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## G = Dic5⋊Dic6order 480 = 25·3·5

### 1st semidirect product of Dic5 and Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic5⋊Dic6
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C30.Q8 — Dic5⋊Dic6
 Lower central C15 — C2×C30 — Dic5⋊Dic6
 Upper central C1 — C22 — C2×C4

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 [×2], C3, C4 [×2], C4 [×8], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×6], Q8 [×4], C10, C10 [×2], Dic3 [×4], C12 [×2], C12 [×4], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30, C30 [×2], C4⋊Q8, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C4⋊Dic3 [×4], C4×C12, C2×Dic6, C2×Dic6, C5×Dic3 [×2], C3×Dic5 [×4], Dic15 [×2], C60 [×2], C2×C30, C4×Dic5, C10.D4 [×4], C2×Dic10, Q8×C10, C122Q8, C6×Dic5 [×2], C5×Dic6 [×2], C10×Dic3 [×2], Dic30 [×2], C2×Dic15 [×2], C2×C60, Dic5⋊Q8, C30.Q8 [×4], C12×Dic5, C10×Dic6, C2×Dic30, Dic5⋊Dic6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D5, D6 [×3], C2×D4, C2×Q8 [×2], D10 [×3], Dic6 [×4], D12 [×2], C22×S3, C4⋊Q8, C5⋊D4 [×2], C22×D5, C2×Dic6 [×2], C2×D12, S3×D5, Q8×D5 [×2], C2×C5⋊D4, C122Q8, C5⋊D12 [×2], C2×S3×D5, Dic5⋊Q8, D5×Dic6 [×2], C2×C5⋊D12, Dic5⋊Dic6

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

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

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

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

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

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