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G = Dic3015C4order 480 = 25·3·5

9th semidirect product of Dic30 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.58D4, C30.6Q16, C12.50D20, Dic3015C4, C10.4Dic12, C30.12SD16, C20.38(C4×S3), (C2×C30).22D4, (C2×C20).53D6, C12.32(C4×D5), C4⋊Dic3.1D5, C60.108(C2×C4), C158(Q8⋊C4), (C2×C10).32D12, C31(C10.Q16), C6.2(D4.D5), C6.2(C5⋊Q16), C52(C2.Dic12), C10.8(C24⋊C2), C10.22(D6⋊C4), (C2×C12).286D10, C20.14(C3⋊D4), C4.15(C3⋊D20), C4.7(D30.C2), C2.2(C5⋊Dic12), C2.8(D304C4), C6.7(D10⋊C4), C30.58(C22⋊C4), (C2×C60).130C22, (C2×Dic30).16C2, C2.2(D12.D5), C22.15(C5⋊D12), (C6×C52C8).3C2, (C2×C52C8).2S3, (C2×C4).136(S3×D5), (C5×C4⋊Dic3).1C2, (C2×C6).27(C5⋊D4), SmallGroup(480,51)

Series: Derived Chief Lower central Upper central

C1C60 — Dic3015C4
C1C5C15C30C60C2×C60C6×C52C8 — Dic3015C4
C15C30C60 — Dic3015C4
C1C22C2×C4

Generators and relations for Dic3015C4
 G = < a,b,c | a60=c4=1, b2=a30, bab-1=a-1, cac-1=a11, cbc-1=a15b >

Subgroups: 412 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 [×3], C12 [×2], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5 [×2], C20 [×2], C20, C2×C10, C24, Dic6 [×3], C2×Dic3 [×2], C2×C12, C30 [×3], Q8⋊C4, C52C8, Dic10 [×3], C2×Dic5, C2×C20, C2×C20, C4⋊Dic3, C2×C24, C2×Dic6, C5×Dic3, Dic15 [×2], C60 [×2], C2×C30, C2×C52C8, C5×C4⋊C4, C2×Dic10, C2.Dic12, C3×C52C8, C10×Dic3, Dic30 [×2], Dic30, C2×Dic15, C2×C60, C10.Q16, C6×C52C8, C5×C4⋊Dic3, C2×Dic30, Dic3015C4
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, C24⋊C2, Dic12, D6⋊C4, S3×D5, D10⋊C4, D4.D5, C5⋊Q16, C2.Dic12, D30.C2, C3⋊D20, C5⋊D12, C10.Q16, D12.D5, C5⋊Dic12, D304C4, Dic3015C4

Smallest permutation representation of Dic3015C4
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)
(1 256 31 286)(2 255 32 285)(3 254 33 284)(4 253 34 283)(5 252 35 282)(6 251 36 281)(7 250 37 280)(8 249 38 279)(9 248 39 278)(10 247 40 277)(11 246 41 276)(12 245 42 275)(13 244 43 274)(14 243 44 273)(15 242 45 272)(16 241 46 271)(17 300 47 270)(18 299 48 269)(19 298 49 268)(20 297 50 267)(21 296 51 266)(22 295 52 265)(23 294 53 264)(24 293 54 263)(25 292 55 262)(26 291 56 261)(27 290 57 260)(28 289 58 259)(29 288 59 258)(30 287 60 257)(61 316 91 346)(62 315 92 345)(63 314 93 344)(64 313 94 343)(65 312 95 342)(66 311 96 341)(67 310 97 340)(68 309 98 339)(69 308 99 338)(70 307 100 337)(71 306 101 336)(72 305 102 335)(73 304 103 334)(74 303 104 333)(75 302 105 332)(76 301 106 331)(77 360 107 330)(78 359 108 329)(79 358 109 328)(80 357 110 327)(81 356 111 326)(82 355 112 325)(83 354 113 324)(84 353 114 323)(85 352 115 322)(86 351 116 321)(87 350 117 320)(88 349 118 319)(89 348 119 318)(90 347 120 317)(121 462 151 432)(122 461 152 431)(123 460 153 430)(124 459 154 429)(125 458 155 428)(126 457 156 427)(127 456 157 426)(128 455 158 425)(129 454 159 424)(130 453 160 423)(131 452 161 422)(132 451 162 421)(133 450 163 480)(134 449 164 479)(135 448 165 478)(136 447 166 477)(137 446 167 476)(138 445 168 475)(139 444 169 474)(140 443 170 473)(141 442 171 472)(142 441 172 471)(143 440 173 470)(144 439 174 469)(145 438 175 468)(146 437 176 467)(147 436 177 466)(148 435 178 465)(149 434 179 464)(150 433 180 463)(181 391 211 361)(182 390 212 420)(183 389 213 419)(184 388 214 418)(185 387 215 417)(186 386 216 416)(187 385 217 415)(188 384 218 414)(189 383 219 413)(190 382 220 412)(191 381 221 411)(192 380 222 410)(193 379 223 409)(194 378 224 408)(195 377 225 407)(196 376 226 406)(197 375 227 405)(198 374 228 404)(199 373 229 403)(200 372 230 402)(201 371 231 401)(202 370 232 400)(203 369 233 399)(204 368 234 398)(205 367 235 397)(206 366 236 396)(207 365 237 395)(208 364 238 394)(209 363 239 393)(210 362 240 392)
(1 111 460 211)(2 62 461 222)(3 73 462 233)(4 84 463 184)(5 95 464 195)(6 106 465 206)(7 117 466 217)(8 68 467 228)(9 79 468 239)(10 90 469 190)(11 101 470 201)(12 112 471 212)(13 63 472 223)(14 74 473 234)(15 85 474 185)(16 96 475 196)(17 107 476 207)(18 118 477 218)(19 69 478 229)(20 80 479 240)(21 91 480 191)(22 102 421 202)(23 113 422 213)(24 64 423 224)(25 75 424 235)(26 86 425 186)(27 97 426 197)(28 108 427 208)(29 119 428 219)(30 70 429 230)(31 81 430 181)(32 92 431 192)(33 103 432 203)(34 114 433 214)(35 65 434 225)(36 76 435 236)(37 87 436 187)(38 98 437 198)(39 109 438 209)(40 120 439 220)(41 71 440 231)(42 82 441 182)(43 93 442 193)(44 104 443 204)(45 115 444 215)(46 66 445 226)(47 77 446 237)(48 88 447 188)(49 99 448 199)(50 110 449 210)(51 61 450 221)(52 72 451 232)(53 83 452 183)(54 94 453 194)(55 105 454 205)(56 116 455 216)(57 67 456 227)(58 78 457 238)(59 89 458 189)(60 100 459 200)(121 414 284 319)(122 365 285 330)(123 376 286 341)(124 387 287 352)(125 398 288 303)(126 409 289 314)(127 420 290 325)(128 371 291 336)(129 382 292 347)(130 393 293 358)(131 404 294 309)(132 415 295 320)(133 366 296 331)(134 377 297 342)(135 388 298 353)(136 399 299 304)(137 410 300 315)(138 361 241 326)(139 372 242 337)(140 383 243 348)(141 394 244 359)(142 405 245 310)(143 416 246 321)(144 367 247 332)(145 378 248 343)(146 389 249 354)(147 400 250 305)(148 411 251 316)(149 362 252 327)(150 373 253 338)(151 384 254 349)(152 395 255 360)(153 406 256 311)(154 417 257 322)(155 368 258 333)(156 379 259 344)(157 390 260 355)(158 401 261 306)(159 412 262 317)(160 363 263 328)(161 374 264 339)(162 385 265 350)(163 396 266 301)(164 407 267 312)(165 418 268 323)(166 369 269 334)(167 380 270 345)(168 391 271 356)(169 402 272 307)(170 413 273 318)(171 364 274 329)(172 375 275 340)(173 386 276 351)(174 397 277 302)(175 408 278 313)(176 419 279 324)(177 370 280 335)(178 381 281 346)(179 392 282 357)(180 403 283 308)

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

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

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

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L24A···24H30A···30F60A···60H
order1222344444455666888810···101212121215152020202020···2024···2430···3060···60
size11112221212606022222101010102···2222244444412···1210···104···44···4

66 irreducible representations

dim11111222222222222222244444444
type+++++++++-+++-+--+++--
imageC1C2C2C2C4S3D4D4D5D6SD16Q16D10C4×S3C3⋊D4D12C4×D5D20C5⋊D4C24⋊C2Dic12S3×D5D4.D5C5⋊Q16D30.C2C3⋊D20C5⋊D12D12.D5C5⋊Dic12
kernelDic3015C4C6×C52C8C5×C4⋊Dic3C2×Dic30Dic30C2×C52C8C60C2×C30C4⋊Dic3C2×C20C30C30C2×C12C20C20C2×C10C12C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps11114111212222224444422222244

Matrix representation of Dic3015C4 in GL4(𝔽241) generated by

14214200
994300
001240
0019151
,
2011000
13022100
0010856
00235133
,
551800
7318600
0044156
0088197
G:=sub<GL(4,GF(241))| [142,99,0,0,142,43,0,0,0,0,1,191,0,0,240,51],[20,130,0,0,110,221,0,0,0,0,108,235,0,0,56,133],[55,73,0,0,18,186,0,0,0,0,44,88,0,0,156,197] >;

Dic3015C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{30}\rtimes_{15}C_4
% in TeX

G:=Group("Dic30:15C4");
// GroupNames label

G:=SmallGroup(480,51);
// by ID

G=gap.SmallGroup(480,51);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,197,176,219,100,1356,18822]);
// 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^11,c*b*c^-1=a^15*b>;
// generators/relations

׿
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