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G = Dic5⋊6Dic6  order 480 = 25·3·5

2nd semidirect product of Dic5 and Dic6 acting through Inn(Dic5)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5⋊6Dic6, Dic30⋊17C4, C15⋊9(C4×Q8), C5⋊4(C4×Dic6), C6.23(Q8×D5), C20.56(C4×S3), (C3×Dic5)⋊5Q8, C12.35(C4×D5), C30.10(C2×Q8), C2.3(D5×Dic6), C60.118(C2×C4), C30.5(C4○D4), (C2×C20).107D6, C4⋊Dic3.14D5, (C4×Dic5).2S3, C10.5(C2×Dic6), (C2×C12).294D10, C3⋊2(Dic5⋊3Q8), (C2×C30).23C23, (C12×Dic5).2C2, Dic15⋊5C4.4C2, C10.47(C4○D12), C2.1(D12⋊5D5), C6.16(D4⋊2D5), C4.10(D30.C2), C30.114(C22×C4), (C2×C60).138C22, Dic15.25(C2×C4), (C2×Dic3).78D10, (C2×Dic30).18C2, (C2×Dic5).155D6, (Dic3×Dic5).10C2, (C10×Dic3).9C22, (C6×Dic5).176C22, (C2×Dic15).30C22, C6.43(C2×C4×D5), C10.76(S3×C2×C4), C22.27(C2×S3×D5), (C2×C4).151(S3×D5), (C5×C4⋊Dic3).4C2, C2.8(C2×D30.C2), (C2×C6).35(C22×D5), (C2×C10).35(C22×S3), SmallGroup(480,409)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — Dic5⋊6Dic6
Chief series
C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic5⋊6Dic6
Lower central
C15 — C30 — Dic5⋊6Dic6
Upper central
C1 — C22 — C2×C4

Generators and relations for Dic5⋊6Dic6
 G = < a,b,c,d | a10=c12=1, b2=a5, d2=c6, bab-1=dad-1=a-1, ac=ca, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 556 in 140 conjugacy classes, 64 normal (34 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×9], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×6], C12 [×2], C12 [×3], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×2], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4×Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3 [×2], C3×Dic5 [×2], C3×Dic5, Dic15 [×4], C60 [×2], C2×C30, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C5×C4⋊C4, C2×Dic10, C4×Dic6, C6×Dic5 [×2], C10×Dic3 [×2], Dic30 [×4], C2×Dic15 [×2], C2×C60, Dic5⋊3Q8, Dic3×Dic5 [×2], Dic15⋊5C4 [×2], C12×Dic5, C5×C4⋊Dic3, C2×Dic30, Dic5⋊6Dic6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D5, D6 [×3], C22×C4, C2×Q8, C4○D4, D10 [×3], Dic6 [×2], C4×S3 [×2], C22×S3, C4×Q8, C4×D5 [×2], C22×D5, C2×Dic6, S3×C2×C4, C4○D12, S3×D5, C2×C4×D5, D4⋊2D5, Q8×D5, C4×Dic6, D30.C2 [×2], C2×S3×D5, Dic5⋊3Q8, D5×Dic6, D12⋊5D5, C2×D30.C2, Dic5⋊6Dic6

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

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

(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)

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

72 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B6A6B6C10A···10F12A12B12C12D12E···12L15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444444444445566610···101212121212···1215152020202020···2030···3060···60
size111122255556666101030303030222222···2222210···1044444412···124···44···4

72 irreducible representations

dim11111112222222222224444444
type+++++++-+++++-+--++--
imageC1C2C2C2C2C2C4S3Q8D5D6D6C4○D4D10D10Dic6C4×S3C4×D5C4○D12S3×D5D4⋊2D5Q8×D5D30.C2C2×S3×D5D5×Dic6D12⋊5D5
kernelDic5⋊6Dic6Dic3×Dic5Dic15⋊5C4C12×Dic5C5×C4⋊Dic3C2×Dic30Dic30C4×Dic5C3×Dic5C4⋊Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5C20C12C10C2×C4C6C6C4C22C2C2
# reps12211181222124244842224244

Matrix representation of Dic5⋊6Dic6 ►in GL5(𝔽61)

600000
060000
006000
000601
0004218
,
110000
011000
001100
0005237
000449
,
600000
0153800
0233800
000600
000060
,
600000
0524300
052900
000924
0001752

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,42,0,0,0,1,18],[11,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,52,44,0,0,0,37,9],[60,0,0,0,0,0,15,23,0,0,0,38,38,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,52,52,0,0,0,43,9,0,0,0,0,0,9,17,0,0,0,24,52] >;
 

Dic5⋊6Dic6 in GAP, Magma, Sage, TeX 

{\rm Dic}_5\rtimes_6{\rm Dic}_6
 
% in TeX
 
G:=Group("Dic5:6Dic6");
 
// GroupNames label
 
G:=SmallGroup(480,409);
 
// by ID
 
G=gap.SmallGroup(480,409);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,219,100,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=c^6,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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