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G = Dic3012C4order 480 = 25·3·5

6th semidirect product of Dic30 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.35D4, C30.5Q16, C20.50D12, C6.4Dic20, Dic3012C4, C30.11SD16, C12.6(C4×D5), C20.64(C4×S3), C60.93(C2×C4), (C2×C6).32D20, (C2×C30).21D4, C4⋊Dic5.1S3, C157(Q8⋊C4), (C2×C12).54D10, (C2×C20).282D6, C6.7(C40⋊C2), C52(C6.SD16), C10.21(D6⋊C4), C12.16(C5⋊D4), C4.15(C5⋊D12), C31(C20.44D4), C4.2(D30.C2), C10.2(D4.S3), C2.2(C3⋊Dic20), C10.2(C3⋊Q16), C2.7(D304C4), 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

C1C60 — Dic3012C4
C1C5C15C30C2×C30C2×C60C3×C4⋊Dic5 — Dic3012C4
C15C30C60 — Dic3012C4
C1C22C2×C4

Generators and relations for Dic3012C4
 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 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C10 [×3], Dic3 [×2], C12 [×2], C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5 [×3], C20 [×2], C2×C10, C3⋊C8, Dic6 [×3], C2×Dic3, C2×C12, C2×C12, C30 [×3], Q8⋊C4, C40, Dic10 [×3], C2×Dic5 [×2], C2×C20, C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C3×Dic5, Dic15 [×2], C60 [×2], C2×C30, C4⋊Dic5, C2×C40, C2×Dic10, C6.SD16, C5×C3⋊C8, C6×Dic5, Dic30 [×2], Dic30, C2×Dic15, C2×C60, C20.44D4, C3×C4⋊Dic5, C10×C3⋊C8, C2×Dic30, Dic3012C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], 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, D304C4, Dic3012C4

Smallest permutation representation of Dic3012C4
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)
(1 381 31 411)(2 380 32 410)(3 379 33 409)(4 378 34 408)(5 377 35 407)(6 376 36 406)(7 375 37 405)(8 374 38 404)(9 373 39 403)(10 372 40 402)(11 371 41 401)(12 370 42 400)(13 369 43 399)(14 368 44 398)(15 367 45 397)(16 366 46 396)(17 365 47 395)(18 364 48 394)(19 363 49 393)(20 362 50 392)(21 361 51 391)(22 420 52 390)(23 419 53 389)(24 418 54 388)(25 417 55 387)(26 416 56 386)(27 415 57 385)(28 414 58 384)(29 413 59 383)(30 412 60 382)(61 457 91 427)(62 456 92 426)(63 455 93 425)(64 454 94 424)(65 453 95 423)(66 452 96 422)(67 451 97 421)(68 450 98 480)(69 449 99 479)(70 448 100 478)(71 447 101 477)(72 446 102 476)(73 445 103 475)(74 444 104 474)(75 443 105 473)(76 442 106 472)(77 441 107 471)(78 440 108 470)(79 439 109 469)(80 438 110 468)(81 437 111 467)(82 436 112 466)(83 435 113 465)(84 434 114 464)(85 433 115 463)(86 432 116 462)(87 431 117 461)(88 430 118 460)(89 429 119 459)(90 428 120 458)(121 350 151 320)(122 349 152 319)(123 348 153 318)(124 347 154 317)(125 346 155 316)(126 345 156 315)(127 344 157 314)(128 343 158 313)(129 342 159 312)(130 341 160 311)(131 340 161 310)(132 339 162 309)(133 338 163 308)(134 337 164 307)(135 336 165 306)(136 335 166 305)(137 334 167 304)(138 333 168 303)(139 332 169 302)(140 331 170 301)(141 330 171 360)(142 329 172 359)(143 328 173 358)(144 327 174 357)(145 326 175 356)(146 325 176 355)(147 324 177 354)(148 323 178 353)(149 322 179 352)(150 321 180 351)(181 299 211 269)(182 298 212 268)(183 297 213 267)(184 296 214 266)(185 295 215 265)(186 294 216 264)(187 293 217 263)(188 292 218 262)(189 291 219 261)(190 290 220 260)(191 289 221 259)(192 288 222 258)(193 287 223 257)(194 286 224 256)(195 285 225 255)(196 284 226 254)(197 283 227 253)(198 282 228 252)(199 281 229 251)(200 280 230 250)(201 279 231 249)(202 278 232 248)(203 277 233 247)(204 276 234 246)(205 275 235 245)(206 274 236 244)(207 273 237 243)(208 272 238 242)(209 271 239 241)(210 270 240 300)
(1 256 67 321)(2 275 68 340)(3 294 69 359)(4 253 70 318)(5 272 71 337)(6 291 72 356)(7 250 73 315)(8 269 74 334)(9 288 75 353)(10 247 76 312)(11 266 77 331)(12 285 78 350)(13 244 79 309)(14 263 80 328)(15 282 81 347)(16 241 82 306)(17 260 83 325)(18 279 84 344)(19 298 85 303)(20 257 86 322)(21 276 87 341)(22 295 88 360)(23 254 89 319)(24 273 90 338)(25 292 91 357)(26 251 92 316)(27 270 93 335)(28 289 94 354)(29 248 95 313)(30 267 96 332)(31 286 97 351)(32 245 98 310)(33 264 99 329)(34 283 100 348)(35 242 101 307)(36 261 102 326)(37 280 103 345)(38 299 104 304)(39 258 105 323)(40 277 106 342)(41 296 107 301)(42 255 108 320)(43 274 109 339)(44 293 110 358)(45 252 111 317)(46 271 112 336)(47 290 113 355)(48 249 114 314)(49 268 115 333)(50 287 116 352)(51 246 117 311)(52 265 118 330)(53 284 119 349)(54 243 120 308)(55 262 61 327)(56 281 62 346)(57 300 63 305)(58 259 64 324)(59 278 65 343)(60 297 66 302)(121 385 195 455)(122 404 196 474)(123 363 197 433)(124 382 198 452)(125 401 199 471)(126 420 200 430)(127 379 201 449)(128 398 202 468)(129 417 203 427)(130 376 204 446)(131 395 205 465)(132 414 206 424)(133 373 207 443)(134 392 208 462)(135 411 209 421)(136 370 210 440)(137 389 211 459)(138 408 212 478)(139 367 213 437)(140 386 214 456)(141 405 215 475)(142 364 216 434)(143 383 217 453)(144 402 218 472)(145 361 219 431)(146 380 220 450)(147 399 221 469)(148 418 222 428)(149 377 223 447)(150 396 224 466)(151 415 225 425)(152 374 226 444)(153 393 227 463)(154 412 228 422)(155 371 229 441)(156 390 230 460)(157 409 231 479)(158 368 232 438)(159 387 233 457)(160 406 234 476)(161 365 235 435)(162 384 236 454)(163 403 237 473)(164 362 238 432)(165 381 239 451)(166 400 240 470)(167 419 181 429)(168 378 182 448)(169 397 183 467)(170 416 184 426)(171 375 185 445)(172 394 186 464)(173 413 187 423)(174 372 188 442)(175 391 189 461)(176 410 190 480)(177 369 191 439)(178 388 192 458)(179 407 193 477)(180 366 194 436)

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

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), (1,381,31,411)(2,380,32,410)(3,379,33,409)(4,378,34,408)(5,377,35,407)(6,376,36,406)(7,375,37,405)(8,374,38,404)(9,373,39,403)(10,372,40,402)(11,371,41,401)(12,370,42,400)(13,369,43,399)(14,368,44,398)(15,367,45,397)(16,366,46,396)(17,365,47,395)(18,364,48,394)(19,363,49,393)(20,362,50,392)(21,361,51,391)(22,420,52,390)(23,419,53,389)(24,418,54,388)(25,417,55,387)(26,416,56,386)(27,415,57,385)(28,414,58,384)(29,413,59,383)(30,412,60,382)(61,457,91,427)(62,456,92,426)(63,455,93,425)(64,454,94,424)(65,453,95,423)(66,452,96,422)(67,451,97,421)(68,450,98,480)(69,449,99,479)(70,448,100,478)(71,447,101,477)(72,446,102,476)(73,445,103,475)(74,444,104,474)(75,443,105,473)(76,442,106,472)(77,441,107,471)(78,440,108,470)(79,439,109,469)(80,438,110,468)(81,437,111,467)(82,436,112,466)(83,435,113,465)(84,434,114,464)(85,433,115,463)(86,432,116,462)(87,431,117,461)(88,430,118,460)(89,429,119,459)(90,428,120,458)(121,350,151,320)(122,349,152,319)(123,348,153,318)(124,347,154,317)(125,346,155,316)(126,345,156,315)(127,344,157,314)(128,343,158,313)(129,342,159,312)(130,341,160,311)(131,340,161,310)(132,339,162,309)(133,338,163,308)(134,337,164,307)(135,336,165,306)(136,335,166,305)(137,334,167,304)(138,333,168,303)(139,332,169,302)(140,331,170,301)(141,330,171,360)(142,329,172,359)(143,328,173,358)(144,327,174,357)(145,326,175,356)(146,325,176,355)(147,324,177,354)(148,323,178,353)(149,322,179,352)(150,321,180,351)(181,299,211,269)(182,298,212,268)(183,297,213,267)(184,296,214,266)(185,295,215,265)(186,294,216,264)(187,293,217,263)(188,292,218,262)(189,291,219,261)(190,290,220,260)(191,289,221,259)(192,288,222,258)(193,287,223,257)(194,286,224,256)(195,285,225,255)(196,284,226,254)(197,283,227,253)(198,282,228,252)(199,281,229,251)(200,280,230,250)(201,279,231,249)(202,278,232,248)(203,277,233,247)(204,276,234,246)(205,275,235,245)(206,274,236,244)(207,273,237,243)(208,272,238,242)(209,271,239,241)(210,270,240,300), (1,256,67,321)(2,275,68,340)(3,294,69,359)(4,253,70,318)(5,272,71,337)(6,291,72,356)(7,250,73,315)(8,269,74,334)(9,288,75,353)(10,247,76,312)(11,266,77,331)(12,285,78,350)(13,244,79,309)(14,263,80,328)(15,282,81,347)(16,241,82,306)(17,260,83,325)(18,279,84,344)(19,298,85,303)(20,257,86,322)(21,276,87,341)(22,295,88,360)(23,254,89,319)(24,273,90,338)(25,292,91,357)(26,251,92,316)(27,270,93,335)(28,289,94,354)(29,248,95,313)(30,267,96,332)(31,286,97,351)(32,245,98,310)(33,264,99,329)(34,283,100,348)(35,242,101,307)(36,261,102,326)(37,280,103,345)(38,299,104,304)(39,258,105,323)(40,277,106,342)(41,296,107,301)(42,255,108,320)(43,274,109,339)(44,293,110,358)(45,252,111,317)(46,271,112,336)(47,290,113,355)(48,249,114,314)(49,268,115,333)(50,287,116,352)(51,246,117,311)(52,265,118,330)(53,284,119,349)(54,243,120,308)(55,262,61,327)(56,281,62,346)(57,300,63,305)(58,259,64,324)(59,278,65,343)(60,297,66,302)(121,385,195,455)(122,404,196,474)(123,363,197,433)(124,382,198,452)(125,401,199,471)(126,420,200,430)(127,379,201,449)(128,398,202,468)(129,417,203,427)(130,376,204,446)(131,395,205,465)(132,414,206,424)(133,373,207,443)(134,392,208,462)(135,411,209,421)(136,370,210,440)(137,389,211,459)(138,408,212,478)(139,367,213,437)(140,386,214,456)(141,405,215,475)(142,364,216,434)(143,383,217,453)(144,402,218,472)(145,361,219,431)(146,380,220,450)(147,399,221,469)(148,418,222,428)(149,377,223,447)(150,396,224,466)(151,415,225,425)(152,374,226,444)(153,393,227,463)(154,412,228,422)(155,371,229,441)(156,390,230,460)(157,409,231,479)(158,368,232,438)(159,387,233,457)(160,406,234,476)(161,365,235,435)(162,384,236,454)(163,403,237,473)(164,362,238,432)(165,381,239,451)(166,400,240,470)(167,419,181,429)(168,378,182,448)(169,397,183,467)(170,416,184,426)(171,375,185,445)(172,394,186,464)(173,413,187,423)(174,372,188,442)(175,391,189,461)(176,410,190,480)(177,369,191,439)(178,388,192,458)(179,407,193,477)(180,366,194,436) );

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

72 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D12E12F15A15B20A···20H30A···30F40A···40P60A···60H
order1222344444455666888810···10121212121212151520···2030···3040···4060···60
size1111222202060602222266662···24420202020442···24···46···64···4

72 irreducible representations

dim11111222222222222222244444444
type+++++++++-+++---++++--
imageC1C2C2C2C4S3D4D4D5D6SD16Q16D10C4×S3D12C3⋊D4C4×D5C5⋊D4D20C40⋊C2Dic20D4.S3C3⋊Q16S3×D5D30.C2C5⋊D12C3⋊D20C6.D20C3⋊Dic20
kernelDic3012C4C3×C4⋊Dic5C10×C3⋊C8C2×Dic30Dic30C4⋊Dic5C60C2×C30C2×C3⋊C8C2×C20C30C30C2×C12C20C20C2×C10C12C12C2×C6C6C6C10C10C2×C4C4C4C22C2C2
# reps11114111212222224448811222244

Matrix representation of Dic3012C4 in GL6(𝔽241)

02400000
110000
0051100
00240000
00008541
0000200122
,
621280000
661790000
006817500
008117300
00008133
0000159160
,
701400000
1011710000
0022712700
001181400
000022564
000017316

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] >;

Dic3012C4 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

׿
×
𝔽