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## G = Dic3.3Dic10order 480 = 25·3·5

### The non-split extension by Dic3 of Dic10 acting via Dic10/C20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic3.3Dic10
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic15⋊5C4 — Dic3.3Dic10
 Lower central C15 — C2×C30 — Dic3.3Dic10
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic3.3Dic10
G = < a,b,c,d | a6=c20=1, b2=a3, d2=c10, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a3b, 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, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C10.D4, C10.D4, C4⋊Dic5, C4×C20, Dic3.Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C20.6Q8, C30.Q8, Dic155C4, C6.Dic10, C3×C10.D4, Dic3×C20, C30.4Q8, Dic3.3Dic10
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, C4○D20, Dic3.Q8, C2×S3×D5, C20.6Q8, S3×Dic10, D6.D10, Dic5.D6, Dic3.3Dic10

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

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

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

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

72 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 ··· 20H 20I ··· 20X 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 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 6 6 6 6 20 20 60 60 2 2 2 2 2 2 ··· 2 4 4 20 20 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 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 C4○D20 D4⋊2S3 S3×Q8 S3×D5 C2×S3×D5 S3×Dic10 D6.D10 Dic5.D6 kernel Dic3.3Dic10 C30.Q8 Dic15⋊5C4 C6.Dic10 C3×C10.D4 Dic3×C20 C30.4Q8 C10.D4 C5×Dic3 C4×Dic3 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 Dic3 C10 C6 C10 C10 C2×C4 C22 C2 C2 C2 # reps 1 1 1 2 1 1 1 1 2 2 2 1 4 4 2 8 4 16 1 1 2 2 4 4 4

Matrix representation of Dic3.3Dic10 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 1 0 0 0 0 60 0
,
 0 11 0 0 0 0 11 0 0 0 0 0 0 0 25 7 0 0 0 0 50 36 0 0 0 0 0 0 48 22 0 0 0 0 9 13
,
 50 0 0 0 0 0 0 50 0 0 0 0 0 0 31 14 0 0 0 0 39 53 0 0 0 0 0 0 48 22 0 0 0 0 9 13
,
 25 57 0 0 0 0 4 36 0 0 0 0 0 0 53 33 0 0 0 0 48 8 0 0 0 0 0 0 60 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[0,11,0,0,0,0,11,0,0,0,0,0,0,0,25,50,0,0,0,0,7,36,0,0,0,0,0,0,48,9,0,0,0,0,22,13],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,31,39,0,0,0,0,14,53,0,0,0,0,0,0,48,9,0,0,0,0,22,13],[25,4,0,0,0,0,57,36,0,0,0,0,0,0,53,48,0,0,0,0,33,8,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

Dic3.3Dic10 in GAP, Magma, Sage, TeX

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

G:=Group("Dic3.3Dic10");
// GroupNames label

G:=SmallGroup(480,455);
// by ID

G=gap.SmallGroup(480,455);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,422,58,1356,18822]);
// Polycyclic

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

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