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## G = Dic3.Dic10order 480 = 25·3·5

### 1st non-split extension by Dic3 of Dic10 acting via Dic10/Dic5=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic3.Dic10
G = < a,b,c,d | a6=c20=1, b2=a3, d2=c10, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a3c-1 >

Subgroups: 460 in 112 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C2×C4, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C4⋊C4, Dic5, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C5×C4⋊C4, Dic3.Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C4.Dic10, Dic3×Dic5, Dic155C4, C6.Dic10, C3×C10.D4, C5×Dic3⋊C4, C605C4, Dic3.Dic10
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, C22×S3, C42.C2, Dic10, C22×D5, C4○D12, D42S3, S3×Q8, S3×D5, C2×Dic10, D42D5, Q82D5, Dic3.Q8, C2×S3×D5, C4.Dic10, S3×Dic10, C12.28D10, C30.C23, Dic3.Dic10

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 12E 12F 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 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 4 6 6 10 10 12 20 30 30 60 2 2 2 2 2 2 ··· 2 4 4 20 20 20 20 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + - - - + - + + - + - image C1 C2 C2 C2 C2 C2 C2 S3 Q8 D5 D6 D6 C4○D4 D10 D10 Dic10 C4○D12 D4⋊2S3 S3×Q8 S3×D5 D4⋊2D5 Q8⋊2D5 C2×S3×D5 S3×Dic10 C12.28D10 C30.C23 kernel Dic3.Dic10 Dic3×Dic5 Dic15⋊5C4 C6.Dic10 C3×C10.D4 C5×Dic3⋊C4 C60⋊5C4 C10.D4 C5×Dic3 Dic3⋊C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 Dic3 C10 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 2 1 1 1 1 2 2 2 1 4 4 2 8 4 1 1 2 2 2 2 4 4 4

Matrix representation of Dic3.Dic10 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 60 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 27 4 0 0 0 0 31 34 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 0 1 0 0 0 0 60 0 0 0 0 0 0 0 23 15 0 0 0 0 46 38 0 0 0 0 0 0 1 60 0 0 0 0 45 17
,
 8 22 0 0 0 0 22 53 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 17 1 0 0 0 0 17 44

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,27,31,0,0,0,0,4,34,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,23,46,0,0,0,0,15,38,0,0,0,0,0,0,1,45,0,0,0,0,60,17],[8,22,0,0,0,0,22,53,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,17,17,0,0,0,0,1,44] >;

Dic3.Dic10 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,419);
// by ID

G=gap.SmallGroup(480,419);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,590,219,58,1356,18822]);
// Polycyclic

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

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