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G = Dic3.2Dic10order 480 = 25·3·5

2nd non-split extension by Dic3 of Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic3.2Dic10, (C2×C20).10D6, C4⋊Dic5.5S3, C30.18(C2×Q8), C10.25(S3×Q8), Dic3⋊C4.9D5, C54(Dic3.Q8), (C5×Dic3).2Q8, C6.7(C2×Dic10), C2.9(S3×Dic10), C33(C4.Dic10), C156(C42.C2), (C2×C12).220D10, (C2×C30).36C23, (C2×Dic5).92D6, C30.4Q8.8C2, C6.Dic10.9C2, C30.Q8.9C2, C10.68(C4○D12), C30.104(C4○D4), C2.7(D20⋊S3), C6.37(D42D5), C10.3(D42S3), (C2×C60).251C22, C6.22(Q82D5), (C2×Dic3).85D10, (Dic3×Dic5).18C2, (C6×Dic5).21C22, C2.11(Dic3.D10), (C2×Dic15).42C22, (C10×Dic3).20C22, (C2×C4).32(S3×D5), C22.126(C2×S3×D5), (C5×Dic3⋊C4).9C2, (C3×C4⋊Dic5).17C2, (C2×C6).48(C22×D5), (C2×C10).48(C22×S3), SmallGroup(480,422)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic3.2Dic10
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic3.2Dic10
C15C2×C30 — Dic3.2Dic10
C1C22C2×C4

Generators and relations for Dic3.2Dic10
 G = < a,b,c,d | a6=c20=1, b2=a3, d2=a3c10, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 460 in 112 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C3, C4 [×8], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], C10 [×3], Dic3 [×2], Dic3 [×3], C12 [×3], C2×C6, C15, C42, C4⋊C4 [×6], Dic5 [×4], C20 [×4], C2×C10, C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C42.C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3, Dic3⋊C4, Dic3⋊C4 [×3], C4⋊Dic3, C3×C4⋊C4, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], Dic15 [×2], C60, C2×C30, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], C5×C4⋊C4, Dic3.Q8, C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, C4.Dic10, Dic3×Dic5, C30.Q8, C6.Dic10 [×2], C3×C4⋊Dic5, C5×Dic3⋊C4, C30.4Q8, Dic3.2Dic10
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D5, D6 [×3], C2×Q8, C4○D4 [×2], D10 [×3], C22×S3, C42.C2, Dic10 [×2], C22×D5, C4○D12, D42S3, S3×Q8, S3×D5, C2×Dic10, D42D5, Q82D5, Dic3.Q8, C2×S3×D5, C4.Dic10, D20⋊S3, S3×Dic10, Dic3.D10, Dic3.2Dic10

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

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

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

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

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

dim11111112222222222444444444
type++++++++-+++++---+-++-
imageC1C2C2C2C2C2C2S3Q8D5D6D6C4○D4D10D10Dic10C4○D12D42S3S3×Q8S3×D5D42D5Q82D5C2×S3×D5D20⋊S3S3×Dic10Dic3.D10
kernelDic3.2Dic10Dic3×Dic5C30.Q8C6.Dic10C3×C4⋊Dic5C5×Dic3⋊C4C30.4Q8C4⋊Dic5C5×Dic3Dic3⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12Dic3C10C10C10C2×C4C6C6C22C2C2C2
# reps11121111222144284112222444

Matrix representation of Dic3.2Dic10 in GL6(𝔽61)

100000
010000
0014600
00495900
0000600
0000060
,
100000
010000
00425000
0051900
0000148
00004447
,
57250000
36340000
001000
000100
0000160
0000260
,
32220000
56290000
0060000
0006000
00003227
00005729

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,49,0,0,0,0,46,59,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,5,0,0,0,0,50,19,0,0,0,0,0,0,14,44,0,0,0,0,8,47],[57,36,0,0,0,0,25,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,60,60],[32,56,0,0,0,0,22,29,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,32,57,0,0,0,0,27,29] >;

Dic3.2Dic10 in GAP, Magma, Sage, TeX

{\rm Dic}_3._2{\rm Dic}_{10}
% in TeX

G:=Group("Dic3.2Dic10");
// GroupNames label

G:=SmallGroup(480,422);
// by ID

G=gap.SmallGroup(480,422);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,590,219,142,1356,18822]);
// Polycyclic

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

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