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G = Dic3.3Dic10order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic3.3Dic10, C30.28(C2×Q8), C10.29(S3×Q8), (C2×C20).260D6, C51(Dic3.Q8), (C5×Dic3).3Q8, (C4×Dic3).9D5, C158(C42.C2), C31(C20.6Q8), C30.43(C4○D4), C6.30(C4○D20), (C2×C12).184D10, (C2×C30).69C23, (C2×Dic5).20D6, C6.11(C2×Dic10), C2.13(S3×Dic10), C10.D4.1S3, C10.33(C4○D12), (C2×C60).404C22, (Dic3×C20).18C2, C30.4Q8.15C2, Dic155C4.11C2, C30.Q8.12C2, C6.Dic10.11C2, C10.41(D42S3), (C2×Dic3).144D10, (C6×Dic5).39C22, C2.22(D6.D10), C2.15(Dic5.D6), (C2×Dic15).63C22, (C10×Dic3).168C22, (C2×C4).72(S3×D5), C22.155(C2×S3×D5), (C2×C6).81(C22×D5), (C2×C10).81(C22×S3), (C3×C10.D4).14C2, SmallGroup(480,455)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic3.3Dic10
C1C5C15C30C2×C30C6×Dic5Dic155C4 — Dic3.3Dic10
C15C2×C30 — Dic3.3Dic10
C1C22C2×C4

Generators and relations for Dic3.3Dic10
 G = < a,b,c,d | a6=c20=1, b2=a3, d2=c10, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a3b, dcd-1=a3c-1 >

Subgroups: 460 in 112 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×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 [×4], C4⋊Dic3, C3×C4⋊C4, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], Dic15 [×2], C60, C2×C30, C10.D4, C10.D4 [×3], C4⋊Dic5 [×2], C4×C20, Dic3.Q8, C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, C20.6Q8, C30.Q8, Dic155C4, C6.Dic10 [×2], C3×C10.D4, Dic3×C20, C30.4Q8, Dic3.3Dic10
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, C4○D20 [×2], Dic3.Q8, C2×S3×D5, C20.6Q8, S3×Dic10, D6.D10, Dic5.D6, Dic3.3Dic10

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

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

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

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

72 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A···20H20I···20X30A···30F60A···60H
order1222344444444445566610···10121212121212151520···2020···2030···3060···60
size1111222666620206060222222···24420202020442···26···64···44···4

72 irreducible representations

dim1111111222222222224444444
type++++++++-+++++---++-
imageC1C2C2C2C2C2C2S3Q8D5D6D6C4○D4D10D10Dic10C4○D12C4○D20D42S3S3×Q8S3×D5C2×S3×D5S3×Dic10D6.D10Dic5.D6
kernelDic3.3Dic10C30.Q8Dic155C4C6.Dic10C3×C10.D4Dic3×C20C30.4Q8C10.D4C5×Dic3C4×Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12Dic3C10C6C10C10C2×C4C22C2C2C2
# reps11121111222144284161122444

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

6000000
0600000
0060000
0006000
0000601
0000600
,
0110000
1100000
0025700
00503600
00004822
0000913
,
5000000
0500000
00311400
00395300
00004822
0000913
,
25570000
4360000
00533300
0048800
0000600
0000060

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

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

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

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

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

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

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

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

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