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G = Dic3⋊Dic10order 480 = 25·3·5

2nd semidirect product of Dic3 and Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic51Dic6, Dic15.5D4, Dic32Dic10, C52(C12⋊Q8), C32(C20⋊Q8), C152(C4⋊Q8), C6.52(D4×D5), (C2×C20).3D6, C6.20(Q8×D5), C30.5(C2×Q8), (C5×Dic3)⋊2Q8, (C3×Dic5)⋊1Q8, C10.54(S3×D4), (C2×C12).1D10, C10.21(S3×Q8), C2.6(D5×Dic6), C30.100(C2×D4), Dic3⋊C4.2D5, (C2×C60).1C22, C6.3(C2×Dic10), C10.2(C2×Dic6), C2.7(S3×Dic10), (C2×C30).18C23, (C2×Dic5).82D6, (C2×Dic30).2C2, C10.D4.2S3, C6.Dic10.3C2, C30.Q8.2C2, C2.8(D10⋊D6), (Dic3×Dic5).5C2, (C2×Dic3).74D10, (C6×Dic5).5C22, (C10×Dic3).5C22, (C2×Dic15).26C22, (C2×C15⋊Q8).5C2, (C2×C4).21(S3×D5), C22.115(C2×S3×D5), (C5×Dic3⋊C4).2C2, (C2×C6).30(C22×D5), (C2×C10).30(C22×S3), (C3×C10.D4).2C2, SmallGroup(480,404)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic3⋊Dic10
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic3⋊Dic10
C15C2×C30 — Dic3⋊Dic10
C1C22C2×C4

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

Subgroups: 652 in 136 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C3, C4 [×10], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×2], Dic3 [×4], C12 [×4], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×2], Dic5 [×4], C20 [×4], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4⋊Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic6 [×2], C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15 [×2], Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×Dic10 [×2], C12⋊Q8, C15⋊Q8 [×2], C6×Dic5 [×2], C10×Dic3 [×2], Dic30 [×2], C2×Dic15 [×2], C2×C60, C20⋊Q8, Dic3×Dic5, C30.Q8, C6.Dic10, C3×C10.D4, C5×Dic3⋊C4, C2×C15⋊Q8, C2×Dic30, Dic3⋊Dic10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D5, D6 [×3], C2×D4, C2×Q8 [×2], D10 [×3], Dic6 [×2], C22×S3, C4⋊Q8, Dic10 [×2], C22×D5, C2×Dic6, S3×D4, S3×Q8, S3×D5, C2×Dic10, D4×D5, Q8×D5, C12⋊Q8, C2×S3×D5, C20⋊Q8, D5×Dic6, S3×Dic10, D10⋊D6, Dic3⋊Dic10

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

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

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

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

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444445566610···1012121212121215152020202020···2030···3060···60
size1111246610101220303060222222···2442020202044444412···124···44···4

60 irreducible representations

dim1111111122222222222444444444
type+++++++++--++++++--+-++-+--+
imageC1C2C2C2C2C2C2C2S3Q8Q8D4D5D6D6D10D10Dic6Dic10S3×D4S3×Q8S3×D5D4×D5Q8×D5C2×S3×D5D5×Dic6S3×Dic10D10⋊D6
kernelDic3⋊Dic10Dic3×Dic5C30.Q8C6.Dic10C3×C10.D4C5×Dic3⋊C4C2×C15⋊Q8C2×Dic30C10.D4C5×Dic3C3×Dic5Dic15Dic3⋊C4C2×Dic5C2×C20C2×Dic3C2×C12Dic5Dic3C10C10C2×C4C6C6C22C2C2C2
# reps1111111112222214248112222444

Matrix representation of Dic3⋊Dic10 in GL6(𝔽61)

100000
010000
0060000
0006000
0000160
000010
,
6000000
0600000
0006000
001000
00002510
00003536
,
5200000
40270000
0060000
000100
00002315
00004638
,
38220000
37230000
0060000
0006000
00002510
00003536

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,60,0,0,0,0,0,0,0,25,35,0,0,0,0,10,36],[52,40,0,0,0,0,0,27,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,23,46,0,0,0,0,15,38],[38,37,0,0,0,0,22,23,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,25,35,0,0,0,0,10,36] >;

Dic3⋊Dic10 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,404);
// by ID

G=gap.SmallGroup(480,404);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,254,219,142,1356,18822]);
// Polycyclic

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

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