metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic30⋊17C4, Dic5⋊6Dic6, C15⋊9(C4×Q8), C5⋊4(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, C3⋊2(Dic5⋊3Q8), (C2×C30).23C23, (C12×Dic5).2C2, Dic15⋊5C4.4C2, C10.47(C4○D12), C2.1(D12⋊5D5), C6.16(D4⋊2D5), 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
Generators and relations for Dic30⋊17C4
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, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C4×Dic5, C10.D4, C5×C4⋊C4, C2×Dic10, C4×Dic6, C6×Dic5, C10×Dic3, Dic30, C2×Dic15, C2×C60, Dic5⋊3Q8, Dic3×Dic5, Dic15⋊5C4, C12×Dic5, C5×C4⋊Dic3, C2×Dic30, Dic30⋊17C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D5, D6, C22×C4, C2×Q8, C4○D4, D10, Dic6, C4×S3, C22×S3, C4×Q8, C4×D5, C22×D5, C2×Dic6, S3×C2×C4, C4○D12, S3×D5, C2×C4×D5, D4⋊2D5, Q8×D5, C4×Dic6, D30.C2, C2×S3×D5, Dic5⋊3Q8, D5×Dic6, D12⋊5D5, C2×D30.C2, Dic30⋊17C4
►Regular action on 480 points
Generators in S480
(1 351 367 314 444 310 328 68 413 431)(2 352 368 315 433 311 329 69 414 432)(3 353 369 316 434 312 330 70 415 421)(4 354 370 317 435 301 331 71 416 422)(5 355 371 318 436 302 332 72 417 423)(6 356 372 319 437 303 333 61 418 424)(7 357 361 320 438 304 334 62 419 425)(8 358 362 321 439 305 335 63 420 426)(9 359 363 322 440 306 336 64 409 427)(10 360 364 323 441 307 325 65 410 428)(11 349 365 324 442 308 326 66 411 429)(12 350 366 313 443 309 327 67 412 430)(13 215 296 33 274 223 101 44 153 89)(14 216 297 34 275 224 102 45 154 90)(15 205 298 35 276 225 103 46 155 91)(16 206 299 36 265 226 104 47 156 92)(17 207 300 25 266 227 105 48 145 93)(18 208 289 26 267 228 106 37 146 94)(19 209 290 27 268 217 107 38 147 95)(20 210 291 28 269 218 108 39 148 96)(21 211 292 29 270 219 97 40 149 85)(22 212 293 30 271 220 98 41 150 86)(23 213 294 31 272 221 99 42 151 87)(24 214 295 32 273 222 100 43 152 88)(49 231 128 389 469 193 384 451 183 284)(50 232 129 390 470 194 373 452 184 285)(51 233 130 391 471 195 374 453 185 286)(52 234 131 392 472 196 375 454 186 287)(53 235 132 393 473 197 376 455 187 288)(54 236 121 394 474 198 377 456 188 277)(55 237 122 395 475 199 378 445 189 278)(56 238 123 396 476 200 379 446 190 279)(57 239 124 385 477 201 380 447 191 280)(58 240 125 386 478 202 381 448 192 281)(59 229 126 387 479 203 382 449 181 282)(60 230 127 388 480 204 383 450 182 283)(73 140 264 159 112 251 460 404 345 174)(74 141 253 160 113 252 461 405 346 175)(75 142 254 161 114 241 462 406 347 176)(76 143 255 162 115 242 463 407 348 177)(77 144 256 163 116 243 464 408 337 178)(78 133 257 164 117 244 465 397 338 179)(79 134 258 165 118 245 466 398 339 180)(80 135 259 166 119 246 467 399 340 169)(81 136 260 167 120 247 468 400 341 170)(82 137 261 168 109 248 457 401 342 171)(83 138 262 157 110 249 458 402 343 172)(84 139 263 158 111 250 459 403 344 173)
(1 242 310 76)(2 243 311 77)(3 244 312 78)(4 245 301 79)(5 246 302 80)(6 247 303 81)(7 248 304 82)(8 249 305 83)(9 250 306 84)(10 251 307 73)(11 252 308 74)(12 241 309 75)(13 387 223 181)(14 388 224 182)(15 389 225 183)(16 390 226 184)(17 391 227 185)(18 392 228 186)(19 393 217 187)(20 394 218 188)(21 395 219 189)(22 396 220 190)(23 385 221 191)(24 386 222 192)(25 51 145 195)(26 52 146 196)(27 53 147 197)(28 54 148 198)(29 55 149 199)(30 56 150 200)(31 57 151 201)(32 58 152 202)(33 59 153 203)(34 60 154 204)(35 49 155 193)(36 50 156 194)(37 375 289 234)(38 376 290 235)(39 377 291 236)(40 378 292 237)(41 379 293 238)(42 380 294 239)(43 381 295 240)(44 382 296 229)(45 383 297 230)(46 384 298 231)(47 373 299 232)(48 374 300 233)(61 341 372 167)(62 342 361 168)(63 343 362 157)(64 344 363 158)(65 345 364 159)(66 346 365 160)(67 347 366 161)(68 348 367 162)(69 337 368 163)(70 338 369 164)(71 339 370 165)(72 340 371 166)(85 475 270 278)(86 476 271 279)(87 477 272 280)(88 478 273 281)(89 479 274 282)(90 480 275 283)(91 469 276 284)(92 470 265 285)(93 471 266 286)(94 472 267 287)(95 473 268 288)(96 474 269 277)(97 445 211 122)(98 446 212 123)(99 447 213 124)(100 448 214 125)(101 449 215 126)(102 450 216 127)(103 451 205 128)(104 452 206 129)(105 453 207 130)(106 454 208 131)(107 455 209 132)(108 456 210 121)(109 334 171 357)(110 335 172 358)(111 336 173 359)(112 325 174 360)(113 326 175 349)(114 327 176 350)(115 328 177 351)(116 329 178 352)(117 330 179 353)(118 331 180 354)(119 332 169 355)(120 333 170 356)(133 421 465 434)(134 422 466 435)(135 423 467 436)(136 424 468 437)(137 425 457 438)(138 426 458 439)(139 427 459 440)(140 428 460 441)(141 429 461 442)(142 430 462 443)(143 431 463 444)(144 432 464 433)(253 411 405 324)(254 412 406 313)(255 413 407 314)(256 414 408 315)(257 415 397 316)(258 416 398 317)(259 417 399 318)(260 418 400 319)(261 419 401 320)(262 420 402 321)(263 409 403 322)(264 410 404 323)
(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)
(1 191 7 185)(2 190 8 184)(3 189 9 183)(4 188 10 182)(5 187 11 181)(6 186 12 192)(13 246 19 252)(14 245 20 251)(15 244 21 250)(16 243 22 249)(17 242 23 248)(18 241 24 247)(25 255 31 261)(26 254 32 260)(27 253 33 259)(28 264 34 258)(29 263 35 257)(30 262 36 256)(37 347 43 341)(38 346 44 340)(39 345 45 339)(40 344 46 338)(41 343 47 337)(42 342 48 348)(49 415 55 409)(50 414 56 420)(51 413 57 419)(52 412 58 418)(53 411 59 417)(54 410 60 416)(61 234 67 240)(62 233 68 239)(63 232 69 238)(64 231 70 237)(65 230 71 236)(66 229 72 235)(73 224 79 218)(74 223 80 217)(75 222 81 228)(76 221 82 227)(77 220 83 226)(78 219 84 225)(85 459 91 465)(86 458 92 464)(87 457 93 463)(88 468 94 462)(89 467 95 461)(90 466 96 460)(97 173 103 179)(98 172 104 178)(99 171 105 177)(100 170 106 176)(101 169 107 175)(102 180 108 174)(109 207 115 213)(110 206 116 212)(111 205 117 211)(112 216 118 210)(113 215 119 209)(114 214 120 208)(121 325 127 331)(122 336 128 330)(123 335 129 329)(124 334 130 328)(125 333 131 327)(126 332 132 326)(133 270 139 276)(134 269 140 275)(135 268 141 274)(136 267 142 273)(137 266 143 272)(138 265 144 271)(145 407 151 401)(146 406 152 400)(147 405 153 399)(148 404 154 398)(149 403 155 397)(150 402 156 408)(157 299 163 293)(158 298 164 292)(159 297 165 291)(160 296 166 290)(161 295 167 289)(162 294 168 300)(193 316 199 322)(194 315 200 321)(195 314 201 320)(196 313 202 319)(197 324 203 318)(198 323 204 317)(277 428 283 422)(278 427 284 421)(279 426 285 432)(280 425 286 431)(281 424 287 430)(282 423 288 429)(301 394 307 388)(302 393 308 387)(303 392 309 386)(304 391 310 385)(305 390 311 396)(306 389 312 395)(349 449 355 455)(350 448 356 454)(351 447 357 453)(352 446 358 452)(353 445 359 451)(354 456 360 450)(361 374 367 380)(362 373 368 379)(363 384 369 378)(364 383 370 377)(365 382 371 376)(366 381 372 375)(433 476 439 470)(434 475 440 469)(435 474 441 480)(436 473 442 479)(437 472 443 478)(438 471 444 477)
G:=sub<Sym(480)| (1,351,367,314,444,310,328,68,413,431)(2,352,368,315,433,311,329,69,414,432)(3,353,369,316,434,312,330,70,415,421)(4,354,370,317,435,301,331,71,416,422)(5,355,371,318,436,302,332,72,417,423)(6,356,372,319,437,303,333,61,418,424)(7,357,361,320,438,304,334,62,419,425)(8,358,362,321,439,305,335,63,420,426)(9,359,363,322,440,306,336,64,409,427)(10,360,364,323,441,307,325,65,410,428)(11,349,365,324,442,308,326,66,411,429)(12,350,366,313,443,309,327,67,412,430)(13,215,296,33,274,223,101,44,153,89)(14,216,297,34,275,224,102,45,154,90)(15,205,298,35,276,225,103,46,155,91)(16,206,299,36,265,226,104,47,156,92)(17,207,300,25,266,227,105,48,145,93)(18,208,289,26,267,228,106,37,146,94)(19,209,290,27,268,217,107,38,147,95)(20,210,291,28,269,218,108,39,148,96)(21,211,292,29,270,219,97,40,149,85)(22,212,293,30,271,220,98,41,150,86)(23,213,294,31,272,221,99,42,151,87)(24,214,295,32,273,222,100,43,152,88)(49,231,128,389,469,193,384,451,183,284)(50,232,129,390,470,194,373,452,184,285)(51,233,130,391,471,195,374,453,185,286)(52,234,131,392,472,196,375,454,186,287)(53,235,132,393,473,197,376,455,187,288)(54,236,121,394,474,198,377,456,188,277)(55,237,122,395,475,199,378,445,189,278)(56,238,123,396,476,200,379,446,190,279)(57,239,124,385,477,201,380,447,191,280)(58,240,125,386,478,202,381,448,192,281)(59,229,126,387,479,203,382,449,181,282)(60,230,127,388,480,204,383,450,182,283)(73,140,264,159,112,251,460,404,345,174)(74,141,253,160,113,252,461,405,346,175)(75,142,254,161,114,241,462,406,347,176)(76,143,255,162,115,242,463,407,348,177)(77,144,256,163,116,243,464,408,337,178)(78,133,257,164,117,244,465,397,338,179)(79,134,258,165,118,245,466,398,339,180)(80,135,259,166,119,246,467,399,340,169)(81,136,260,167,120,247,468,400,341,170)(82,137,261,168,109,248,457,401,342,171)(83,138,262,157,110,249,458,402,343,172)(84,139,263,158,111,250,459,403,344,173), (1,242,310,76)(2,243,311,77)(3,244,312,78)(4,245,301,79)(5,246,302,80)(6,247,303,81)(7,248,304,82)(8,249,305,83)(9,250,306,84)(10,251,307,73)(11,252,308,74)(12,241,309,75)(13,387,223,181)(14,388,224,182)(15,389,225,183)(16,390,226,184)(17,391,227,185)(18,392,228,186)(19,393,217,187)(20,394,218,188)(21,395,219,189)(22,396,220,190)(23,385,221,191)(24,386,222,192)(25,51,145,195)(26,52,146,196)(27,53,147,197)(28,54,148,198)(29,55,149,199)(30,56,150,200)(31,57,151,201)(32,58,152,202)(33,59,153,203)(34,60,154,204)(35,49,155,193)(36,50,156,194)(37,375,289,234)(38,376,290,235)(39,377,291,236)(40,378,292,237)(41,379,293,238)(42,380,294,239)(43,381,295,240)(44,382,296,229)(45,383,297,230)(46,384,298,231)(47,373,299,232)(48,374,300,233)(61,341,372,167)(62,342,361,168)(63,343,362,157)(64,344,363,158)(65,345,364,159)(66,346,365,160)(67,347,366,161)(68,348,367,162)(69,337,368,163)(70,338,369,164)(71,339,370,165)(72,340,371,166)(85,475,270,278)(86,476,271,279)(87,477,272,280)(88,478,273,281)(89,479,274,282)(90,480,275,283)(91,469,276,284)(92,470,265,285)(93,471,266,286)(94,472,267,287)(95,473,268,288)(96,474,269,277)(97,445,211,122)(98,446,212,123)(99,447,213,124)(100,448,214,125)(101,449,215,126)(102,450,216,127)(103,451,205,128)(104,452,206,129)(105,453,207,130)(106,454,208,131)(107,455,209,132)(108,456,210,121)(109,334,171,357)(110,335,172,358)(111,336,173,359)(112,325,174,360)(113,326,175,349)(114,327,176,350)(115,328,177,351)(116,329,178,352)(117,330,179,353)(118,331,180,354)(119,332,169,355)(120,333,170,356)(133,421,465,434)(134,422,466,435)(135,423,467,436)(136,424,468,437)(137,425,457,438)(138,426,458,439)(139,427,459,440)(140,428,460,441)(141,429,461,442)(142,430,462,443)(143,431,463,444)(144,432,464,433)(253,411,405,324)(254,412,406,313)(255,413,407,314)(256,414,408,315)(257,415,397,316)(258,416,398,317)(259,417,399,318)(260,418,400,319)(261,419,401,320)(262,420,402,321)(263,409,403,322)(264,410,404,323), (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), (1,191,7,185)(2,190,8,184)(3,189,9,183)(4,188,10,182)(5,187,11,181)(6,186,12,192)(13,246,19,252)(14,245,20,251)(15,244,21,250)(16,243,22,249)(17,242,23,248)(18,241,24,247)(25,255,31,261)(26,254,32,260)(27,253,33,259)(28,264,34,258)(29,263,35,257)(30,262,36,256)(37,347,43,341)(38,346,44,340)(39,345,45,339)(40,344,46,338)(41,343,47,337)(42,342,48,348)(49,415,55,409)(50,414,56,420)(51,413,57,419)(52,412,58,418)(53,411,59,417)(54,410,60,416)(61,234,67,240)(62,233,68,239)(63,232,69,238)(64,231,70,237)(65,230,71,236)(66,229,72,235)(73,224,79,218)(74,223,80,217)(75,222,81,228)(76,221,82,227)(77,220,83,226)(78,219,84,225)(85,459,91,465)(86,458,92,464)(87,457,93,463)(88,468,94,462)(89,467,95,461)(90,466,96,460)(97,173,103,179)(98,172,104,178)(99,171,105,177)(100,170,106,176)(101,169,107,175)(102,180,108,174)(109,207,115,213)(110,206,116,212)(111,205,117,211)(112,216,118,210)(113,215,119,209)(114,214,120,208)(121,325,127,331)(122,336,128,330)(123,335,129,329)(124,334,130,328)(125,333,131,327)(126,332,132,326)(133,270,139,276)(134,269,140,275)(135,268,141,274)(136,267,142,273)(137,266,143,272)(138,265,144,271)(145,407,151,401)(146,406,152,400)(147,405,153,399)(148,404,154,398)(149,403,155,397)(150,402,156,408)(157,299,163,293)(158,298,164,292)(159,297,165,291)(160,296,166,290)(161,295,167,289)(162,294,168,300)(193,316,199,322)(194,315,200,321)(195,314,201,320)(196,313,202,319)(197,324,203,318)(198,323,204,317)(277,428,283,422)(278,427,284,421)(279,426,285,432)(280,425,286,431)(281,424,287,430)(282,423,288,429)(301,394,307,388)(302,393,308,387)(303,392,309,386)(304,391,310,385)(305,390,311,396)(306,389,312,395)(349,449,355,455)(350,448,356,454)(351,447,357,453)(352,446,358,452)(353,445,359,451)(354,456,360,450)(361,374,367,380)(362,373,368,379)(363,384,369,378)(364,383,370,377)(365,382,371,376)(366,381,372,375)(433,476,439,470)(434,475,440,469)(435,474,441,480)(436,473,442,479)(437,472,443,478)(438,471,444,477)>;
G:=Group( (1,351,367,314,444,310,328,68,413,431)(2,352,368,315,433,311,329,69,414,432)(3,353,369,316,434,312,330,70,415,421)(4,354,370,317,435,301,331,71,416,422)(5,355,371,318,436,302,332,72,417,423)(6,356,372,319,437,303,333,61,418,424)(7,357,361,320,438,304,334,62,419,425)(8,358,362,321,439,305,335,63,420,426)(9,359,363,322,440,306,336,64,409,427)(10,360,364,323,441,307,325,65,410,428)(11,349,365,324,442,308,326,66,411,429)(12,350,366,313,443,309,327,67,412,430)(13,215,296,33,274,223,101,44,153,89)(14,216,297,34,275,224,102,45,154,90)(15,205,298,35,276,225,103,46,155,91)(16,206,299,36,265,226,104,47,156,92)(17,207,300,25,266,227,105,48,145,93)(18,208,289,26,267,228,106,37,146,94)(19,209,290,27,268,217,107,38,147,95)(20,210,291,28,269,218,108,39,148,96)(21,211,292,29,270,219,97,40,149,85)(22,212,293,30,271,220,98,41,150,86)(23,213,294,31,272,221,99,42,151,87)(24,214,295,32,273,222,100,43,152,88)(49,231,128,389,469,193,384,451,183,284)(50,232,129,390,470,194,373,452,184,285)(51,233,130,391,471,195,374,453,185,286)(52,234,131,392,472,196,375,454,186,287)(53,235,132,393,473,197,376,455,187,288)(54,236,121,394,474,198,377,456,188,277)(55,237,122,395,475,199,378,445,189,278)(56,238,123,396,476,200,379,446,190,279)(57,239,124,385,477,201,380,447,191,280)(58,240,125,386,478,202,381,448,192,281)(59,229,126,387,479,203,382,449,181,282)(60,230,127,388,480,204,383,450,182,283)(73,140,264,159,112,251,460,404,345,174)(74,141,253,160,113,252,461,405,346,175)(75,142,254,161,114,241,462,406,347,176)(76,143,255,162,115,242,463,407,348,177)(77,144,256,163,116,243,464,408,337,178)(78,133,257,164,117,244,465,397,338,179)(79,134,258,165,118,245,466,398,339,180)(80,135,259,166,119,246,467,399,340,169)(81,136,260,167,120,247,468,400,341,170)(82,137,261,168,109,248,457,401,342,171)(83,138,262,157,110,249,458,402,343,172)(84,139,263,158,111,250,459,403,344,173), (1,242,310,76)(2,243,311,77)(3,244,312,78)(4,245,301,79)(5,246,302,80)(6,247,303,81)(7,248,304,82)(8,249,305,83)(9,250,306,84)(10,251,307,73)(11,252,308,74)(12,241,309,75)(13,387,223,181)(14,388,224,182)(15,389,225,183)(16,390,226,184)(17,391,227,185)(18,392,228,186)(19,393,217,187)(20,394,218,188)(21,395,219,189)(22,396,220,190)(23,385,221,191)(24,386,222,192)(25,51,145,195)(26,52,146,196)(27,53,147,197)(28,54,148,198)(29,55,149,199)(30,56,150,200)(31,57,151,201)(32,58,152,202)(33,59,153,203)(34,60,154,204)(35,49,155,193)(36,50,156,194)(37,375,289,234)(38,376,290,235)(39,377,291,236)(40,378,292,237)(41,379,293,238)(42,380,294,239)(43,381,295,240)(44,382,296,229)(45,383,297,230)(46,384,298,231)(47,373,299,232)(48,374,300,233)(61,341,372,167)(62,342,361,168)(63,343,362,157)(64,344,363,158)(65,345,364,159)(66,346,365,160)(67,347,366,161)(68,348,367,162)(69,337,368,163)(70,338,369,164)(71,339,370,165)(72,340,371,166)(85,475,270,278)(86,476,271,279)(87,477,272,280)(88,478,273,281)(89,479,274,282)(90,480,275,283)(91,469,276,284)(92,470,265,285)(93,471,266,286)(94,472,267,287)(95,473,268,288)(96,474,269,277)(97,445,211,122)(98,446,212,123)(99,447,213,124)(100,448,214,125)(101,449,215,126)(102,450,216,127)(103,451,205,128)(104,452,206,129)(105,453,207,130)(106,454,208,131)(107,455,209,132)(108,456,210,121)(109,334,171,357)(110,335,172,358)(111,336,173,359)(112,325,174,360)(113,326,175,349)(114,327,176,350)(115,328,177,351)(116,329,178,352)(117,330,179,353)(118,331,180,354)(119,332,169,355)(120,333,170,356)(133,421,465,434)(134,422,466,435)(135,423,467,436)(136,424,468,437)(137,425,457,438)(138,426,458,439)(139,427,459,440)(140,428,460,441)(141,429,461,442)(142,430,462,443)(143,431,463,444)(144,432,464,433)(253,411,405,324)(254,412,406,313)(255,413,407,314)(256,414,408,315)(257,415,397,316)(258,416,398,317)(259,417,399,318)(260,418,400,319)(261,419,401,320)(262,420,402,321)(263,409,403,322)(264,410,404,323), (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), (1,191,7,185)(2,190,8,184)(3,189,9,183)(4,188,10,182)(5,187,11,181)(6,186,12,192)(13,246,19,252)(14,245,20,251)(15,244,21,250)(16,243,22,249)(17,242,23,248)(18,241,24,247)(25,255,31,261)(26,254,32,260)(27,253,33,259)(28,264,34,258)(29,263,35,257)(30,262,36,256)(37,347,43,341)(38,346,44,340)(39,345,45,339)(40,344,46,338)(41,343,47,337)(42,342,48,348)(49,415,55,409)(50,414,56,420)(51,413,57,419)(52,412,58,418)(53,411,59,417)(54,410,60,416)(61,234,67,240)(62,233,68,239)(63,232,69,238)(64,231,70,237)(65,230,71,236)(66,229,72,235)(73,224,79,218)(74,223,80,217)(75,222,81,228)(76,221,82,227)(77,220,83,226)(78,219,84,225)(85,459,91,465)(86,458,92,464)(87,457,93,463)(88,468,94,462)(89,467,95,461)(90,466,96,460)(97,173,103,179)(98,172,104,178)(99,171,105,177)(100,170,106,176)(101,169,107,175)(102,180,108,174)(109,207,115,213)(110,206,116,212)(111,205,117,211)(112,216,118,210)(113,215,119,209)(114,214,120,208)(121,325,127,331)(122,336,128,330)(123,335,129,329)(124,334,130,328)(125,333,131,327)(126,332,132,326)(133,270,139,276)(134,269,140,275)(135,268,141,274)(136,267,142,273)(137,266,143,272)(138,265,144,271)(145,407,151,401)(146,406,152,400)(147,405,153,399)(148,404,154,398)(149,403,155,397)(150,402,156,408)(157,299,163,293)(158,298,164,292)(159,297,165,291)(160,296,166,290)(161,295,167,289)(162,294,168,300)(193,316,199,322)(194,315,200,321)(195,314,201,320)(196,313,202,319)(197,324,203,318)(198,323,204,317)(277,428,283,422)(278,427,284,421)(279,426,285,432)(280,425,286,431)(281,424,287,430)(282,423,288,429)(301,394,307,388)(302,393,308,387)(303,392,309,386)(304,391,310,385)(305,390,311,396)(306,389,312,395)(349,449,355,455)(350,448,356,454)(351,447,357,453)(352,446,358,452)(353,445,359,451)(354,456,360,450)(361,374,367,380)(362,373,368,379)(363,384,369,378)(364,383,370,377)(365,382,371,376)(366,381,372,375)(433,476,439,470)(434,475,440,469)(435,474,441,480)(436,473,442,479)(437,472,443,478)(438,471,444,477) );
G=PermutationGroup([[(1,351,367,314,444,310,328,68,413,431),(2,352,368,315,433,311,329,69,414,432),(3,353,369,316,434,312,330,70,415,421),(4,354,370,317,435,301,331,71,416,422),(5,355,371,318,436,302,332,72,417,423),(6,356,372,319,437,303,333,61,418,424),(7,357,361,320,438,304,334,62,419,425),(8,358,362,321,439,305,335,63,420,426),(9,359,363,322,440,306,336,64,409,427),(10,360,364,323,441,307,325,65,410,428),(11,349,365,324,442,308,326,66,411,429),(12,350,366,313,443,309,327,67,412,430),(13,215,296,33,274,223,101,44,153,89),(14,216,297,34,275,224,102,45,154,90),(15,205,298,35,276,225,103,46,155,91),(16,206,299,36,265,226,104,47,156,92),(17,207,300,25,266,227,105,48,145,93),(18,208,289,26,267,228,106,37,146,94),(19,209,290,27,268,217,107,38,147,95),(20,210,291,28,269,218,108,39,148,96),(21,211,292,29,270,219,97,40,149,85),(22,212,293,30,271,220,98,41,150,86),(23,213,294,31,272,221,99,42,151,87),(24,214,295,32,273,222,100,43,152,88),(49,231,128,389,469,193,384,451,183,284),(50,232,129,390,470,194,373,452,184,285),(51,233,130,391,471,195,374,453,185,286),(52,234,131,392,472,196,375,454,186,287),(53,235,132,393,473,197,376,455,187,288),(54,236,121,394,474,198,377,456,188,277),(55,237,122,395,475,199,378,445,189,278),(56,238,123,396,476,200,379,446,190,279),(57,239,124,385,477,201,380,447,191,280),(58,240,125,386,478,202,381,448,192,281),(59,229,126,387,479,203,382,449,181,282),(60,230,127,388,480,204,383,450,182,283),(73,140,264,159,112,251,460,404,345,174),(74,141,253,160,113,252,461,405,346,175),(75,142,254,161,114,241,462,406,347,176),(76,143,255,162,115,242,463,407,348,177),(77,144,256,163,116,243,464,408,337,178),(78,133,257,164,117,244,465,397,338,179),(79,134,258,165,118,245,466,398,339,180),(80,135,259,166,119,246,467,399,340,169),(81,136,260,167,120,247,468,400,341,170),(82,137,261,168,109,248,457,401,342,171),(83,138,262,157,110,249,458,402,343,172),(84,139,263,158,111,250,459,403,344,173)], [(1,242,310,76),(2,243,311,77),(3,244,312,78),(4,245,301,79),(5,246,302,80),(6,247,303,81),(7,248,304,82),(8,249,305,83),(9,250,306,84),(10,251,307,73),(11,252,308,74),(12,241,309,75),(13,387,223,181),(14,388,224,182),(15,389,225,183),(16,390,226,184),(17,391,227,185),(18,392,228,186),(19,393,217,187),(20,394,218,188),(21,395,219,189),(22,396,220,190),(23,385,221,191),(24,386,222,192),(25,51,145,195),(26,52,146,196),(27,53,147,197),(28,54,148,198),(29,55,149,199),(30,56,150,200),(31,57,151,201),(32,58,152,202),(33,59,153,203),(34,60,154,204),(35,49,155,193),(36,50,156,194),(37,375,289,234),(38,376,290,235),(39,377,291,236),(40,378,292,237),(41,379,293,238),(42,380,294,239),(43,381,295,240),(44,382,296,229),(45,383,297,230),(46,384,298,231),(47,373,299,232),(48,374,300,233),(61,341,372,167),(62,342,361,168),(63,343,362,157),(64,344,363,158),(65,345,364,159),(66,346,365,160),(67,347,366,161),(68,348,367,162),(69,337,368,163),(70,338,369,164),(71,339,370,165),(72,340,371,166),(85,475,270,278),(86,476,271,279),(87,477,272,280),(88,478,273,281),(89,479,274,282),(90,480,275,283),(91,469,276,284),(92,470,265,285),(93,471,266,286),(94,472,267,287),(95,473,268,288),(96,474,269,277),(97,445,211,122),(98,446,212,123),(99,447,213,124),(100,448,214,125),(101,449,215,126),(102,450,216,127),(103,451,205,128),(104,452,206,129),(105,453,207,130),(106,454,208,131),(107,455,209,132),(108,456,210,121),(109,334,171,357),(110,335,172,358),(111,336,173,359),(112,325,174,360),(113,326,175,349),(114,327,176,350),(115,328,177,351),(116,329,178,352),(117,330,179,353),(118,331,180,354),(119,332,169,355),(120,333,170,356),(133,421,465,434),(134,422,466,435),(135,423,467,436),(136,424,468,437),(137,425,457,438),(138,426,458,439),(139,427,459,440),(140,428,460,441),(141,429,461,442),(142,430,462,443),(143,431,463,444),(144,432,464,433),(253,411,405,324),(254,412,406,313),(255,413,407,314),(256,414,408,315),(257,415,397,316),(258,416,398,317),(259,417,399,318),(260,418,400,319),(261,419,401,320),(262,420,402,321),(263,409,403,322),(264,410,404,323)], [(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)], [(1,191,7,185),(2,190,8,184),(3,189,9,183),(4,188,10,182),(5,187,11,181),(6,186,12,192),(13,246,19,252),(14,245,20,251),(15,244,21,250),(16,243,22,249),(17,242,23,248),(18,241,24,247),(25,255,31,261),(26,254,32,260),(27,253,33,259),(28,264,34,258),(29,263,35,257),(30,262,36,256),(37,347,43,341),(38,346,44,340),(39,345,45,339),(40,344,46,338),(41,343,47,337),(42,342,48,348),(49,415,55,409),(50,414,56,420),(51,413,57,419),(52,412,58,418),(53,411,59,417),(54,410,60,416),(61,234,67,240),(62,233,68,239),(63,232,69,238),(64,231,70,237),(65,230,71,236),(66,229,72,235),(73,224,79,218),(74,223,80,217),(75,222,81,228),(76,221,82,227),(77,220,83,226),(78,219,84,225),(85,459,91,465),(86,458,92,464),(87,457,93,463),(88,468,94,462),(89,467,95,461),(90,466,96,460),(97,173,103,179),(98,172,104,178),(99,171,105,177),(100,170,106,176),(101,169,107,175),(102,180,108,174),(109,207,115,213),(110,206,116,212),(111,205,117,211),(112,216,118,210),(113,215,119,209),(114,214,120,208),(121,325,127,331),(122,336,128,330),(123,335,129,329),(124,334,130,328),(125,333,131,327),(126,332,132,326),(133,270,139,276),(134,269,140,275),(135,268,141,274),(136,267,142,273),(137,266,143,272),(138,265,144,271),(145,407,151,401),(146,406,152,400),(147,405,153,399),(148,404,154,398),(149,403,155,397),(150,402,156,408),(157,299,163,293),(158,298,164,292),(159,297,165,291),(160,296,166,290),(161,295,167,289),(162,294,168,300),(193,316,199,322),(194,315,200,321),(195,314,201,320),(196,313,202,319),(197,324,203,318),(198,323,204,317),(277,428,283,422),(278,427,284,421),(279,426,285,432),(280,425,286,431),(281,424,287,430),(282,423,288,429),(301,394,307,388),(302,393,308,387),(303,392,309,386),(304,391,310,385),(305,390,311,396),(306,389,312,395),(349,449,355,455),(350,448,356,454),(351,447,357,453),(352,446,358,452),(353,445,359,451),(354,456,360,450),(361,374,367,380),(362,373,368,379),(363,384,369,378),(364,383,370,377),(365,382,371,376),(366,381,372,375),(433,476,439,470),(434,475,440,469),(435,474,441,480),(436,473,442,479),(437,472,443,478),(438,471,444,477)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | - | - | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | Dic6 | C4×S3 | C4×D5 | C4○D12 | S3×D5 | D4⋊2D5 | Q8×D5 | D30.C2 | C2×S3×D5 | D5×Dic6 | D12⋊5D5 |
| kernel | Dic30⋊17C4 | Dic3×Dic5 | Dic15⋊5C4 | C12×Dic5 | C5×C4⋊Dic3 | C2×Dic30 | Dic30 | C4×Dic5 | C3×Dic5 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | Dic5 | C20 | C12 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic30⋊17C4 ►in GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 42 | 18 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 52 | 37 |
| 0 | 0 | 0 | 44 | 9 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 15 | 38 | 0 | 0 |
| 0 | 23 | 38 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 52 | 43 | 0 | 0 |
| 0 | 52 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 24 |
| 0 | 0 | 0 | 17 | 52 |
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] >;
Dic30⋊17C4 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