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## G = Dic3×Dic10order 480 = 25·3·5

### Direct product of Dic3 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic3×Dic10
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic3×Dic10
 Lower central C15 — 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, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 508 in 140 conjugacy classes, 72 normal (34 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C30, C4×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, C4×Dic3, C4⋊Dic3, C6×Q8, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×Dic10, Q8×Dic3, C3×Dic10, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C4×Dic10, Dic3×Dic5, C30.Q8, Dic3×C20, C605C4, C6×Dic10, Dic3×Dic10
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D5, Dic3, D6, C22×C4, C2×Q8, C4○D4, D10, C2×Dic3, C22×S3, C4×Q8, Dic10, C4×D5, C22×D5, S3×Q8, Q83S3, C22×Dic3, S3×D5, C2×Dic10, C2×C4×D5, C4○D20, Q8×Dic3, D5×Dic3, C2×S3×D5, C4×Dic10, S3×Dic10, D60⋊C2, C2×D5×Dic3, Dic3×Dic10

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

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A ··· 20H 20I ··· 20X 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 12 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 3 3 3 3 6 6 10 10 10 10 30 30 30 30 2 2 2 2 2 2 ··· 2 4 4 20 20 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

78 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + - + - + + + + - - + + - + - + image C1 C2 C2 C2 C2 C2 C4 S3 Q8 D5 Dic3 D6 D6 C4○D4 D10 D10 Dic10 C4×D5 C4○D20 S3×Q8 Q8⋊3S3 S3×D5 D5×Dic3 C2×S3×D5 S3×Dic10 D60⋊C2 kernel Dic3×Dic10 Dic3×Dic5 C30.Q8 Dic3×C20 C60⋊5C4 C6×Dic10 C3×Dic10 C2×Dic10 C5×Dic3 C4×Dic3 Dic10 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 Dic3 C12 C6 C10 C10 C2×C4 C4 C22 C2 C2 # reps 1 2 2 1 1 1 8 1 2 2 4 2 1 2 4 2 8 8 8 1 1 2 4 2 4 4

Matrix representation of Dic3×Dic10 in GL6(𝔽61)

 60 1 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 53 25 0 0 0 0 17 8 0 0 0 0 0 0 11 0 0 0 0 0 0 11 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 41 0 0 0 0 0 19 3 0 0 0 0 0 0 24 0 0 0 0 0 0 28
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 21 19 0 0 0 0 25 40 0 0 0 0 0 0 0 1 0 0 0 0 60 0

G:=sub<GL(6,GF(61))| [60,60,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[53,17,0,0,0,0,25,8,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,41,19,0,0,0,0,0,3,0,0,0,0,0,0,24,0,0,0,0,0,0,28],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,21,25,0,0,0,0,19,40,0,0,0,0,0,0,0,60,0,0,0,0,1,0] >;

Dic3×Dic10 in GAP, Magma, Sage, TeX

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

G:=Group("Dic3xDic10");
// GroupNames label

G:=SmallGroup(480,406);
// by ID

G=gap.SmallGroup(480,406);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,422,100,1356,18822]);
// 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,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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