470 108 440)(79 469 109 439)(80 468 110 438)(81 467 111 437)(82 466 112 436)(83 465 113 435)(84 464 114 434)(85 463 115 433)(86 462 116 432)(87 461 117 431)(88 460 118 430)(89 459 119 429)(90 458 120 428)(121 352 151 322)(122 351 152 321)(123 350 153 320)(124 349 154 319)(125 348 155 318)(126 347 156 317)(127 346 157 316)(128 345 158 315)(129 344 159 314)(130 343 160 313)(131 342 161 312)(132 341 162 311)(133 340 163 310)(134 339 164 309)(135 338 165 308)(136 337 166 307)(137 336 167 306)(138 335 168 305)(139 334 169 304)(140 333 170 303)(141 332 171 302)(142 331 172 301)(143 330 173 360)(144 329 174 359)(145 328 175 358)(146 327 176 357)(147 326 177 356)(148 325 178 355)(149 324 179 354)(150 323 180 353)(181 272 211 242)(182 271 212 241)(183 270 213 300)(184 269 214 299)(185 268 215 298)(186 267 216 297)(187 266 217 296)(188 265 218 295)(189 264 219 294)(190 263 220 293)(191 262 221 292)(192 261 222 291)(193 260 223 290)(194 259 224 289)(195 258 225 288)(196 257 226 287)(197 256 227 286)(198 255 228 285)(199 254 229 284)(200 253 230 283)(201 252 231 282)(202 251 232 281)(203 250 233 280)(204 249 234 279)(205 248 235 278)(206 247 236 277)(207 246 237 276)(208 245 238 275)(209 244 239 274)(210 243 240 273)
(1 266 112 311)(2 285 113 330)(3 244 114 349)(4 263 115 308)(5 282 116 327)(6 241 117 346)(7 260 118 305)(8 279 119 324)(9 298 120 343)(10 257 61 302)(11 276 62 321)(12 295 63 340)(13 254 64 359)(14 273 65 318)(15 292 66 337)(16 251 67 356)(17 270 68 315)(18 289 69 334)(19 248 70 353)(20 267 71 312)(21 286 72 331)(22 245 73 350)(23 264 74 309)(24 283 75 328)(25 242 76 347)(26 261 77 306)(27 280 78 325)(28 299 79 344)(29 258 80 303)(30 277 81 322)(31 296 82 341)(32 255 83 360)(33 274 84 319)(34 293 85 338)(35 252 86 357)(36 271 87 316)(37 290 88 335)(38 249 89 354)(39 268 90 313)(40 287 91 332)(41 246 92 351)(42 265 93 310)(43 284 94 329)(44 243 95 348)(45 262 96 307)(46 281 97 326)(47 300 98 345)(48 259 99 304)(49 278 100 323)(50 297 101 342)(51 256 102 301)(52 275 103 320)(53 294 104 339)(54 253 105 358)(55 272 106 317)(56 291 107 336)(57 250 108 355)(58 269 109 314)(59 288 110 333)(60 247 111 352)(121 382 206 452)(122 401 207 471)(123 420 208 430)(124 379 209 449)(125 398 210 468)(126 417 211 427)(127 376 212 446)(128 395 213 465)(129 414 214 424)(130 373 215 443)(131 392 216 462)(132 411 217 421)(133 370 218 440)(134 389 219 459)(135 408 220 478)(136 367 221 437)(137 386 222 456)(138 405 223 475)(139 364 224 434)(140 383 225 453)(141 402 226 472)(142 361 227 431)(143 380 228 450)(144 399 229 469)(145 418 230 428)(146 377 231 447)(147 396 232 466)(148 415 233 425)(149 374 234 444)(150 393 235 463)(151 412 236 422)(152 371 237 441)(153 390 238 460)(154 409 239 479)(155 368 240 438)(156 387 181 457)(157 406 182 476)(158 365 183 435)(159 384 184 454)(160 403 185 473)(161 362 186 432)(162 381 187 451)(163 400 188 470)(164 419 189 429)(165 378 190 448)(166 397 191 467)(167 416 192 426)(168 375 193 445)(169 394 194 464)(170 413 195 423)(171 372 196 442)(172 391 197 461)(173 410 198 480)(174 369 199 439)(175 388 200 458)(176 407 201 477)(177 366 202 436)(178 385 203 455)(179 404 204 474)(180 363 205 433)
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), (1,366,31,396)(2,365,32,395)(3,364,33,394)(4,363,34,393)(5,362,35,392)(6,361,36,391)(7,420,37,390)(8,419,38,389)(9,418,39,388)(10,417,40,387)(11,416,41,386)(12,415,42,385)(13,414,43,384)(14,413,44,383)(15,412,45,382)(16,411,46,381)(17,410,47,380)(18,409,48,379)(19,408,49,378)(20,407,50,377)(21,406,51,376)(22,405,52,375)(23,404,53,374)(24,403,54,373)(25,402,55,372)(26,401,56,371)(27,400,57,370)(28,399,58,369)(29,398,59,368)(30,397,60,367)(61,427,91,457)(62,426,92,456)(63,425,93,455)(64,424,94,454)(65,423,95,453)(66,422,96,452)(67,421,97,451)(68,480,98,450)(69,479,99,449)(70,478,100,448)(71,477,101,447)(72,476,102,446)(73,475,103,445)(74,474,104,444)(75,473,105,443)(76,472,106,442)(77,471,107,441)(78,470,108,440)(79,469,109,439)(80,468,110,438)(81,467,111,437)(82,466,112,436)(83,465,113,435)(84,464,114,434)(85,463,115,433)(86,462,116,432)(87,461,117,431)(88,460,118,430)(89,459,119,429)(90,458,120,428)(121,352,151,322)(122,351,152,321)(123,350,153,320)(124,349,154,319)(125,348,155,318)(126,347,156,317)(127,346,157,316)(128,345,158,315)(129,344,159,314)(130,343,160,313)(131,342,161,312)(132,341,162,311)(133,340,163,310)(134,339,164,309)(135,338,165,308)(136,337,166,307)(137,336,167,306)(138,335,168,305)(139,334,169,304)(140,333,170,303)(141,332,171,302)(142,331,172,301)(143,330,173,360)(144,329,174,359)(145,328,175,358)(146,327,176,357)(147,326,177,356)(148,325,178,355)(149,324,179,354)(150,323,180,353)(181,272,211,242)(182,271,212,241)(183,270,213,300)(184,269,214,299)(185,268,215,298)(186,267,216,297)(187,266,217,296)(188,265,218,295)(189,264,219,294)(190,263,220,293)(191,262,221,292)(192,261,222,291)(193,260,223,290)(194,259,224,289)(195,258,225,288)(196,257,226,287)(197,256,227,286)(198,255,228,285)(199,254,229,284)(200,253,230,283)(201,252,231,282)(202,251,232,281)(203,250,233,280)(204,249,234,279)(205,248,235,278)(206,247,236,277)(207,246,237,276)(208,245,238,275)(209,244,239,274)(210,243,240,273), (1,266,112,311)(2,285,113,330)(3,244,114,349)(4,263,115,308)(5,282,116,327)(6,241,117,346)(7,260,118,305)(8,279,119,324)(9,298,120,343)(10,257,61,302)(11,276,62,321)(12,295,63,340)(13,254,64,359)(14,273,65,318)(15,292,66,337)(16,251,67,356)(17,270,68,315)(18,289,69,334)(19,248,70,353)(20,267,71,312)(21,286,72,331)(22,245,73,350)(23,264,74,309)(24,283,75,328)(25,242,76,347)(26,261,77,306)(27,280,78,325)(28,299,79,344)(29,258,80,303)(30,277,81,322)(31,296,82,341)(32,255,83,360)(33,274,84,319)(34,293,85,338)(35,252,86,357)(36,271,87,316)(37,290,88,335)(38,249,89,354)(39,268,90,313)(40,287,91,332)(41,246,92,351)(42,265,93,310)(43,284,94,329)(44,243,95,348)(45,262,96,307)(46,281,97,326)(47,300,98,345)(48,259,99,304)(49,278,100,323)(50,297,101,342)(51,256,102,301)(52,275,103,320)(53,294,104,339)(54,253,105,358)(55,272,106,317)(56,291,107,336)(57,250,108,355)(58,269,109,314)(59,288,110,333)(60,247,111,352)(121,382,206,452)(122,401,207,471)(123,420,208,430)(124,379,209,449)(125,398,210,468)(126,417,211,427)(127,376,212,446)(128,395,213,465)(129,414,214,424)(130,373,215,443)(131,392,216,462)(132,411,217,421)(133,370,218,440)(134,389,219,459)(135,408,220,478)(136,367,221,437)(137,386,222,456)(138,405,223,475)(139,364,224,434)(140,383,225,453)(141,402,226,472)(142,361,227,431)(143,380,228,450)(144,399,229,469)(145,418,230,428)(146,377,231,447)(147,396,232,466)(148,415,233,425)(149,374,234,444)(150,393,235,463)(151,412,236,422)(152,371,237,441)(153,390,238,460)(154,409,239,479)(155,368,240,438)(156,387,181,457)(157,406,182,476)(158,365,183,435)(159,384,184,454)(160,403,185,473)(161,362,186,432)(162,381,187,451)(163,400,188,470)(164,419,189,429)(165,378,190,448)(166,397,191,467)(167,416,192,426)(168,375,193,445)(169,394,194,464)(170,413,195,423)(171,372,196,442)(172,391,197,461)(173,410,198,480)(174,369,199,439)(175,388,200,458)(176,407,201,477)(177,366,202,436)(178,385,203,455)(179,404,204,474)(180,363,205,433)>;
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), (1,366,31,396)(2,365,32,395)(3,364,33,394)(4,363,34,393)(5,362,35,392)(6,361,36,391)(7,420,37,390)(8,419,38,389)(9,418,39,388)(10,417,40,387)(11,416,41,386)(12,415,42,385)(13,414,43,384)(14,413,44,383)(15,412,45,382)(16,411,46,381)(17,410,47,380)(18,409,48,379)(19,408,49,378)(20,407,50,377)(21,406,51,376)(22,405,52,375)(23,404,53,374)(24,403,54,373)(25,402,55,372)(26,401,56,371)(27,400,57,370)(28,399,58,369)(29,398,59,368)(30,397,60,367)(61,427,91,457)(62,426,92,456)(63,425,93,455)(64,424,94,454)(65,423,95,453)(66,422,96,452)(67,421,97,451)(68,480,98,450)(69,479,99,449)(70,478,100,448)(71,477,101,447)(72,476,102,446)(73,475,103,445)(74,474,104,444)(75,473,105,443)(76,472,106,442)(77,471,107,441)(78,470,108,440)(79,469,109,439)(80,468,110,438)(81,467,111,437)(82,466,112,436)(83,465,113,435)(84,464,114,434)(85,463,115,433)(86,462,116,432)(87,461,117,431)(88,460,118,430)(89,459,119,429)(90,458,120,428)(121,352,151,322)(122,351,152,321)(123,350,153,320)(124,349,154,319)(125,348,155,318)(126,347,156,317)(127,346,157,316)(128,345,158,315)(129,344,159,314)(130,343,160,313)(131,342,161,312)(132,341,162,311)(133,340,163,310)(134,339,164,309)(135,338,165,308)(136,337,166,307)(137,336,167,306)(138,335,168,305)(139,334,169,304)(140,333,170,303)(141,332,171,302)(142,331,172,301)(143,330,173,360)(144,329,174,359)(145,328,175,358)(146,327,176,357)(147,326,177,356)(148,325,178,355)(149,324,179,354)(150,323,180,353)(181,272,211,242)(182,271,212,241)(183,270,213,300)(184,269,214,299)(185,268,215,298)(186,267,216,297)(187,266,217,296)(188,265,218,295)(189,264,219,294)(190,263,220,293)(191,262,221,292)(192,261,222,291)(193,260,223,290)(194,259,224,289)(195,258,225,288)(196,257,226,287)(197,256,227,286)(198,255,228,285)(199,254,229,284)(200,253,230,283)(201,252,231,282)(202,251,232,281)(203,250,233,280)(204,249,234,279)(205,248,235,278)(206,247,236,277)(207,246,237,276)(208,245,238,275)(209,244,239,274)(210,243,240,273), (1,266,112,311)(2,285,113,330)(3,244,114,349)(4,263,115,308)(5,282,116,327)(6,241,117,346)(7,260,118,305)(8,279,119,324)(9,298,120,343)(10,257,61,302)(11,276,62,321)(12,295,63,340)(13,254,64,359)(14,273,65,318)(15,292,66,337)(16,251,67,356)(17,270,68,315)(18,289,69,334)(19,248,70,353)(20,267,71,312)(21,286,72,331)(22,245,73,350)(23,264,74,309)(24,283,75,328)(25,242,76,347)(26,261,77,306)(27,280,78,325)(28,299,79,344)(29,258,80,303)(30,277,81,322)(31,296,82,341)(32,255,83,360)(33,274,84,319)(34,293,85,338)(35,252,86,357)(36,271,87,316)(37,290,88,335)(38,249,89,354)(39,268,90,313)(40,287,91,332)(41,246,92,351)(42,265,93,310)(43,284,94,329)(44,243,95,348)(45,262,96,307)(46,281,97,326)(47,300,98,345)(48,259,99,304)(49,278,100,323)(50,297,101,342)(51,256,102,301)(52,275,103,320)(53,294,104,339)(54,253,105,358)(55,272,106,317)(56,291,107,336)(57,250,108,355)(58,269,109,314)(59,288,110,333)(60,247,111,352)(121,382,206,452)(122,401,207,471)(123,420,208,430)(124,379,209,449)(125,398,210,468)(126,417,211,427)(127,376,212,446)(128,395,213,465)(129,414,214,424)(130,373,215,443)(131,392,216,462)(132,411,217,421)(133,370,218,440)(134,389,219,459)(135,408,220,478)(136,367,221,437)(137,386,222,456)(138,405,223,475)(139,364,224,434)(140,383,225,453)(141,402,226,472)(142,361,227,431)(143,380,228,450)(144,399,229,469)(145,418,230,428)(146,377,231,447)(147,396,232,466)(148,415,233,425)(149,374,234,444)(150,393,235,463)(151,412,236,422)(152,371,237,441)(153,390,238,460)(154,409,239,479)(155,368,240,438)(156,387,181,457)(157,406,182,476)(158,365,183,435)(159,384,184,454)(160,403,185,473)(161,362,186,432)(162,381,187,451)(163,400,188,470)(164,419,189,429)(165,378,190,448)(166,397,191,467)(167,416,192,426)(168,375,193,445)(169,394,194,464)(170,413,195,423)(171,372,196,442)(172,391,197,461)(173,410,198,480)(174,369,199,439)(175,388,200,458)(176,407,201,477)(177,366,202,436)(178,385,203,455)(179,404,204,474)(180,363,205,433) );
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)], [(1,366,31,396),(2,365,32,395),(3,364,33,394),(4,363,34,393),(5,362,35,392),(6,361,36,391),(7,420,37,390),(8,419,38,389),(9,418,39,388),(10,417,40,387),(11,416,41,386),(12,415,42,385),(13,414,43,384),(14,413,44,383),(15,412,45,382),(16,411,46,381),(17,410,47,380),(18,409,48,379),(19,408,49,378),(20,407,50,377),(21,406,51,376),(22,405,52,375),(23,404,53,374),(24,403,54,373),(25,402,55,372),(26,401,56,371),(27,400,57,370),(28,399,58,369),(29,398,59,368),(30,397,60,367),(61,427,91,457),(62,426,92,456),(63,425,93,455),(64,424,94,454),(65,423,95,453),(66,422,96,452),(67,421,97,451),(68,480,98,450),(69,479,99,449),(70,478,100,448),(71,477,101,447),(72,476,102,446),(73,475,103,445),(74,474,104,444),(75,473,105,443),(76,472,106,442),(77,471,107,441),(78,470,108,440),(79,469,109,439),(80,468,110,438),(81,467,111,437),(82,466,112,436),(83,465,113,435),(84,464,114,434),(85,463,115,433),(86,462,116,432),(87,461,117,431),(88,460,118,430),(89,459,119,429),(90,458,120,428),(121,352,151,322),(122,351,152,321),(123,350,153,320),(124,349,154,319),(125,348,155,318),(126,347,156,317),(127,346,157,316),(128,345,158,315),(129,344,159,314),(130,343,160,313),(131,342,161,312),(132,341,162,311),(133,340,163,310),(134,339,164,309),(135,338,165,308),(136,337,166,307),(137,336,167,306),(138,335,168,305),(139,334,169,304),(140,333,170,303),(141,332,171,302),(142,331,172,301),(143,330,173,360),(144,329,174,359),(145,328,175,358),(146,327,176,357),(147,326,177,356),(148,325,178,355),(149,324,179,354),(150,323,180,353),(181,272,211,242),(182,271,212,241),(183,270,213,300),(184,269,214,299),(185,268,215,298),(186,267,216,297),(187,266,217,296),(188,265,218,295),(189,264,219,294),(190,263,220,293),(191,262,221,292),(192,261,222,291),(193,260,223,290),(194,259,224,289),(195,258,225,288),(196,257,226,287),(197,256,227,286),(198,255,228,285),(199,254,229,284),(200,253,230,283),(201,252,231,282),(202,251,232,281),(203,250,233,280),(204,249,234,279),(205,248,235,278),(206,247,236,277),(207,246,237,276),(208,245,238,275),(209,244,239,274),(210,243,240,273)], [(1,266,112,311),(2,285,113,330),(3,244,114,349),(4,263,115,308),(5,282,116,327),(6,241,117,346),(7,260,118,305),(8,279,119,324),(9,298,120,343),(10,257,61,302),(11,276,62,321),(12,295,63,340),(13,254,64,359),(14,273,65,318),(15,292,66,337),(16,251,67,356),(17,270,68,315),(18,289,69,334),(19,248,70,353),(20,267,71,312),(21,286,72,331),(22,245,73,350),(23,264,74,309),(24,283,75,328),(25,242,76,347),(26,261,77,306),(27,280,78,325),(28,299,79,344),(29,258,80,303),(30,277,81,322),(31,296,82,341),(32,255,83,360),(33,274,84,319),(34,293,85,338),(35,252,86,357),(36,271,87,316),(37,290,88,335),(38,249,89,354),(39,268,90,313),(40,287,91,332),(41,246,92,351),(42,265,93,310),(43,284,94,329),(44,243,95,348),(45,262,96,307),(46,281,97,326),(47,300,98,345),(48,259,99,304),(49,278,100,323),(50,297,101,342),(51,256,102,301),(52,275,103,320),(53,294,104,339),(54,253,105,358),(55,272,106,317),(56,291,107,336),(57,250,108,355),(58,269,109,314),(59,288,110,333),(60,247,111,352),(121,382,206,452),(122,401,207,471),(123,420,208,430),(124,379,209,449),(125,398,210,468),(126,417,211,427),(127,376,212,446),(128,395,213,465),(129,414,214,424),(130,373,215,443),(131,392,216,462),(132,411,217,421),(133,370,218,440),(134,389,219,459),(135,408,220,478),(136,367,221,437),(137,386,222,456),(138,405,223,475),(139,364,224,434),(140,383,225,453),(141,402,226,472),(142,361,227,431),(143,380,228,450),(144,399,229,469),(145,418,230,428),(146,377,231,447),(147,396,232,466),(148,415,233,425),(149,374,234,444),(150,393,235,463),(151,412,236,422),(152,371,237,441),(153,390,238,460),(154,409,239,479),(155,368,240,438),(156,387,181,457),(157,406,182,476),(158,365,183,435),(159,384,184,454),(160,403,185,473),(161,362,186,432),(162,381,187,451),(163,400,188,470),(164,419,189,429),(165,378,190,448),(166,397,191,467),(167,416,192,426),(168,375,193,445),(169,394,194,464),(170,413,195,423),(171,372,196,442),(172,391,197,461),(173,410,198,480),(174,369,199,439),(175,388,200,458),(176,407,201,477),(177,366,202,436),(178,385,203,455),(179,404,204,474),(180,363,205,433)]])
72 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 | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F | 40A | ··· | 40P | 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 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
72 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 | D12 | C3⋊D4 | C4×D5 | C5⋊D4 | D20 | C40⋊C2 | Dic20 | D4.S3 | C3⋊Q16 | S3×D5 | D30.C2 | C5⋊D12 | C3⋊D20 | C6.D20 | C3⋊Dic20 |
| kernel | Dic30⋊12C4 | C3×C4⋊Dic5 | C10×C3⋊C8 | C2×Dic30 | Dic30 | C4⋊Dic5 | C60 | C2×C30 | C2×C3⋊C8 | C2×C20 | C30 | C30 | C2×C12 | C20 | C20 | C2×C10 | C12 | C12 | C2×C6 | C6 | C6 | C10 | C10 | C2×C4 | 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 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of Dic30⋊12C4 ►in GL6(𝔽241)
| 0 | 240 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 85 | 41 |
| 0 | 0 | 0 | 0 | 200 | 122 |
| 62 | 128 | 0 | 0 | 0 | 0 |
| 66 | 179 | 0 | 0 | 0 | 0 |
| 0 | 0 | 68 | 175 | 0 | 0 |
| 0 | 0 | 81 | 173 | 0 | 0 |
| 0 | 0 | 0 | 0 | 81 | 33 |
| 0 | 0 | 0 | 0 | 159 | 160 |
| 70 | 140 | 0 | 0 | 0 | 0 |
| 101 | 171 | 0 | 0 | 0 | 0 |
| 0 | 0 | 227 | 127 | 0 | 0 |
| 0 | 0 | 118 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 225 | 64 |
| 0 | 0 | 0 | 0 | 173 | 16 |
G:=sub<GL(6,GF(241))| [0,1,0,0,0,0,240,1,0,0,0,0,0,0,51,240,0,0,0,0,1,0,0,0,0,0,0,0,85,200,0,0,0,0,41,122],[62,66,0,0,0,0,128,179,0,0,0,0,0,0,68,81,0,0,0,0,175,173,0,0,0,0,0,0,81,159,0,0,0,0,33,160],[70,101,0,0,0,0,140,171,0,0,0,0,0,0,227,118,0,0,0,0,127,14,0,0,0,0,0,0,225,173,0,0,0,0,64,16] >;
Dic30⋊12C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{30}\rtimes_{12}C_4 % in TeX
G:=Group("Dic30:12C4"); // GroupNames label
G:=SmallGroup(480,50);
// by ID
G=gap.SmallGroup(480,50);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,92,675,80,1356,18822]);
// 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^19,c*b*c^-1=a^15*b>;
// generators/relations