metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.2D4, C4.10D60, Dic30⋊9C4, C12.10D20, C20.10D12, C30.17Q16, C30.26SD16, C4⋊C4.3D15, C4.2(C4×D15), C20.48(C4×S3), C60.78(C2×C4), (C2×C20).67D6, (C2×C4).35D30, C12.16(C4×D5), (C2×C12).68D10, (C2×C30).138D4, C5⋊3(C6.SD16), C3⋊2(C10.Q16), C6.8(D4.D5), C6.8(C5⋊Q16), C10.30(D6⋊C4), C15⋊11(Q8⋊C4), C2.2(D4.D15), (C2×C60).53C22, C10.8(D4.S3), (C2×Dic30).6C2, C2.2(C15⋊7Q16), C10.8(C3⋊Q16), C2.6(D30⋊3C4), C30.72(C22⋊C4), C6.15(D10⋊C4), C22.15(C15⋊7D4), (C5×C4⋊C4).3S3, (C3×C4⋊C4).3D5, (C15×C4⋊C4).3C2, (C2×C15⋊3C8).3C2, (C2×C6).70(C5⋊D4), (C2×C10).70(C3⋊D4), SmallGroup(480,170)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic30⋊9C4
G = < a,b,c | a60=c4=1, b2=a30, bab-1=a-1, cac-1=a31, cbc-1=a45b >
Subgroups: 420 in 84 conjugacy classes, 41 normal (39 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, C20, C2×C10, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C30, Q8⋊C4, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, Dic15, C60, C60, C2×C30, C2×C5⋊2C8, C5×C4⋊C4, C2×Dic10, C6.SD16, C15⋊3C8, Dic30, Dic30, C2×Dic15, C2×C60, C2×C60, C10.Q16, C2×C15⋊3C8, C15×C4⋊C4, C2×Dic30, Dic30⋊9C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, SD16, Q16, D10, C4×S3, D12, C3⋊D4, D15, Q8⋊C4, C4×D5, D20, C5⋊D4, D6⋊C4, D4.S3, C3⋊Q16, D30, D10⋊C4, D4.D5, C5⋊Q16, C6.SD16, C4×D15, D60, C15⋊7D4, C10.Q16, D30⋊3C4, D4.D15, C15⋊7Q16, Dic30⋊9C4
►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)
(1 306 31 336)(2 305 32 335)(3 304 33 334)(4 303 34 333)(5 302 35 332)(6 301 36 331)(7 360 37 330)(8 359 38 329)(9 358 39 328)(10 357 40 327)(11 356 41 326)(12 355 42 325)(13 354 43 324)(14 353 44 323)(15 352 45 322)(16 351 46 321)(17 350 47 320)(18 349 48 319)(19 348 49 318)(20 347 50 317)(21 346 51 316)(22 345 52 315)(23 344 53 314)(24 343 54 313)(25 342 55 312)(26 341 56 311)(27 340 57 310)(28 339 58 309)(29 338 59 308)(30 337 60 307)(61 273 91 243)(62 272 92 242)(63 271 93 241)(64 270 94 300)(65 269 95 299)(66 268 96 298)(67 267 97 297)(68 266 98 296)(69 265 99 295)(70 264 100 294)(71 263 101 293)(72 262 102 292)(73 261 103 291)(74 260 104 290)(75 259 105 289)(76 258 106 288)(77 257 107 287)(78 256 108 286)(79 255 109 285)(80 254 110 284)(81 253 111 283)(82 252 112 282)(83 251 113 281)(84 250 114 280)(85 249 115 279)(86 248 116 278)(87 247 117 277)(88 246 118 276)(89 245 119 275)(90 244 120 274)(121 429 151 459)(122 428 152 458)(123 427 153 457)(124 426 154 456)(125 425 155 455)(126 424 156 454)(127 423 157 453)(128 422 158 452)(129 421 159 451)(130 480 160 450)(131 479 161 449)(132 478 162 448)(133 477 163 447)(134 476 164 446)(135 475 165 445)(136 474 166 444)(137 473 167 443)(138 472 168 442)(139 471 169 441)(140 470 170 440)(141 469 171 439)(142 468 172 438)(143 467 173 437)(144 466 174 436)(145 465 175 435)(146 464 176 434)(147 463 177 433)(148 462 178 432)(149 461 179 431)(150 460 180 430)(181 366 211 396)(182 365 212 395)(183 364 213 394)(184 363 214 393)(185 362 215 392)(186 361 216 391)(187 420 217 390)(188 419 218 389)(189 418 219 388)(190 417 220 387)(191 416 221 386)(192 415 222 385)(193 414 223 384)(194 413 224 383)(195 412 225 382)(196 411 226 381)(197 410 227 380)(198 409 228 379)(199 408 229 378)(200 407 230 377)(201 406 231 376)(202 405 232 375)(203 404 233 374)(204 403 234 373)(205 402 235 372)(206 401 236 371)(207 400 237 370)(208 399 238 369)(209 398 239 368)(210 397 240 367)
(1 226 98 149)(2 197 99 180)(3 228 100 151)(4 199 101 122)(5 230 102 153)(6 201 103 124)(7 232 104 155)(8 203 105 126)(9 234 106 157)(10 205 107 128)(11 236 108 159)(12 207 109 130)(13 238 110 161)(14 209 111 132)(15 240 112 163)(16 211 113 134)(17 182 114 165)(18 213 115 136)(19 184 116 167)(20 215 117 138)(21 186 118 169)(22 217 119 140)(23 188 120 171)(24 219 61 142)(25 190 62 173)(26 221 63 144)(27 192 64 175)(28 223 65 146)(29 194 66 177)(30 225 67 148)(31 196 68 179)(32 227 69 150)(33 198 70 121)(34 229 71 152)(35 200 72 123)(36 231 73 154)(37 202 74 125)(38 233 75 156)(39 204 76 127)(40 235 77 158)(41 206 78 129)(42 237 79 160)(43 208 80 131)(44 239 81 162)(45 210 82 133)(46 181 83 164)(47 212 84 135)(48 183 85 166)(49 214 86 137)(50 185 87 168)(51 216 88 139)(52 187 89 170)(53 218 90 141)(54 189 91 172)(55 220 92 143)(56 191 93 174)(57 222 94 145)(58 193 95 176)(59 224 96 147)(60 195 97 178)(241 451 311 371)(242 422 312 402)(243 453 313 373)(244 424 314 404)(245 455 315 375)(246 426 316 406)(247 457 317 377)(248 428 318 408)(249 459 319 379)(250 430 320 410)(251 461 321 381)(252 432 322 412)(253 463 323 383)(254 434 324 414)(255 465 325 385)(256 436 326 416)(257 467 327 387)(258 438 328 418)(259 469 329 389)(260 440 330 420)(261 471 331 391)(262 442 332 362)(263 473 333 393)(264 444 334 364)(265 475 335 395)(266 446 336 366)(267 477 337 397)(268 448 338 368)(269 479 339 399)(270 450 340 370)(271 421 341 401)(272 452 342 372)(273 423 343 403)(274 454 344 374)(275 425 345 405)(276 456 346 376)(277 427 347 407)(278 458 348 378)(279 429 349 409)(280 460 350 380)(281 431 351 411)(282 462 352 382)(283 433 353 413)(284 464 354 384)(285 435 355 415)(286 466 356 386)(287 437 357 417)(288 468 358 388)(289 439 359 419)(290 470 360 390)(291 441 301 361)(292 472 302 392)(293 443 303 363)(294 474 304 394)(295 445 305 365)(296 476 306 396)(297 447 307 367)(298 478 308 398)(299 449 309 369)(300 480 310 400)
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), (1,306,31,336)(2,305,32,335)(3,304,33,334)(4,303,34,333)(5,302,35,332)(6,301,36,331)(7,360,37,330)(8,359,38,329)(9,358,39,328)(10,357,40,327)(11,356,41,326)(12,355,42,325)(13,354,43,324)(14,353,44,323)(15,352,45,322)(16,351,46,321)(17,350,47,320)(18,349,48,319)(19,348,49,318)(20,347,50,317)(21,346,51,316)(22,345,52,315)(23,344,53,314)(24,343,54,313)(25,342,55,312)(26,341,56,311)(27,340,57,310)(28,339,58,309)(29,338,59,308)(30,337,60,307)(61,273,91,243)(62,272,92,242)(63,271,93,241)(64,270,94,300)(65,269,95,299)(66,268,96,298)(67,267,97,297)(68,266,98,296)(69,265,99,295)(70,264,100,294)(71,263,101,293)(72,262,102,292)(73,261,103,291)(74,260,104,290)(75,259,105,289)(76,258,106,288)(77,257,107,287)(78,256,108,286)(79,255,109,285)(80,254,110,284)(81,253,111,283)(82,252,112,282)(83,251,113,281)(84,250,114,280)(85,249,115,279)(86,248,116,278)(87,247,117,277)(88,246,118,276)(89,245,119,275)(90,244,120,274)(121,429,151,459)(122,428,152,458)(123,427,153,457)(124,426,154,456)(125,425,155,455)(126,424,156,454)(127,423,157,453)(128,422,158,452)(129,421,159,451)(130,480,160,450)(131,479,161,449)(132,478,162,448)(133,477,163,447)(134,476,164,446)(135,475,165,445)(136,474,166,444)(137,473,167,443)(138,472,168,442)(139,471,169,441)(140,470,170,440)(141,469,171,439)(142,468,172,438)(143,467,173,437)(144,466,174,436)(145,465,175,435)(146,464,176,434)(147,463,177,433)(148,462,178,432)(149,461,179,431)(150,460,180,430)(181,366,211,396)(182,365,212,395)(183,364,213,394)(184,363,214,393)(185,362,215,392)(186,361,216,391)(187,420,217,390)(188,419,218,389)(189,418,219,388)(190,417,220,387)(191,416,221,386)(192,415,222,385)(193,414,223,384)(194,413,224,383)(195,412,225,382)(196,411,226,381)(197,410,227,380)(198,409,228,379)(199,408,229,378)(200,407,230,377)(201,406,231,376)(202,405,232,375)(203,404,233,374)(204,403,234,373)(205,402,235,372)(206,401,236,371)(207,400,237,370)(208,399,238,369)(209,398,239,368)(210,397,240,367), (1,226,98,149)(2,197,99,180)(3,228,100,151)(4,199,101,122)(5,230,102,153)(6,201,103,124)(7,232,104,155)(8,203,105,126)(9,234,106,157)(10,205,107,128)(11,236,108,159)(12,207,109,130)(13,238,110,161)(14,209,111,132)(15,240,112,163)(16,211,113,134)(17,182,114,165)(18,213,115,136)(19,184,116,167)(20,215,117,138)(21,186,118,169)(22,217,119,140)(23,188,120,171)(24,219,61,142)(25,190,62,173)(26,221,63,144)(27,192,64,175)(28,223,65,146)(29,194,66,177)(30,225,67,148)(31,196,68,179)(32,227,69,150)(33,198,70,121)(34,229,71,152)(35,200,72,123)(36,231,73,154)(37,202,74,125)(38,233,75,156)(39,204,76,127)(40,235,77,158)(41,206,78,129)(42,237,79,160)(43,208,80,131)(44,239,81,162)(45,210,82,133)(46,181,83,164)(47,212,84,135)(48,183,85,166)(49,214,86,137)(50,185,87,168)(51,216,88,139)(52,187,89,170)(53,218,90,141)(54,189,91,172)(55,220,92,143)(56,191,93,174)(57,222,94,145)(58,193,95,176)(59,224,96,147)(60,195,97,178)(241,451,311,371)(242,422,312,402)(243,453,313,373)(244,424,314,404)(245,455,315,375)(246,426,316,406)(247,457,317,377)(248,428,318,408)(249,459,319,379)(250,430,320,410)(251,461,321,381)(252,432,322,412)(253,463,323,383)(254,434,324,414)(255,465,325,385)(256,436,326,416)(257,467,327,387)(258,438,328,418)(259,469,329,389)(260,440,330,420)(261,471,331,391)(262,442,332,362)(263,473,333,393)(264,444,334,364)(265,475,335,395)(266,446,336,366)(267,477,337,397)(268,448,338,368)(269,479,339,399)(270,450,340,370)(271,421,341,401)(272,452,342,372)(273,423,343,403)(274,454,344,374)(275,425,345,405)(276,456,346,376)(277,427,347,407)(278,458,348,378)(279,429,349,409)(280,460,350,380)(281,431,351,411)(282,462,352,382)(283,433,353,413)(284,464,354,384)(285,435,355,415)(286,466,356,386)(287,437,357,417)(288,468,358,388)(289,439,359,419)(290,470,360,390)(291,441,301,361)(292,472,302,392)(293,443,303,363)(294,474,304,394)(295,445,305,365)(296,476,306,396)(297,447,307,367)(298,478,308,398)(299,449,309,369)(300,480,310,400)>;
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), (1,306,31,336)(2,305,32,335)(3,304,33,334)(4,303,34,333)(5,302,35,332)(6,301,36,331)(7,360,37,330)(8,359,38,329)(9,358,39,328)(10,357,40,327)(11,356,41,326)(12,355,42,325)(13,354,43,324)(14,353,44,323)(15,352,45,322)(16,351,46,321)(17,350,47,320)(18,349,48,319)(19,348,49,318)(20,347,50,317)(21,346,51,316)(22,345,52,315)(23,344,53,314)(24,343,54,313)(25,342,55,312)(26,341,56,311)(27,340,57,310)(28,339,58,309)(29,338,59,308)(30,337,60,307)(61,273,91,243)(62,272,92,242)(63,271,93,241)(64,270,94,300)(65,269,95,299)(66,268,96,298)(67,267,97,297)(68,266,98,296)(69,265,99,295)(70,264,100,294)(71,263,101,293)(72,262,102,292)(73,261,103,291)(74,260,104,290)(75,259,105,289)(76,258,106,288)(77,257,107,287)(78,256,108,286)(79,255,109,285)(80,254,110,284)(81,253,111,283)(82,252,112,282)(83,251,113,281)(84,250,114,280)(85,249,115,279)(86,248,116,278)(87,247,117,277)(88,246,118,276)(89,245,119,275)(90,244,120,274)(121,429,151,459)(122,428,152,458)(123,427,153,457)(124,426,154,456)(125,425,155,455)(126,424,156,454)(127,423,157,453)(128,422,158,452)(129,421,159,451)(130,480,160,450)(131,479,161,449)(132,478,162,448)(133,477,163,447)(134,476,164,446)(135,475,165,445)(136,474,166,444)(137,473,167,443)(138,472,168,442)(139,471,169,441)(140,470,170,440)(141,469,171,439)(142,468,172,438)(143,467,173,437)(144,466,174,436)(145,465,175,435)(146,464,176,434)(147,463,177,433)(148,462,178,432)(149,461,179,431)(150,460,180,430)(181,366,211,396)(182,365,212,395)(183,364,213,394)(184,363,214,393)(185,362,215,392)(186,361,216,391)(187,420,217,390)(188,419,218,389)(189,418,219,388)(190,417,220,387)(191,416,221,386)(192,415,222,385)(193,414,223,384)(194,413,224,383)(195,412,225,382)(196,411,226,381)(197,410,227,380)(198,409,228,379)(199,408,229,378)(200,407,230,377)(201,406,231,376)(202,405,232,375)(203,404,233,374)(204,403,234,373)(205,402,235,372)(206,401,236,371)(207,400,237,370)(208,399,238,369)(209,398,239,368)(210,397,240,367), (1,226,98,149)(2,197,99,180)(3,228,100,151)(4,199,101,122)(5,230,102,153)(6,201,103,124)(7,232,104,155)(8,203,105,126)(9,234,106,157)(10,205,107,128)(11,236,108,159)(12,207,109,130)(13,238,110,161)(14,209,111,132)(15,240,112,163)(16,211,113,134)(17,182,114,165)(18,213,115,136)(19,184,116,167)(20,215,117,138)(21,186,118,169)(22,217,119,140)(23,188,120,171)(24,219,61,142)(25,190,62,173)(26,221,63,144)(27,192,64,175)(28,223,65,146)(29,194,66,177)(30,225,67,148)(31,196,68,179)(32,227,69,150)(33,198,70,121)(34,229,71,152)(35,200,72,123)(36,231,73,154)(37,202,74,125)(38,233,75,156)(39,204,76,127)(40,235,77,158)(41,206,78,129)(42,237,79,160)(43,208,80,131)(44,239,81,162)(45,210,82,133)(46,181,83,164)(47,212,84,135)(48,183,85,166)(49,214,86,137)(50,185,87,168)(51,216,88,139)(52,187,89,170)(53,218,90,141)(54,189,91,172)(55,220,92,143)(56,191,93,174)(57,222,94,145)(58,193,95,176)(59,224,96,147)(60,195,97,178)(241,451,311,371)(242,422,312,402)(243,453,313,373)(244,424,314,404)(245,455,315,375)(246,426,316,406)(247,457,317,377)(248,428,318,408)(249,459,319,379)(250,430,320,410)(251,461,321,381)(252,432,322,412)(253,463,323,383)(254,434,324,414)(255,465,325,385)(256,436,326,416)(257,467,327,387)(258,438,328,418)(259,469,329,389)(260,440,330,420)(261,471,331,391)(262,442,332,362)(263,473,333,393)(264,444,334,364)(265,475,335,395)(266,446,336,366)(267,477,337,397)(268,448,338,368)(269,479,339,399)(270,450,340,370)(271,421,341,401)(272,452,342,372)(273,423,343,403)(274,454,344,374)(275,425,345,405)(276,456,346,376)(277,427,347,407)(278,458,348,378)(279,429,349,409)(280,460,350,380)(281,431,351,411)(282,462,352,382)(283,433,353,413)(284,464,354,384)(285,435,355,415)(286,466,356,386)(287,437,357,417)(288,468,358,388)(289,439,359,419)(290,470,360,390)(291,441,301,361)(292,472,302,392)(293,443,303,363)(294,474,304,394)(295,445,305,365)(296,476,306,396)(297,447,307,367)(298,478,308,398)(299,449,309,369)(300,480,310,400) );
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)], [(1,306,31,336),(2,305,32,335),(3,304,33,334),(4,303,34,333),(5,302,35,332),(6,301,36,331),(7,360,37,330),(8,359,38,329),(9,358,39,328),(10,357,40,327),(11,356,41,326),(12,355,42,325),(13,354,43,324),(14,353,44,323),(15,352,45,322),(16,351,46,321),(17,350,47,320),(18,349,48,319),(19,348,49,318),(20,347,50,317),(21,346,51,316),(22,345,52,315),(23,344,53,314),(24,343,54,313),(25,342,55,312),(26,341,56,311),(27,340,57,310),(28,339,58,309),(29,338,59,308),(30,337,60,307),(61,273,91,243),(62,272,92,242),(63,271,93,241),(64,270,94,300),(65,269,95,299),(66,268,96,298),(67,267,97,297),(68,266,98,296),(69,265,99,295),(70,264,100,294),(71,263,101,293),(72,262,102,292),(73,261,103,291),(74,260,104,290),(75,259,105,289),(76,258,106,288),(77,257,107,287),(78,256,108,286),(79,255,109,285),(80,254,110,284),(81,253,111,283),(82,252,112,282),(83,251,113,281),(84,250,114,280),(85,249,115,279),(86,248,116,278),(87,247,117,277),(88,246,118,276),(89,245,119,275),(90,244,120,274),(121,429,151,459),(122,428,152,458),(123,427,153,457),(124,426,154,456),(125,425,155,455),(126,424,156,454),(127,423,157,453),(128,422,158,452),(129,421,159,451),(130,480,160,450),(131,479,161,449),(132,478,162,448),(133,477,163,447),(134,476,164,446),(135,475,165,445),(136,474,166,444),(137,473,167,443),(138,472,168,442),(139,471,169,441),(140,470,170,440),(141,469,171,439),(142,468,172,438),(143,467,173,437),(144,466,174,436),(145,465,175,435),(146,464,176,434),(147,463,177,433),(148,462,178,432),(149,461,179,431),(150,460,180,430),(181,366,211,396),(182,365,212,395),(183,364,213,394),(184,363,214,393),(185,362,215,392),(186,361,216,391),(187,420,217,390),(188,419,218,389),(189,418,219,388),(190,417,220,387),(191,416,221,386),(192,415,222,385),(193,414,223,384),(194,413,224,383),(195,412,225,382),(196,411,226,381),(197,410,227,380),(198,409,228,379),(199,408,229,378),(200,407,230,377),(201,406,231,376),(202,405,232,375),(203,404,233,374),(204,403,234,373),(205,402,235,372),(206,401,236,371),(207,400,237,370),(208,399,238,369),(209,398,239,368),(210,397,240,367)], [(1,226,98,149),(2,197,99,180),(3,228,100,151),(4,199,101,122),(5,230,102,153),(6,201,103,124),(7,232,104,155),(8,203,105,126),(9,234,106,157),(10,205,107,128),(11,236,108,159),(12,207,109,130),(13,238,110,161),(14,209,111,132),(15,240,112,163),(16,211,113,134),(17,182,114,165),(18,213,115,136),(19,184,116,167),(20,215,117,138),(21,186,118,169),(22,217,119,140),(23,188,120,171),(24,219,61,142),(25,190,62,173),(26,221,63,144),(27,192,64,175),(28,223,65,146),(29,194,66,177),(30,225,67,148),(31,196,68,179),(32,227,69,150),(33,198,70,121),(34,229,71,152),(35,200,72,123),(36,231,73,154),(37,202,74,125),(38,233,75,156),(39,204,76,127),(40,235,77,158),(41,206,78,129),(42,237,79,160),(43,208,80,131),(44,239,81,162),(45,210,82,133),(46,181,83,164),(47,212,84,135),(48,183,85,166),(49,214,86,137),(50,185,87,168),(51,216,88,139),(52,187,89,170),(53,218,90,141),(54,189,91,172),(55,220,92,143),(56,191,93,174),(57,222,94,145),(58,193,95,176),(59,224,96,147),(60,195,97,178),(241,451,311,371),(242,422,312,402),(243,453,313,373),(244,424,314,404),(245,455,315,375),(246,426,316,406),(247,457,317,377),(248,428,318,408),(249,459,319,379),(250,430,320,410),(251,461,321,381),(252,432,322,412),(253,463,323,383),(254,434,324,414),(255,465,325,385),(256,436,326,416),(257,467,327,387),(258,438,328,418),(259,469,329,389),(260,440,330,420),(261,471,331,391),(262,442,332,362),(263,473,333,393),(264,444,334,364),(265,475,335,395),(266,446,336,366),(267,477,337,397),(268,448,338,368),(269,479,339,399),(270,450,340,370),(271,421,341,401),(272,452,342,372),(273,423,343,403),(274,454,344,374),(275,425,345,405),(276,456,346,376),(277,427,347,407),(278,458,348,378),(279,429,349,409),(280,460,350,380),(281,431,351,411),(282,462,352,382),(283,433,353,413),(284,464,354,384),(285,435,355,415),(286,466,356,386),(287,437,357,417),(288,468,358,388),(289,439,359,419),(290,470,360,390),(291,441,301,361),(292,472,302,392),(293,443,303,363),(294,474,304,394),(295,445,305,365),(296,476,306,396),(297,447,307,367),(298,478,308,398),(299,449,309,369),(300,480,310,400)]])
84 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 | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| 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 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | D15 | C4×D5 | D20 | C5⋊D4 | D30 | C4×D15 | D60 | C15⋊7D4 | D4.S3 | C3⋊Q16 | D4.D5 | C5⋊Q16 | D4.D15 | C15⋊7Q16 |
| kernel | Dic30⋊9C4 | C2×C15⋊3C8 | C15×C4⋊C4 | C2×Dic30 | Dic30 | C5×C4⋊C4 | C60 | C2×C30 | C3×C4⋊C4 | C2×C20 | C30 | C30 | C2×C12 | C20 | C20 | C2×C10 | C4⋊C4 | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic30⋊9C4 ►in GL4(𝔽241) generated by
| 84 | 147 | 0 | 0 |
| 94 | 110 | 0 | 0 |
| 0 | 0 | 103 | 5 |
| 0 | 0 | 47 | 138 |
| 10 | 182 | 0 | 0 |
| 210 | 231 | 0 | 0 |
| 0 | 0 | 2 | 43 |
| 0 | 0 | 140 | 239 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 65 | 240 |
| 0 | 0 | 127 | 176 |
G:=sub<GL(4,GF(241))| [84,94,0,0,147,110,0,0,0,0,103,47,0,0,5,138],[10,210,0,0,182,231,0,0,0,0,2,140,0,0,43,239],[64,0,0,0,0,64,0,0,0,0,65,127,0,0,240,176] >;
Dic30⋊9C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{30}\rtimes_9C_4 % in TeX
G:=Group("Dic30:9C4"); // GroupNames label
G:=SmallGroup(480,170);
// by ID
G=gap.SmallGroup(480,170);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,36,675,346,80,2693,18822]);
// 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,c*b*c^-1=a^45*b>;
// generators/relations