metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊1Dic6, Dic15.5D4, Dic3⋊2Dic10, C5⋊2(C12⋊Q8), C3⋊2(C20⋊Q8), C15⋊2(C4⋊Q8), C6.52(D4×D5), (C2×C20).3D6, C6.20(Q8×D5), C30.5(C2×Q8), (C5×Dic3)⋊2Q8, (C3×Dic5)⋊1Q8, C10.54(S3×D4), (C2×C12).1D10, C10.21(S3×Q8), C2.6(D5×Dic6), C30.100(C2×D4), Dic3⋊C4.2D5, (C2×C60).1C22, C6.3(C2×Dic10), C10.2(C2×Dic6), C2.7(S3×Dic10), (C2×C30).18C23, (C2×Dic5).82D6, (C2×Dic30).2C2, C10.D4.2S3, C6.Dic10.3C2, C30.Q8.2C2, C2.8(D10⋊D6), (Dic3×Dic5).5C2, (C2×Dic3).74D10, (C6×Dic5).5C22, (C10×Dic3).5C22, (C2×Dic15).26C22, (C2×C15⋊Q8).5C2, (C2×C4).21(S3×D5), C22.115(C2×S3×D5), (C5×Dic3⋊C4).2C2, (C2×C6).30(C22×D5), (C2×C10).30(C22×S3), (C3×C10.D4).2C2, SmallGroup(480,404)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊Dic10
G = < a,b,c,d | a6=c20=1, b2=a3, d2=c10, bab-1=dad-1=a-1, ac=ca, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 652 in 136 conjugacy classes, 54 normal (44 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C5×Dic3, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×Dic10, C12⋊Q8, C15⋊Q8, C6×Dic5, C10×Dic3, Dic30, C2×Dic15, C2×C60, C20⋊Q8, Dic3×Dic5, C30.Q8, C6.Dic10, C3×C10.D4, C5×Dic3⋊C4, C2×C15⋊Q8, C2×Dic30, Dic3⋊Dic10
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, D10, Dic6, C22×S3, C4⋊Q8, Dic10, C22×D5, C2×Dic6, S3×D4, S3×Q8, S3×D5, C2×Dic10, D4×D5, Q8×D5, C12⋊Q8, C2×S3×D5, C20⋊Q8, D5×Dic6, S3×Dic10, D10⋊D6, Dic3⋊Dic10
►Regular action on 480 points
Generators in S480
(1 241 401 346 80 384)(2 242 402 347 61 385)(3 243 403 348 62 386)(4 244 404 349 63 387)(5 245 405 350 64 388)(6 246 406 351 65 389)(7 247 407 352 66 390)(8 248 408 353 67 391)(9 249 409 354 68 392)(10 250 410 355 69 393)(11 251 411 356 70 394)(12 252 412 357 71 395)(13 253 413 358 72 396)(14 254 414 359 73 397)(15 255 415 360 74 398)(16 256 416 341 75 399)(17 257 417 342 76 400)(18 258 418 343 77 381)(19 259 419 344 78 382)(20 260 420 345 79 383)(21 182 261 137 332 99)(22 183 262 138 333 100)(23 184 263 139 334 81)(24 185 264 140 335 82)(25 186 265 121 336 83)(26 187 266 122 337 84)(27 188 267 123 338 85)(28 189 268 124 339 86)(29 190 269 125 340 87)(30 191 270 126 321 88)(31 192 271 127 322 89)(32 193 272 128 323 90)(33 194 273 129 324 91)(34 195 274 130 325 92)(35 196 275 131 326 93)(36 197 276 132 327 94)(37 198 277 133 328 95)(38 199 278 134 329 96)(39 200 279 135 330 97)(40 181 280 136 331 98)(41 367 159 446 466 118)(42 368 160 447 467 119)(43 369 141 448 468 120)(44 370 142 449 469 101)(45 371 143 450 470 102)(46 372 144 451 471 103)(47 373 145 452 472 104)(48 374 146 453 473 105)(49 375 147 454 474 106)(50 376 148 455 475 107)(51 377 149 456 476 108)(52 378 150 457 477 109)(53 379 151 458 478 110)(54 380 152 459 479 111)(55 361 153 460 480 112)(56 362 154 441 461 113)(57 363 155 442 462 114)(58 364 156 443 463 115)(59 365 157 444 464 116)(60 366 158 445 465 117)(161 437 281 229 204 313)(162 438 282 230 205 314)(163 439 283 231 206 315)(164 440 284 232 207 316)(165 421 285 233 208 317)(166 422 286 234 209 318)(167 423 287 235 210 319)(168 424 288 236 211 320)(169 425 289 237 212 301)(170 426 290 238 213 302)(171 427 291 239 214 303)(172 428 292 240 215 304)(173 429 293 221 216 305)(174 430 294 222 217 306)(175 431 295 223 218 307)(176 432 296 224 219 308)(177 433 297 225 220 309)(178 434 298 226 201 310)(179 435 299 227 202 311)(180 436 300 228 203 312)
(1 41 346 446)(2 447 347 42)(3 43 348 448)(4 449 349 44)(5 45 350 450)(6 451 351 46)(7 47 352 452)(8 453 353 48)(9 49 354 454)(10 455 355 50)(11 51 356 456)(12 457 357 52)(13 53 358 458)(14 459 359 54)(15 55 360 460)(16 441 341 56)(17 57 342 442)(18 443 343 58)(19 59 344 444)(20 445 345 60)(21 303 137 291)(22 292 138 304)(23 305 139 293)(24 294 140 306)(25 307 121 295)(26 296 122 308)(27 309 123 297)(28 298 124 310)(29 311 125 299)(30 300 126 312)(31 313 127 281)(32 282 128 314)(33 315 129 283)(34 284 130 316)(35 317 131 285)(36 286 132 318)(37 319 133 287)(38 288 134 320)(39 301 135 289)(40 290 136 302)(61 119 242 160)(62 141 243 120)(63 101 244 142)(64 143 245 102)(65 103 246 144)(66 145 247 104)(67 105 248 146)(68 147 249 106)(69 107 250 148)(70 149 251 108)(71 109 252 150)(72 151 253 110)(73 111 254 152)(74 153 255 112)(75 113 256 154)(76 155 257 114)(77 115 258 156)(78 157 259 116)(79 117 260 158)(80 159 241 118)(81 173 263 221)(82 222 264 174)(83 175 265 223)(84 224 266 176)(85 177 267 225)(86 226 268 178)(87 179 269 227)(88 228 270 180)(89 161 271 229)(90 230 272 162)(91 163 273 231)(92 232 274 164)(93 165 275 233)(94 234 276 166)(95 167 277 235)(96 236 278 168)(97 169 279 237)(98 238 280 170)(99 171 261 239)(100 240 262 172)(181 426 331 213)(182 214 332 427)(183 428 333 215)(184 216 334 429)(185 430 335 217)(186 218 336 431)(187 432 337 219)(188 220 338 433)(189 434 339 201)(190 202 340 435)(191 436 321 203)(192 204 322 437)(193 438 323 205)(194 206 324 439)(195 440 325 207)(196 208 326 421)(197 422 327 209)(198 210 328 423)(199 424 329 211)(200 212 330 425)(361 415 480 398)(362 399 461 416)(363 417 462 400)(364 381 463 418)(365 419 464 382)(366 383 465 420)(367 401 466 384)(368 385 467 402)(369 403 468 386)(370 387 469 404)(371 405 470 388)(372 389 471 406)(373 407 472 390)(374 391 473 408)(375 409 474 392)(376 393 475 410)(377 411 476 394)(378 395 477 412)(379 413 478 396)(380 397 479 414)
(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)
(1 198 11 188)(2 197 12 187)(3 196 13 186)(4 195 14 185)(5 194 15 184)(6 193 16 183)(7 192 17 182)(8 191 18 181)(9 190 19 200)(10 189 20 199)(21 247 31 257)(22 246 32 256)(23 245 33 255)(24 244 34 254)(25 243 35 253)(26 242 36 252)(27 241 37 251)(28 260 38 250)(29 259 39 249)(30 258 40 248)(41 210 51 220)(42 209 52 219)(43 208 53 218)(44 207 54 217)(45 206 55 216)(46 205 56 215)(47 204 57 214)(48 203 58 213)(49 202 59 212)(50 201 60 211)(61 132 71 122)(62 131 72 121)(63 130 73 140)(64 129 74 139)(65 128 75 138)(66 127 76 137)(67 126 77 136)(68 125 78 135)(69 124 79 134)(70 123 80 133)(81 405 91 415)(82 404 92 414)(83 403 93 413)(84 402 94 412)(85 401 95 411)(86 420 96 410)(87 419 97 409)(88 418 98 408)(89 417 99 407)(90 416 100 406)(101 316 111 306)(102 315 112 305)(103 314 113 304)(104 313 114 303)(105 312 115 302)(106 311 116 301)(107 310 117 320)(108 309 118 319)(109 308 119 318)(110 307 120 317)(141 285 151 295)(142 284 152 294)(143 283 153 293)(144 282 154 292)(145 281 155 291)(146 300 156 290)(147 299 157 289)(148 298 158 288)(149 297 159 287)(150 296 160 286)(161 462 171 472)(162 461 172 471)(163 480 173 470)(164 479 174 469)(165 478 175 468)(166 477 176 467)(167 476 177 466)(168 475 178 465)(169 474 179 464)(170 473 180 463)(221 371 231 361)(222 370 232 380)(223 369 233 379)(224 368 234 378)(225 367 235 377)(226 366 236 376)(227 365 237 375)(228 364 238 374)(229 363 239 373)(230 362 240 372)(261 390 271 400)(262 389 272 399)(263 388 273 398)(264 387 274 397)(265 386 275 396)(266 385 276 395)(267 384 277 394)(268 383 278 393)(269 382 279 392)(270 381 280 391)(321 343 331 353)(322 342 332 352)(323 341 333 351)(324 360 334 350)(325 359 335 349)(326 358 336 348)(327 357 337 347)(328 356 338 346)(329 355 339 345)(330 354 340 344)(421 458 431 448)(422 457 432 447)(423 456 433 446)(424 455 434 445)(425 454 435 444)(426 453 436 443)(427 452 437 442)(428 451 438 441)(429 450 439 460)(430 449 440 459)
G:=sub<Sym(480)| (1,241,401,346,80,384)(2,242,402,347,61,385)(3,243,403,348,62,386)(4,244,404,349,63,387)(5,245,405,350,64,388)(6,246,406,351,65,389)(7,247,407,352,66,390)(8,248,408,353,67,391)(9,249,409,354,68,392)(10,250,410,355,69,393)(11,251,411,356,70,394)(12,252,412,357,71,395)(13,253,413,358,72,396)(14,254,414,359,73,397)(15,255,415,360,74,398)(16,256,416,341,75,399)(17,257,417,342,76,400)(18,258,418,343,77,381)(19,259,419,344,78,382)(20,260,420,345,79,383)(21,182,261,137,332,99)(22,183,262,138,333,100)(23,184,263,139,334,81)(24,185,264,140,335,82)(25,186,265,121,336,83)(26,187,266,122,337,84)(27,188,267,123,338,85)(28,189,268,124,339,86)(29,190,269,125,340,87)(30,191,270,126,321,88)(31,192,271,127,322,89)(32,193,272,128,323,90)(33,194,273,129,324,91)(34,195,274,130,325,92)(35,196,275,131,326,93)(36,197,276,132,327,94)(37,198,277,133,328,95)(38,199,278,134,329,96)(39,200,279,135,330,97)(40,181,280,136,331,98)(41,367,159,446,466,118)(42,368,160,447,467,119)(43,369,141,448,468,120)(44,370,142,449,469,101)(45,371,143,450,470,102)(46,372,144,451,471,103)(47,373,145,452,472,104)(48,374,146,453,473,105)(49,375,147,454,474,106)(50,376,148,455,475,107)(51,377,149,456,476,108)(52,378,150,457,477,109)(53,379,151,458,478,110)(54,380,152,459,479,111)(55,361,153,460,480,112)(56,362,154,441,461,113)(57,363,155,442,462,114)(58,364,156,443,463,115)(59,365,157,444,464,116)(60,366,158,445,465,117)(161,437,281,229,204,313)(162,438,282,230,205,314)(163,439,283,231,206,315)(164,440,284,232,207,316)(165,421,285,233,208,317)(166,422,286,234,209,318)(167,423,287,235,210,319)(168,424,288,236,211,320)(169,425,289,237,212,301)(170,426,290,238,213,302)(171,427,291,239,214,303)(172,428,292,240,215,304)(173,429,293,221,216,305)(174,430,294,222,217,306)(175,431,295,223,218,307)(176,432,296,224,219,308)(177,433,297,225,220,309)(178,434,298,226,201,310)(179,435,299,227,202,311)(180,436,300,228,203,312), (1,41,346,446)(2,447,347,42)(3,43,348,448)(4,449,349,44)(5,45,350,450)(6,451,351,46)(7,47,352,452)(8,453,353,48)(9,49,354,454)(10,455,355,50)(11,51,356,456)(12,457,357,52)(13,53,358,458)(14,459,359,54)(15,55,360,460)(16,441,341,56)(17,57,342,442)(18,443,343,58)(19,59,344,444)(20,445,345,60)(21,303,137,291)(22,292,138,304)(23,305,139,293)(24,294,140,306)(25,307,121,295)(26,296,122,308)(27,309,123,297)(28,298,124,310)(29,311,125,299)(30,300,126,312)(31,313,127,281)(32,282,128,314)(33,315,129,283)(34,284,130,316)(35,317,131,285)(36,286,132,318)(37,319,133,287)(38,288,134,320)(39,301,135,289)(40,290,136,302)(61,119,242,160)(62,141,243,120)(63,101,244,142)(64,143,245,102)(65,103,246,144)(66,145,247,104)(67,105,248,146)(68,147,249,106)(69,107,250,148)(70,149,251,108)(71,109,252,150)(72,151,253,110)(73,111,254,152)(74,153,255,112)(75,113,256,154)(76,155,257,114)(77,115,258,156)(78,157,259,116)(79,117,260,158)(80,159,241,118)(81,173,263,221)(82,222,264,174)(83,175,265,223)(84,224,266,176)(85,177,267,225)(86,226,268,178)(87,179,269,227)(88,228,270,180)(89,161,271,229)(90,230,272,162)(91,163,273,231)(92,232,274,164)(93,165,275,233)(94,234,276,166)(95,167,277,235)(96,236,278,168)(97,169,279,237)(98,238,280,170)(99,171,261,239)(100,240,262,172)(181,426,331,213)(182,214,332,427)(183,428,333,215)(184,216,334,429)(185,430,335,217)(186,218,336,431)(187,432,337,219)(188,220,338,433)(189,434,339,201)(190,202,340,435)(191,436,321,203)(192,204,322,437)(193,438,323,205)(194,206,324,439)(195,440,325,207)(196,208,326,421)(197,422,327,209)(198,210,328,423)(199,424,329,211)(200,212,330,425)(361,415,480,398)(362,399,461,416)(363,417,462,400)(364,381,463,418)(365,419,464,382)(366,383,465,420)(367,401,466,384)(368,385,467,402)(369,403,468,386)(370,387,469,404)(371,405,470,388)(372,389,471,406)(373,407,472,390)(374,391,473,408)(375,409,474,392)(376,393,475,410)(377,411,476,394)(378,395,477,412)(379,413,478,396)(380,397,479,414), (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), (1,198,11,188)(2,197,12,187)(3,196,13,186)(4,195,14,185)(5,194,15,184)(6,193,16,183)(7,192,17,182)(8,191,18,181)(9,190,19,200)(10,189,20,199)(21,247,31,257)(22,246,32,256)(23,245,33,255)(24,244,34,254)(25,243,35,253)(26,242,36,252)(27,241,37,251)(28,260,38,250)(29,259,39,249)(30,258,40,248)(41,210,51,220)(42,209,52,219)(43,208,53,218)(44,207,54,217)(45,206,55,216)(46,205,56,215)(47,204,57,214)(48,203,58,213)(49,202,59,212)(50,201,60,211)(61,132,71,122)(62,131,72,121)(63,130,73,140)(64,129,74,139)(65,128,75,138)(66,127,76,137)(67,126,77,136)(68,125,78,135)(69,124,79,134)(70,123,80,133)(81,405,91,415)(82,404,92,414)(83,403,93,413)(84,402,94,412)(85,401,95,411)(86,420,96,410)(87,419,97,409)(88,418,98,408)(89,417,99,407)(90,416,100,406)(101,316,111,306)(102,315,112,305)(103,314,113,304)(104,313,114,303)(105,312,115,302)(106,311,116,301)(107,310,117,320)(108,309,118,319)(109,308,119,318)(110,307,120,317)(141,285,151,295)(142,284,152,294)(143,283,153,293)(144,282,154,292)(145,281,155,291)(146,300,156,290)(147,299,157,289)(148,298,158,288)(149,297,159,287)(150,296,160,286)(161,462,171,472)(162,461,172,471)(163,480,173,470)(164,479,174,469)(165,478,175,468)(166,477,176,467)(167,476,177,466)(168,475,178,465)(169,474,179,464)(170,473,180,463)(221,371,231,361)(222,370,232,380)(223,369,233,379)(224,368,234,378)(225,367,235,377)(226,366,236,376)(227,365,237,375)(228,364,238,374)(229,363,239,373)(230,362,240,372)(261,390,271,400)(262,389,272,399)(263,388,273,398)(264,387,274,397)(265,386,275,396)(266,385,276,395)(267,384,277,394)(268,383,278,393)(269,382,279,392)(270,381,280,391)(321,343,331,353)(322,342,332,352)(323,341,333,351)(324,360,334,350)(325,359,335,349)(326,358,336,348)(327,357,337,347)(328,356,338,346)(329,355,339,345)(330,354,340,344)(421,458,431,448)(422,457,432,447)(423,456,433,446)(424,455,434,445)(425,454,435,444)(426,453,436,443)(427,452,437,442)(428,451,438,441)(429,450,439,460)(430,449,440,459)>;
G:=Group( (1,241,401,346,80,384)(2,242,402,347,61,385)(3,243,403,348,62,386)(4,244,404,349,63,387)(5,245,405,350,64,388)(6,246,406,351,65,389)(7,247,407,352,66,390)(8,248,408,353,67,391)(9,249,409,354,68,392)(10,250,410,355,69,393)(11,251,411,356,70,394)(12,252,412,357,71,395)(13,253,413,358,72,396)(14,254,414,359,73,397)(15,255,415,360,74,398)(16,256,416,341,75,399)(17,257,417,342,76,400)(18,258,418,343,77,381)(19,259,419,344,78,382)(20,260,420,345,79,383)(21,182,261,137,332,99)(22,183,262,138,333,100)(23,184,263,139,334,81)(24,185,264,140,335,82)(25,186,265,121,336,83)(26,187,266,122,337,84)(27,188,267,123,338,85)(28,189,268,124,339,86)(29,190,269,125,340,87)(30,191,270,126,321,88)(31,192,271,127,322,89)(32,193,272,128,323,90)(33,194,273,129,324,91)(34,195,274,130,325,92)(35,196,275,131,326,93)(36,197,276,132,327,94)(37,198,277,133,328,95)(38,199,278,134,329,96)(39,200,279,135,330,97)(40,181,280,136,331,98)(41,367,159,446,466,118)(42,368,160,447,467,119)(43,369,141,448,468,120)(44,370,142,449,469,101)(45,371,143,450,470,102)(46,372,144,451,471,103)(47,373,145,452,472,104)(48,374,146,453,473,105)(49,375,147,454,474,106)(50,376,148,455,475,107)(51,377,149,456,476,108)(52,378,150,457,477,109)(53,379,151,458,478,110)(54,380,152,459,479,111)(55,361,153,460,480,112)(56,362,154,441,461,113)(57,363,155,442,462,114)(58,364,156,443,463,115)(59,365,157,444,464,116)(60,366,158,445,465,117)(161,437,281,229,204,313)(162,438,282,230,205,314)(163,439,283,231,206,315)(164,440,284,232,207,316)(165,421,285,233,208,317)(166,422,286,234,209,318)(167,423,287,235,210,319)(168,424,288,236,211,320)(169,425,289,237,212,301)(170,426,290,238,213,302)(171,427,291,239,214,303)(172,428,292,240,215,304)(173,429,293,221,216,305)(174,430,294,222,217,306)(175,431,295,223,218,307)(176,432,296,224,219,308)(177,433,297,225,220,309)(178,434,298,226,201,310)(179,435,299,227,202,311)(180,436,300,228,203,312), (1,41,346,446)(2,447,347,42)(3,43,348,448)(4,449,349,44)(5,45,350,450)(6,451,351,46)(7,47,352,452)(8,453,353,48)(9,49,354,454)(10,455,355,50)(11,51,356,456)(12,457,357,52)(13,53,358,458)(14,459,359,54)(15,55,360,460)(16,441,341,56)(17,57,342,442)(18,443,343,58)(19,59,344,444)(20,445,345,60)(21,303,137,291)(22,292,138,304)(23,305,139,293)(24,294,140,306)(25,307,121,295)(26,296,122,308)(27,309,123,297)(28,298,124,310)(29,311,125,299)(30,300,126,312)(31,313,127,281)(32,282,128,314)(33,315,129,283)(34,284,130,316)(35,317,131,285)(36,286,132,318)(37,319,133,287)(38,288,134,320)(39,301,135,289)(40,290,136,302)(61,119,242,160)(62,141,243,120)(63,101,244,142)(64,143,245,102)(65,103,246,144)(66,145,247,104)(67,105,248,146)(68,147,249,106)(69,107,250,148)(70,149,251,108)(71,109,252,150)(72,151,253,110)(73,111,254,152)(74,153,255,112)(75,113,256,154)(76,155,257,114)(77,115,258,156)(78,157,259,116)(79,117,260,158)(80,159,241,118)(81,173,263,221)(82,222,264,174)(83,175,265,223)(84,224,266,176)(85,177,267,225)(86,226,268,178)(87,179,269,227)(88,228,270,180)(89,161,271,229)(90,230,272,162)(91,163,273,231)(92,232,274,164)(93,165,275,233)(94,234,276,166)(95,167,277,235)(96,236,278,168)(97,169,279,237)(98,238,280,170)(99,171,261,239)(100,240,262,172)(181,426,331,213)(182,214,332,427)(183,428,333,215)(184,216,334,429)(185,430,335,217)(186,218,336,431)(187,432,337,219)(188,220,338,433)(189,434,339,201)(190,202,340,435)(191,436,321,203)(192,204,322,437)(193,438,323,205)(194,206,324,439)(195,440,325,207)(196,208,326,421)(197,422,327,209)(198,210,328,423)(199,424,329,211)(200,212,330,425)(361,415,480,398)(362,399,461,416)(363,417,462,400)(364,381,463,418)(365,419,464,382)(366,383,465,420)(367,401,466,384)(368,385,467,402)(369,403,468,386)(370,387,469,404)(371,405,470,388)(372,389,471,406)(373,407,472,390)(374,391,473,408)(375,409,474,392)(376,393,475,410)(377,411,476,394)(378,395,477,412)(379,413,478,396)(380,397,479,414), (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), (1,198,11,188)(2,197,12,187)(3,196,13,186)(4,195,14,185)(5,194,15,184)(6,193,16,183)(7,192,17,182)(8,191,18,181)(9,190,19,200)(10,189,20,199)(21,247,31,257)(22,246,32,256)(23,245,33,255)(24,244,34,254)(25,243,35,253)(26,242,36,252)(27,241,37,251)(28,260,38,250)(29,259,39,249)(30,258,40,248)(41,210,51,220)(42,209,52,219)(43,208,53,218)(44,207,54,217)(45,206,55,216)(46,205,56,215)(47,204,57,214)(48,203,58,213)(49,202,59,212)(50,201,60,211)(61,132,71,122)(62,131,72,121)(63,130,73,140)(64,129,74,139)(65,128,75,138)(66,127,76,137)(67,126,77,136)(68,125,78,135)(69,124,79,134)(70,123,80,133)(81,405,91,415)(82,404,92,414)(83,403,93,413)(84,402,94,412)(85,401,95,411)(86,420,96,410)(87,419,97,409)(88,418,98,408)(89,417,99,407)(90,416,100,406)(101,316,111,306)(102,315,112,305)(103,314,113,304)(104,313,114,303)(105,312,115,302)(106,311,116,301)(107,310,117,320)(108,309,118,319)(109,308,119,318)(110,307,120,317)(141,285,151,295)(142,284,152,294)(143,283,153,293)(144,282,154,292)(145,281,155,291)(146,300,156,290)(147,299,157,289)(148,298,158,288)(149,297,159,287)(150,296,160,286)(161,462,171,472)(162,461,172,471)(163,480,173,470)(164,479,174,469)(165,478,175,468)(166,477,176,467)(167,476,177,466)(168,475,178,465)(169,474,179,464)(170,473,180,463)(221,371,231,361)(222,370,232,380)(223,369,233,379)(224,368,234,378)(225,367,235,377)(226,366,236,376)(227,365,237,375)(228,364,238,374)(229,363,239,373)(230,362,240,372)(261,390,271,400)(262,389,272,399)(263,388,273,398)(264,387,274,397)(265,386,275,396)(266,385,276,395)(267,384,277,394)(268,383,278,393)(269,382,279,392)(270,381,280,391)(321,343,331,353)(322,342,332,352)(323,341,333,351)(324,360,334,350)(325,359,335,349)(326,358,336,348)(327,357,337,347)(328,356,338,346)(329,355,339,345)(330,354,340,344)(421,458,431,448)(422,457,432,447)(423,456,433,446)(424,455,434,445)(425,454,435,444)(426,453,436,443)(427,452,437,442)(428,451,438,441)(429,450,439,460)(430,449,440,459) );
G=PermutationGroup([[(1,241,401,346,80,384),(2,242,402,347,61,385),(3,243,403,348,62,386),(4,244,404,349,63,387),(5,245,405,350,64,388),(6,246,406,351,65,389),(7,247,407,352,66,390),(8,248,408,353,67,391),(9,249,409,354,68,392),(10,250,410,355,69,393),(11,251,411,356,70,394),(12,252,412,357,71,395),(13,253,413,358,72,396),(14,254,414,359,73,397),(15,255,415,360,74,398),(16,256,416,341,75,399),(17,257,417,342,76,400),(18,258,418,343,77,381),(19,259,419,344,78,382),(20,260,420,345,79,383),(21,182,261,137,332,99),(22,183,262,138,333,100),(23,184,263,139,334,81),(24,185,264,140,335,82),(25,186,265,121,336,83),(26,187,266,122,337,84),(27,188,267,123,338,85),(28,189,268,124,339,86),(29,190,269,125,340,87),(30,191,270,126,321,88),(31,192,271,127,322,89),(32,193,272,128,323,90),(33,194,273,129,324,91),(34,195,274,130,325,92),(35,196,275,131,326,93),(36,197,276,132,327,94),(37,198,277,133,328,95),(38,199,278,134,329,96),(39,200,279,135,330,97),(40,181,280,136,331,98),(41,367,159,446,466,118),(42,368,160,447,467,119),(43,369,141,448,468,120),(44,370,142,449,469,101),(45,371,143,450,470,102),(46,372,144,451,471,103),(47,373,145,452,472,104),(48,374,146,453,473,105),(49,375,147,454,474,106),(50,376,148,455,475,107),(51,377,149,456,476,108),(52,378,150,457,477,109),(53,379,151,458,478,110),(54,380,152,459,479,111),(55,361,153,460,480,112),(56,362,154,441,461,113),(57,363,155,442,462,114),(58,364,156,443,463,115),(59,365,157,444,464,116),(60,366,158,445,465,117),(161,437,281,229,204,313),(162,438,282,230,205,314),(163,439,283,231,206,315),(164,440,284,232,207,316),(165,421,285,233,208,317),(166,422,286,234,209,318),(167,423,287,235,210,319),(168,424,288,236,211,320),(169,425,289,237,212,301),(170,426,290,238,213,302),(171,427,291,239,214,303),(172,428,292,240,215,304),(173,429,293,221,216,305),(174,430,294,222,217,306),(175,431,295,223,218,307),(176,432,296,224,219,308),(177,433,297,225,220,309),(178,434,298,226,201,310),(179,435,299,227,202,311),(180,436,300,228,203,312)], [(1,41,346,446),(2,447,347,42),(3,43,348,448),(4,449,349,44),(5,45,350,450),(6,451,351,46),(7,47,352,452),(8,453,353,48),(9,49,354,454),(10,455,355,50),(11,51,356,456),(12,457,357,52),(13,53,358,458),(14,459,359,54),(15,55,360,460),(16,441,341,56),(17,57,342,442),(18,443,343,58),(19,59,344,444),(20,445,345,60),(21,303,137,291),(22,292,138,304),(23,305,139,293),(24,294,140,306),(25,307,121,295),(26,296,122,308),(27,309,123,297),(28,298,124,310),(29,311,125,299),(30,300,126,312),(31,313,127,281),(32,282,128,314),(33,315,129,283),(34,284,130,316),(35,317,131,285),(36,286,132,318),(37,319,133,287),(38,288,134,320),(39,301,135,289),(40,290,136,302),(61,119,242,160),(62,141,243,120),(63,101,244,142),(64,143,245,102),(65,103,246,144),(66,145,247,104),(67,105,248,146),(68,147,249,106),(69,107,250,148),(70,149,251,108),(71,109,252,150),(72,151,253,110),(73,111,254,152),(74,153,255,112),(75,113,256,154),(76,155,257,114),(77,115,258,156),(78,157,259,116),(79,117,260,158),(80,159,241,118),(81,173,263,221),(82,222,264,174),(83,175,265,223),(84,224,266,176),(85,177,267,225),(86,226,268,178),(87,179,269,227),(88,228,270,180),(89,161,271,229),(90,230,272,162),(91,163,273,231),(92,232,274,164),(93,165,275,233),(94,234,276,166),(95,167,277,235),(96,236,278,168),(97,169,279,237),(98,238,280,170),(99,171,261,239),(100,240,262,172),(181,426,331,213),(182,214,332,427),(183,428,333,215),(184,216,334,429),(185,430,335,217),(186,218,336,431),(187,432,337,219),(188,220,338,433),(189,434,339,201),(190,202,340,435),(191,436,321,203),(192,204,322,437),(193,438,323,205),(194,206,324,439),(195,440,325,207),(196,208,326,421),(197,422,327,209),(198,210,328,423),(199,424,329,211),(200,212,330,425),(361,415,480,398),(362,399,461,416),(363,417,462,400),(364,381,463,418),(365,419,464,382),(366,383,465,420),(367,401,466,384),(368,385,467,402),(369,403,468,386),(370,387,469,404),(371,405,470,388),(372,389,471,406),(373,407,472,390),(374,391,473,408),(375,409,474,392),(376,393,475,410),(377,411,476,394),(378,395,477,412),(379,413,478,396),(380,397,479,414)], [(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)], [(1,198,11,188),(2,197,12,187),(3,196,13,186),(4,195,14,185),(5,194,15,184),(6,193,16,183),(7,192,17,182),(8,191,18,181),(9,190,19,200),(10,189,20,199),(21,247,31,257),(22,246,32,256),(23,245,33,255),(24,244,34,254),(25,243,35,253),(26,242,36,252),(27,241,37,251),(28,260,38,250),(29,259,39,249),(30,258,40,248),(41,210,51,220),(42,209,52,219),(43,208,53,218),(44,207,54,217),(45,206,55,216),(46,205,56,215),(47,204,57,214),(48,203,58,213),(49,202,59,212),(50,201,60,211),(61,132,71,122),(62,131,72,121),(63,130,73,140),(64,129,74,139),(65,128,75,138),(66,127,76,137),(67,126,77,136),(68,125,78,135),(69,124,79,134),(70,123,80,133),(81,405,91,415),(82,404,92,414),(83,403,93,413),(84,402,94,412),(85,401,95,411),(86,420,96,410),(87,419,97,409),(88,418,98,408),(89,417,99,407),(90,416,100,406),(101,316,111,306),(102,315,112,305),(103,314,113,304),(104,313,114,303),(105,312,115,302),(106,311,116,301),(107,310,117,320),(108,309,118,319),(109,308,119,318),(110,307,120,317),(141,285,151,295),(142,284,152,294),(143,283,153,293),(144,282,154,292),(145,281,155,291),(146,300,156,290),(147,299,157,289),(148,298,158,288),(149,297,159,287),(150,296,160,286),(161,462,171,472),(162,461,172,471),(163,480,173,470),(164,479,174,469),(165,478,175,468),(166,477,176,467),(167,476,177,466),(168,475,178,465),(169,474,179,464),(170,473,180,463),(221,371,231,361),(222,370,232,380),(223,369,233,379),(224,368,234,378),(225,367,235,377),(226,366,236,376),(227,365,237,375),(228,364,238,374),(229,363,239,373),(230,362,240,372),(261,390,271,400),(262,389,272,399),(263,388,273,398),(264,387,274,397),(265,386,275,396),(266,385,276,395),(267,384,277,394),(268,383,278,393),(269,382,279,392),(270,381,280,391),(321,343,331,353),(322,342,332,352),(323,341,333,351),(324,360,334,350),(325,359,335,349),(326,358,336,348),(327,357,337,347),(328,356,338,346),(329,355,339,345),(330,354,340,344),(421,458,431,448),(422,457,432,447),(423,456,433,446),(424,455,434,445),(425,454,435,444),(426,453,436,443),(427,452,437,442),(428,451,438,441),(429,450,439,460),(430,449,440,459)]])
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 | 1 | 2 | 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 | C2 | S3 | Q8 | Q8 | D4 | D5 | D6 | D6 | D10 | D10 | Dic6 | Dic10 | S3×D4 | S3×Q8 | S3×D5 | D4×D5 | Q8×D5 | C2×S3×D5 | D5×Dic6 | S3×Dic10 | D10⋊D6 |
| kernel | Dic3⋊Dic10 | Dic3×Dic5 | C30.Q8 | C6.Dic10 | C3×C10.D4 | C5×Dic3⋊C4 | C2×C15⋊Q8 | C2×Dic30 | C10.D4 | C5×Dic3 | C3×Dic5 | Dic15 | Dic3⋊C4 | C2×Dic5 | C2×C20 | C2×Dic3 | C2×C12 | Dic5 | Dic3 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic3⋊Dic10 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 10 |
| 0 | 0 | 0 | 0 | 35 | 36 |
| 52 | 0 | 0 | 0 | 0 | 0 |
| 40 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 15 |
| 0 | 0 | 0 | 0 | 46 | 38 |
| 38 | 22 | 0 | 0 | 0 | 0 |
| 37 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 10 |
| 0 | 0 | 0 | 0 | 35 | 36 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,60,0,0,0,0,0,0,0,25,35,0,0,0,0,10,36],[52,40,0,0,0,0,0,27,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,23,46,0,0,0,0,15,38],[38,37,0,0,0,0,22,23,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,25,35,0,0,0,0,10,36] >;
Dic3⋊Dic10 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes {\rm Dic}_{10} % in TeX
G:=Group("Dic3:Dic10"); // GroupNames label
G:=SmallGroup(480,404);
// by ID
G=gap.SmallGroup(480,404);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,254,219,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^20=1,b^2=a^3,d^2=c^10,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations