metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic30⋊14C4, Dic3⋊6Dic10, C15⋊10(C4×Q8), C3⋊3(C4×Dic10), C20.67(C4×S3), (C5×Dic3)⋊5Q8, C12.25(C4×D5), C30.15(C2×Q8), C10.23(S3×Q8), C60.104(C2×C4), C30.9(C4○D4), (C2×C20).291D6, C4⋊Dic5.14S3, (C4×Dic3).2D5, C2.3(S3×Dic10), C6.5(C2×Dic10), C6.46(C4○D20), (C2×C12).109D10, C5⋊3(Dic6⋊C4), C4.5(D30.C2), (C2×C30).30C23, (C2×Dic5).89D6, (Dic3×C20).2C2, Dic15⋊5C4.8C2, C2.2(D20⋊5S3), C30.116(C22×C4), (C2×C60).110C22, Dic15.26(C2×C4), (C2×Dic30).14C2, C10.19(D4⋊2S3), (Dic3×Dic5).15C2, (C2×Dic3).174D10, (C6×Dic5).15C22, (C2×Dic15).37C22, (C10×Dic3).157C22, C6.44(C2×C4×D5), C10.77(S3×C2×C4), C22.29(C2×S3×D5), (C2×C4).101(S3×D5), (C3×C4⋊Dic5).4C2, C2.9(C2×D30.C2), (C2×C6).42(C22×D5), (C2×C10).42(C22×S3), SmallGroup(480,416)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic30⋊14C4
G = < a,b,c | a60=c4=1, b2=a30, bab-1=a-1, cac-1=a41, bc=cb >
Subgroups: 556 in 140 conjugacy classes, 64 normal (34 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×Dic10, Dic6⋊C4, C6×Dic5, C10×Dic3, Dic30, C2×Dic15, C2×C60, C4×Dic10, Dic3×Dic5, Dic15⋊5C4, C3×C4⋊Dic5, Dic3×C20, C2×Dic30, Dic30⋊14C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D5, D6, C22×C4, C2×Q8, C4○D4, D10, C4×S3, C22×S3, C4×Q8, Dic10, C4×D5, C22×D5, S3×C2×C4, D4⋊2S3, S3×Q8, S3×D5, C2×Dic10, C2×C4×D5, C4○D20, Dic6⋊C4, D30.C2, C2×S3×D5, C4×Dic10, D20⋊5S3, S3×Dic10, C2×D30.C2, Dic30⋊14C4
►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)
(1 470 31 440)(2 469 32 439)(3 468 33 438)(4 467 34 437)(5 466 35 436)(6 465 36 435)(7 464 37 434)(8 463 38 433)(9 462 39 432)(10 461 40 431)(11 460 41 430)(12 459 42 429)(13 458 43 428)(14 457 44 427)(15 456 45 426)(16 455 46 425)(17 454 47 424)(18 453 48 423)(19 452 49 422)(20 451 50 421)(21 450 51 480)(22 449 52 479)(23 448 53 478)(24 447 54 477)(25 446 55 476)(26 445 56 475)(27 444 57 474)(28 443 58 473)(29 442 59 472)(30 441 60 471)(61 401 91 371)(62 400 92 370)(63 399 93 369)(64 398 94 368)(65 397 95 367)(66 396 96 366)(67 395 97 365)(68 394 98 364)(69 393 99 363)(70 392 100 362)(71 391 101 361)(72 390 102 420)(73 389 103 419)(74 388 104 418)(75 387 105 417)(76 386 106 416)(77 385 107 415)(78 384 108 414)(79 383 109 413)(80 382 110 412)(81 381 111 411)(82 380 112 410)(83 379 113 409)(84 378 114 408)(85 377 115 407)(86 376 116 406)(87 375 117 405)(88 374 118 404)(89 373 119 403)(90 372 120 402)(121 263 151 293)(122 262 152 292)(123 261 153 291)(124 260 154 290)(125 259 155 289)(126 258 156 288)(127 257 157 287)(128 256 158 286)(129 255 159 285)(130 254 160 284)(131 253 161 283)(132 252 162 282)(133 251 163 281)(134 250 164 280)(135 249 165 279)(136 248 166 278)(137 247 167 277)(138 246 168 276)(139 245 169 275)(140 244 170 274)(141 243 171 273)(142 242 172 272)(143 241 173 271)(144 300 174 270)(145 299 175 269)(146 298 176 268)(147 297 177 267)(148 296 178 266)(149 295 179 265)(150 294 180 264)(181 360 211 330)(182 359 212 329)(183 358 213 328)(184 357 214 327)(185 356 215 326)(186 355 216 325)(187 354 217 324)(188 353 218 323)(189 352 219 322)(190 351 220 321)(191 350 221 320)(192 349 222 319)(193 348 223 318)(194 347 224 317)(195 346 225 316)(196 345 226 315)(197 344 227 314)(198 343 228 313)(199 342 229 312)(200 341 230 311)(201 340 231 310)(202 339 232 309)(203 338 233 308)(204 337 234 307)(205 336 235 306)(206 335 236 305)(207 334 237 304)(208 333 238 303)(209 332 239 302)(210 331 240 301)
(1 118 184 245)(2 99 185 286)(3 80 186 267)(4 61 187 248)(5 102 188 289)(6 83 189 270)(7 64 190 251)(8 105 191 292)(9 86 192 273)(10 67 193 254)(11 108 194 295)(12 89 195 276)(13 70 196 257)(14 111 197 298)(15 92 198 279)(16 73 199 260)(17 114 200 241)(18 95 201 282)(19 76 202 263)(20 117 203 244)(21 98 204 285)(22 79 205 266)(23 120 206 247)(24 101 207 288)(25 82 208 269)(26 63 209 250)(27 104 210 291)(28 85 211 272)(29 66 212 253)(30 107 213 294)(31 88 214 275)(32 69 215 256)(33 110 216 297)(34 91 217 278)(35 72 218 259)(36 113 219 300)(37 94 220 281)(38 75 221 262)(39 116 222 243)(40 97 223 284)(41 78 224 265)(42 119 225 246)(43 100 226 287)(44 81 227 268)(45 62 228 249)(46 103 229 290)(47 84 230 271)(48 65 231 252)(49 106 232 293)(50 87 233 274)(51 68 234 255)(52 109 235 296)(53 90 236 277)(54 71 237 258)(55 112 238 299)(56 93 239 280)(57 74 240 261)(58 115 181 242)(59 96 182 283)(60 77 183 264)(121 422 416 309)(122 463 417 350)(123 444 418 331)(124 425 419 312)(125 466 420 353)(126 447 361 334)(127 428 362 315)(128 469 363 356)(129 450 364 337)(130 431 365 318)(131 472 366 359)(132 453 367 340)(133 434 368 321)(134 475 369 302)(135 456 370 343)(136 437 371 324)(137 478 372 305)(138 459 373 346)(139 440 374 327)(140 421 375 308)(141 462 376 349)(142 443 377 330)(143 424 378 311)(144 465 379 352)(145 446 380 333)(146 427 381 314)(147 468 382 355)(148 449 383 336)(149 430 384 317)(150 471 385 358)(151 452 386 339)(152 433 387 320)(153 474 388 301)(154 455 389 342)(155 436 390 323)(156 477 391 304)(157 458 392 345)(158 439 393 326)(159 480 394 307)(160 461 395 348)(161 442 396 329)(162 423 397 310)(163 464 398 351)(164 445 399 332)(165 426 400 313)(166 467 401 354)(167 448 402 335)(168 429 403 316)(169 470 404 357)(170 451 405 338)(171 432 406 319)(172 473 407 360)(173 454 408 341)(174 435 409 322)(175 476 410 303)(176 457 411 344)(177 438 412 325)(178 479 413 306)(179 460 414 347)(180 441 415 328)
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), (1,470,31,440)(2,469,32,439)(3,468,33,438)(4,467,34,437)(5,466,35,436)(6,465,36,435)(7,464,37,434)(8,463,38,433)(9,462,39,432)(10,461,40,431)(11,460,41,430)(12,459,42,429)(13,458,43,428)(14,457,44,427)(15,456,45,426)(16,455,46,425)(17,454,47,424)(18,453,48,423)(19,452,49,422)(20,451,50,421)(21,450,51,480)(22,449,52,479)(23,448,53,478)(24,447,54,477)(25,446,55,476)(26,445,56,475)(27,444,57,474)(28,443,58,473)(29,442,59,472)(30,441,60,471)(61,401,91,371)(62,400,92,370)(63,399,93,369)(64,398,94,368)(65,397,95,367)(66,396,96,366)(67,395,97,365)(68,394,98,364)(69,393,99,363)(70,392,100,362)(71,391,101,361)(72,390,102,420)(73,389,103,419)(74,388,104,418)(75,387,105,417)(76,386,106,416)(77,385,107,415)(78,384,108,414)(79,383,109,413)(80,382,110,412)(81,381,111,411)(82,380,112,410)(83,379,113,409)(84,378,114,408)(85,377,115,407)(86,376,116,406)(87,375,117,405)(88,374,118,404)(89,373,119,403)(90,372,120,402)(121,263,151,293)(122,262,152,292)(123,261,153,291)(124,260,154,290)(125,259,155,289)(126,258,156,288)(127,257,157,287)(128,256,158,286)(129,255,159,285)(130,254,160,284)(131,253,161,283)(132,252,162,282)(133,251,163,281)(134,250,164,280)(135,249,165,279)(136,248,166,278)(137,247,167,277)(138,246,168,276)(139,245,169,275)(140,244,170,274)(141,243,171,273)(142,242,172,272)(143,241,173,271)(144,300,174,270)(145,299,175,269)(146,298,176,268)(147,297,177,267)(148,296,178,266)(149,295,179,265)(150,294,180,264)(181,360,211,330)(182,359,212,329)(183,358,213,328)(184,357,214,327)(185,356,215,326)(186,355,216,325)(187,354,217,324)(188,353,218,323)(189,352,219,322)(190,351,220,321)(191,350,221,320)(192,349,222,319)(193,348,223,318)(194,347,224,317)(195,346,225,316)(196,345,226,315)(197,344,227,314)(198,343,228,313)(199,342,229,312)(200,341,230,311)(201,340,231,310)(202,339,232,309)(203,338,233,308)(204,337,234,307)(205,336,235,306)(206,335,236,305)(207,334,237,304)(208,333,238,303)(209,332,239,302)(210,331,240,301), (1,118,184,245)(2,99,185,286)(3,80,186,267)(4,61,187,248)(5,102,188,289)(6,83,189,270)(7,64,190,251)(8,105,191,292)(9,86,192,273)(10,67,193,254)(11,108,194,295)(12,89,195,276)(13,70,196,257)(14,111,197,298)(15,92,198,279)(16,73,199,260)(17,114,200,241)(18,95,201,282)(19,76,202,263)(20,117,203,244)(21,98,204,285)(22,79,205,266)(23,120,206,247)(24,101,207,288)(25,82,208,269)(26,63,209,250)(27,104,210,291)(28,85,211,272)(29,66,212,253)(30,107,213,294)(31,88,214,275)(32,69,215,256)(33,110,216,297)(34,91,217,278)(35,72,218,259)(36,113,219,300)(37,94,220,281)(38,75,221,262)(39,116,222,243)(40,97,223,284)(41,78,224,265)(42,119,225,246)(43,100,226,287)(44,81,227,268)(45,62,228,249)(46,103,229,290)(47,84,230,271)(48,65,231,252)(49,106,232,293)(50,87,233,274)(51,68,234,255)(52,109,235,296)(53,90,236,277)(54,71,237,258)(55,112,238,299)(56,93,239,280)(57,74,240,261)(58,115,181,242)(59,96,182,283)(60,77,183,264)(121,422,416,309)(122,463,417,350)(123,444,418,331)(124,425,419,312)(125,466,420,353)(126,447,361,334)(127,428,362,315)(128,469,363,356)(129,450,364,337)(130,431,365,318)(131,472,366,359)(132,453,367,340)(133,434,368,321)(134,475,369,302)(135,456,370,343)(136,437,371,324)(137,478,372,305)(138,459,373,346)(139,440,374,327)(140,421,375,308)(141,462,376,349)(142,443,377,330)(143,424,378,311)(144,465,379,352)(145,446,380,333)(146,427,381,314)(147,468,382,355)(148,449,383,336)(149,430,384,317)(150,471,385,358)(151,452,386,339)(152,433,387,320)(153,474,388,301)(154,455,389,342)(155,436,390,323)(156,477,391,304)(157,458,392,345)(158,439,393,326)(159,480,394,307)(160,461,395,348)(161,442,396,329)(162,423,397,310)(163,464,398,351)(164,445,399,332)(165,426,400,313)(166,467,401,354)(167,448,402,335)(168,429,403,316)(169,470,404,357)(170,451,405,338)(171,432,406,319)(172,473,407,360)(173,454,408,341)(174,435,409,322)(175,476,410,303)(176,457,411,344)(177,438,412,325)(178,479,413,306)(179,460,414,347)(180,441,415,328)>;
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), (1,470,31,440)(2,469,32,439)(3,468,33,438)(4,467,34,437)(5,466,35,436)(6,465,36,435)(7,464,37,434)(8,463,38,433)(9,462,39,432)(10,461,40,431)(11,460,41,430)(12,459,42,429)(13,458,43,428)(14,457,44,427)(15,456,45,426)(16,455,46,425)(17,454,47,424)(18,453,48,423)(19,452,49,422)(20,451,50,421)(21,450,51,480)(22,449,52,479)(23,448,53,478)(24,447,54,477)(25,446,55,476)(26,445,56,475)(27,444,57,474)(28,443,58,473)(29,442,59,472)(30,441,60,471)(61,401,91,371)(62,400,92,370)(63,399,93,369)(64,398,94,368)(65,397,95,367)(66,396,96,366)(67,395,97,365)(68,394,98,364)(69,393,99,363)(70,392,100,362)(71,391,101,361)(72,390,102,420)(73,389,103,419)(74,388,104,418)(75,387,105,417)(76,386,106,416)(77,385,107,415)(78,384,108,414)(79,383,109,413)(80,382,110,412)(81,381,111,411)(82,380,112,410)(83,379,113,409)(84,378,114,408)(85,377,115,407)(86,376,116,406)(87,375,117,405)(88,374,118,404)(89,373,119,403)(90,372,120,402)(121,263,151,293)(122,262,152,292)(123,261,153,291)(124,260,154,290)(125,259,155,289)(126,258,156,288)(127,257,157,287)(128,256,158,286)(129,255,159,285)(130,254,160,284)(131,253,161,283)(132,252,162,282)(133,251,163,281)(134,250,164,280)(135,249,165,279)(136,248,166,278)(137,247,167,277)(138,246,168,276)(139,245,169,275)(140,244,170,274)(141,243,171,273)(142,242,172,272)(143,241,173,271)(144,300,174,270)(145,299,175,269)(146,298,176,268)(147,297,177,267)(148,296,178,266)(149,295,179,265)(150,294,180,264)(181,360,211,330)(182,359,212,329)(183,358,213,328)(184,357,214,327)(185,356,215,326)(186,355,216,325)(187,354,217,324)(188,353,218,323)(189,352,219,322)(190,351,220,321)(191,350,221,320)(192,349,222,319)(193,348,223,318)(194,347,224,317)(195,346,225,316)(196,345,226,315)(197,344,227,314)(198,343,228,313)(199,342,229,312)(200,341,230,311)(201,340,231,310)(202,339,232,309)(203,338,233,308)(204,337,234,307)(205,336,235,306)(206,335,236,305)(207,334,237,304)(208,333,238,303)(209,332,239,302)(210,331,240,301), (1,118,184,245)(2,99,185,286)(3,80,186,267)(4,61,187,248)(5,102,188,289)(6,83,189,270)(7,64,190,251)(8,105,191,292)(9,86,192,273)(10,67,193,254)(11,108,194,295)(12,89,195,276)(13,70,196,257)(14,111,197,298)(15,92,198,279)(16,73,199,260)(17,114,200,241)(18,95,201,282)(19,76,202,263)(20,117,203,244)(21,98,204,285)(22,79,205,266)(23,120,206,247)(24,101,207,288)(25,82,208,269)(26,63,209,250)(27,104,210,291)(28,85,211,272)(29,66,212,253)(30,107,213,294)(31,88,214,275)(32,69,215,256)(33,110,216,297)(34,91,217,278)(35,72,218,259)(36,113,219,300)(37,94,220,281)(38,75,221,262)(39,116,222,243)(40,97,223,284)(41,78,224,265)(42,119,225,246)(43,100,226,287)(44,81,227,268)(45,62,228,249)(46,103,229,290)(47,84,230,271)(48,65,231,252)(49,106,232,293)(50,87,233,274)(51,68,234,255)(52,109,235,296)(53,90,236,277)(54,71,237,258)(55,112,238,299)(56,93,239,280)(57,74,240,261)(58,115,181,242)(59,96,182,283)(60,77,183,264)(121,422,416,309)(122,463,417,350)(123,444,418,331)(124,425,419,312)(125,466,420,353)(126,447,361,334)(127,428,362,315)(128,469,363,356)(129,450,364,337)(130,431,365,318)(131,472,366,359)(132,453,367,340)(133,434,368,321)(134,475,369,302)(135,456,370,343)(136,437,371,324)(137,478,372,305)(138,459,373,346)(139,440,374,327)(140,421,375,308)(141,462,376,349)(142,443,377,330)(143,424,378,311)(144,465,379,352)(145,446,380,333)(146,427,381,314)(147,468,382,355)(148,449,383,336)(149,430,384,317)(150,471,385,358)(151,452,386,339)(152,433,387,320)(153,474,388,301)(154,455,389,342)(155,436,390,323)(156,477,391,304)(157,458,392,345)(158,439,393,326)(159,480,394,307)(160,461,395,348)(161,442,396,329)(162,423,397,310)(163,464,398,351)(164,445,399,332)(165,426,400,313)(166,467,401,354)(167,448,402,335)(168,429,403,316)(169,470,404,357)(170,451,405,338)(171,432,406,319)(172,473,407,360)(173,454,408,341)(174,435,409,322)(175,476,410,303)(176,457,411,344)(177,438,412,325)(178,479,413,306)(179,460,414,347)(180,441,415,328) );
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)], [(1,470,31,440),(2,469,32,439),(3,468,33,438),(4,467,34,437),(5,466,35,436),(6,465,36,435),(7,464,37,434),(8,463,38,433),(9,462,39,432),(10,461,40,431),(11,460,41,430),(12,459,42,429),(13,458,43,428),(14,457,44,427),(15,456,45,426),(16,455,46,425),(17,454,47,424),(18,453,48,423),(19,452,49,422),(20,451,50,421),(21,450,51,480),(22,449,52,479),(23,448,53,478),(24,447,54,477),(25,446,55,476),(26,445,56,475),(27,444,57,474),(28,443,58,473),(29,442,59,472),(30,441,60,471),(61,401,91,371),(62,400,92,370),(63,399,93,369),(64,398,94,368),(65,397,95,367),(66,396,96,366),(67,395,97,365),(68,394,98,364),(69,393,99,363),(70,392,100,362),(71,391,101,361),(72,390,102,420),(73,389,103,419),(74,388,104,418),(75,387,105,417),(76,386,106,416),(77,385,107,415),(78,384,108,414),(79,383,109,413),(80,382,110,412),(81,381,111,411),(82,380,112,410),(83,379,113,409),(84,378,114,408),(85,377,115,407),(86,376,116,406),(87,375,117,405),(88,374,118,404),(89,373,119,403),(90,372,120,402),(121,263,151,293),(122,262,152,292),(123,261,153,291),(124,260,154,290),(125,259,155,289),(126,258,156,288),(127,257,157,287),(128,256,158,286),(129,255,159,285),(130,254,160,284),(131,253,161,283),(132,252,162,282),(133,251,163,281),(134,250,164,280),(135,249,165,279),(136,248,166,278),(137,247,167,277),(138,246,168,276),(139,245,169,275),(140,244,170,274),(141,243,171,273),(142,242,172,272),(143,241,173,271),(144,300,174,270),(145,299,175,269),(146,298,176,268),(147,297,177,267),(148,296,178,266),(149,295,179,265),(150,294,180,264),(181,360,211,330),(182,359,212,329),(183,358,213,328),(184,357,214,327),(185,356,215,326),(186,355,216,325),(187,354,217,324),(188,353,218,323),(189,352,219,322),(190,351,220,321),(191,350,221,320),(192,349,222,319),(193,348,223,318),(194,347,224,317),(195,346,225,316),(196,345,226,315),(197,344,227,314),(198,343,228,313),(199,342,229,312),(200,341,230,311),(201,340,231,310),(202,339,232,309),(203,338,233,308),(204,337,234,307),(205,336,235,306),(206,335,236,305),(207,334,237,304),(208,333,238,303),(209,332,239,302),(210,331,240,301)], [(1,118,184,245),(2,99,185,286),(3,80,186,267),(4,61,187,248),(5,102,188,289),(6,83,189,270),(7,64,190,251),(8,105,191,292),(9,86,192,273),(10,67,193,254),(11,108,194,295),(12,89,195,276),(13,70,196,257),(14,111,197,298),(15,92,198,279),(16,73,199,260),(17,114,200,241),(18,95,201,282),(19,76,202,263),(20,117,203,244),(21,98,204,285),(22,79,205,266),(23,120,206,247),(24,101,207,288),(25,82,208,269),(26,63,209,250),(27,104,210,291),(28,85,211,272),(29,66,212,253),(30,107,213,294),(31,88,214,275),(32,69,215,256),(33,110,216,297),(34,91,217,278),(35,72,218,259),(36,113,219,300),(37,94,220,281),(38,75,221,262),(39,116,222,243),(40,97,223,284),(41,78,224,265),(42,119,225,246),(43,100,226,287),(44,81,227,268),(45,62,228,249),(46,103,229,290),(47,84,230,271),(48,65,231,252),(49,106,232,293),(50,87,233,274),(51,68,234,255),(52,109,235,296),(53,90,236,277),(54,71,237,258),(55,112,238,299),(56,93,239,280),(57,74,240,261),(58,115,181,242),(59,96,182,283),(60,77,183,264),(121,422,416,309),(122,463,417,350),(123,444,418,331),(124,425,419,312),(125,466,420,353),(126,447,361,334),(127,428,362,315),(128,469,363,356),(129,450,364,337),(130,431,365,318),(131,472,366,359),(132,453,367,340),(133,434,368,321),(134,475,369,302),(135,456,370,343),(136,437,371,324),(137,478,372,305),(138,459,373,346),(139,440,374,327),(140,421,375,308),(141,462,376,349),(142,443,377,330),(143,424,378,311),(144,465,379,352),(145,446,380,333),(146,427,381,314),(147,468,382,355),(148,449,383,336),(149,430,384,317),(150,471,385,358),(151,452,386,339),(152,433,387,320),(153,474,388,301),(154,455,389,342),(155,436,390,323),(156,477,391,304),(157,458,392,345),(158,439,393,326),(159,480,394,307),(160,461,395,348),(161,442,396,329),(162,423,397,310),(163,464,398,351),(164,445,399,332),(165,426,400,313),(166,467,401,354),(167,448,402,335),(168,429,403,316),(169,470,404,357),(170,451,405,338),(171,432,406,319),(172,473,407,360),(173,454,408,341),(174,435,409,322),(175,476,410,303),(176,457,411,344),(177,438,412,325),(178,479,413,306),(179,460,414,347),(180,441,415,328)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 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 | 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 | 3 | 3 | 3 | 3 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C4×S3 | Dic10 | C4×D5 | C4○D20 | D4⋊2S3 | S3×Q8 | S3×D5 | D30.C2 | C2×S3×D5 | D20⋊5S3 | S3×Dic10 |
| kernel | Dic30⋊14C4 | Dic3×Dic5 | Dic15⋊5C4 | C3×C4⋊Dic5 | Dic3×C20 | C2×Dic30 | Dic30 | C4⋊Dic5 | C5×Dic3 | C4×Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C20 | Dic3 | C12 | C6 | C10 | C10 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic30⋊14C4 ►in GL4(𝔽61) generated by
| 23 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 48 | 8 |
| 0 | 0 | 0 | 14 |
| 0 | 50 | 0 | 0 |
| 50 | 0 | 0 | 0 |
| 0 | 0 | 35 | 3 |
| 0 | 0 | 19 | 26 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 42 | 28 |
| 0 | 0 | 35 | 19 |
G:=sub<GL(4,GF(61))| [23,0,0,0,0,8,0,0,0,0,48,0,0,0,8,14],[0,50,0,0,50,0,0,0,0,0,35,19,0,0,3,26],[50,0,0,0,0,50,0,0,0,0,42,35,0,0,28,19] >;
Dic30⋊14C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{30}\rtimes_{14}C_4 % in TeX
G:=Group("Dic30:14C4"); // GroupNames label
G:=SmallGroup(480,416);
// by ID
G=gap.SmallGroup(480,416);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,422,100,1356,18822]);
// 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^41,b*c=c*b>;
// generators/relations