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G = Dic3017C4order 480 = 25·3·5

11st semidirect product of Dic30 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic3017C4, Dic56Dic6, C159(C4×Q8), C54(C4×Dic6), C6.23(Q8×D5), C20.56(C4×S3), (C3×Dic5)⋊5Q8, C12.35(C4×D5), C30.10(C2×Q8), C2.3(D5×Dic6), C60.118(C2×C4), C30.5(C4○D4), (C2×C20).107D6, C4⋊Dic3.14D5, (C4×Dic5).2S3, C10.5(C2×Dic6), (C2×C12).294D10, C32(Dic53Q8), (C2×C30).23C23, (C12×Dic5).2C2, Dic155C4.4C2, C10.47(C4○D12), C2.1(D125D5), C6.16(D42D5), C30.114(C22×C4), C4.10(D30.C2), (C2×C60).138C22, (C2×Dic30).18C2, (C2×Dic3).78D10, Dic15.25(C2×C4), (C2×Dic5).155D6, (Dic3×Dic5).10C2, (C10×Dic3).9C22, (C2×Dic15).30C22, (C6×Dic5).176C22, C6.43(C2×C4×D5), C10.76(S3×C2×C4), C22.27(C2×S3×D5), (C2×C4).151(S3×D5), (C5×C4⋊Dic3).4C2, C2.8(C2×D30.C2), (C2×C6).35(C22×D5), (C2×C10).35(C22×S3), SmallGroup(480,409)

Series: Derived Chief Lower central Upper central

C1C30 — Dic3017C4
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic3017C4
C15C30 — Dic3017C4
C1C22C2×C4

Generators and relations for Dic3017C4
 G = < a,b,c,d | a10=c12=1, b2=a5, d2=c6, bab-1=dad-1=a-1, ac=ca, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 556 in 140 conjugacy classes, 64 normal (34 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×9], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×6], C12 [×2], C12 [×3], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×2], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4×Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3 [×2], C3×Dic5 [×2], C3×Dic5, Dic15 [×4], C60 [×2], C2×C30, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C5×C4⋊C4, C2×Dic10, C4×Dic6, C6×Dic5 [×2], C10×Dic3 [×2], Dic30 [×4], C2×Dic15 [×2], C2×C60, Dic53Q8, Dic3×Dic5 [×2], Dic155C4 [×2], C12×Dic5, C5×C4⋊Dic3, C2×Dic30, Dic3017C4
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], Dic6 [×2], C4×S3 [×2], C22×S3, C4×Q8, C4×D5 [×2], C22×D5, C2×Dic6, S3×C2×C4, C4○D12, S3×D5, C2×C4×D5, D42D5, Q8×D5, C4×Dic6, D30.C2 [×2], C2×S3×D5, Dic53Q8, D5×Dic6, D125D5, C2×D30.C2, Dic3017C4

Smallest permutation representation of Dic3017C4
Regular action on 480 points
Generators in S480
(1 286 17 421 228 370 449 37 250 384)(2 287 18 422 217 371 450 38 251 373)(3 288 19 423 218 372 451 39 252 374)(4 277 20 424 219 361 452 40 241 375)(5 278 21 425 220 362 453 41 242 376)(6 279 22 426 221 363 454 42 243 377)(7 280 23 427 222 364 455 43 244 378)(8 281 24 428 223 365 456 44 245 379)(9 282 13 429 224 366 445 45 246 380)(10 283 14 430 225 367 446 46 247 381)(11 284 15 431 226 368 447 47 248 382)(12 285 16 432 227 369 448 48 249 383)(25 171 127 350 100 473 269 49 113 210)(26 172 128 351 101 474 270 50 114 211)(27 173 129 352 102 475 271 51 115 212)(28 174 130 353 103 476 272 52 116 213)(29 175 131 354 104 477 273 53 117 214)(30 176 132 355 105 478 274 54 118 215)(31 177 121 356 106 479 275 55 119 216)(32 178 122 357 107 480 276 56 120 205)(33 179 123 358 108 469 265 57 109 206)(34 180 124 359 97 470 266 58 110 207)(35 169 125 360 98 471 267 59 111 208)(36 170 126 349 99 472 268 60 112 209)(61 264 340 413 91 230 153 397 457 441)(62 253 341 414 92 231 154 398 458 442)(63 254 342 415 93 232 155 399 459 443)(64 255 343 416 94 233 156 400 460 444)(65 256 344 417 95 234 145 401 461 433)(66 257 345 418 96 235 146 402 462 434)(67 258 346 419 85 236 147 403 463 435)(68 259 347 420 86 237 148 404 464 436)(69 260 348 409 87 238 149 405 465 437)(70 261 337 410 88 239 150 406 466 438)(71 262 338 411 89 240 151 407 467 439)(72 263 339 412 90 229 152 408 468 440)(73 183 135 299 302 320 325 166 392 194)(74 184 136 300 303 321 326 167 393 195)(75 185 137 289 304 322 327 168 394 196)(76 186 138 290 305 323 328 157 395 197)(77 187 139 291 306 324 329 158 396 198)(78 188 140 292 307 313 330 159 385 199)(79 189 141 293 308 314 331 160 386 200)(80 190 142 294 309 315 332 161 387 201)(81 191 143 295 310 316 333 162 388 202)(82 192 144 296 311 317 334 163 389 203)(83 181 133 297 312 318 335 164 390 204)(84 182 134 298 301 319 336 165 391 193)
(1 105 370 215)(2 106 371 216)(3 107 372 205)(4 108 361 206)(5 97 362 207)(6 98 363 208)(7 99 364 209)(8 100 365 210)(9 101 366 211)(10 102 367 212)(11 103 368 213)(12 104 369 214)(13 128 45 50)(14 129 46 51)(15 130 47 52)(16 131 48 53)(17 132 37 54)(18 121 38 55)(19 122 39 56)(20 123 40 57)(21 124 41 58)(22 125 42 59)(23 126 43 60)(24 127 44 49)(25 379 473 223)(26 380 474 224)(27 381 475 225)(28 382 476 226)(29 383 477 227)(30 384 478 228)(31 373 479 217)(32 374 480 218)(33 375 469 219)(34 376 470 220)(35 377 471 221)(36 378 472 222)(61 75 230 322)(62 76 231 323)(63 77 232 324)(64 78 233 313)(65 79 234 314)(66 80 235 315)(67 81 236 316)(68 82 237 317)(69 83 238 318)(70 84 239 319)(71 73 240 320)(72 74 229 321)(85 333 435 191)(86 334 436 192)(87 335 437 181)(88 336 438 182)(89 325 439 183)(90 326 440 184)(91 327 441 185)(92 328 442 186)(93 329 443 187)(94 330 444 188)(95 331 433 189)(96 332 434 190)(109 277 358 452)(110 278 359 453)(111 279 360 454)(112 280 349 455)(113 281 350 456)(114 282 351 445)(115 283 352 446)(116 284 353 447)(117 285 354 448)(118 286 355 449)(119 287 356 450)(120 288 357 451)(133 409 164 465)(134 410 165 466)(135 411 166 467)(136 412 167 468)(137 413 168 457)(138 414 157 458)(139 415 158 459)(140 416 159 460)(141 417 160 461)(142 418 161 462)(143 419 162 463)(144 420 163 464)(145 308 256 200)(146 309 257 201)(147 310 258 202)(148 311 259 203)(149 312 260 204)(150 301 261 193)(151 302 262 194)(152 303 263 195)(153 304 264 196)(154 305 253 197)(155 306 254 198)(156 307 255 199)(169 243 267 426)(170 244 268 427)(171 245 269 428)(172 246 270 429)(173 247 271 430)(174 248 272 431)(175 249 273 432)(176 250 274 421)(177 251 275 422)(178 252 276 423)(179 241 265 424)(180 242 266 425)(289 340 394 397)(290 341 395 398)(291 342 396 399)(292 343 385 400)(293 344 386 401)(294 345 387 402)(295 346 388 403)(296 347 389 404)(297 348 390 405)(298 337 391 406)(299 338 392 407)(300 339 393 408)
(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)
(1 321 7 315)(2 320 8 314)(3 319 9 313)(4 318 10 324)(5 317 11 323)(6 316 12 322)(13 292 19 298)(14 291 20 297)(15 290 21 296)(16 289 22 295)(17 300 23 294)(18 299 24 293)(25 95 31 89)(26 94 32 88)(27 93 33 87)(28 92 34 86)(29 91 35 85)(30 90 36 96)(37 393 43 387)(38 392 44 386)(39 391 45 385)(40 390 46 396)(41 389 47 395)(42 388 48 394)(49 401 55 407)(50 400 56 406)(51 399 57 405)(52 398 58 404)(53 397 59 403)(54 408 60 402)(61 98 67 104)(62 97 68 103)(63 108 69 102)(64 107 70 101)(65 106 71 100)(66 105 72 99)(73 365 79 371)(74 364 80 370)(75 363 81 369)(76 362 82 368)(77 361 83 367)(78 372 84 366)(109 149 115 155)(110 148 116 154)(111 147 117 153)(112 146 118 152)(113 145 119 151)(114 156 120 150)(121 338 127 344)(122 337 128 343)(123 348 129 342)(124 347 130 341)(125 346 131 340)(126 345 132 339)(133 430 139 424)(134 429 140 423)(135 428 141 422)(136 427 142 421)(137 426 143 432)(138 425 144 431)(157 242 163 248)(158 241 164 247)(159 252 165 246)(160 251 166 245)(161 250 167 244)(162 249 168 243)(169 419 175 413)(170 418 176 412)(171 417 177 411)(172 416 178 410)(173 415 179 409)(174 414 180 420)(181 225 187 219)(182 224 188 218)(183 223 189 217)(184 222 190 228)(185 221 191 227)(186 220 192 226)(193 445 199 451)(194 456 200 450)(195 455 201 449)(196 454 202 448)(197 453 203 447)(198 452 204 446)(205 239 211 233)(206 238 212 232)(207 237 213 231)(208 236 214 230)(209 235 215 229)(210 234 216 240)(253 359 259 353)(254 358 260 352)(255 357 261 351)(256 356 262 350)(257 355 263 349)(258 354 264 360)(265 465 271 459)(266 464 272 458)(267 463 273 457)(268 462 274 468)(269 461 275 467)(270 460 276 466)(277 312 283 306)(278 311 284 305)(279 310 285 304)(280 309 286 303)(281 308 287 302)(282 307 288 301)(325 379 331 373)(326 378 332 384)(327 377 333 383)(328 376 334 382)(329 375 335 381)(330 374 336 380)(433 479 439 473)(434 478 440 472)(435 477 441 471)(436 476 442 470)(437 475 443 469)(438 474 444 480)

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

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

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

72 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B6A6B6C10A···10F12A12B12C12D12E···12L15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444444444445566610···101212121212···1215152020202020···2030···3060···60
size111122255556666101030303030222222···2222210···1044444412···124···44···4

72 irreducible representations

dim11111112222222222224444444
type+++++++-+++++-+--++--
imageC1C2C2C2C2C2C4S3Q8D5D6D6C4○D4D10D10Dic6C4×S3C4×D5C4○D12S3×D5D42D5Q8×D5D30.C2C2×S3×D5D5×Dic6D125D5
kernelDic3017C4Dic3×Dic5Dic155C4C12×Dic5C5×C4⋊Dic3C2×Dic30Dic30C4×Dic5C3×Dic5C4⋊Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5C20C12C10C2×C4C6C6C4C22C2C2
# reps12211181222124244842224244

Matrix representation of Dic3017C4 in GL5(𝔽61)

600000
060000
006000
000601
0004218
,
110000
011000
001100
0005237
000449
,
600000
0153800
0233800
000600
000060
,
600000
0524300
052900
000924
0001752

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,42,0,0,0,1,18],[11,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,52,44,0,0,0,37,9],[60,0,0,0,0,0,15,23,0,0,0,38,38,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,52,52,0,0,0,43,9,0,0,0,0,0,9,17,0,0,0,24,52] >;

Dic3017C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{30}\rtimes_{17}C_4
% in TeX

G:=Group("Dic30:17C4");
// GroupNames label

G:=SmallGroup(480,409);
// by ID

G=gap.SmallGroup(480,409);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=c^6,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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