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G = Dic3.Dic10order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic3.1Dic10, (C2×C20).8D6, (C2×C12).8D10, C10.24(S3×Q8), C30.16(C2×Q8), C605C4.5C2, Dic3⋊C4.4D5, (C2×C60).4C22, C53(Dic3.Q8), (C5×Dic3).1Q8, C2.8(S3×Dic10), C6.6(C2×Dic10), C155(C42.C2), C32(C4.Dic10), C30.12(C4○D4), C10.3(C4○D12), (C2×C30).33C23, C6.3(Q82D5), (C2×Dic5).90D6, C10.D4.4S3, C6.Dic10.7C2, Dic155C4.9C2, C6.64(D42D5), C2.7(C12.28D10), (C2×Dic3).83D10, C10.64(D42S3), (Dic3×Dic5).16C2, (C6×Dic5).18C22, C2.10(C30.C23), (C10×Dic3).17C22, (C2×Dic15).40C22, (C2×C4).30(S3×D5), C22.124(C2×S3×D5), (C5×Dic3⋊C4).4C2, (C2×C6).45(C22×D5), (C2×C10).45(C22×S3), (C3×C10.D4).4C2, SmallGroup(480,419)

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=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a3c-1 >

Subgroups: 460 in 112 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×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, C10.D4, C4⋊Dic5 [×3], C5×C4⋊C4, Dic3.Q8, C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, C4.Dic10, Dic3×Dic5, Dic155C4, C6.Dic10 [×2], C3×C10.D4, C5×Dic3⋊C4, C605C4, Dic3.Dic10
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, S3×Dic10, C12.28D10, C30.C23, Dic3.Dic10

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

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

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

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

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×D5S3×Dic10C12.28D10C30.C23
kernelDic3.Dic10Dic3×Dic5Dic155C4C6.Dic10C3×C10.D4C5×Dic3⋊C4C605C4C10.D4C5×Dic3Dic3⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12Dic3C10C10C10C2×C4C6C6C22C2C2C2
# reps11121111222144284112222444

Matrix representation of Dic3.Dic10 in GL6(𝔽61)

100000
010000
0016000
001000
000010
000001
,
6000000
0600000
0027400
00313400
0000600
0000060
,
010000
6000000
00231500
00463800
0000160
00004517
,
8220000
22530000
0011000
0001100
0000171
00001744

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

Dic3.Dic10 in GAP, Magma, Sage, TeX

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

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

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

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

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

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

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