G = Dic3.2Dic10 order 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, C5⋊4(Dic3.Q8), (C5×Dic3).2Q8, C6.7(C2×Dic10), C2.9(S3×Dic10), C3⋊3(C4.Dic10), C15⋊6(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(D4⋊2D5), C10.3(D4⋊2S3), (C2×C60).251C22, C6.22(Q8⋊2D5), (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
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, C3, C4, C22, C5, C6, C2×C4, C2×C4, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C4⋊C4, Dic5, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, Dic3.Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C4.Dic10, Dic3×Dic5, C30.Q8, C6.Dic10, C3×C4⋊Dic5, C5×Dic3⋊C4, C30.4Q8, Dic3.2Dic10
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, C22×S3, C42.C2, Dic10, C22×D5, C4○D12, D4⋊2S3, S3×Q8, S3×D5, C2×Dic10, D4⋊2D5, Q8⋊2D5, Dic3.Q8, C2×S3×D5, C4.Dic10, D20⋊S3, S3×Dic10, Dic3.D10, Dic3.2Dic10
►Regular action on 480 points
Generators in S480
(1 326 223 110 410 161)(2 327 224 111 411 162)(3 328 225 112 412 163)(4 329 226 113 413 164)(5 330 227 114 414 165)(6 331 228 115 415 166)(7 332 229 116 416 167)(8 333 230 117 417 168)(9 334 231 118 418 169)(10 335 232 119 419 170)(11 336 233 120 420 171)(12 337 234 101 401 172)(13 338 235 102 402 173)(14 339 236 103 403 174)(15 340 237 104 404 175)(16 321 238 105 405 176)(17 322 239 106 406 177)(18 323 240 107 407 178)(19 324 221 108 408 179)(20 325 222 109 409 180)(21 382 467 183 129 218)(22 383 468 184 130 219)(23 384 469 185 131 220)(24 385 470 186 132 201)(25 386 471 187 133 202)(26 387 472 188 134 203)(27 388 473 189 135 204)(28 389 474 190 136 205)(29 390 475 191 137 206)(30 391 476 192 138 207)(31 392 477 193 139 208)(32 393 478 194 140 209)(33 394 479 195 121 210)(34 395 480 196 122 211)(35 396 461 197 123 212)(36 397 462 198 124 213)(37 398 463 199 125 214)(38 399 464 200 126 215)(39 400 465 181 127 216)(40 381 466 182 128 217)(41 446 284 63 148 313)(42 447 285 64 149 314)(43 448 286 65 150 315)(44 449 287 66 151 316)(45 450 288 67 152 317)(46 451 289 68 153 318)(47 452 290 69 154 319)(48 453 291 70 155 320)(49 454 292 71 156 301)(50 455 293 72 157 302)(51 456 294 73 158 303)(52 457 295 74 159 304)(53 458 296 75 160 305)(54 459 297 76 141 306)(55 460 298 77 142 307)(56 441 299 78 143 308)(57 442 300 79 144 309)(58 443 281 80 145 310)(59 444 282 61 146 311)(60 445 283 62 147 312)(81 358 244 373 263 427)(82 359 245 374 264 428)(83 360 246 375 265 429)(84 341 247 376 266 430)(85 342 248 377 267 431)(86 343 249 378 268 432)(87 344 250 379 269 433)(88 345 251 380 270 434)(89 346 252 361 271 435)(90 347 253 362 272 436)(91 348 254 363 273 437)(92 349 255 364 274 438)(93 350 256 365 275 439)(94 351 257 366 276 440)(95 352 258 367 277 421)(96 353 259 368 278 422)(97 354 260 369 279 423)(98 355 241 370 280 424)(99 356 242 371 261 425)(100 357 243 372 262 426)
(1 91 110 363)(2 364 111 92)(3 93 112 365)(4 366 113 94)(5 95 114 367)(6 368 115 96)(7 97 116 369)(8 370 117 98)(9 99 118 371)(10 372 119 100)(11 81 120 373)(12 374 101 82)(13 83 102 375)(14 376 103 84)(15 85 104 377)(16 378 105 86)(17 87 106 379)(18 380 107 88)(19 89 108 361)(20 362 109 90)(21 79 183 57)(22 58 184 80)(23 61 185 59)(24 60 186 62)(25 63 187 41)(26 42 188 64)(27 65 189 43)(28 44 190 66)(29 67 191 45)(30 46 192 68)(31 69 193 47)(32 48 194 70)(33 71 195 49)(34 50 196 72)(35 73 197 51)(36 52 198 74)(37 75 199 53)(38 54 200 76)(39 77 181 55)(40 56 182 78)(121 301 394 292)(122 293 395 302)(123 303 396 294)(124 295 397 304)(125 305 398 296)(126 297 399 306)(127 307 400 298)(128 299 381 308)(129 309 382 300)(130 281 383 310)(131 311 384 282)(132 283 385 312)(133 313 386 284)(134 285 387 314)(135 315 388 286)(136 287 389 316)(137 317 390 288)(138 289 391 318)(139 319 392 290)(140 291 393 320)(141 215 459 464)(142 465 460 216)(143 217 441 466)(144 467 442 218)(145 219 443 468)(146 469 444 220)(147 201 445 470)(148 471 446 202)(149 203 447 472)(150 473 448 204)(151 205 449 474)(152 475 450 206)(153 207 451 476)(154 477 452 208)(155 209 453 478)(156 479 454 210)(157 211 455 480)(158 461 456 212)(159 213 457 462)(160 463 458 214)(161 348 223 273)(162 274 224 349)(163 350 225 275)(164 276 226 351)(165 352 227 277)(166 278 228 353)(167 354 229 279)(168 280 230 355)(169 356 231 261)(170 262 232 357)(171 358 233 263)(172 264 234 359)(173 360 235 265)(174 266 236 341)(175 342 237 267)(176 268 238 343)(177 344 239 269)(178 270 240 345)(179 346 221 271)(180 272 222 347)(241 417 424 333)(242 334 425 418)(243 419 426 335)(244 336 427 420)(245 401 428 337)(246 338 429 402)(247 403 430 339)(248 340 431 404)(249 405 432 321)(250 322 433 406)(251 407 434 323)(252 324 435 408)(253 409 436 325)(254 326 437 410)(255 411 438 327)(256 328 439 412)(257 413 440 329)(258 330 421 414)(259 415 422 331)(260 332 423 416)
(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 68 120 56)(2 67 101 55)(3 66 102 54)(4 65 103 53)(5 64 104 52)(6 63 105 51)(7 62 106 50)(8 61 107 49)(9 80 108 48)(10 79 109 47)(11 78 110 46)(12 77 111 45)(13 76 112 44)(14 75 113 43)(15 74 114 42)(16 73 115 41)(17 72 116 60)(18 71 117 59)(19 70 118 58)(20 69 119 57)(21 362 193 100)(22 361 194 99)(23 380 195 98)(24 379 196 97)(25 378 197 96)(26 377 198 95)(27 376 199 94)(28 375 200 93)(29 374 181 92)(30 373 182 91)(31 372 183 90)(32 371 184 89)(33 370 185 88)(34 369 186 87)(35 368 187 86)(36 367 188 85)(37 366 189 84)(38 365 190 83)(39 364 191 82)(40 363 192 81)(121 355 384 270)(122 354 385 269)(123 353 386 268)(124 352 387 267)(125 351 388 266)(126 350 389 265)(127 349 390 264)(128 348 391 263)(129 347 392 262)(130 346 393 261)(131 345 394 280)(132 344 395 279)(133 343 396 278)(134 342 397 277)(135 341 398 276)(136 360 399 275)(137 359 400 274)(138 358 381 273)(139 357 382 272)(140 356 383 271)(141 412 449 338)(142 411 450 337)(143 410 451 336)(144 409 452 335)(145 408 453 334)(146 407 454 333)(147 406 455 332)(148 405 456 331)(149 404 457 330)(150 403 458 329)(151 402 459 328)(152 401 460 327)(153 420 441 326)(154 419 442 325)(155 418 443 324)(156 417 444 323)(157 416 445 322)(158 415 446 321)(159 414 447 340)(160 413 448 339)(161 289 233 308)(162 288 234 307)(163 287 235 306)(164 286 236 305)(165 285 237 304)(166 284 238 303)(167 283 239 302)(168 282 240 301)(169 281 221 320)(170 300 222 319)(171 299 223 318)(172 298 224 317)(173 297 225 316)(174 296 226 315)(175 295 227 314)(176 294 228 313)(177 293 229 312)(178 292 230 311)(179 291 231 310)(180 290 232 309)(201 250 480 423)(202 249 461 422)(203 248 462 421)(204 247 463 440)(205 246 464 439)(206 245 465 438)(207 244 466 437)(208 243 467 436)(209 242 468 435)(210 241 469 434)(211 260 470 433)(212 259 471 432)(213 258 472 431)(214 257 473 430)(215 256 474 429)(216 255 475 428)(217 254 476 427)(218 253 477 426)(219 252 478 425)(220 251 479 424)
G:=sub<Sym(480)| (1,326,223,110,410,161)(2,327,224,111,411,162)(3,328,225,112,412,163)(4,329,226,113,413,164)(5,330,227,114,414,165)(6,331,228,115,415,166)(7,332,229,116,416,167)(8,333,230,117,417,168)(9,334,231,118,418,169)(10,335,232,119,419,170)(11,336,233,120,420,171)(12,337,234,101,401,172)(13,338,235,102,402,173)(14,339,236,103,403,174)(15,340,237,104,404,175)(16,321,238,105,405,176)(17,322,239,106,406,177)(18,323,240,107,407,178)(19,324,221,108,408,179)(20,325,222,109,409,180)(21,382,467,183,129,218)(22,383,468,184,130,219)(23,384,469,185,131,220)(24,385,470,186,132,201)(25,386,471,187,133,202)(26,387,472,188,134,203)(27,388,473,189,135,204)(28,389,474,190,136,205)(29,390,475,191,137,206)(30,391,476,192,138,207)(31,392,477,193,139,208)(32,393,478,194,140,209)(33,394,479,195,121,210)(34,395,480,196,122,211)(35,396,461,197,123,212)(36,397,462,198,124,213)(37,398,463,199,125,214)(38,399,464,200,126,215)(39,400,465,181,127,216)(40,381,466,182,128,217)(41,446,284,63,148,313)(42,447,285,64,149,314)(43,448,286,65,150,315)(44,449,287,66,151,316)(45,450,288,67,152,317)(46,451,289,68,153,318)(47,452,290,69,154,319)(48,453,291,70,155,320)(49,454,292,71,156,301)(50,455,293,72,157,302)(51,456,294,73,158,303)(52,457,295,74,159,304)(53,458,296,75,160,305)(54,459,297,76,141,306)(55,460,298,77,142,307)(56,441,299,78,143,308)(57,442,300,79,144,309)(58,443,281,80,145,310)(59,444,282,61,146,311)(60,445,283,62,147,312)(81,358,244,373,263,427)(82,359,245,374,264,428)(83,360,246,375,265,429)(84,341,247,376,266,430)(85,342,248,377,267,431)(86,343,249,378,268,432)(87,344,250,379,269,433)(88,345,251,380,270,434)(89,346,252,361,271,435)(90,347,253,362,272,436)(91,348,254,363,273,437)(92,349,255,364,274,438)(93,350,256,365,275,439)(94,351,257,366,276,440)(95,352,258,367,277,421)(96,353,259,368,278,422)(97,354,260,369,279,423)(98,355,241,370,280,424)(99,356,242,371,261,425)(100,357,243,372,262,426), (1,91,110,363)(2,364,111,92)(3,93,112,365)(4,366,113,94)(5,95,114,367)(6,368,115,96)(7,97,116,369)(8,370,117,98)(9,99,118,371)(10,372,119,100)(11,81,120,373)(12,374,101,82)(13,83,102,375)(14,376,103,84)(15,85,104,377)(16,378,105,86)(17,87,106,379)(18,380,107,88)(19,89,108,361)(20,362,109,90)(21,79,183,57)(22,58,184,80)(23,61,185,59)(24,60,186,62)(25,63,187,41)(26,42,188,64)(27,65,189,43)(28,44,190,66)(29,67,191,45)(30,46,192,68)(31,69,193,47)(32,48,194,70)(33,71,195,49)(34,50,196,72)(35,73,197,51)(36,52,198,74)(37,75,199,53)(38,54,200,76)(39,77,181,55)(40,56,182,78)(121,301,394,292)(122,293,395,302)(123,303,396,294)(124,295,397,304)(125,305,398,296)(126,297,399,306)(127,307,400,298)(128,299,381,308)(129,309,382,300)(130,281,383,310)(131,311,384,282)(132,283,385,312)(133,313,386,284)(134,285,387,314)(135,315,388,286)(136,287,389,316)(137,317,390,288)(138,289,391,318)(139,319,392,290)(140,291,393,320)(141,215,459,464)(142,465,460,216)(143,217,441,466)(144,467,442,218)(145,219,443,468)(146,469,444,220)(147,201,445,470)(148,471,446,202)(149,203,447,472)(150,473,448,204)(151,205,449,474)(152,475,450,206)(153,207,451,476)(154,477,452,208)(155,209,453,478)(156,479,454,210)(157,211,455,480)(158,461,456,212)(159,213,457,462)(160,463,458,214)(161,348,223,273)(162,274,224,349)(163,350,225,275)(164,276,226,351)(165,352,227,277)(166,278,228,353)(167,354,229,279)(168,280,230,355)(169,356,231,261)(170,262,232,357)(171,358,233,263)(172,264,234,359)(173,360,235,265)(174,266,236,341)(175,342,237,267)(176,268,238,343)(177,344,239,269)(178,270,240,345)(179,346,221,271)(180,272,222,347)(241,417,424,333)(242,334,425,418)(243,419,426,335)(244,336,427,420)(245,401,428,337)(246,338,429,402)(247,403,430,339)(248,340,431,404)(249,405,432,321)(250,322,433,406)(251,407,434,323)(252,324,435,408)(253,409,436,325)(254,326,437,410)(255,411,438,327)(256,328,439,412)(257,413,440,329)(258,330,421,414)(259,415,422,331)(260,332,423,416), (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), (1,68,120,56)(2,67,101,55)(3,66,102,54)(4,65,103,53)(5,64,104,52)(6,63,105,51)(7,62,106,50)(8,61,107,49)(9,80,108,48)(10,79,109,47)(11,78,110,46)(12,77,111,45)(13,76,112,44)(14,75,113,43)(15,74,114,42)(16,73,115,41)(17,72,116,60)(18,71,117,59)(19,70,118,58)(20,69,119,57)(21,362,193,100)(22,361,194,99)(23,380,195,98)(24,379,196,97)(25,378,197,96)(26,377,198,95)(27,376,199,94)(28,375,200,93)(29,374,181,92)(30,373,182,91)(31,372,183,90)(32,371,184,89)(33,370,185,88)(34,369,186,87)(35,368,187,86)(36,367,188,85)(37,366,189,84)(38,365,190,83)(39,364,191,82)(40,363,192,81)(121,355,384,270)(122,354,385,269)(123,353,386,268)(124,352,387,267)(125,351,388,266)(126,350,389,265)(127,349,390,264)(128,348,391,263)(129,347,392,262)(130,346,393,261)(131,345,394,280)(132,344,395,279)(133,343,396,278)(134,342,397,277)(135,341,398,276)(136,360,399,275)(137,359,400,274)(138,358,381,273)(139,357,382,272)(140,356,383,271)(141,412,449,338)(142,411,450,337)(143,410,451,336)(144,409,452,335)(145,408,453,334)(146,407,454,333)(147,406,455,332)(148,405,456,331)(149,404,457,330)(150,403,458,329)(151,402,459,328)(152,401,460,327)(153,420,441,326)(154,419,442,325)(155,418,443,324)(156,417,444,323)(157,416,445,322)(158,415,446,321)(159,414,447,340)(160,413,448,339)(161,289,233,308)(162,288,234,307)(163,287,235,306)(164,286,236,305)(165,285,237,304)(166,284,238,303)(167,283,239,302)(168,282,240,301)(169,281,221,320)(170,300,222,319)(171,299,223,318)(172,298,224,317)(173,297,225,316)(174,296,226,315)(175,295,227,314)(176,294,228,313)(177,293,229,312)(178,292,230,311)(179,291,231,310)(180,290,232,309)(201,250,480,423)(202,249,461,422)(203,248,462,421)(204,247,463,440)(205,246,464,439)(206,245,465,438)(207,244,466,437)(208,243,467,436)(209,242,468,435)(210,241,469,434)(211,260,470,433)(212,259,471,432)(213,258,472,431)(214,257,473,430)(215,256,474,429)(216,255,475,428)(217,254,476,427)(218,253,477,426)(219,252,478,425)(220,251,479,424)>;
G:=Group( (1,326,223,110,410,161)(2,327,224,111,411,162)(3,328,225,112,412,163)(4,329,226,113,413,164)(5,330,227,114,414,165)(6,331,228,115,415,166)(7,332,229,116,416,167)(8,333,230,117,417,168)(9,334,231,118,418,169)(10,335,232,119,419,170)(11,336,233,120,420,171)(12,337,234,101,401,172)(13,338,235,102,402,173)(14,339,236,103,403,174)(15,340,237,104,404,175)(16,321,238,105,405,176)(17,322,239,106,406,177)(18,323,240,107,407,178)(19,324,221,108,408,179)(20,325,222,109,409,180)(21,382,467,183,129,218)(22,383,468,184,130,219)(23,384,469,185,131,220)(24,385,470,186,132,201)(25,386,471,187,133,202)(26,387,472,188,134,203)(27,388,473,189,135,204)(28,389,474,190,136,205)(29,390,475,191,137,206)(30,391,476,192,138,207)(31,392,477,193,139,208)(32,393,478,194,140,209)(33,394,479,195,121,210)(34,395,480,196,122,211)(35,396,461,197,123,212)(36,397,462,198,124,213)(37,398,463,199,125,214)(38,399,464,200,126,215)(39,400,465,181,127,216)(40,381,466,182,128,217)(41,446,284,63,148,313)(42,447,285,64,149,314)(43,448,286,65,150,315)(44,449,287,66,151,316)(45,450,288,67,152,317)(46,451,289,68,153,318)(47,452,290,69,154,319)(48,453,291,70,155,320)(49,454,292,71,156,301)(50,455,293,72,157,302)(51,456,294,73,158,303)(52,457,295,74,159,304)(53,458,296,75,160,305)(54,459,297,76,141,306)(55,460,298,77,142,307)(56,441,299,78,143,308)(57,442,300,79,144,309)(58,443,281,80,145,310)(59,444,282,61,146,311)(60,445,283,62,147,312)(81,358,244,373,263,427)(82,359,245,374,264,428)(83,360,246,375,265,429)(84,341,247,376,266,430)(85,342,248,377,267,431)(86,343,249,378,268,432)(87,344,250,379,269,433)(88,345,251,380,270,434)(89,346,252,361,271,435)(90,347,253,362,272,436)(91,348,254,363,273,437)(92,349,255,364,274,438)(93,350,256,365,275,439)(94,351,257,366,276,440)(95,352,258,367,277,421)(96,353,259,368,278,422)(97,354,260,369,279,423)(98,355,241,370,280,424)(99,356,242,371,261,425)(100,357,243,372,262,426), (1,91,110,363)(2,364,111,92)(3,93,112,365)(4,366,113,94)(5,95,114,367)(6,368,115,96)(7,97,116,369)(8,370,117,98)(9,99,118,371)(10,372,119,100)(11,81,120,373)(12,374,101,82)(13,83,102,375)(14,376,103,84)(15,85,104,377)(16,378,105,86)(17,87,106,379)(18,380,107,88)(19,89,108,361)(20,362,109,90)(21,79,183,57)(22,58,184,80)(23,61,185,59)(24,60,186,62)(25,63,187,41)(26,42,188,64)(27,65,189,43)(28,44,190,66)(29,67,191,45)(30,46,192,68)(31,69,193,47)(32,48,194,70)(33,71,195,49)(34,50,196,72)(35,73,197,51)(36,52,198,74)(37,75,199,53)(38,54,200,76)(39,77,181,55)(40,56,182,78)(121,301,394,292)(122,293,395,302)(123,303,396,294)(124,295,397,304)(125,305,398,296)(126,297,399,306)(127,307,400,298)(128,299,381,308)(129,309,382,300)(130,281,383,310)(131,311,384,282)(132,283,385,312)(133,313,386,284)(134,285,387,314)(135,315,388,286)(136,287,389,316)(137,317,390,288)(138,289,391,318)(139,319,392,290)(140,291,393,320)(141,215,459,464)(142,465,460,216)(143,217,441,466)(144,467,442,218)(145,219,443,468)(146,469,444,220)(147,201,445,470)(148,471,446,202)(149,203,447,472)(150,473,448,204)(151,205,449,474)(152,475,450,206)(153,207,451,476)(154,477,452,208)(155,209,453,478)(156,479,454,210)(157,211,455,480)(158,461,456,212)(159,213,457,462)(160,463,458,214)(161,348,223,273)(162,274,224,349)(163,350,225,275)(164,276,226,351)(165,352,227,277)(166,278,228,353)(167,354,229,279)(168,280,230,355)(169,356,231,261)(170,262,232,357)(171,358,233,263)(172,264,234,359)(173,360,235,265)(174,266,236,341)(175,342,237,267)(176,268,238,343)(177,344,239,269)(178,270,240,345)(179,346,221,271)(180,272,222,347)(241,417,424,333)(242,334,425,418)(243,419,426,335)(244,336,427,420)(245,401,428,337)(246,338,429,402)(247,403,430,339)(248,340,431,404)(249,405,432,321)(250,322,433,406)(251,407,434,323)(252,324,435,408)(253,409,436,325)(254,326,437,410)(255,411,438,327)(256,328,439,412)(257,413,440,329)(258,330,421,414)(259,415,422,331)(260,332,423,416), (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), (1,68,120,56)(2,67,101,55)(3,66,102,54)(4,65,103,53)(5,64,104,52)(6,63,105,51)(7,62,106,50)(8,61,107,49)(9,80,108,48)(10,79,109,47)(11,78,110,46)(12,77,111,45)(13,76,112,44)(14,75,113,43)(15,74,114,42)(16,73,115,41)(17,72,116,60)(18,71,117,59)(19,70,118,58)(20,69,119,57)(21,362,193,100)(22,361,194,99)(23,380,195,98)(24,379,196,97)(25,378,197,96)(26,377,198,95)(27,376,199,94)(28,375,200,93)(29,374,181,92)(30,373,182,91)(31,372,183,90)(32,371,184,89)(33,370,185,88)(34,369,186,87)(35,368,187,86)(36,367,188,85)(37,366,189,84)(38,365,190,83)(39,364,191,82)(40,363,192,81)(121,355,384,270)(122,354,385,269)(123,353,386,268)(124,352,387,267)(125,351,388,266)(126,350,389,265)(127,349,390,264)(128,348,391,263)(129,347,392,262)(130,346,393,261)(131,345,394,280)(132,344,395,279)(133,343,396,278)(134,342,397,277)(135,341,398,276)(136,360,399,275)(137,359,400,274)(138,358,381,273)(139,357,382,272)(140,356,383,271)(141,412,449,338)(142,411,450,337)(143,410,451,336)(144,409,452,335)(145,408,453,334)(146,407,454,333)(147,406,455,332)(148,405,456,331)(149,404,457,330)(150,403,458,329)(151,402,459,328)(152,401,460,327)(153,420,441,326)(154,419,442,325)(155,418,443,324)(156,417,444,323)(157,416,445,322)(158,415,446,321)(159,414,447,340)(160,413,448,339)(161,289,233,308)(162,288,234,307)(163,287,235,306)(164,286,236,305)(165,285,237,304)(166,284,238,303)(167,283,239,302)(168,282,240,301)(169,281,221,320)(170,300,222,319)(171,299,223,318)(172,298,224,317)(173,297,225,316)(174,296,226,315)(175,295,227,314)(176,294,228,313)(177,293,229,312)(178,292,230,311)(179,291,231,310)(180,290,232,309)(201,250,480,423)(202,249,461,422)(203,248,462,421)(204,247,463,440)(205,246,464,439)(206,245,465,438)(207,244,466,437)(208,243,467,436)(209,242,468,435)(210,241,469,434)(211,260,470,433)(212,259,471,432)(213,258,472,431)(214,257,473,430)(215,256,474,429)(216,255,475,428)(217,254,476,427)(218,253,477,426)(219,252,478,425)(220,251,479,424) );
G=PermutationGroup([[(1,326,223,110,410,161),(2,327,224,111,411,162),(3,328,225,112,412,163),(4,329,226,113,413,164),(5,330,227,114,414,165),(6,331,228,115,415,166),(7,332,229,116,416,167),(8,333,230,117,417,168),(9,334,231,118,418,169),(10,335,232,119,419,170),(11,336,233,120,420,171),(12,337,234,101,401,172),(13,338,235,102,402,173),(14,339,236,103,403,174),(15,340,237,104,404,175),(16,321,238,105,405,176),(17,322,239,106,406,177),(18,323,240,107,407,178),(19,324,221,108,408,179),(20,325,222,109,409,180),(21,382,467,183,129,218),(22,383,468,184,130,219),(23,384,469,185,131,220),(24,385,470,186,132,201),(25,386,471,187,133,202),(26,387,472,188,134,203),(27,388,473,189,135,204),(28,389,474,190,136,205),(29,390,475,191,137,206),(30,391,476,192,138,207),(31,392,477,193,139,208),(32,393,478,194,140,209),(33,394,479,195,121,210),(34,395,480,196,122,211),(35,396,461,197,123,212),(36,397,462,198,124,213),(37,398,463,199,125,214),(38,399,464,200,126,215),(39,400,465,181,127,216),(40,381,466,182,128,217),(41,446,284,63,148,313),(42,447,285,64,149,314),(43,448,286,65,150,315),(44,449,287,66,151,316),(45,450,288,67,152,317),(46,451,289,68,153,318),(47,452,290,69,154,319),(48,453,291,70,155,320),(49,454,292,71,156,301),(50,455,293,72,157,302),(51,456,294,73,158,303),(52,457,295,74,159,304),(53,458,296,75,160,305),(54,459,297,76,141,306),(55,460,298,77,142,307),(56,441,299,78,143,308),(57,442,300,79,144,309),(58,443,281,80,145,310),(59,444,282,61,146,311),(60,445,283,62,147,312),(81,358,244,373,263,427),(82,359,245,374,264,428),(83,360,246,375,265,429),(84,341,247,376,266,430),(85,342,248,377,267,431),(86,343,249,378,268,432),(87,344,250,379,269,433),(88,345,251,380,270,434),(89,346,252,361,271,435),(90,347,253,362,272,436),(91,348,254,363,273,437),(92,349,255,364,274,438),(93,350,256,365,275,439),(94,351,257,366,276,440),(95,352,258,367,277,421),(96,353,259,368,278,422),(97,354,260,369,279,423),(98,355,241,370,280,424),(99,356,242,371,261,425),(100,357,243,372,262,426)], [(1,91,110,363),(2,364,111,92),(3,93,112,365),(4,366,113,94),(5,95,114,367),(6,368,115,96),(7,97,116,369),(8,370,117,98),(9,99,118,371),(10,372,119,100),(11,81,120,373),(12,374,101,82),(13,83,102,375),(14,376,103,84),(15,85,104,377),(16,378,105,86),(17,87,106,379),(18,380,107,88),(19,89,108,361),(20,362,109,90),(21,79,183,57),(22,58,184,80),(23,61,185,59),(24,60,186,62),(25,63,187,41),(26,42,188,64),(27,65,189,43),(28,44,190,66),(29,67,191,45),(30,46,192,68),(31,69,193,47),(32,48,194,70),(33,71,195,49),(34,50,196,72),(35,73,197,51),(36,52,198,74),(37,75,199,53),(38,54,200,76),(39,77,181,55),(40,56,182,78),(121,301,394,292),(122,293,395,302),(123,303,396,294),(124,295,397,304),(125,305,398,296),(126,297,399,306),(127,307,400,298),(128,299,381,308),(129,309,382,300),(130,281,383,310),(131,311,384,282),(132,283,385,312),(133,313,386,284),(134,285,387,314),(135,315,388,286),(136,287,389,316),(137,317,390,288),(138,289,391,318),(139,319,392,290),(140,291,393,320),(141,215,459,464),(142,465,460,216),(143,217,441,466),(144,467,442,218),(145,219,443,468),(146,469,444,220),(147,201,445,470),(148,471,446,202),(149,203,447,472),(150,473,448,204),(151,205,449,474),(152,475,450,206),(153,207,451,476),(154,477,452,208),(155,209,453,478),(156,479,454,210),(157,211,455,480),(158,461,456,212),(159,213,457,462),(160,463,458,214),(161,348,223,273),(162,274,224,349),(163,350,225,275),(164,276,226,351),(165,352,227,277),(166,278,228,353),(167,354,229,279),(168,280,230,355),(169,356,231,261),(170,262,232,357),(171,358,233,263),(172,264,234,359),(173,360,235,265),(174,266,236,341),(175,342,237,267),(176,268,238,343),(177,344,239,269),(178,270,240,345),(179,346,221,271),(180,272,222,347),(241,417,424,333),(242,334,425,418),(243,419,426,335),(244,336,427,420),(245,401,428,337),(246,338,429,402),(247,403,430,339),(248,340,431,404),(249,405,432,321),(250,322,433,406),(251,407,434,323),(252,324,435,408),(253,409,436,325),(254,326,437,410),(255,411,438,327),(256,328,439,412),(257,413,440,329),(258,330,421,414),(259,415,422,331),(260,332,423,416)], [(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)], [(1,68,120,56),(2,67,101,55),(3,66,102,54),(4,65,103,53),(5,64,104,52),(6,63,105,51),(7,62,106,50),(8,61,107,49),(9,80,108,48),(10,79,109,47),(11,78,110,46),(12,77,111,45),(13,76,112,44),(14,75,113,43),(15,74,114,42),(16,73,115,41),(17,72,116,60),(18,71,117,59),(19,70,118,58),(20,69,119,57),(21,362,193,100),(22,361,194,99),(23,380,195,98),(24,379,196,97),(25,378,197,96),(26,377,198,95),(27,376,199,94),(28,375,200,93),(29,374,181,92),(30,373,182,91),(31,372,183,90),(32,371,184,89),(33,370,185,88),(34,369,186,87),(35,368,187,86),(36,367,188,85),(37,366,189,84),(38,365,190,83),(39,364,191,82),(40,363,192,81),(121,355,384,270),(122,354,385,269),(123,353,386,268),(124,352,387,267),(125,351,388,266),(126,350,389,265),(127,349,390,264),(128,348,391,263),(129,347,392,262),(130,346,393,261),(131,345,394,280),(132,344,395,279),(133,343,396,278),(134,342,397,277),(135,341,398,276),(136,360,399,275),(137,359,400,274),(138,358,381,273),(139,357,382,272),(140,356,383,271),(141,412,449,338),(142,411,450,337),(143,410,451,336),(144,409,452,335),(145,408,453,334),(146,407,454,333),(147,406,455,332),(148,405,456,331),(149,404,457,330),(150,403,458,329),(151,402,459,328),(152,401,460,327),(153,420,441,326),(154,419,442,325),(155,418,443,324),(156,417,444,323),(157,416,445,322),(158,415,446,321),(159,414,447,340),(160,413,448,339),(161,289,233,308),(162,288,234,307),(163,287,235,306),(164,286,236,305),(165,285,237,304),(166,284,238,303),(167,283,239,302),(168,282,240,301),(169,281,221,320),(170,300,222,319),(171,299,223,318),(172,298,224,317),(173,297,225,316),(174,296,226,315),(175,295,227,314),(176,294,228,313),(177,293,229,312),(178,292,230,311),(179,291,231,310),(180,290,232,309),(201,250,480,423),(202,249,461,422),(203,248,462,421),(204,247,463,440),(205,246,464,439),(206,245,465,438),(207,244,466,437),(208,243,467,436),(209,242,468,435),(210,241,469,434),(211,260,470,433),(212,259,471,432),(213,258,472,431),(214,257,473,430),(215,256,474,429),(216,255,475,428),(217,254,476,427),(218,253,477,426),(219,252,478,425),(220,251,479,424)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 4 | 6 | 6 | 10 | 10 | 12 | 20 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | Dic10 | C4○D12 | D4⋊2S3 | S3×Q8 | S3×D5 | D4⋊2D5 | Q8⋊2D5 | C2×S3×D5 | D20⋊S3 | S3×Dic10 | Dic3.D10 |
| kernel | Dic3.2Dic10 | Dic3×Dic5 | C30.Q8 | C6.Dic10 | C3×C4⋊Dic5 | C5×Dic3⋊C4 | C30.4Q8 | C4⋊Dic5 | C5×Dic3 | Dic3⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | Dic3 | C10 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 8 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic3.2Dic10 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 46 | 0 | 0 |
| 0 | 0 | 49 | 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 | 50 | 0 | 0 |
| 0 | 0 | 5 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 8 |
| 0 | 0 | 0 | 0 | 44 | 47 |
| 57 | 25 | 0 | 0 | 0 | 0 |
| 36 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 2 | 60 |
| 32 | 22 | 0 | 0 | 0 | 0 |
| 56 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 27 |
| 0 | 0 | 0 | 0 | 57 | 29 |
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