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G = Dic35Dic10order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic35Dic10, C15⋊Q84C4, C154(C4×Q8), C32(C4×Dic10), C30.2(C2×Q8), (C5×Dic3)⋊3Q8, C10.19(S3×Q8), (C2×C20).255D6, (C4×Dic3).8D5, Dic5.7(C4×S3), Dic3.7(C4×D5), C2.1(S3×Dic10), C6.1(C2×Dic10), C6.60(C4○D20), C30.99(C4○D4), (C2×C12).173D10, C52(Dic6⋊C4), (C2×C30).14C23, C30.41(C22×C4), (C2×Dic5).78D6, C6.Dic10.1C2, (C2×C60).399C22, Dic15.24(C2×C4), (Dic3×C20).17C2, (Dic3×Dic5).2C2, C10.D4.10S3, C30.4Q8.14C2, C10.35(D42S3), (C6×Dic5).1C22, (C2×Dic3).171D10, C2.1(Dic5.D6), (C2×Dic15).23C22, (C10×Dic3).154C22, C2.13(C4×S3×D5), C6.10(C2×C4×D5), C10.41(S3×C2×C4), (C2×C15⋊Q8).2C2, (C2×C4).67(S3×D5), C22.22(C2×S3×D5), (C3×Dic5).7(C2×C4), (C2×C6).26(C22×D5), (C2×C10).26(C22×S3), (C5×Dic3).33(C2×C4), (C3×C10.D4).13C2, SmallGroup(480,400)

Series: Derived Chief Lower central Upper central

C1C30 — Dic35Dic10
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic35Dic10
C15C30 — Dic35Dic10
C1C22C2×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 [×3], C3, C4 [×11], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×4], Dic3 [×3], C12 [×4], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×2], Dic5 [×4], C20 [×5], 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, C4×Dic3 [×2], Dic3⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C5×Dic3 [×4], C3×Dic5 [×2], C3×Dic5, Dic15 [×2], Dic15, C60, C2×C30, C4×Dic5 [×2], C10.D4, C10.D4, C4⋊Dic5, C4×C20, C2×Dic10, Dic6⋊C4, C15⋊Q8 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, C4×Dic10, Dic3×Dic5 [×2], C6.Dic10, C3×C10.D4, Dic3×C20, C30.4Q8, C2×C15⋊Q8, Dic35Dic10
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], C4×S3 [×2], C22×S3, C4×Q8, Dic10 [×2], C4×D5 [×2], 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
(1 309 99 299 103 343)(2 344 104 300 100 310)(3 311 81 281 105 345)(4 346 106 282 82 312)(5 313 83 283 107 347)(6 348 108 284 84 314)(7 315 85 285 109 349)(8 350 110 286 86 316)(9 317 87 287 111 351)(10 352 112 288 88 318)(11 319 89 289 113 353)(12 354 114 290 90 320)(13 301 91 291 115 355)(14 356 116 292 92 302)(15 303 93 293 117 357)(16 358 118 294 94 304)(17 305 95 295 119 359)(18 360 120 296 96 306)(19 307 97 297 101 341)(20 342 102 298 98 308)(21 79 474 160 60 419)(22 420 41 141 475 80)(23 61 476 142 42 401)(24 402 43 143 477 62)(25 63 478 144 44 403)(26 404 45 145 479 64)(27 65 480 146 46 405)(28 406 47 147 461 66)(29 67 462 148 48 407)(30 408 49 149 463 68)(31 69 464 150 50 409)(32 410 51 151 465 70)(33 71 466 152 52 411)(34 412 53 153 467 72)(35 73 468 154 54 413)(36 414 55 155 469 74)(37 75 470 156 56 415)(38 416 57 157 471 76)(39 77 472 158 58 417)(40 418 59 159 473 78)(121 164 337 376 217 262)(122 263 218 377 338 165)(123 166 339 378 219 264)(124 265 220 379 340 167)(125 168 321 380 201 266)(126 267 202 361 322 169)(127 170 323 362 203 268)(128 269 204 363 324 171)(129 172 325 364 205 270)(130 271 206 365 326 173)(131 174 327 366 207 272)(132 273 208 367 328 175)(133 176 329 368 209 274)(134 275 210 369 330 177)(135 178 331 370 211 276)(136 277 212 371 332 179)(137 180 333 372 213 278)(138 279 214 373 334 161)(139 162 335 374 215 280)(140 261 216 375 336 163)(181 392 247 230 425 456)(182 457 426 231 248 393)(183 394 249 232 427 458)(184 459 428 233 250 395)(185 396 251 234 429 460)(186 441 430 235 252 397)(187 398 253 236 431 442)(188 443 432 237 254 399)(189 400 255 238 433 444)(190 445 434 239 256 381)(191 382 257 240 435 446)(192 447 436 221 258 383)(193 384 259 222 437 448)(194 449 438 223 260 385)(195 386 241 224 439 450)(196 451 440 225 242 387)(197 388 243 226 421 452)(198 453 422 227 244 389)(199 390 245 228 423 454)(200 455 424 229 246 391)
(1 461 299 406)(2 462 300 407)(3 463 281 408)(4 464 282 409)(5 465 283 410)(6 466 284 411)(7 467 285 412)(8 468 286 413)(9 469 287 414)(10 470 288 415)(11 471 289 416)(12 472 290 417)(13 473 291 418)(14 474 292 419)(15 475 293 420)(16 476 294 401)(17 477 295 402)(18 478 296 403)(19 479 297 404)(20 480 298 405)(21 302 160 116)(22 303 141 117)(23 304 142 118)(24 305 143 119)(25 306 144 120)(26 307 145 101)(27 308 146 102)(28 309 147 103)(29 310 148 104)(30 311 149 105)(31 312 150 106)(32 313 151 107)(33 314 152 108)(34 315 153 109)(35 316 154 110)(36 317 155 111)(37 318 156 112)(38 319 157 113)(39 320 158 114)(40 301 159 115)(41 357 80 93)(42 358 61 94)(43 359 62 95)(44 360 63 96)(45 341 64 97)(46 342 65 98)(47 343 66 99)(48 344 67 100)(49 345 68 81)(50 346 69 82)(51 347 70 83)(52 348 71 84)(53 349 72 85)(54 350 73 86)(55 351 74 87)(56 352 75 88)(57 353 76 89)(58 354 77 90)(59 355 78 91)(60 356 79 92)(121 438 376 385)(122 439 377 386)(123 440 378 387)(124 421 379 388)(125 422 380 389)(126 423 361 390)(127 424 362 391)(128 425 363 392)(129 426 364 393)(130 427 365 394)(131 428 366 395)(132 429 367 396)(133 430 368 397)(134 431 369 398)(135 432 370 399)(136 433 371 400)(137 434 372 381)(138 435 373 382)(139 436 374 383)(140 437 375 384)(161 446 214 257)(162 447 215 258)(163 448 216 259)(164 449 217 260)(165 450 218 241)(166 451 219 242)(167 452 220 243)(168 453 201 244)(169 454 202 245)(170 455 203 246)(171 456 204 247)(172 457 205 248)(173 458 206 249)(174 459 207 250)(175 460 208 251)(176 441 209 252)(177 442 210 253)(178 443 211 254)(179 444 212 255)(180 445 213 256)(181 269 230 324)(182 270 231 325)(183 271 232 326)(184 272 233 327)(185 273 234 328)(186 274 235 329)(187 275 236 330)(188 276 237 331)(189 277 238 332)(190 278 239 333)(191 279 240 334)(192 280 221 335)(193 261 222 336)(194 262 223 337)(195 263 224 338)(196 264 225 339)(197 265 226 340)(198 266 227 321)(199 267 228 322)(200 268 229 323)
(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)
(1 170 11 180)(2 169 12 179)(3 168 13 178)(4 167 14 177)(5 166 15 176)(6 165 16 175)(7 164 17 174)(8 163 18 173)(9 162 19 172)(10 161 20 171)(21 236 31 226)(22 235 32 225)(23 234 33 224)(24 233 34 223)(25 232 35 222)(26 231 36 221)(27 230 37 240)(28 229 38 239)(29 228 39 238)(30 227 40 237)(41 397 51 387)(42 396 52 386)(43 395 53 385)(44 394 54 384)(45 393 55 383)(46 392 56 382)(47 391 57 381)(48 390 58 400)(49 389 59 399)(50 388 60 398)(61 429 71 439)(62 428 72 438)(63 427 73 437)(64 426 74 436)(65 425 75 435)(66 424 76 434)(67 423 77 433)(68 422 78 432)(69 421 79 431)(70 440 80 430)(81 380 91 370)(82 379 92 369)(83 378 93 368)(84 377 94 367)(85 376 95 366)(86 375 96 365)(87 374 97 364)(88 373 98 363)(89 372 99 362)(90 371 100 361)(101 270 111 280)(102 269 112 279)(103 268 113 278)(104 267 114 277)(105 266 115 276)(106 265 116 275)(107 264 117 274)(108 263 118 273)(109 262 119 272)(110 261 120 271)(121 359 131 349)(122 358 132 348)(123 357 133 347)(124 356 134 346)(125 355 135 345)(126 354 136 344)(127 353 137 343)(128 352 138 342)(129 351 139 341)(130 350 140 360)(141 186 151 196)(142 185 152 195)(143 184 153 194)(144 183 154 193)(145 182 155 192)(146 181 156 191)(147 200 157 190)(148 199 158 189)(149 198 159 188)(150 197 160 187)(201 291 211 281)(202 290 212 300)(203 289 213 299)(204 288 214 298)(205 287 215 297)(206 286 216 296)(207 285 217 295)(208 284 218 294)(209 283 219 293)(210 282 220 292)(241 401 251 411)(242 420 252 410)(243 419 253 409)(244 418 254 408)(245 417 255 407)(246 416 256 406)(247 415 257 405)(248 414 258 404)(249 413 259 403)(250 412 260 402)(301 331 311 321)(302 330 312 340)(303 329 313 339)(304 328 314 338)(305 327 315 337)(306 326 316 336)(307 325 317 335)(308 324 318 334)(309 323 319 333)(310 322 320 332)(441 465 451 475)(442 464 452 474)(443 463 453 473)(444 462 454 472)(445 461 455 471)(446 480 456 470)(447 479 457 469)(448 478 458 468)(449 477 459 467)(450 476 460 466)

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

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

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

78 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A···20H20I···20X30A···30F60A···60H
order1222344444444444444445566610···10121212121212151520···2020···2030···3060···60
size11112223333661010101030303030222222···24420202020442···26···64···44···4

78 irreducible representations

dim111111112222222222224444444
type++++++++-+++++---++-
imageC1C2C2C2C2C2C2C4S3Q8D5D6D6C4○D4D10D10C4×S3Dic10C4×D5C4○D20D42S3S3×Q8S3×D5C2×S3×D5S3×Dic10C4×S3×D5Dic5.D6
kernelDic35Dic10Dic3×Dic5C6.Dic10C3×C10.D4Dic3×C20C30.4Q8C2×C15⋊Q8C15⋊Q8C10.D4C5×Dic3C4×Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5Dic3Dic3C6C10C10C2×C4C22C2C2C2
# reps121111181222124248881122444

Matrix representation of Dic35Dic10 in GL5(𝔽61)

600000
01000
00100
00001
0006060
,
110000
060000
006000
000825
0001753
,
10000
0542900
0325900
000825
0001753
,
600000
0292000
0253200
00010
00001

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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