metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic30⋊11C4, Dic15⋊10Q8, C15⋊18(C4×Q8), C4⋊C4.7D15, C4.4(C4×D15), C2.1(Q8×D15), C6.38(Q8×D5), C20.59(C4×S3), C60.84(C2×C4), (C2×C4).28D30, C12.27(C4×D5), C30.91(C2×Q8), C10.38(S3×Q8), (C2×C20).208D6, (C2×C12).206D10, C5⋊5(Dic6⋊C4), C3⋊4(Dic5⋊3Q8), (C2×Dic30).8C2, C30.4Q8.5C2, C30.218(C4○D4), C2.3(D4⋊2D15), C6.95(D4⋊2D5), C30.159(C22×C4), (C2×C60).176C22, (C2×C30).284C23, (C4×Dic15).10C2, Dic15.30(C2×C4), C10.95(D4⋊2S3), C22.15(C22×D15), (C2×Dic15).160C22, C6.64(C2×C4×D5), (C5×C4⋊C4).4S3, C10.96(S3×C2×C4), (C3×C4⋊C4).4D5, C2.10(C2×C4×D15), (C15×C4⋊C4).4C2, (C2×C6).280(C22×D5), (C2×C10).279(C22×S3), SmallGroup(480,852)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic30⋊11C4
G = < a,b,c | a60=c4=1, b2=a30, bab-1=a-1, cac-1=a31, bc=cb >
Subgroups: 612 in 140 conjugacy classes, 65 normal (33 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×9], C22, C5, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10 [×3], Dic3 [×7], C12 [×2], C12 [×2], C2×C6, C15, C42 [×3], C4⋊C4, C4⋊C4 [×2], C2×Q8, Dic5 [×7], C20 [×2], C20 [×2], C2×C10, Dic6 [×4], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C30 [×3], C4×Q8, Dic10 [×4], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C4×Dic3 [×3], Dic3⋊C4 [×2], C3×C4⋊C4, C2×Dic6, Dic15 [×6], Dic15, C60 [×2], C60 [×2], C2×C30, C4×Dic5 [×3], C10.D4 [×2], C5×C4⋊C4, C2×Dic10, Dic6⋊C4, Dic30 [×4], C2×Dic15 [×2], C2×Dic15 [×2], C2×C60, C2×C60 [×2], Dic5⋊3Q8, C4×Dic15, C4×Dic15 [×2], C30.4Q8 [×2], C15×C4⋊C4, C2×Dic30, Dic30⋊11C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D5, D6 [×3], C22×C4, C2×Q8, C4○D4, D10 [×3], C4×S3 [×2], C22×S3, D15, C4×Q8, C4×D5 [×2], C22×D5, S3×C2×C4, D4⋊2S3, S3×Q8, D30 [×3], C2×C4×D5, D4⋊2D5, Q8×D5, Dic6⋊C4, C4×D15 [×2], C22×D15, Dic5⋊3Q8, C2×C4×D15, D4⋊2D15, Q8×D15, Dic30⋊11C4
►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)
(1 412 31 382)(2 411 32 381)(3 410 33 380)(4 409 34 379)(5 408 35 378)(6 407 36 377)(7 406 37 376)(8 405 38 375)(9 404 39 374)(10 403 40 373)(11 402 41 372)(12 401 42 371)(13 400 43 370)(14 399 44 369)(15 398 45 368)(16 397 46 367)(17 396 47 366)(18 395 48 365)(19 394 49 364)(20 393 50 363)(21 392 51 362)(22 391 52 361)(23 390 53 420)(24 389 54 419)(25 388 55 418)(26 387 56 417)(27 386 57 416)(28 385 58 415)(29 384 59 414)(30 383 60 413)(61 124 91 154)(62 123 92 153)(63 122 93 152)(64 121 94 151)(65 180 95 150)(66 179 96 149)(67 178 97 148)(68 177 98 147)(69 176 99 146)(70 175 100 145)(71 174 101 144)(72 173 102 143)(73 172 103 142)(74 171 104 141)(75 170 105 140)(76 169 106 139)(77 168 107 138)(78 167 108 137)(79 166 109 136)(80 165 110 135)(81 164 111 134)(82 163 112 133)(83 162 113 132)(84 161 114 131)(85 160 115 130)(86 159 116 129)(87 158 117 128)(88 157 118 127)(89 156 119 126)(90 155 120 125)(181 280 211 250)(182 279 212 249)(183 278 213 248)(184 277 214 247)(185 276 215 246)(186 275 216 245)(187 274 217 244)(188 273 218 243)(189 272 219 242)(190 271 220 241)(191 270 221 300)(192 269 222 299)(193 268 223 298)(194 267 224 297)(195 266 225 296)(196 265 226 295)(197 264 227 294)(198 263 228 293)(199 262 229 292)(200 261 230 291)(201 260 231 290)(202 259 232 289)(203 258 233 288)(204 257 234 287)(205 256 235 286)(206 255 236 285)(207 254 237 284)(208 253 238 283)(209 252 239 282)(210 251 240 281)(301 432 331 462)(302 431 332 461)(303 430 333 460)(304 429 334 459)(305 428 335 458)(306 427 336 457)(307 426 337 456)(308 425 338 455)(309 424 339 454)(310 423 340 453)(311 422 341 452)(312 421 342 451)(313 480 343 450)(314 479 344 449)(315 478 345 448)(316 477 346 447)(317 476 347 446)(318 475 348 445)(319 474 349 444)(320 473 350 443)(321 472 351 442)(322 471 352 441)(323 470 353 440)(324 469 354 439)(325 468 355 438)(326 467 356 437)(327 466 357 436)(328 465 358 435)(329 464 359 434)(330 463 360 433)
(1 62 468 257)(2 93 469 288)(3 64 470 259)(4 95 471 290)(5 66 472 261)(6 97 473 292)(7 68 474 263)(8 99 475 294)(9 70 476 265)(10 101 477 296)(11 72 478 267)(12 103 479 298)(13 74 480 269)(14 105 421 300)(15 76 422 271)(16 107 423 242)(17 78 424 273)(18 109 425 244)(19 80 426 275)(20 111 427 246)(21 82 428 277)(22 113 429 248)(23 84 430 279)(24 115 431 250)(25 86 432 281)(26 117 433 252)(27 88 434 283)(28 119 435 254)(29 90 436 285)(30 61 437 256)(31 92 438 287)(32 63 439 258)(33 94 440 289)(34 65 441 260)(35 96 442 291)(36 67 443 262)(37 98 444 293)(38 69 445 264)(39 100 446 295)(40 71 447 266)(41 102 448 297)(42 73 449 268)(43 104 450 299)(44 75 451 270)(45 106 452 241)(46 77 453 272)(47 108 454 243)(48 79 455 274)(49 110 456 245)(50 81 457 276)(51 112 458 247)(52 83 459 278)(53 114 460 249)(54 85 461 280)(55 116 462 251)(56 87 463 282)(57 118 464 253)(58 89 465 284)(59 120 466 255)(60 91 467 286)(121 353 232 410)(122 324 233 381)(123 355 234 412)(124 326 235 383)(125 357 236 414)(126 328 237 385)(127 359 238 416)(128 330 239 387)(129 301 240 418)(130 332 181 389)(131 303 182 420)(132 334 183 391)(133 305 184 362)(134 336 185 393)(135 307 186 364)(136 338 187 395)(137 309 188 366)(138 340 189 397)(139 311 190 368)(140 342 191 399)(141 313 192 370)(142 344 193 401)(143 315 194 372)(144 346 195 403)(145 317 196 374)(146 348 197 405)(147 319 198 376)(148 350 199 407)(149 321 200 378)(150 352 201 409)(151 323 202 380)(152 354 203 411)(153 325 204 382)(154 356 205 413)(155 327 206 384)(156 358 207 415)(157 329 208 386)(158 360 209 417)(159 331 210 388)(160 302 211 419)(161 333 212 390)(162 304 213 361)(163 335 214 392)(164 306 215 363)(165 337 216 394)(166 308 217 365)(167 339 218 396)(168 310 219 367)(169 341 220 398)(170 312 221 369)(171 343 222 400)(172 314 223 371)(173 345 224 402)(174 316 225 373)(175 347 226 404)(176 318 227 375)(177 349 228 406)(178 320 229 377)(179 351 230 408)(180 322 231 379)
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), (1,412,31,382)(2,411,32,381)(3,410,33,380)(4,409,34,379)(5,408,35,378)(6,407,36,377)(7,406,37,376)(8,405,38,375)(9,404,39,374)(10,403,40,373)(11,402,41,372)(12,401,42,371)(13,400,43,370)(14,399,44,369)(15,398,45,368)(16,397,46,367)(17,396,47,366)(18,395,48,365)(19,394,49,364)(20,393,50,363)(21,392,51,362)(22,391,52,361)(23,390,53,420)(24,389,54,419)(25,388,55,418)(26,387,56,417)(27,386,57,416)(28,385,58,415)(29,384,59,414)(30,383,60,413)(61,124,91,154)(62,123,92,153)(63,122,93,152)(64,121,94,151)(65,180,95,150)(66,179,96,149)(67,178,97,148)(68,177,98,147)(69,176,99,146)(70,175,100,145)(71,174,101,144)(72,173,102,143)(73,172,103,142)(74,171,104,141)(75,170,105,140)(76,169,106,139)(77,168,107,138)(78,167,108,137)(79,166,109,136)(80,165,110,135)(81,164,111,134)(82,163,112,133)(83,162,113,132)(84,161,114,131)(85,160,115,130)(86,159,116,129)(87,158,117,128)(88,157,118,127)(89,156,119,126)(90,155,120,125)(181,280,211,250)(182,279,212,249)(183,278,213,248)(184,277,214,247)(185,276,215,246)(186,275,216,245)(187,274,217,244)(188,273,218,243)(189,272,219,242)(190,271,220,241)(191,270,221,300)(192,269,222,299)(193,268,223,298)(194,267,224,297)(195,266,225,296)(196,265,226,295)(197,264,227,294)(198,263,228,293)(199,262,229,292)(200,261,230,291)(201,260,231,290)(202,259,232,289)(203,258,233,288)(204,257,234,287)(205,256,235,286)(206,255,236,285)(207,254,237,284)(208,253,238,283)(209,252,239,282)(210,251,240,281)(301,432,331,462)(302,431,332,461)(303,430,333,460)(304,429,334,459)(305,428,335,458)(306,427,336,457)(307,426,337,456)(308,425,338,455)(309,424,339,454)(310,423,340,453)(311,422,341,452)(312,421,342,451)(313,480,343,450)(314,479,344,449)(315,478,345,448)(316,477,346,447)(317,476,347,446)(318,475,348,445)(319,474,349,444)(320,473,350,443)(321,472,351,442)(322,471,352,441)(323,470,353,440)(324,469,354,439)(325,468,355,438)(326,467,356,437)(327,466,357,436)(328,465,358,435)(329,464,359,434)(330,463,360,433), (1,62,468,257)(2,93,469,288)(3,64,470,259)(4,95,471,290)(5,66,472,261)(6,97,473,292)(7,68,474,263)(8,99,475,294)(9,70,476,265)(10,101,477,296)(11,72,478,267)(12,103,479,298)(13,74,480,269)(14,105,421,300)(15,76,422,271)(16,107,423,242)(17,78,424,273)(18,109,425,244)(19,80,426,275)(20,111,427,246)(21,82,428,277)(22,113,429,248)(23,84,430,279)(24,115,431,250)(25,86,432,281)(26,117,433,252)(27,88,434,283)(28,119,435,254)(29,90,436,285)(30,61,437,256)(31,92,438,287)(32,63,439,258)(33,94,440,289)(34,65,441,260)(35,96,442,291)(36,67,443,262)(37,98,444,293)(38,69,445,264)(39,100,446,295)(40,71,447,266)(41,102,448,297)(42,73,449,268)(43,104,450,299)(44,75,451,270)(45,106,452,241)(46,77,453,272)(47,108,454,243)(48,79,455,274)(49,110,456,245)(50,81,457,276)(51,112,458,247)(52,83,459,278)(53,114,460,249)(54,85,461,280)(55,116,462,251)(56,87,463,282)(57,118,464,253)(58,89,465,284)(59,120,466,255)(60,91,467,286)(121,353,232,410)(122,324,233,381)(123,355,234,412)(124,326,235,383)(125,357,236,414)(126,328,237,385)(127,359,238,416)(128,330,239,387)(129,301,240,418)(130,332,181,389)(131,303,182,420)(132,334,183,391)(133,305,184,362)(134,336,185,393)(135,307,186,364)(136,338,187,395)(137,309,188,366)(138,340,189,397)(139,311,190,368)(140,342,191,399)(141,313,192,370)(142,344,193,401)(143,315,194,372)(144,346,195,403)(145,317,196,374)(146,348,197,405)(147,319,198,376)(148,350,199,407)(149,321,200,378)(150,352,201,409)(151,323,202,380)(152,354,203,411)(153,325,204,382)(154,356,205,413)(155,327,206,384)(156,358,207,415)(157,329,208,386)(158,360,209,417)(159,331,210,388)(160,302,211,419)(161,333,212,390)(162,304,213,361)(163,335,214,392)(164,306,215,363)(165,337,216,394)(166,308,217,365)(167,339,218,396)(168,310,219,367)(169,341,220,398)(170,312,221,369)(171,343,222,400)(172,314,223,371)(173,345,224,402)(174,316,225,373)(175,347,226,404)(176,318,227,375)(177,349,228,406)(178,320,229,377)(179,351,230,408)(180,322,231,379)>;
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), (1,412,31,382)(2,411,32,381)(3,410,33,380)(4,409,34,379)(5,408,35,378)(6,407,36,377)(7,406,37,376)(8,405,38,375)(9,404,39,374)(10,403,40,373)(11,402,41,372)(12,401,42,371)(13,400,43,370)(14,399,44,369)(15,398,45,368)(16,397,46,367)(17,396,47,366)(18,395,48,365)(19,394,49,364)(20,393,50,363)(21,392,51,362)(22,391,52,361)(23,390,53,420)(24,389,54,419)(25,388,55,418)(26,387,56,417)(27,386,57,416)(28,385,58,415)(29,384,59,414)(30,383,60,413)(61,124,91,154)(62,123,92,153)(63,122,93,152)(64,121,94,151)(65,180,95,150)(66,179,96,149)(67,178,97,148)(68,177,98,147)(69,176,99,146)(70,175,100,145)(71,174,101,144)(72,173,102,143)(73,172,103,142)(74,171,104,141)(75,170,105,140)(76,169,106,139)(77,168,107,138)(78,167,108,137)(79,166,109,136)(80,165,110,135)(81,164,111,134)(82,163,112,133)(83,162,113,132)(84,161,114,131)(85,160,115,130)(86,159,116,129)(87,158,117,128)(88,157,118,127)(89,156,119,126)(90,155,120,125)(181,280,211,250)(182,279,212,249)(183,278,213,248)(184,277,214,247)(185,276,215,246)(186,275,216,245)(187,274,217,244)(188,273,218,243)(189,272,219,242)(190,271,220,241)(191,270,221,300)(192,269,222,299)(193,268,223,298)(194,267,224,297)(195,266,225,296)(196,265,226,295)(197,264,227,294)(198,263,228,293)(199,262,229,292)(200,261,230,291)(201,260,231,290)(202,259,232,289)(203,258,233,288)(204,257,234,287)(205,256,235,286)(206,255,236,285)(207,254,237,284)(208,253,238,283)(209,252,239,282)(210,251,240,281)(301,432,331,462)(302,431,332,461)(303,430,333,460)(304,429,334,459)(305,428,335,458)(306,427,336,457)(307,426,337,456)(308,425,338,455)(309,424,339,454)(310,423,340,453)(311,422,341,452)(312,421,342,451)(313,480,343,450)(314,479,344,449)(315,478,345,448)(316,477,346,447)(317,476,347,446)(318,475,348,445)(319,474,349,444)(320,473,350,443)(321,472,351,442)(322,471,352,441)(323,470,353,440)(324,469,354,439)(325,468,355,438)(326,467,356,437)(327,466,357,436)(328,465,358,435)(329,464,359,434)(330,463,360,433), (1,62,468,257)(2,93,469,288)(3,64,470,259)(4,95,471,290)(5,66,472,261)(6,97,473,292)(7,68,474,263)(8,99,475,294)(9,70,476,265)(10,101,477,296)(11,72,478,267)(12,103,479,298)(13,74,480,269)(14,105,421,300)(15,76,422,271)(16,107,423,242)(17,78,424,273)(18,109,425,244)(19,80,426,275)(20,111,427,246)(21,82,428,277)(22,113,429,248)(23,84,430,279)(24,115,431,250)(25,86,432,281)(26,117,433,252)(27,88,434,283)(28,119,435,254)(29,90,436,285)(30,61,437,256)(31,92,438,287)(32,63,439,258)(33,94,440,289)(34,65,441,260)(35,96,442,291)(36,67,443,262)(37,98,444,293)(38,69,445,264)(39,100,446,295)(40,71,447,266)(41,102,448,297)(42,73,449,268)(43,104,450,299)(44,75,451,270)(45,106,452,241)(46,77,453,272)(47,108,454,243)(48,79,455,274)(49,110,456,245)(50,81,457,276)(51,112,458,247)(52,83,459,278)(53,114,460,249)(54,85,461,280)(55,116,462,251)(56,87,463,282)(57,118,464,253)(58,89,465,284)(59,120,466,255)(60,91,467,286)(121,353,232,410)(122,324,233,381)(123,355,234,412)(124,326,235,383)(125,357,236,414)(126,328,237,385)(127,359,238,416)(128,330,239,387)(129,301,240,418)(130,332,181,389)(131,303,182,420)(132,334,183,391)(133,305,184,362)(134,336,185,393)(135,307,186,364)(136,338,187,395)(137,309,188,366)(138,340,189,397)(139,311,190,368)(140,342,191,399)(141,313,192,370)(142,344,193,401)(143,315,194,372)(144,346,195,403)(145,317,196,374)(146,348,197,405)(147,319,198,376)(148,350,199,407)(149,321,200,378)(150,352,201,409)(151,323,202,380)(152,354,203,411)(153,325,204,382)(154,356,205,413)(155,327,206,384)(156,358,207,415)(157,329,208,386)(158,360,209,417)(159,331,210,388)(160,302,211,419)(161,333,212,390)(162,304,213,361)(163,335,214,392)(164,306,215,363)(165,337,216,394)(166,308,217,365)(167,339,218,396)(168,310,219,367)(169,341,220,398)(170,312,221,369)(171,343,222,400)(172,314,223,371)(173,345,224,402)(174,316,225,373)(175,347,226,404)(176,318,227,375)(177,349,228,406)(178,320,229,377)(179,351,230,408)(180,322,231,379) );
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)], [(1,412,31,382),(2,411,32,381),(3,410,33,380),(4,409,34,379),(5,408,35,378),(6,407,36,377),(7,406,37,376),(8,405,38,375),(9,404,39,374),(10,403,40,373),(11,402,41,372),(12,401,42,371),(13,400,43,370),(14,399,44,369),(15,398,45,368),(16,397,46,367),(17,396,47,366),(18,395,48,365),(19,394,49,364),(20,393,50,363),(21,392,51,362),(22,391,52,361),(23,390,53,420),(24,389,54,419),(25,388,55,418),(26,387,56,417),(27,386,57,416),(28,385,58,415),(29,384,59,414),(30,383,60,413),(61,124,91,154),(62,123,92,153),(63,122,93,152),(64,121,94,151),(65,180,95,150),(66,179,96,149),(67,178,97,148),(68,177,98,147),(69,176,99,146),(70,175,100,145),(71,174,101,144),(72,173,102,143),(73,172,103,142),(74,171,104,141),(75,170,105,140),(76,169,106,139),(77,168,107,138),(78,167,108,137),(79,166,109,136),(80,165,110,135),(81,164,111,134),(82,163,112,133),(83,162,113,132),(84,161,114,131),(85,160,115,130),(86,159,116,129),(87,158,117,128),(88,157,118,127),(89,156,119,126),(90,155,120,125),(181,280,211,250),(182,279,212,249),(183,278,213,248),(184,277,214,247),(185,276,215,246),(186,275,216,245),(187,274,217,244),(188,273,218,243),(189,272,219,242),(190,271,220,241),(191,270,221,300),(192,269,222,299),(193,268,223,298),(194,267,224,297),(195,266,225,296),(196,265,226,295),(197,264,227,294),(198,263,228,293),(199,262,229,292),(200,261,230,291),(201,260,231,290),(202,259,232,289),(203,258,233,288),(204,257,234,287),(205,256,235,286),(206,255,236,285),(207,254,237,284),(208,253,238,283),(209,252,239,282),(210,251,240,281),(301,432,331,462),(302,431,332,461),(303,430,333,460),(304,429,334,459),(305,428,335,458),(306,427,336,457),(307,426,337,456),(308,425,338,455),(309,424,339,454),(310,423,340,453),(311,422,341,452),(312,421,342,451),(313,480,343,450),(314,479,344,449),(315,478,345,448),(316,477,346,447),(317,476,347,446),(318,475,348,445),(319,474,349,444),(320,473,350,443),(321,472,351,442),(322,471,352,441),(323,470,353,440),(324,469,354,439),(325,468,355,438),(326,467,356,437),(327,466,357,436),(328,465,358,435),(329,464,359,434),(330,463,360,433)], [(1,62,468,257),(2,93,469,288),(3,64,470,259),(4,95,471,290),(5,66,472,261),(6,97,473,292),(7,68,474,263),(8,99,475,294),(9,70,476,265),(10,101,477,296),(11,72,478,267),(12,103,479,298),(13,74,480,269),(14,105,421,300),(15,76,422,271),(16,107,423,242),(17,78,424,273),(18,109,425,244),(19,80,426,275),(20,111,427,246),(21,82,428,277),(22,113,429,248),(23,84,430,279),(24,115,431,250),(25,86,432,281),(26,117,433,252),(27,88,434,283),(28,119,435,254),(29,90,436,285),(30,61,437,256),(31,92,438,287),(32,63,439,258),(33,94,440,289),(34,65,441,260),(35,96,442,291),(36,67,443,262),(37,98,444,293),(38,69,445,264),(39,100,446,295),(40,71,447,266),(41,102,448,297),(42,73,449,268),(43,104,450,299),(44,75,451,270),(45,106,452,241),(46,77,453,272),(47,108,454,243),(48,79,455,274),(49,110,456,245),(50,81,457,276),(51,112,458,247),(52,83,459,278),(53,114,460,249),(54,85,461,280),(55,116,462,251),(56,87,463,282),(57,118,464,253),(58,89,465,284),(59,120,466,255),(60,91,467,286),(121,353,232,410),(122,324,233,381),(123,355,234,412),(124,326,235,383),(125,357,236,414),(126,328,237,385),(127,359,238,416),(128,330,239,387),(129,301,240,418),(130,332,181,389),(131,303,182,420),(132,334,183,391),(133,305,184,362),(134,336,185,393),(135,307,186,364),(136,338,187,395),(137,309,188,366),(138,340,189,397),(139,311,190,368),(140,342,191,399),(141,313,192,370),(142,344,193,401),(143,315,194,372),(144,346,195,403),(145,317,196,374),(146,348,197,405),(147,319,198,376),(148,350,199,407),(149,321,200,378),(150,352,201,409),(151,323,202,380),(152,354,203,411),(153,325,204,382),(154,356,205,413),(155,327,206,384),(156,358,207,415),(157,329,208,386),(158,360,209,417),(159,331,210,388),(160,302,211,419),(161,333,212,390),(162,304,213,361),(163,335,214,392),(164,306,215,363),(165,337,216,394),(166,308,217,365),(167,339,218,396),(168,310,219,367),(169,341,220,398),(170,312,221,369),(171,343,222,400),(172,314,223,371),(173,345,224,402),(174,316,225,373),(175,347,226,404),(176,318,227,375),(177,349,228,406),(178,320,229,377),(179,351,230,408),(180,322,231,379)])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 15 | 15 | 15 | 15 | 30 | ··· | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | - | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D5 | D6 | C4○D4 | D10 | C4×S3 | D15 | C4×D5 | D30 | C4×D15 | D4⋊2S3 | S3×Q8 | D4⋊2D5 | Q8×D5 | D4⋊2D15 | Q8×D15 |
| kernel | Dic30⋊11C4 | C4×Dic15 | C30.4Q8 | C15×C4⋊C4 | C2×Dic30 | Dic30 | C5×C4⋊C4 | Dic15 | C3×C4⋊C4 | C2×C20 | C30 | C2×C12 | C20 | C4⋊C4 | C12 | C2×C4 | C4 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 3 | 2 | 1 | 1 | 8 | 1 | 2 | 2 | 3 | 2 | 6 | 4 | 4 | 8 | 12 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic30⋊11C4 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 37 | 0 | 0 |
| 0 | 24 | 47 | 0 | 0 |
| 0 | 0 | 0 | 27 | 32 |
| 0 | 0 | 0 | 42 | 34 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 44 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 21 |
| 0 | 0 | 0 | 58 | 1 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 | 13 |
| 0 | 0 | 0 | 33 | 50 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,28,24,0,0,0,37,47,0,0,0,0,0,27,42,0,0,0,32,34],[1,0,0,0,0,0,1,0,0,0,0,44,60,0,0,0,0,0,60,58,0,0,0,21,1],[11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,33,0,0,0,13,50] >;
Dic30⋊11C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{30}\rtimes_{11}C_4 % in TeX
G:=Group("Dic30:11C4"); // GroupNames label
G:=SmallGroup(480,852);
// by ID
G=gap.SmallGroup(480,852);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,135,142,2693,18822]);
// 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^31,b*c=c*b>;
// generators/relations