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G = Dic3⋊5Dic10order 480 = 25·3·5

1st semidirect product of Dic3 and Dic10 acting through Inn(Dic3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic3⋊5Dic10
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic3⋊5Dic10
 Lower central C15 — C30 — Dic3⋊5Dic10
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic35Dic10
G = < a,b,c,d | a6=c20=1, b2=a3, d2=c10, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 556 in 140 conjugacy classes, 62 normal (44 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, 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, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4×C20, C2×Dic10, Dic6⋊C4, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C4×Dic10, Dic3×Dic5, C6.Dic10, C3×C10.D4, Dic3×C20, C30.4Q8, C2×C15⋊Q8, Dic35Dic10
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D5, D6, C22×C4, C2×Q8, C4○D4, D10, C4×S3, C22×S3, C4×Q8, Dic10, C4×D5, C22×D5, S3×C2×C4, D42S3, S3×Q8, S3×D5, C2×Dic10, C2×C4×D5, C4○D20, Dic6⋊C4, C2×S3×D5, C4×Dic10, S3×Dic10, C4×S3×D5, Dic5.D6, Dic35Dic10

Smallest permutation representation of Dic35Dic10
Regular action on 480 points
Generators in S480
(1 247 347 191 479 454)(2 455 480 192 348 248)(3 249 349 193 461 456)(4 457 462 194 350 250)(5 251 351 195 463 458)(6 459 464 196 352 252)(7 253 353 197 465 460)(8 441 466 198 354 254)(9 255 355 199 467 442)(10 443 468 200 356 256)(11 257 357 181 469 444)(12 445 470 182 358 258)(13 259 359 183 471 446)(14 447 472 184 360 260)(15 241 341 185 473 448)(16 449 474 186 342 242)(17 243 343 187 475 450)(18 451 476 188 344 244)(19 245 345 189 477 452)(20 453 478 190 346 246)(21 163 427 52 228 308)(22 309 229 53 428 164)(23 165 429 54 230 310)(24 311 231 55 430 166)(25 167 431 56 232 312)(26 313 233 57 432 168)(27 169 433 58 234 314)(28 315 235 59 434 170)(29 171 435 60 236 316)(30 317 237 41 436 172)(31 173 437 42 238 318)(32 319 239 43 438 174)(33 175 439 44 240 320)(34 301 221 45 440 176)(35 177 421 46 222 302)(36 303 223 47 422 178)(37 179 423 48 224 304)(38 305 225 49 424 180)(39 161 425 50 226 306)(40 307 227 51 426 162)(61 136 291 142 406 204)(62 205 407 143 292 137)(63 138 293 144 408 206)(64 207 409 145 294 139)(65 140 295 146 410 208)(66 209 411 147 296 121)(67 122 297 148 412 210)(68 211 413 149 298 123)(69 124 299 150 414 212)(70 213 415 151 300 125)(71 126 281 152 416 214)(72 215 417 153 282 127)(73 128 283 154 418 216)(74 217 419 155 284 129)(75 130 285 156 420 218)(76 219 401 157 286 131)(77 132 287 158 402 220)(78 201 403 159 288 133)(79 134 289 160 404 202)(80 203 405 141 290 135)(81 106 280 325 392 380)(82 361 393 326 261 107)(83 108 262 327 394 362)(84 363 395 328 263 109)(85 110 264 329 396 364)(86 365 397 330 265 111)(87 112 266 331 398 366)(88 367 399 332 267 113)(89 114 268 333 400 368)(90 369 381 334 269 115)(91 116 270 335 382 370)(92 371 383 336 271 117)(93 118 272 337 384 372)(94 373 385 338 273 119)(95 120 274 339 386 374)(96 375 387 340 275 101)(97 102 276 321 388 376)(98 377 389 322 277 103)(99 104 278 323 390 378)(100 379 391 324 279 105)
(1 365 191 265)(2 366 192 266)(3 367 193 267)(4 368 194 268)(5 369 195 269)(6 370 196 270)(7 371 197 271)(8 372 198 272)(9 373 199 273)(10 374 200 274)(11 375 181 275)(12 376 182 276)(13 377 183 277)(14 378 184 278)(15 379 185 279)(16 380 186 280)(17 361 187 261)(18 362 188 262)(19 363 189 263)(20 364 190 264)(21 74 52 155)(22 75 53 156)(23 76 54 157)(24 77 55 158)(25 78 56 159)(26 79 57 160)(27 80 58 141)(28 61 59 142)(29 62 60 143)(30 63 41 144)(31 64 42 145)(32 65 43 146)(33 66 44 147)(34 67 45 148)(35 68 46 149)(36 69 47 150)(37 70 48 151)(38 71 49 152)(39 72 50 153)(40 73 51 154)(81 474 325 242)(82 475 326 243)(83 476 327 244)(84 477 328 245)(85 478 329 246)(86 479 330 247)(87 480 331 248)(88 461 332 249)(89 462 333 250)(90 463 334 251)(91 464 335 252)(92 465 336 253)(93 466 337 254)(94 467 338 255)(95 468 339 256)(96 469 340 257)(97 470 321 258)(98 471 322 259)(99 472 323 260)(100 473 324 241)(101 444 387 357)(102 445 388 358)(103 446 389 359)(104 447 390 360)(105 448 391 341)(106 449 392 342)(107 450 393 343)(108 451 394 344)(109 452 395 345)(110 453 396 346)(111 454 397 347)(112 455 398 348)(113 456 399 349)(114 457 400 350)(115 458 381 351)(116 459 382 352)(117 460 383 353)(118 441 384 354)(119 442 385 355)(120 443 386 356)(121 240 411 175)(122 221 412 176)(123 222 413 177)(124 223 414 178)(125 224 415 179)(126 225 416 180)(127 226 417 161)(128 227 418 162)(129 228 419 163)(130 229 420 164)(131 230 401 165)(132 231 402 166)(133 232 403 167)(134 233 404 168)(135 234 405 169)(136 235 406 170)(137 236 407 171)(138 237 408 172)(139 238 409 173)(140 239 410 174)(201 431 288 312)(202 432 289 313)(203 433 290 314)(204 434 291 315)(205 435 292 316)(206 436 293 317)(207 437 294 318)(208 438 295 319)(209 439 296 320)(210 440 297 301)(211 421 298 302)(212 422 299 303)(213 423 300 304)(214 424 281 305)(215 425 282 306)(216 426 283 307)(217 427 284 308)(218 428 285 309)(219 429 286 310)(220 430 287 311)
(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)
(1 293 11 283)(2 292 12 282)(3 291 13 281)(4 290 14 300)(5 289 15 299)(6 288 16 298)(7 287 17 297)(8 286 18 296)(9 285 19 295)(10 284 20 294)(21 85 31 95)(22 84 32 94)(23 83 33 93)(24 82 34 92)(25 81 35 91)(26 100 36 90)(27 99 37 89)(28 98 38 88)(29 97 39 87)(30 96 40 86)(41 340 51 330)(42 339 52 329)(43 338 53 328)(44 337 54 327)(45 336 55 326)(46 335 56 325)(47 334 57 324)(48 333 58 323)(49 332 59 322)(50 331 60 321)(61 471 71 461)(62 470 72 480)(63 469 73 479)(64 468 74 478)(65 467 75 477)(66 466 76 476)(67 465 77 475)(68 464 78 474)(69 463 79 473)(70 462 80 472)(101 162 111 172)(102 161 112 171)(103 180 113 170)(104 179 114 169)(105 178 115 168)(106 177 116 167)(107 176 117 166)(108 175 118 165)(109 174 119 164)(110 173 120 163)(121 441 131 451)(122 460 132 450)(123 459 133 449)(124 458 134 448)(125 457 135 447)(126 456 136 446)(127 455 137 445)(128 454 138 444)(129 453 139 443)(130 452 140 442)(141 260 151 250)(142 259 152 249)(143 258 153 248)(144 257 154 247)(145 256 155 246)(146 255 156 245)(147 254 157 244)(148 253 158 243)(149 252 159 242)(150 251 160 241)(181 216 191 206)(182 215 192 205)(183 214 193 204)(184 213 194 203)(185 212 195 202)(186 211 196 201)(187 210 197 220)(188 209 198 219)(189 208 199 218)(190 207 200 217)(221 383 231 393)(222 382 232 392)(223 381 233 391)(224 400 234 390)(225 399 235 389)(226 398 236 388)(227 397 237 387)(228 396 238 386)(229 395 239 385)(230 394 240 384)(261 440 271 430)(262 439 272 429)(263 438 273 428)(264 437 274 427)(265 436 275 426)(266 435 276 425)(267 434 277 424)(268 433 278 423)(269 432 279 422)(270 431 280 421)(301 371 311 361)(302 370 312 380)(303 369 313 379)(304 368 314 378)(305 367 315 377)(306 366 316 376)(307 365 317 375)(308 364 318 374)(309 363 319 373)(310 362 320 372)(341 414 351 404)(342 413 352 403)(343 412 353 402)(344 411 354 401)(345 410 355 420)(346 409 356 419)(347 408 357 418)(348 407 358 417)(349 406 359 416)(350 405 360 415)

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

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

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

78 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 12F 15A 15B 20A ··· 20H 20I ··· 20X 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 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 3 3 3 3 6 6 10 10 10 10 30 30 30 30 2 2 2 2 2 2 ··· 2 4 4 20 20 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

78 irreducible representations

 dim 1 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 C2 C4 S3 Q8 D5 D6 D6 C4○D4 D10 D10 C4×S3 Dic10 C4×D5 C4○D20 D4⋊2S3 S3×Q8 S3×D5 C2×S3×D5 S3×Dic10 C4×S3×D5 Dic5.D6 kernel Dic3⋊5Dic10 Dic3×Dic5 C6.Dic10 C3×C10.D4 Dic3×C20 C30.4Q8 C2×C15⋊Q8 C15⋊Q8 C10.D4 C5×Dic3 C4×Dic3 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 Dic5 Dic3 Dic3 C6 C10 C10 C2×C4 C22 C2 C2 C2 # reps 1 2 1 1 1 1 1 8 1 2 2 2 1 2 4 2 4 8 8 8 1 1 2 2 4 4 4

Matrix representation of Dic35Dic10 in GL5(𝔽61)

 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 60 60
,
 11 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 8 25 0 0 0 17 53
,
 1 0 0 0 0 0 54 29 0 0 0 32 59 0 0 0 0 0 8 25 0 0 0 17 53
,
 60 0 0 0 0 0 29 20 0 0 0 25 32 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,1,60],[11,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,8,17,0,0,0,25,53],[1,0,0,0,0,0,54,32,0,0,0,29,59,0,0,0,0,0,8,17,0,0,0,25,53],[60,0,0,0,0,0,29,25,0,0,0,20,32,0,0,0,0,0,1,0,0,0,0,0,1] >;

Dic35Dic10 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_5{\rm Dic}_{10}
% in TeX

G:=Group("Dic3:5Dic10");
// GroupNames label

G:=SmallGroup(480,400);
// by ID

G=gap.SmallGroup(480,400);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,254,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^20=1,b^2=a^3,d^2=c^10,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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