direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5×Dic6, C15⋊8(C4×Q8), C3⋊1(Q8×Dic5), C5⋊6(C4×Dic6), C6.22(Q8×D5), C30.9(C2×Q8), C20.55(C4×S3), (C5×Dic6)⋊9C4, (C3×Dic5)⋊4Q8, C4.5(S3×Dic5), C2.2(D5×Dic6), C60.117(C2×C4), C30.4(C4○D4), (C2×C20).106D6, (C4×Dic5).1S3, C60⋊5C4.24C2, C10.4(C2×Dic6), C10.1(C4○D12), (C2×C12).293D10, (C2×C30).22C23, C6.1(Q8⋊2D5), (C2×Dic6).10D5, (C12×Dic5).1C2, (C10×Dic6).4C2, C12.20(C2×Dic5), C6.Dic10.4C2, C2.1(C12.28D10), C6.6(C22×Dic5), C30.113(C22×C4), (C2×C60).137C22, (C2×Dic3).77D10, (Dic3×Dic5).9C2, (C2×Dic5).154D6, Dic3.3(C2×Dic5), (C10×Dic3).8C22, (C2×Dic15).29C22, (C6×Dic5).175C22, C2.8(C2×S3×Dic5), C10.114(S3×C2×C4), C22.26(C2×S3×D5), (C2×C4).150(S3×D5), (C2×C6).34(C22×D5), (C2×C10).34(C22×S3), (C5×Dic3).25(C2×C4), SmallGroup(480,408)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5×Dic6
G = < a,b,c,d | a10=c12=1, b2=a5, d2=c6, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 492 in 140 conjugacy classes, 72 normal (34 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4×Q8, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C4×Dic5, C4⋊Dic5, Q8×C10, C4×Dic6, C6×Dic5, C5×Dic6, C10×Dic3, C2×Dic15, C2×C60, Q8×Dic5, Dic3×Dic5, C6.Dic10, C12×Dic5, C60⋊5C4, C10×Dic6, Dic5×Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D5, D6, C22×C4, C2×Q8, C4○D4, Dic5, D10, Dic6, C4×S3, C22×S3, C4×Q8, C2×Dic5, C22×D5, C2×Dic6, S3×C2×C4, C4○D12, S3×D5, Q8×D5, Q8⋊2D5, C22×Dic5, C4×Dic6, S3×Dic5, C2×S3×D5, Q8×Dic5, D5×Dic6, C12.28D10, C2×S3×Dic5, Dic5×Dic6
►Regular action on 480 points
Generators in S480
(1 385 225 117 425 239 92 257 135 63)(2 386 226 118 426 240 93 258 136 64)(3 387 227 119 427 229 94 259 137 65)(4 388 228 120 428 230 95 260 138 66)(5 389 217 109 429 231 96 261 139 67)(6 390 218 110 430 232 85 262 140 68)(7 391 219 111 431 233 86 263 141 69)(8 392 220 112 432 234 87 264 142 70)(9 393 221 113 421 235 88 253 143 71)(10 394 222 114 422 236 89 254 144 72)(11 395 223 115 423 237 90 255 133 61)(12 396 224 116 424 238 91 256 134 62)(13 304 203 411 344 372 399 292 444 174)(14 305 204 412 345 361 400 293 433 175)(15 306 193 413 346 362 401 294 434 176)(16 307 194 414 347 363 402 295 435 177)(17 308 195 415 348 364 403 296 436 178)(18 309 196 416 337 365 404 297 437 179)(19 310 197 417 338 366 405 298 438 180)(20 311 198 418 339 367 406 299 439 169)(21 312 199 419 340 368 407 300 440 170)(22 301 200 420 341 369 408 289 441 171)(23 302 201 409 342 370 397 290 442 172)(24 303 202 410 343 371 398 291 443 173)(25 446 48 463 375 76 267 243 213 360)(26 447 37 464 376 77 268 244 214 349)(27 448 38 465 377 78 269 245 215 350)(28 449 39 466 378 79 270 246 216 351)(29 450 40 467 379 80 271 247 205 352)(30 451 41 468 380 81 272 248 206 353)(31 452 42 457 381 82 273 249 207 354)(32 453 43 458 382 83 274 250 208 355)(33 454 44 459 383 84 275 251 209 356)(34 455 45 460 384 73 276 252 210 357)(35 456 46 461 373 74 265 241 211 358)(36 445 47 462 374 75 266 242 212 359)(49 185 283 166 330 322 124 145 101 469)(50 186 284 167 331 323 125 146 102 470)(51 187 285 168 332 324 126 147 103 471)(52 188 286 157 333 313 127 148 104 472)(53 189 287 158 334 314 128 149 105 473)(54 190 288 159 335 315 129 150 106 474)(55 191 277 160 336 316 130 151 107 475)(56 192 278 161 325 317 131 152 108 476)(57 181 279 162 326 318 132 153 97 477)(58 182 280 163 327 319 121 154 98 478)(59 183 281 164 328 320 122 155 99 479)(60 184 282 165 329 321 123 156 100 480)
(1 380 239 353)(2 381 240 354)(3 382 229 355)(4 383 230 356)(5 384 231 357)(6 373 232 358)(7 374 233 359)(8 375 234 360)(9 376 235 349)(10 377 236 350)(11 378 237 351)(12 379 238 352)(13 188 372 127)(14 189 361 128)(15 190 362 129)(16 191 363 130)(17 192 364 131)(18 181 365 132)(19 182 366 121)(20 183 367 122)(21 184 368 123)(22 185 369 124)(23 186 370 125)(24 187 371 126)(25 70 76 432)(26 71 77 421)(27 72 78 422)(28 61 79 423)(29 62 80 424)(30 63 81 425)(31 64 82 426)(32 65 83 427)(33 66 84 428)(34 67 73 429)(35 68 74 430)(36 69 75 431)(37 253 244 221)(38 254 245 222)(39 255 246 223)(40 256 247 224)(41 257 248 225)(42 258 249 226)(43 259 250 227)(44 260 251 228)(45 261 252 217)(46 262 241 218)(47 263 242 219)(48 264 243 220)(49 408 322 301)(50 397 323 302)(51 398 324 303)(52 399 313 304)(53 400 314 305)(54 401 315 306)(55 402 316 307)(56 403 317 308)(57 404 318 309)(58 405 319 310)(59 406 320 311)(60 407 321 312)(85 211 390 461)(86 212 391 462)(87 213 392 463)(88 214 393 464)(89 215 394 465)(90 216 395 466)(91 205 396 467)(92 206 385 468)(93 207 386 457)(94 208 387 458)(95 209 388 459)(96 210 389 460)(97 437 162 416)(98 438 163 417)(99 439 164 418)(100 440 165 419)(101 441 166 420)(102 442 167 409)(103 443 168 410)(104 444 157 411)(105 433 158 412)(106 434 159 413)(107 435 160 414)(108 436 161 415)(109 455 139 276)(110 456 140 265)(111 445 141 266)(112 446 142 267)(113 447 143 268)(114 448 144 269)(115 449 133 270)(116 450 134 271)(117 451 135 272)(118 452 136 273)(119 453 137 274)(120 454 138 275)(145 171 283 341)(146 172 284 342)(147 173 285 343)(148 174 286 344)(149 175 287 345)(150 176 288 346)(151 177 277 347)(152 178 278 348)(153 179 279 337)(154 180 280 338)(155 169 281 339)(156 170 282 340)(193 474 294 335)(194 475 295 336)(195 476 296 325)(196 477 297 326)(197 478 298 327)(198 479 299 328)(199 480 300 329)(200 469 289 330)(201 470 290 331)(202 471 291 332)(203 472 292 333)(204 473 293 334)
(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)
(1 199 7 193)(2 198 8 204)(3 197 9 203)(4 196 10 202)(5 195 11 201)(6 194 12 200)(13 137 19 143)(14 136 20 142)(15 135 21 141)(16 134 22 140)(17 133 23 139)(18 144 24 138)(25 314 31 320)(26 313 32 319)(27 324 33 318)(28 323 34 317)(29 322 35 316)(30 321 36 315)(37 148 43 154)(38 147 44 153)(39 146 45 152)(40 145 46 151)(41 156 47 150)(42 155 48 149)(49 74 55 80)(50 73 56 79)(51 84 57 78)(52 83 58 77)(53 82 59 76)(54 81 60 75)(61 302 67 308)(62 301 68 307)(63 312 69 306)(64 311 70 305)(65 310 71 304)(66 309 72 303)(85 435 91 441)(86 434 92 440)(87 433 93 439)(88 444 94 438)(89 443 95 437)(90 442 96 436)(97 465 103 459)(98 464 104 458)(99 463 105 457)(100 462 106 468)(101 461 107 467)(102 460 108 466)(109 364 115 370)(110 363 116 369)(111 362 117 368)(112 361 118 367)(113 372 119 366)(114 371 120 365)(121 447 127 453)(122 446 128 452)(123 445 129 451)(124 456 130 450)(125 455 131 449)(126 454 132 448)(157 208 163 214)(158 207 164 213)(159 206 165 212)(160 205 166 211)(161 216 167 210)(162 215 168 209)(169 264 175 258)(170 263 176 257)(171 262 177 256)(172 261 178 255)(173 260 179 254)(174 259 180 253)(181 269 187 275)(182 268 188 274)(183 267 189 273)(184 266 190 272)(185 265 191 271)(186 276 192 270)(217 348 223 342)(218 347 224 341)(219 346 225 340)(220 345 226 339)(221 344 227 338)(222 343 228 337)(229 298 235 292)(230 297 236 291)(231 296 237 290)(232 295 238 289)(233 294 239 300)(234 293 240 299)(241 277 247 283)(242 288 248 282)(243 287 249 281)(244 286 250 280)(245 285 251 279)(246 284 252 278)(325 351 331 357)(326 350 332 356)(327 349 333 355)(328 360 334 354)(329 359 335 353)(330 358 336 352)(373 475 379 469)(374 474 380 480)(375 473 381 479)(376 472 382 478)(377 471 383 477)(378 470 384 476)(385 419 391 413)(386 418 392 412)(387 417 393 411)(388 416 394 410)(389 415 395 409)(390 414 396 420)(397 429 403 423)(398 428 404 422)(399 427 405 421)(400 426 406 432)(401 425 407 431)(402 424 408 430)
G:=sub<Sym(480)| (1,385,225,117,425,239,92,257,135,63)(2,386,226,118,426,240,93,258,136,64)(3,387,227,119,427,229,94,259,137,65)(4,388,228,120,428,230,95,260,138,66)(5,389,217,109,429,231,96,261,139,67)(6,390,218,110,430,232,85,262,140,68)(7,391,219,111,431,233,86,263,141,69)(8,392,220,112,432,234,87,264,142,70)(9,393,221,113,421,235,88,253,143,71)(10,394,222,114,422,236,89,254,144,72)(11,395,223,115,423,237,90,255,133,61)(12,396,224,116,424,238,91,256,134,62)(13,304,203,411,344,372,399,292,444,174)(14,305,204,412,345,361,400,293,433,175)(15,306,193,413,346,362,401,294,434,176)(16,307,194,414,347,363,402,295,435,177)(17,308,195,415,348,364,403,296,436,178)(18,309,196,416,337,365,404,297,437,179)(19,310,197,417,338,366,405,298,438,180)(20,311,198,418,339,367,406,299,439,169)(21,312,199,419,340,368,407,300,440,170)(22,301,200,420,341,369,408,289,441,171)(23,302,201,409,342,370,397,290,442,172)(24,303,202,410,343,371,398,291,443,173)(25,446,48,463,375,76,267,243,213,360)(26,447,37,464,376,77,268,244,214,349)(27,448,38,465,377,78,269,245,215,350)(28,449,39,466,378,79,270,246,216,351)(29,450,40,467,379,80,271,247,205,352)(30,451,41,468,380,81,272,248,206,353)(31,452,42,457,381,82,273,249,207,354)(32,453,43,458,382,83,274,250,208,355)(33,454,44,459,383,84,275,251,209,356)(34,455,45,460,384,73,276,252,210,357)(35,456,46,461,373,74,265,241,211,358)(36,445,47,462,374,75,266,242,212,359)(49,185,283,166,330,322,124,145,101,469)(50,186,284,167,331,323,125,146,102,470)(51,187,285,168,332,324,126,147,103,471)(52,188,286,157,333,313,127,148,104,472)(53,189,287,158,334,314,128,149,105,473)(54,190,288,159,335,315,129,150,106,474)(55,191,277,160,336,316,130,151,107,475)(56,192,278,161,325,317,131,152,108,476)(57,181,279,162,326,318,132,153,97,477)(58,182,280,163,327,319,121,154,98,478)(59,183,281,164,328,320,122,155,99,479)(60,184,282,165,329,321,123,156,100,480), (1,380,239,353)(2,381,240,354)(3,382,229,355)(4,383,230,356)(5,384,231,357)(6,373,232,358)(7,374,233,359)(8,375,234,360)(9,376,235,349)(10,377,236,350)(11,378,237,351)(12,379,238,352)(13,188,372,127)(14,189,361,128)(15,190,362,129)(16,191,363,130)(17,192,364,131)(18,181,365,132)(19,182,366,121)(20,183,367,122)(21,184,368,123)(22,185,369,124)(23,186,370,125)(24,187,371,126)(25,70,76,432)(26,71,77,421)(27,72,78,422)(28,61,79,423)(29,62,80,424)(30,63,81,425)(31,64,82,426)(32,65,83,427)(33,66,84,428)(34,67,73,429)(35,68,74,430)(36,69,75,431)(37,253,244,221)(38,254,245,222)(39,255,246,223)(40,256,247,224)(41,257,248,225)(42,258,249,226)(43,259,250,227)(44,260,251,228)(45,261,252,217)(46,262,241,218)(47,263,242,219)(48,264,243,220)(49,408,322,301)(50,397,323,302)(51,398,324,303)(52,399,313,304)(53,400,314,305)(54,401,315,306)(55,402,316,307)(56,403,317,308)(57,404,318,309)(58,405,319,310)(59,406,320,311)(60,407,321,312)(85,211,390,461)(86,212,391,462)(87,213,392,463)(88,214,393,464)(89,215,394,465)(90,216,395,466)(91,205,396,467)(92,206,385,468)(93,207,386,457)(94,208,387,458)(95,209,388,459)(96,210,389,460)(97,437,162,416)(98,438,163,417)(99,439,164,418)(100,440,165,419)(101,441,166,420)(102,442,167,409)(103,443,168,410)(104,444,157,411)(105,433,158,412)(106,434,159,413)(107,435,160,414)(108,436,161,415)(109,455,139,276)(110,456,140,265)(111,445,141,266)(112,446,142,267)(113,447,143,268)(114,448,144,269)(115,449,133,270)(116,450,134,271)(117,451,135,272)(118,452,136,273)(119,453,137,274)(120,454,138,275)(145,171,283,341)(146,172,284,342)(147,173,285,343)(148,174,286,344)(149,175,287,345)(150,176,288,346)(151,177,277,347)(152,178,278,348)(153,179,279,337)(154,180,280,338)(155,169,281,339)(156,170,282,340)(193,474,294,335)(194,475,295,336)(195,476,296,325)(196,477,297,326)(197,478,298,327)(198,479,299,328)(199,480,300,329)(200,469,289,330)(201,470,290,331)(202,471,291,332)(203,472,292,333)(204,473,293,334), (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), (1,199,7,193)(2,198,8,204)(3,197,9,203)(4,196,10,202)(5,195,11,201)(6,194,12,200)(13,137,19,143)(14,136,20,142)(15,135,21,141)(16,134,22,140)(17,133,23,139)(18,144,24,138)(25,314,31,320)(26,313,32,319)(27,324,33,318)(28,323,34,317)(29,322,35,316)(30,321,36,315)(37,148,43,154)(38,147,44,153)(39,146,45,152)(40,145,46,151)(41,156,47,150)(42,155,48,149)(49,74,55,80)(50,73,56,79)(51,84,57,78)(52,83,58,77)(53,82,59,76)(54,81,60,75)(61,302,67,308)(62,301,68,307)(63,312,69,306)(64,311,70,305)(65,310,71,304)(66,309,72,303)(85,435,91,441)(86,434,92,440)(87,433,93,439)(88,444,94,438)(89,443,95,437)(90,442,96,436)(97,465,103,459)(98,464,104,458)(99,463,105,457)(100,462,106,468)(101,461,107,467)(102,460,108,466)(109,364,115,370)(110,363,116,369)(111,362,117,368)(112,361,118,367)(113,372,119,366)(114,371,120,365)(121,447,127,453)(122,446,128,452)(123,445,129,451)(124,456,130,450)(125,455,131,449)(126,454,132,448)(157,208,163,214)(158,207,164,213)(159,206,165,212)(160,205,166,211)(161,216,167,210)(162,215,168,209)(169,264,175,258)(170,263,176,257)(171,262,177,256)(172,261,178,255)(173,260,179,254)(174,259,180,253)(181,269,187,275)(182,268,188,274)(183,267,189,273)(184,266,190,272)(185,265,191,271)(186,276,192,270)(217,348,223,342)(218,347,224,341)(219,346,225,340)(220,345,226,339)(221,344,227,338)(222,343,228,337)(229,298,235,292)(230,297,236,291)(231,296,237,290)(232,295,238,289)(233,294,239,300)(234,293,240,299)(241,277,247,283)(242,288,248,282)(243,287,249,281)(244,286,250,280)(245,285,251,279)(246,284,252,278)(325,351,331,357)(326,350,332,356)(327,349,333,355)(328,360,334,354)(329,359,335,353)(330,358,336,352)(373,475,379,469)(374,474,380,480)(375,473,381,479)(376,472,382,478)(377,471,383,477)(378,470,384,476)(385,419,391,413)(386,418,392,412)(387,417,393,411)(388,416,394,410)(389,415,395,409)(390,414,396,420)(397,429,403,423)(398,428,404,422)(399,427,405,421)(400,426,406,432)(401,425,407,431)(402,424,408,430)>;
G:=Group( (1,385,225,117,425,239,92,257,135,63)(2,386,226,118,426,240,93,258,136,64)(3,387,227,119,427,229,94,259,137,65)(4,388,228,120,428,230,95,260,138,66)(5,389,217,109,429,231,96,261,139,67)(6,390,218,110,430,232,85,262,140,68)(7,391,219,111,431,233,86,263,141,69)(8,392,220,112,432,234,87,264,142,70)(9,393,221,113,421,235,88,253,143,71)(10,394,222,114,422,236,89,254,144,72)(11,395,223,115,423,237,90,255,133,61)(12,396,224,116,424,238,91,256,134,62)(13,304,203,411,344,372,399,292,444,174)(14,305,204,412,345,361,400,293,433,175)(15,306,193,413,346,362,401,294,434,176)(16,307,194,414,347,363,402,295,435,177)(17,308,195,415,348,364,403,296,436,178)(18,309,196,416,337,365,404,297,437,179)(19,310,197,417,338,366,405,298,438,180)(20,311,198,418,339,367,406,299,439,169)(21,312,199,419,340,368,407,300,440,170)(22,301,200,420,341,369,408,289,441,171)(23,302,201,409,342,370,397,290,442,172)(24,303,202,410,343,371,398,291,443,173)(25,446,48,463,375,76,267,243,213,360)(26,447,37,464,376,77,268,244,214,349)(27,448,38,465,377,78,269,245,215,350)(28,449,39,466,378,79,270,246,216,351)(29,450,40,467,379,80,271,247,205,352)(30,451,41,468,380,81,272,248,206,353)(31,452,42,457,381,82,273,249,207,354)(32,453,43,458,382,83,274,250,208,355)(33,454,44,459,383,84,275,251,209,356)(34,455,45,460,384,73,276,252,210,357)(35,456,46,461,373,74,265,241,211,358)(36,445,47,462,374,75,266,242,212,359)(49,185,283,166,330,322,124,145,101,469)(50,186,284,167,331,323,125,146,102,470)(51,187,285,168,332,324,126,147,103,471)(52,188,286,157,333,313,127,148,104,472)(53,189,287,158,334,314,128,149,105,473)(54,190,288,159,335,315,129,150,106,474)(55,191,277,160,336,316,130,151,107,475)(56,192,278,161,325,317,131,152,108,476)(57,181,279,162,326,318,132,153,97,477)(58,182,280,163,327,319,121,154,98,478)(59,183,281,164,328,320,122,155,99,479)(60,184,282,165,329,321,123,156,100,480), (1,380,239,353)(2,381,240,354)(3,382,229,355)(4,383,230,356)(5,384,231,357)(6,373,232,358)(7,374,233,359)(8,375,234,360)(9,376,235,349)(10,377,236,350)(11,378,237,351)(12,379,238,352)(13,188,372,127)(14,189,361,128)(15,190,362,129)(16,191,363,130)(17,192,364,131)(18,181,365,132)(19,182,366,121)(20,183,367,122)(21,184,368,123)(22,185,369,124)(23,186,370,125)(24,187,371,126)(25,70,76,432)(26,71,77,421)(27,72,78,422)(28,61,79,423)(29,62,80,424)(30,63,81,425)(31,64,82,426)(32,65,83,427)(33,66,84,428)(34,67,73,429)(35,68,74,430)(36,69,75,431)(37,253,244,221)(38,254,245,222)(39,255,246,223)(40,256,247,224)(41,257,248,225)(42,258,249,226)(43,259,250,227)(44,260,251,228)(45,261,252,217)(46,262,241,218)(47,263,242,219)(48,264,243,220)(49,408,322,301)(50,397,323,302)(51,398,324,303)(52,399,313,304)(53,400,314,305)(54,401,315,306)(55,402,316,307)(56,403,317,308)(57,404,318,309)(58,405,319,310)(59,406,320,311)(60,407,321,312)(85,211,390,461)(86,212,391,462)(87,213,392,463)(88,214,393,464)(89,215,394,465)(90,216,395,466)(91,205,396,467)(92,206,385,468)(93,207,386,457)(94,208,387,458)(95,209,388,459)(96,210,389,460)(97,437,162,416)(98,438,163,417)(99,439,164,418)(100,440,165,419)(101,441,166,420)(102,442,167,409)(103,443,168,410)(104,444,157,411)(105,433,158,412)(106,434,159,413)(107,435,160,414)(108,436,161,415)(109,455,139,276)(110,456,140,265)(111,445,141,266)(112,446,142,267)(113,447,143,268)(114,448,144,269)(115,449,133,270)(116,450,134,271)(117,451,135,272)(118,452,136,273)(119,453,137,274)(120,454,138,275)(145,171,283,341)(146,172,284,342)(147,173,285,343)(148,174,286,344)(149,175,287,345)(150,176,288,346)(151,177,277,347)(152,178,278,348)(153,179,279,337)(154,180,280,338)(155,169,281,339)(156,170,282,340)(193,474,294,335)(194,475,295,336)(195,476,296,325)(196,477,297,326)(197,478,298,327)(198,479,299,328)(199,480,300,329)(200,469,289,330)(201,470,290,331)(202,471,291,332)(203,472,292,333)(204,473,293,334), (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), (1,199,7,193)(2,198,8,204)(3,197,9,203)(4,196,10,202)(5,195,11,201)(6,194,12,200)(13,137,19,143)(14,136,20,142)(15,135,21,141)(16,134,22,140)(17,133,23,139)(18,144,24,138)(25,314,31,320)(26,313,32,319)(27,324,33,318)(28,323,34,317)(29,322,35,316)(30,321,36,315)(37,148,43,154)(38,147,44,153)(39,146,45,152)(40,145,46,151)(41,156,47,150)(42,155,48,149)(49,74,55,80)(50,73,56,79)(51,84,57,78)(52,83,58,77)(53,82,59,76)(54,81,60,75)(61,302,67,308)(62,301,68,307)(63,312,69,306)(64,311,70,305)(65,310,71,304)(66,309,72,303)(85,435,91,441)(86,434,92,440)(87,433,93,439)(88,444,94,438)(89,443,95,437)(90,442,96,436)(97,465,103,459)(98,464,104,458)(99,463,105,457)(100,462,106,468)(101,461,107,467)(102,460,108,466)(109,364,115,370)(110,363,116,369)(111,362,117,368)(112,361,118,367)(113,372,119,366)(114,371,120,365)(121,447,127,453)(122,446,128,452)(123,445,129,451)(124,456,130,450)(125,455,131,449)(126,454,132,448)(157,208,163,214)(158,207,164,213)(159,206,165,212)(160,205,166,211)(161,216,167,210)(162,215,168,209)(169,264,175,258)(170,263,176,257)(171,262,177,256)(172,261,178,255)(173,260,179,254)(174,259,180,253)(181,269,187,275)(182,268,188,274)(183,267,189,273)(184,266,190,272)(185,265,191,271)(186,276,192,270)(217,348,223,342)(218,347,224,341)(219,346,225,340)(220,345,226,339)(221,344,227,338)(222,343,228,337)(229,298,235,292)(230,297,236,291)(231,296,237,290)(232,295,238,289)(233,294,239,300)(234,293,240,299)(241,277,247,283)(242,288,248,282)(243,287,249,281)(244,286,250,280)(245,285,251,279)(246,284,252,278)(325,351,331,357)(326,350,332,356)(327,349,333,355)(328,360,334,354)(329,359,335,353)(330,358,336,352)(373,475,379,469)(374,474,380,480)(375,473,381,479)(376,472,382,478)(377,471,383,477)(378,470,384,476)(385,419,391,413)(386,418,392,412)(387,417,393,411)(388,416,394,410)(389,415,395,409)(390,414,396,420)(397,429,403,423)(398,428,404,422)(399,427,405,421)(400,426,406,432)(401,425,407,431)(402,424,408,430) );
G=PermutationGroup([[(1,385,225,117,425,239,92,257,135,63),(2,386,226,118,426,240,93,258,136,64),(3,387,227,119,427,229,94,259,137,65),(4,388,228,120,428,230,95,260,138,66),(5,389,217,109,429,231,96,261,139,67),(6,390,218,110,430,232,85,262,140,68),(7,391,219,111,431,233,86,263,141,69),(8,392,220,112,432,234,87,264,142,70),(9,393,221,113,421,235,88,253,143,71),(10,394,222,114,422,236,89,254,144,72),(11,395,223,115,423,237,90,255,133,61),(12,396,224,116,424,238,91,256,134,62),(13,304,203,411,344,372,399,292,444,174),(14,305,204,412,345,361,400,293,433,175),(15,306,193,413,346,362,401,294,434,176),(16,307,194,414,347,363,402,295,435,177),(17,308,195,415,348,364,403,296,436,178),(18,309,196,416,337,365,404,297,437,179),(19,310,197,417,338,366,405,298,438,180),(20,311,198,418,339,367,406,299,439,169),(21,312,199,419,340,368,407,300,440,170),(22,301,200,420,341,369,408,289,441,171),(23,302,201,409,342,370,397,290,442,172),(24,303,202,410,343,371,398,291,443,173),(25,446,48,463,375,76,267,243,213,360),(26,447,37,464,376,77,268,244,214,349),(27,448,38,465,377,78,269,245,215,350),(28,449,39,466,378,79,270,246,216,351),(29,450,40,467,379,80,271,247,205,352),(30,451,41,468,380,81,272,248,206,353),(31,452,42,457,381,82,273,249,207,354),(32,453,43,458,382,83,274,250,208,355),(33,454,44,459,383,84,275,251,209,356),(34,455,45,460,384,73,276,252,210,357),(35,456,46,461,373,74,265,241,211,358),(36,445,47,462,374,75,266,242,212,359),(49,185,283,166,330,322,124,145,101,469),(50,186,284,167,331,323,125,146,102,470),(51,187,285,168,332,324,126,147,103,471),(52,188,286,157,333,313,127,148,104,472),(53,189,287,158,334,314,128,149,105,473),(54,190,288,159,335,315,129,150,106,474),(55,191,277,160,336,316,130,151,107,475),(56,192,278,161,325,317,131,152,108,476),(57,181,279,162,326,318,132,153,97,477),(58,182,280,163,327,319,121,154,98,478),(59,183,281,164,328,320,122,155,99,479),(60,184,282,165,329,321,123,156,100,480)], [(1,380,239,353),(2,381,240,354),(3,382,229,355),(4,383,230,356),(5,384,231,357),(6,373,232,358),(7,374,233,359),(8,375,234,360),(9,376,235,349),(10,377,236,350),(11,378,237,351),(12,379,238,352),(13,188,372,127),(14,189,361,128),(15,190,362,129),(16,191,363,130),(17,192,364,131),(18,181,365,132),(19,182,366,121),(20,183,367,122),(21,184,368,123),(22,185,369,124),(23,186,370,125),(24,187,371,126),(25,70,76,432),(26,71,77,421),(27,72,78,422),(28,61,79,423),(29,62,80,424),(30,63,81,425),(31,64,82,426),(32,65,83,427),(33,66,84,428),(34,67,73,429),(35,68,74,430),(36,69,75,431),(37,253,244,221),(38,254,245,222),(39,255,246,223),(40,256,247,224),(41,257,248,225),(42,258,249,226),(43,259,250,227),(44,260,251,228),(45,261,252,217),(46,262,241,218),(47,263,242,219),(48,264,243,220),(49,408,322,301),(50,397,323,302),(51,398,324,303),(52,399,313,304),(53,400,314,305),(54,401,315,306),(55,402,316,307),(56,403,317,308),(57,404,318,309),(58,405,319,310),(59,406,320,311),(60,407,321,312),(85,211,390,461),(86,212,391,462),(87,213,392,463),(88,214,393,464),(89,215,394,465),(90,216,395,466),(91,205,396,467),(92,206,385,468),(93,207,386,457),(94,208,387,458),(95,209,388,459),(96,210,389,460),(97,437,162,416),(98,438,163,417),(99,439,164,418),(100,440,165,419),(101,441,166,420),(102,442,167,409),(103,443,168,410),(104,444,157,411),(105,433,158,412),(106,434,159,413),(107,435,160,414),(108,436,161,415),(109,455,139,276),(110,456,140,265),(111,445,141,266),(112,446,142,267),(113,447,143,268),(114,448,144,269),(115,449,133,270),(116,450,134,271),(117,451,135,272),(118,452,136,273),(119,453,137,274),(120,454,138,275),(145,171,283,341),(146,172,284,342),(147,173,285,343),(148,174,286,344),(149,175,287,345),(150,176,288,346),(151,177,277,347),(152,178,278,348),(153,179,279,337),(154,180,280,338),(155,169,281,339),(156,170,282,340),(193,474,294,335),(194,475,295,336),(195,476,296,325),(196,477,297,326),(197,478,298,327),(198,479,299,328),(199,480,300,329),(200,469,289,330),(201,470,290,331),(202,471,291,332),(203,472,292,333),(204,473,293,334)], [(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)], [(1,199,7,193),(2,198,8,204),(3,197,9,203),(4,196,10,202),(5,195,11,201),(6,194,12,200),(13,137,19,143),(14,136,20,142),(15,135,21,141),(16,134,22,140),(17,133,23,139),(18,144,24,138),(25,314,31,320),(26,313,32,319),(27,324,33,318),(28,323,34,317),(29,322,35,316),(30,321,36,315),(37,148,43,154),(38,147,44,153),(39,146,45,152),(40,145,46,151),(41,156,47,150),(42,155,48,149),(49,74,55,80),(50,73,56,79),(51,84,57,78),(52,83,58,77),(53,82,59,76),(54,81,60,75),(61,302,67,308),(62,301,68,307),(63,312,69,306),(64,311,70,305),(65,310,71,304),(66,309,72,303),(85,435,91,441),(86,434,92,440),(87,433,93,439),(88,444,94,438),(89,443,95,437),(90,442,96,436),(97,465,103,459),(98,464,104,458),(99,463,105,457),(100,462,106,468),(101,461,107,467),(102,460,108,466),(109,364,115,370),(110,363,116,369),(111,362,117,368),(112,361,118,367),(113,372,119,366),(114,371,120,365),(121,447,127,453),(122,446,128,452),(123,445,129,451),(124,456,130,450),(125,455,131,449),(126,454,132,448),(157,208,163,214),(158,207,164,213),(159,206,165,212),(160,205,166,211),(161,216,167,210),(162,215,168,209),(169,264,175,258),(170,263,176,257),(171,262,177,256),(172,261,178,255),(173,260,179,254),(174,259,180,253),(181,269,187,275),(182,268,188,274),(183,267,189,273),(184,266,190,272),(185,265,191,271),(186,276,192,270),(217,348,223,342),(218,347,224,341),(219,346,225,340),(220,345,226,339),(221,344,227,338),(222,343,228,337),(229,298,235,292),(230,297,236,291),(231,296,237,290),(232,295,238,289),(233,294,239,300),(234,293,240,299),(241,277,247,283),(242,288,248,282),(243,287,249,281),(244,286,250,280),(245,285,251,279),(246,284,252,278),(325,351,331,357),(326,350,332,356),(327,349,333,355),(328,360,334,354),(329,359,335,353),(330,358,336,352),(373,475,379,469),(374,474,380,480),(375,473,381,479),(376,472,382,478),(377,471,383,477),(378,470,384,476),(385,419,391,413),(386,418,392,412),(387,417,393,411),(388,416,394,410),(389,415,395,409),(390,414,396,420),(397,429,403,423),(398,428,404,422),(399,427,405,421),(400,426,406,432),(401,425,407,431),(402,424,408,430)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 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 | 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 | 2 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | + | + | - | + | - | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D5 | D6 | D6 | C4○D4 | Dic5 | D10 | D10 | Dic6 | C4×S3 | C4○D12 | S3×D5 | Q8×D5 | Q8⋊2D5 | S3×Dic5 | C2×S3×D5 | D5×Dic6 | C12.28D10 |
| kernel | Dic5×Dic6 | Dic3×Dic5 | C6.Dic10 | C12×Dic5 | C60⋊5C4 | C10×Dic6 | C5×Dic6 | C4×Dic5 | C3×Dic5 | C2×Dic6 | C2×Dic5 | C2×C20 | C30 | Dic6 | C2×Dic3 | C2×C12 | Dic5 | C20 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic5×Dic6 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 60 |
| 0 | 0 | 19 | 43 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 5 | 51 |
| 0 | 0 | 27 | 56 |
| 46 | 38 | 0 | 0 |
| 23 | 23 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 50 | 0 | 0 | 0 |
| 50 | 11 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,1,19,0,0,60,43],[60,0,0,0,0,60,0,0,0,0,5,27,0,0,51,56],[46,23,0,0,38,23,0,0,0,0,1,0,0,0,0,1],[50,50,0,0,0,11,0,0,0,0,1,0,0,0,0,1] >;
Dic5×Dic6 in GAP, Magma, Sage, TeX
{\rm Dic}_5\times {\rm Dic}_6 % in TeX
G:=Group("Dic5xDic6"); // GroupNames label
G:=SmallGroup(480,408);
// by ID
G=gap.SmallGroup(480,408);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=c^6,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations