metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.71D4, C20.21D12, Dic5⋊3Dic6, C15⋊4(C4⋊Q8), C6.29(Q8×D5), (C3×Dic5)⋊6Q8, C5⋊3(C12⋊2Q8), C30.25(C2×Q8), (C2×C20).117D6, C10.59(C2×D12), C30.118(C2×D4), (C4×Dic5).3S3, (C2×Dic6).4D5, C2.13(D5×Dic6), (C2×C12).300D10, C3⋊1(Dic5⋊Q8), C12.61(C5⋊D4), C4.11(C5⋊D12), (C2×C30).66C23, (C10×Dic6).5C2, (C12×Dic5).3C2, C10.11(C2×Dic6), (C2×C60).144C22, (C2×Dic30).19C2, (C2×Dic3).19D10, (C2×Dic5).166D6, C30.Q8.10C2, (C6×Dic5).188C22, (C2×Dic15).61C22, (C10×Dic3).38C22, C6.13(C2×C5⋊D4), (C2×C4).157(S3×D5), C2.17(C2×C5⋊D12), C22.152(C2×S3×D5), (C2×C6).78(C22×D5), (C2×C10).78(C22×S3), SmallGroup(480,452)
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, dbd-1=a5b, dcd-1=c-1 >
Subgroups: 588 in 136 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C10, C10, 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, C30, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4⋊Dic3, C4×C12, C2×Dic6, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C2×Dic10, Q8×C10, C12⋊2Q8, C6×Dic5, C5×Dic6, C10×Dic3, Dic30, C2×Dic15, C2×C60, Dic5⋊Q8, C30.Q8, C12×Dic5, C10×Dic6, C2×Dic30, Dic5⋊Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, D10, Dic6, D12, C22×S3, C4⋊Q8, C5⋊D4, C22×D5, C2×Dic6, C2×D12, S3×D5, Q8×D5, C2×C5⋊D4, C12⋊2Q8, C5⋊D12, C2×S3×D5, Dic5⋊Q8, D5×Dic6, C2×C5⋊D12, Dic5⋊Dic6
►Regular action on 480 points
Generators in S480
(1 226 196 267 162 313 474 337 176 99)(2 227 197 268 163 314 475 338 177 100)(3 228 198 269 164 315 476 339 178 101)(4 217 199 270 165 316 477 340 179 102)(5 218 200 271 166 317 478 341 180 103)(6 219 201 272 167 318 479 342 169 104)(7 220 202 273 168 319 480 343 170 105)(8 221 203 274 157 320 469 344 171 106)(9 222 204 275 158 321 470 345 172 107)(10 223 193 276 159 322 471 346 173 108)(11 224 194 265 160 323 472 347 174 97)(12 225 195 266 161 324 473 348 175 98)(13 294 335 53 433 260 27 460 65 286)(14 295 336 54 434 261 28 461 66 287)(15 296 325 55 435 262 29 462 67 288)(16 297 326 56 436 263 30 463 68 277)(17 298 327 57 437 264 31 464 69 278)(18 299 328 58 438 253 32 465 70 279)(19 300 329 59 439 254 33 466 71 280)(20 289 330 60 440 255 34 467 72 281)(21 290 331 49 441 256 35 468 61 282)(22 291 332 50 442 257 36 457 62 283)(23 292 333 51 443 258 25 458 63 284)(24 293 334 52 444 259 26 459 64 285)(37 420 357 244 302 430 389 86 454 119)(38 409 358 245 303 431 390 87 455 120)(39 410 359 246 304 432 391 88 456 109)(40 411 360 247 305 421 392 89 445 110)(41 412 349 248 306 422 393 90 446 111)(42 413 350 249 307 423 394 91 447 112)(43 414 351 250 308 424 395 92 448 113)(44 415 352 251 309 425 396 93 449 114)(45 416 353 252 310 426 385 94 450 115)(46 417 354 241 311 427 386 95 451 116)(47 418 355 242 312 428 387 96 452 117)(48 419 356 243 301 429 388 85 453 118)(73 372 212 122 399 185 153 373 234 142)(74 361 213 123 400 186 154 374 235 143)(75 362 214 124 401 187 155 375 236 144)(76 363 215 125 402 188 156 376 237 133)(77 364 216 126 403 189 145 377 238 134)(78 365 205 127 404 190 146 378 239 135)(79 366 206 128 405 191 147 379 240 136)(80 367 207 129 406 192 148 380 229 137)(81 368 208 130 407 181 149 381 230 138)(82 369 209 131 408 182 150 382 231 139)(83 370 210 132 397 183 151 383 232 140)(84 371 211 121 398 184 152 384 233 141)
(1 279 313 438)(2 280 314 439)(3 281 315 440)(4 282 316 441)(5 283 317 442)(6 284 318 443)(7 285 319 444)(8 286 320 433)(9 287 321 434)(10 288 322 435)(11 277 323 436)(12 278 324 437)(13 157 260 106)(14 158 261 107)(15 159 262 108)(16 160 263 97)(17 161 264 98)(18 162 253 99)(19 163 254 100)(20 164 255 101)(21 165 256 102)(22 166 257 103)(23 167 258 104)(24 168 259 105)(25 169 292 272)(26 170 293 273)(27 171 294 274)(28 172 295 275)(29 173 296 276)(30 174 297 265)(31 175 298 266)(32 176 299 267)(33 177 300 268)(34 178 289 269)(35 179 290 270)(36 180 291 271)(37 183 430 83)(38 184 431 84)(39 185 432 73)(40 186 421 74)(41 187 422 75)(42 188 423 76)(43 189 424 77)(44 190 425 78)(45 191 426 79)(46 192 427 80)(47 181 428 81)(48 182 429 82)(49 217 61 477)(50 218 62 478)(51 219 63 479)(52 220 64 480)(53 221 65 469)(54 222 66 470)(55 223 67 471)(56 224 68 472)(57 225 69 473)(58 226 70 474)(59 227 71 475)(60 228 72 476)(85 231 356 131)(86 232 357 132)(87 233 358 121)(88 234 359 122)(89 235 360 123)(90 236 349 124)(91 237 350 125)(92 238 351 126)(93 239 352 127)(94 240 353 128)(95 229 354 129)(96 230 355 130)(109 153 304 372)(110 154 305 361)(111 155 306 362)(112 156 307 363)(113 145 308 364)(114 146 309 365)(115 147 310 366)(116 148 311 367)(117 149 312 368)(118 150 301 369)(119 151 302 370)(120 152 303 371)(133 413 402 394)(134 414 403 395)(135 415 404 396)(136 416 405 385)(137 417 406 386)(138 418 407 387)(139 419 408 388)(140 420 397 389)(141 409 398 390)(142 410 399 391)(143 411 400 392)(144 412 401 393)(193 462 346 325)(194 463 347 326)(195 464 348 327)(196 465 337 328)(197 466 338 329)(198 467 339 330)(199 468 340 331)(200 457 341 332)(201 458 342 333)(202 459 343 334)(203 460 344 335)(204 461 345 336)(205 449 378 251)(206 450 379 252)(207 451 380 241)(208 452 381 242)(209 453 382 243)(210 454 383 244)(211 455 384 245)(212 456 373 246)(213 445 374 247)(214 446 375 248)(215 447 376 249)(216 448 377 250)
(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)
(1 305 7 311)(2 304 8 310)(3 303 9 309)(4 302 10 308)(5 301 11 307)(6 312 12 306)(13 379 19 373)(14 378 20 384)(15 377 21 383)(16 376 22 382)(17 375 23 381)(18 374 24 380)(25 130 31 124)(26 129 32 123)(27 128 33 122)(28 127 34 121)(29 126 35 132)(30 125 36 131)(37 471 43 477)(38 470 44 476)(39 469 45 475)(40 480 46 474)(41 479 47 473)(42 478 48 472)(49 83 55 77)(50 82 56 76)(51 81 57 75)(52 80 58 74)(53 79 59 73)(54 78 60 84)(61 183 67 189)(62 182 68 188)(63 181 69 187)(64 192 70 186)(65 191 71 185)(66 190 72 184)(85 265 91 271)(86 276 92 270)(87 275 93 269)(88 274 94 268)(89 273 95 267)(90 272 96 266)(97 249 103 243)(98 248 104 242)(99 247 105 241)(100 246 106 252)(101 245 107 251)(102 244 108 250)(109 320 115 314)(110 319 116 313)(111 318 117 324)(112 317 118 323)(113 316 119 322)(114 315 120 321)(133 332 139 326)(134 331 140 325)(135 330 141 336)(136 329 142 335)(137 328 143 334)(138 327 144 333)(145 282 151 288)(146 281 152 287)(147 280 153 286)(148 279 154 285)(149 278 155 284)(150 277 156 283)(157 450 163 456)(158 449 164 455)(159 448 165 454)(160 447 166 453)(161 446 167 452)(162 445 168 451)(169 355 175 349)(170 354 176 360)(171 353 177 359)(172 352 178 358)(173 351 179 357)(174 350 180 356)(193 395 199 389)(194 394 200 388)(195 393 201 387)(196 392 202 386)(197 391 203 385)(198 390 204 396)(205 255 211 261)(206 254 212 260)(207 253 213 259)(208 264 214 258)(209 263 215 257)(210 262 216 256)(217 430 223 424)(218 429 224 423)(219 428 225 422)(220 427 226 421)(221 426 227 432)(222 425 228 431)(229 299 235 293)(230 298 236 292)(231 297 237 291)(232 296 238 290)(233 295 239 289)(234 294 240 300)(337 411 343 417)(338 410 344 416)(339 409 345 415)(340 420 346 414)(341 419 347 413)(342 418 348 412)(361 444 367 438)(362 443 368 437)(363 442 369 436)(364 441 370 435)(365 440 371 434)(366 439 372 433)(397 462 403 468)(398 461 404 467)(399 460 405 466)(400 459 406 465)(401 458 407 464)(402 457 408 463)
G:=sub<Sym(480)| (1,226,196,267,162,313,474,337,176,99)(2,227,197,268,163,314,475,338,177,100)(3,228,198,269,164,315,476,339,178,101)(4,217,199,270,165,316,477,340,179,102)(5,218,200,271,166,317,478,341,180,103)(6,219,201,272,167,318,479,342,169,104)(7,220,202,273,168,319,480,343,170,105)(8,221,203,274,157,320,469,344,171,106)(9,222,204,275,158,321,470,345,172,107)(10,223,193,276,159,322,471,346,173,108)(11,224,194,265,160,323,472,347,174,97)(12,225,195,266,161,324,473,348,175,98)(13,294,335,53,433,260,27,460,65,286)(14,295,336,54,434,261,28,461,66,287)(15,296,325,55,435,262,29,462,67,288)(16,297,326,56,436,263,30,463,68,277)(17,298,327,57,437,264,31,464,69,278)(18,299,328,58,438,253,32,465,70,279)(19,300,329,59,439,254,33,466,71,280)(20,289,330,60,440,255,34,467,72,281)(21,290,331,49,441,256,35,468,61,282)(22,291,332,50,442,257,36,457,62,283)(23,292,333,51,443,258,25,458,63,284)(24,293,334,52,444,259,26,459,64,285)(37,420,357,244,302,430,389,86,454,119)(38,409,358,245,303,431,390,87,455,120)(39,410,359,246,304,432,391,88,456,109)(40,411,360,247,305,421,392,89,445,110)(41,412,349,248,306,422,393,90,446,111)(42,413,350,249,307,423,394,91,447,112)(43,414,351,250,308,424,395,92,448,113)(44,415,352,251,309,425,396,93,449,114)(45,416,353,252,310,426,385,94,450,115)(46,417,354,241,311,427,386,95,451,116)(47,418,355,242,312,428,387,96,452,117)(48,419,356,243,301,429,388,85,453,118)(73,372,212,122,399,185,153,373,234,142)(74,361,213,123,400,186,154,374,235,143)(75,362,214,124,401,187,155,375,236,144)(76,363,215,125,402,188,156,376,237,133)(77,364,216,126,403,189,145,377,238,134)(78,365,205,127,404,190,146,378,239,135)(79,366,206,128,405,191,147,379,240,136)(80,367,207,129,406,192,148,380,229,137)(81,368,208,130,407,181,149,381,230,138)(82,369,209,131,408,182,150,382,231,139)(83,370,210,132,397,183,151,383,232,140)(84,371,211,121,398,184,152,384,233,141), (1,279,313,438)(2,280,314,439)(3,281,315,440)(4,282,316,441)(5,283,317,442)(6,284,318,443)(7,285,319,444)(8,286,320,433)(9,287,321,434)(10,288,322,435)(11,277,323,436)(12,278,324,437)(13,157,260,106)(14,158,261,107)(15,159,262,108)(16,160,263,97)(17,161,264,98)(18,162,253,99)(19,163,254,100)(20,164,255,101)(21,165,256,102)(22,166,257,103)(23,167,258,104)(24,168,259,105)(25,169,292,272)(26,170,293,273)(27,171,294,274)(28,172,295,275)(29,173,296,276)(30,174,297,265)(31,175,298,266)(32,176,299,267)(33,177,300,268)(34,178,289,269)(35,179,290,270)(36,180,291,271)(37,183,430,83)(38,184,431,84)(39,185,432,73)(40,186,421,74)(41,187,422,75)(42,188,423,76)(43,189,424,77)(44,190,425,78)(45,191,426,79)(46,192,427,80)(47,181,428,81)(48,182,429,82)(49,217,61,477)(50,218,62,478)(51,219,63,479)(52,220,64,480)(53,221,65,469)(54,222,66,470)(55,223,67,471)(56,224,68,472)(57,225,69,473)(58,226,70,474)(59,227,71,475)(60,228,72,476)(85,231,356,131)(86,232,357,132)(87,233,358,121)(88,234,359,122)(89,235,360,123)(90,236,349,124)(91,237,350,125)(92,238,351,126)(93,239,352,127)(94,240,353,128)(95,229,354,129)(96,230,355,130)(109,153,304,372)(110,154,305,361)(111,155,306,362)(112,156,307,363)(113,145,308,364)(114,146,309,365)(115,147,310,366)(116,148,311,367)(117,149,312,368)(118,150,301,369)(119,151,302,370)(120,152,303,371)(133,413,402,394)(134,414,403,395)(135,415,404,396)(136,416,405,385)(137,417,406,386)(138,418,407,387)(139,419,408,388)(140,420,397,389)(141,409,398,390)(142,410,399,391)(143,411,400,392)(144,412,401,393)(193,462,346,325)(194,463,347,326)(195,464,348,327)(196,465,337,328)(197,466,338,329)(198,467,339,330)(199,468,340,331)(200,457,341,332)(201,458,342,333)(202,459,343,334)(203,460,344,335)(204,461,345,336)(205,449,378,251)(206,450,379,252)(207,451,380,241)(208,452,381,242)(209,453,382,243)(210,454,383,244)(211,455,384,245)(212,456,373,246)(213,445,374,247)(214,446,375,248)(215,447,376,249)(216,448,377,250), (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), (1,305,7,311)(2,304,8,310)(3,303,9,309)(4,302,10,308)(5,301,11,307)(6,312,12,306)(13,379,19,373)(14,378,20,384)(15,377,21,383)(16,376,22,382)(17,375,23,381)(18,374,24,380)(25,130,31,124)(26,129,32,123)(27,128,33,122)(28,127,34,121)(29,126,35,132)(30,125,36,131)(37,471,43,477)(38,470,44,476)(39,469,45,475)(40,480,46,474)(41,479,47,473)(42,478,48,472)(49,83,55,77)(50,82,56,76)(51,81,57,75)(52,80,58,74)(53,79,59,73)(54,78,60,84)(61,183,67,189)(62,182,68,188)(63,181,69,187)(64,192,70,186)(65,191,71,185)(66,190,72,184)(85,265,91,271)(86,276,92,270)(87,275,93,269)(88,274,94,268)(89,273,95,267)(90,272,96,266)(97,249,103,243)(98,248,104,242)(99,247,105,241)(100,246,106,252)(101,245,107,251)(102,244,108,250)(109,320,115,314)(110,319,116,313)(111,318,117,324)(112,317,118,323)(113,316,119,322)(114,315,120,321)(133,332,139,326)(134,331,140,325)(135,330,141,336)(136,329,142,335)(137,328,143,334)(138,327,144,333)(145,282,151,288)(146,281,152,287)(147,280,153,286)(148,279,154,285)(149,278,155,284)(150,277,156,283)(157,450,163,456)(158,449,164,455)(159,448,165,454)(160,447,166,453)(161,446,167,452)(162,445,168,451)(169,355,175,349)(170,354,176,360)(171,353,177,359)(172,352,178,358)(173,351,179,357)(174,350,180,356)(193,395,199,389)(194,394,200,388)(195,393,201,387)(196,392,202,386)(197,391,203,385)(198,390,204,396)(205,255,211,261)(206,254,212,260)(207,253,213,259)(208,264,214,258)(209,263,215,257)(210,262,216,256)(217,430,223,424)(218,429,224,423)(219,428,225,422)(220,427,226,421)(221,426,227,432)(222,425,228,431)(229,299,235,293)(230,298,236,292)(231,297,237,291)(232,296,238,290)(233,295,239,289)(234,294,240,300)(337,411,343,417)(338,410,344,416)(339,409,345,415)(340,420,346,414)(341,419,347,413)(342,418,348,412)(361,444,367,438)(362,443,368,437)(363,442,369,436)(364,441,370,435)(365,440,371,434)(366,439,372,433)(397,462,403,468)(398,461,404,467)(399,460,405,466)(400,459,406,465)(401,458,407,464)(402,457,408,463)>;
G:=Group( (1,226,196,267,162,313,474,337,176,99)(2,227,197,268,163,314,475,338,177,100)(3,228,198,269,164,315,476,339,178,101)(4,217,199,270,165,316,477,340,179,102)(5,218,200,271,166,317,478,341,180,103)(6,219,201,272,167,318,479,342,169,104)(7,220,202,273,168,319,480,343,170,105)(8,221,203,274,157,320,469,344,171,106)(9,222,204,275,158,321,470,345,172,107)(10,223,193,276,159,322,471,346,173,108)(11,224,194,265,160,323,472,347,174,97)(12,225,195,266,161,324,473,348,175,98)(13,294,335,53,433,260,27,460,65,286)(14,295,336,54,434,261,28,461,66,287)(15,296,325,55,435,262,29,462,67,288)(16,297,326,56,436,263,30,463,68,277)(17,298,327,57,437,264,31,464,69,278)(18,299,328,58,438,253,32,465,70,279)(19,300,329,59,439,254,33,466,71,280)(20,289,330,60,440,255,34,467,72,281)(21,290,331,49,441,256,35,468,61,282)(22,291,332,50,442,257,36,457,62,283)(23,292,333,51,443,258,25,458,63,284)(24,293,334,52,444,259,26,459,64,285)(37,420,357,244,302,430,389,86,454,119)(38,409,358,245,303,431,390,87,455,120)(39,410,359,246,304,432,391,88,456,109)(40,411,360,247,305,421,392,89,445,110)(41,412,349,248,306,422,393,90,446,111)(42,413,350,249,307,423,394,91,447,112)(43,414,351,250,308,424,395,92,448,113)(44,415,352,251,309,425,396,93,449,114)(45,416,353,252,310,426,385,94,450,115)(46,417,354,241,311,427,386,95,451,116)(47,418,355,242,312,428,387,96,452,117)(48,419,356,243,301,429,388,85,453,118)(73,372,212,122,399,185,153,373,234,142)(74,361,213,123,400,186,154,374,235,143)(75,362,214,124,401,187,155,375,236,144)(76,363,215,125,402,188,156,376,237,133)(77,364,216,126,403,189,145,377,238,134)(78,365,205,127,404,190,146,378,239,135)(79,366,206,128,405,191,147,379,240,136)(80,367,207,129,406,192,148,380,229,137)(81,368,208,130,407,181,149,381,230,138)(82,369,209,131,408,182,150,382,231,139)(83,370,210,132,397,183,151,383,232,140)(84,371,211,121,398,184,152,384,233,141), (1,279,313,438)(2,280,314,439)(3,281,315,440)(4,282,316,441)(5,283,317,442)(6,284,318,443)(7,285,319,444)(8,286,320,433)(9,287,321,434)(10,288,322,435)(11,277,323,436)(12,278,324,437)(13,157,260,106)(14,158,261,107)(15,159,262,108)(16,160,263,97)(17,161,264,98)(18,162,253,99)(19,163,254,100)(20,164,255,101)(21,165,256,102)(22,166,257,103)(23,167,258,104)(24,168,259,105)(25,169,292,272)(26,170,293,273)(27,171,294,274)(28,172,295,275)(29,173,296,276)(30,174,297,265)(31,175,298,266)(32,176,299,267)(33,177,300,268)(34,178,289,269)(35,179,290,270)(36,180,291,271)(37,183,430,83)(38,184,431,84)(39,185,432,73)(40,186,421,74)(41,187,422,75)(42,188,423,76)(43,189,424,77)(44,190,425,78)(45,191,426,79)(46,192,427,80)(47,181,428,81)(48,182,429,82)(49,217,61,477)(50,218,62,478)(51,219,63,479)(52,220,64,480)(53,221,65,469)(54,222,66,470)(55,223,67,471)(56,224,68,472)(57,225,69,473)(58,226,70,474)(59,227,71,475)(60,228,72,476)(85,231,356,131)(86,232,357,132)(87,233,358,121)(88,234,359,122)(89,235,360,123)(90,236,349,124)(91,237,350,125)(92,238,351,126)(93,239,352,127)(94,240,353,128)(95,229,354,129)(96,230,355,130)(109,153,304,372)(110,154,305,361)(111,155,306,362)(112,156,307,363)(113,145,308,364)(114,146,309,365)(115,147,310,366)(116,148,311,367)(117,149,312,368)(118,150,301,369)(119,151,302,370)(120,152,303,371)(133,413,402,394)(134,414,403,395)(135,415,404,396)(136,416,405,385)(137,417,406,386)(138,418,407,387)(139,419,408,388)(140,420,397,389)(141,409,398,390)(142,410,399,391)(143,411,400,392)(144,412,401,393)(193,462,346,325)(194,463,347,326)(195,464,348,327)(196,465,337,328)(197,466,338,329)(198,467,339,330)(199,468,340,331)(200,457,341,332)(201,458,342,333)(202,459,343,334)(203,460,344,335)(204,461,345,336)(205,449,378,251)(206,450,379,252)(207,451,380,241)(208,452,381,242)(209,453,382,243)(210,454,383,244)(211,455,384,245)(212,456,373,246)(213,445,374,247)(214,446,375,248)(215,447,376,249)(216,448,377,250), (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), (1,305,7,311)(2,304,8,310)(3,303,9,309)(4,302,10,308)(5,301,11,307)(6,312,12,306)(13,379,19,373)(14,378,20,384)(15,377,21,383)(16,376,22,382)(17,375,23,381)(18,374,24,380)(25,130,31,124)(26,129,32,123)(27,128,33,122)(28,127,34,121)(29,126,35,132)(30,125,36,131)(37,471,43,477)(38,470,44,476)(39,469,45,475)(40,480,46,474)(41,479,47,473)(42,478,48,472)(49,83,55,77)(50,82,56,76)(51,81,57,75)(52,80,58,74)(53,79,59,73)(54,78,60,84)(61,183,67,189)(62,182,68,188)(63,181,69,187)(64,192,70,186)(65,191,71,185)(66,190,72,184)(85,265,91,271)(86,276,92,270)(87,275,93,269)(88,274,94,268)(89,273,95,267)(90,272,96,266)(97,249,103,243)(98,248,104,242)(99,247,105,241)(100,246,106,252)(101,245,107,251)(102,244,108,250)(109,320,115,314)(110,319,116,313)(111,318,117,324)(112,317,118,323)(113,316,119,322)(114,315,120,321)(133,332,139,326)(134,331,140,325)(135,330,141,336)(136,329,142,335)(137,328,143,334)(138,327,144,333)(145,282,151,288)(146,281,152,287)(147,280,153,286)(148,279,154,285)(149,278,155,284)(150,277,156,283)(157,450,163,456)(158,449,164,455)(159,448,165,454)(160,447,166,453)(161,446,167,452)(162,445,168,451)(169,355,175,349)(170,354,176,360)(171,353,177,359)(172,352,178,358)(173,351,179,357)(174,350,180,356)(193,395,199,389)(194,394,200,388)(195,393,201,387)(196,392,202,386)(197,391,203,385)(198,390,204,396)(205,255,211,261)(206,254,212,260)(207,253,213,259)(208,264,214,258)(209,263,215,257)(210,262,216,256)(217,430,223,424)(218,429,224,423)(219,428,225,422)(220,427,226,421)(221,426,227,432)(222,425,228,431)(229,299,235,293)(230,298,236,292)(231,297,237,291)(232,296,238,290)(233,295,239,289)(234,294,240,300)(337,411,343,417)(338,410,344,416)(339,409,345,415)(340,420,346,414)(341,419,347,413)(342,418,348,412)(361,444,367,438)(362,443,368,437)(363,442,369,436)(364,441,370,435)(365,440,371,434)(366,439,372,433)(397,462,403,468)(398,461,404,467)(399,460,405,466)(400,459,406,465)(401,458,407,464)(402,457,408,463) );
G=PermutationGroup([[(1,226,196,267,162,313,474,337,176,99),(2,227,197,268,163,314,475,338,177,100),(3,228,198,269,164,315,476,339,178,101),(4,217,199,270,165,316,477,340,179,102),(5,218,200,271,166,317,478,341,180,103),(6,219,201,272,167,318,479,342,169,104),(7,220,202,273,168,319,480,343,170,105),(8,221,203,274,157,320,469,344,171,106),(9,222,204,275,158,321,470,345,172,107),(10,223,193,276,159,322,471,346,173,108),(11,224,194,265,160,323,472,347,174,97),(12,225,195,266,161,324,473,348,175,98),(13,294,335,53,433,260,27,460,65,286),(14,295,336,54,434,261,28,461,66,287),(15,296,325,55,435,262,29,462,67,288),(16,297,326,56,436,263,30,463,68,277),(17,298,327,57,437,264,31,464,69,278),(18,299,328,58,438,253,32,465,70,279),(19,300,329,59,439,254,33,466,71,280),(20,289,330,60,440,255,34,467,72,281),(21,290,331,49,441,256,35,468,61,282),(22,291,332,50,442,257,36,457,62,283),(23,292,333,51,443,258,25,458,63,284),(24,293,334,52,444,259,26,459,64,285),(37,420,357,244,302,430,389,86,454,119),(38,409,358,245,303,431,390,87,455,120),(39,410,359,246,304,432,391,88,456,109),(40,411,360,247,305,421,392,89,445,110),(41,412,349,248,306,422,393,90,446,111),(42,413,350,249,307,423,394,91,447,112),(43,414,351,250,308,424,395,92,448,113),(44,415,352,251,309,425,396,93,449,114),(45,416,353,252,310,426,385,94,450,115),(46,417,354,241,311,427,386,95,451,116),(47,418,355,242,312,428,387,96,452,117),(48,419,356,243,301,429,388,85,453,118),(73,372,212,122,399,185,153,373,234,142),(74,361,213,123,400,186,154,374,235,143),(75,362,214,124,401,187,155,375,236,144),(76,363,215,125,402,188,156,376,237,133),(77,364,216,126,403,189,145,377,238,134),(78,365,205,127,404,190,146,378,239,135),(79,366,206,128,405,191,147,379,240,136),(80,367,207,129,406,192,148,380,229,137),(81,368,208,130,407,181,149,381,230,138),(82,369,209,131,408,182,150,382,231,139),(83,370,210,132,397,183,151,383,232,140),(84,371,211,121,398,184,152,384,233,141)], [(1,279,313,438),(2,280,314,439),(3,281,315,440),(4,282,316,441),(5,283,317,442),(6,284,318,443),(7,285,319,444),(8,286,320,433),(9,287,321,434),(10,288,322,435),(11,277,323,436),(12,278,324,437),(13,157,260,106),(14,158,261,107),(15,159,262,108),(16,160,263,97),(17,161,264,98),(18,162,253,99),(19,163,254,100),(20,164,255,101),(21,165,256,102),(22,166,257,103),(23,167,258,104),(24,168,259,105),(25,169,292,272),(26,170,293,273),(27,171,294,274),(28,172,295,275),(29,173,296,276),(30,174,297,265),(31,175,298,266),(32,176,299,267),(33,177,300,268),(34,178,289,269),(35,179,290,270),(36,180,291,271),(37,183,430,83),(38,184,431,84),(39,185,432,73),(40,186,421,74),(41,187,422,75),(42,188,423,76),(43,189,424,77),(44,190,425,78),(45,191,426,79),(46,192,427,80),(47,181,428,81),(48,182,429,82),(49,217,61,477),(50,218,62,478),(51,219,63,479),(52,220,64,480),(53,221,65,469),(54,222,66,470),(55,223,67,471),(56,224,68,472),(57,225,69,473),(58,226,70,474),(59,227,71,475),(60,228,72,476),(85,231,356,131),(86,232,357,132),(87,233,358,121),(88,234,359,122),(89,235,360,123),(90,236,349,124),(91,237,350,125),(92,238,351,126),(93,239,352,127),(94,240,353,128),(95,229,354,129),(96,230,355,130),(109,153,304,372),(110,154,305,361),(111,155,306,362),(112,156,307,363),(113,145,308,364),(114,146,309,365),(115,147,310,366),(116,148,311,367),(117,149,312,368),(118,150,301,369),(119,151,302,370),(120,152,303,371),(133,413,402,394),(134,414,403,395),(135,415,404,396),(136,416,405,385),(137,417,406,386),(138,418,407,387),(139,419,408,388),(140,420,397,389),(141,409,398,390),(142,410,399,391),(143,411,400,392),(144,412,401,393),(193,462,346,325),(194,463,347,326),(195,464,348,327),(196,465,337,328),(197,466,338,329),(198,467,339,330),(199,468,340,331),(200,457,341,332),(201,458,342,333),(202,459,343,334),(203,460,344,335),(204,461,345,336),(205,449,378,251),(206,450,379,252),(207,451,380,241),(208,452,381,242),(209,453,382,243),(210,454,383,244),(211,455,384,245),(212,456,373,246),(213,445,374,247),(214,446,375,248),(215,447,376,249),(216,448,377,250)], [(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)], [(1,305,7,311),(2,304,8,310),(3,303,9,309),(4,302,10,308),(5,301,11,307),(6,312,12,306),(13,379,19,373),(14,378,20,384),(15,377,21,383),(16,376,22,382),(17,375,23,381),(18,374,24,380),(25,130,31,124),(26,129,32,123),(27,128,33,122),(28,127,34,121),(29,126,35,132),(30,125,36,131),(37,471,43,477),(38,470,44,476),(39,469,45,475),(40,480,46,474),(41,479,47,473),(42,478,48,472),(49,83,55,77),(50,82,56,76),(51,81,57,75),(52,80,58,74),(53,79,59,73),(54,78,60,84),(61,183,67,189),(62,182,68,188),(63,181,69,187),(64,192,70,186),(65,191,71,185),(66,190,72,184),(85,265,91,271),(86,276,92,270),(87,275,93,269),(88,274,94,268),(89,273,95,267),(90,272,96,266),(97,249,103,243),(98,248,104,242),(99,247,105,241),(100,246,106,252),(101,245,107,251),(102,244,108,250),(109,320,115,314),(110,319,116,313),(111,318,117,324),(112,317,118,323),(113,316,119,322),(114,315,120,321),(133,332,139,326),(134,331,140,325),(135,330,141,336),(136,329,142,335),(137,328,143,334),(138,327,144,333),(145,282,151,288),(146,281,152,287),(147,280,153,286),(148,279,154,285),(149,278,155,284),(150,277,156,283),(157,450,163,456),(158,449,164,455),(159,448,165,454),(160,447,166,453),(161,446,167,452),(162,445,168,451),(169,355,175,349),(170,354,176,360),(171,353,177,359),(172,352,178,358),(173,351,179,357),(174,350,180,356),(193,395,199,389),(194,394,200,388),(195,393,201,387),(196,392,202,386),(197,391,203,385),(198,390,204,396),(205,255,211,261),(206,254,212,260),(207,253,213,259),(208,264,214,258),(209,263,215,257),(210,262,216,256),(217,430,223,424),(218,429,224,423),(219,428,225,422),(220,427,226,421),(221,426,227,432),(222,425,228,431),(229,299,235,293),(230,298,236,292),(231,297,237,291),(232,296,238,290),(233,295,239,289),(234,294,240,300),(337,411,343,417),(338,410,344,416),(339,409,345,415),(340,420,346,414),(341,419,347,413),(342,418,348,412),(361,444,367,438),(362,443,368,437),(363,442,369,436),(364,441,370,435),(365,440,371,434),(366,439,372,433),(397,462,403,468),(398,461,404,467),(399,460,405,466),(400,459,406,465),(401,458,407,464),(402,457,408,463)]])
66 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 | ··· | 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 | 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 | 10 | 10 | 10 | 10 | 12 | 12 | 60 | 60 | 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 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D5 | D6 | D6 | D10 | D10 | Dic6 | D12 | C5⋊D4 | S3×D5 | Q8×D5 | C5⋊D12 | C2×S3×D5 | D5×Dic6 |
| kernel | Dic5⋊Dic6 | C30.Q8 | C12×Dic5 | C10×Dic6 | C2×Dic30 | C4×Dic5 | C3×Dic5 | C60 | C2×Dic6 | C2×Dic5 | C2×C20 | C2×Dic3 | C2×C12 | Dic5 | C20 | C12 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 4 | 2 | 8 | 4 | 8 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of Dic5⋊Dic6 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 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 | 45 | 17 |
| 23 | 46 | 0 | 0 | 0 | 0 |
| 15 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 58 |
| 0 | 0 | 0 | 0 | 29 | 56 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 17 | 53 | 0 | 0 | 0 | 0 |
| 36 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 53 | 0 | 0 |
| 0 | 0 | 53 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 44 |
| 0 | 0 | 0 | 0 | 33 | 14 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,45,0,0,0,0,60,17],[23,15,0,0,0,0,46,38,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,5,29,0,0,0,0,58,56],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,36,0,0,0,0,53,44,0,0,0,0,0,0,39,53,0,0,0,0,53,22,0,0,0,0,0,0,47,33,0,0,0,0,44,14] >;
Dic5⋊Dic6 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes {\rm Dic}_6 % in TeX
G:=Group("Dic5:Dic6"); // GroupNames label
G:=SmallGroup(480,452);
// by ID
G=gap.SmallGroup(480,452);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,120,219,100,1356,18822]);
// 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,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations