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## G = Dic5⋊5Dic6order 480 = 25·3·5

### 1st semidirect product of Dic5 and Dic6 acting through Inn(Dic5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic5⋊5Dic6
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic5⋊5Dic6
 Lower central C15 — C30 — Dic5⋊5Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic55Dic6
G = < a,b,c,d | a10=c12=1, b2=a5, d2=c6, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 556 in 140 conjugacy classes, 62 normal (44 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C2×C30, C4×Dic5, C4×Dic5, C10.D4, C5×C4⋊C4, C2×Dic10, C4×Dic6, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, Dic53Q8, Dic3×Dic5, C30.Q8, C12×Dic5, C5×Dic3⋊C4, C30.4Q8, C2×C15⋊Q8, Dic55Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D5, D6, C22×C4, C2×Q8, C4○D4, D10, Dic6, C4×S3, C22×S3, C4×Q8, C4×D5, C22×D5, C2×Dic6, S3×C2×C4, C4○D12, S3×D5, C2×C4×D5, D42D5, Q8×D5, C4×Dic6, C2×S3×D5, Dic53Q8, D5×Dic6, C4×S3×D5, Dic3.D10, Dic55Dic6

Smallest permutation representation of Dic55Dic6
Regular action on 480 points
Generators in S480
(1 441 255 377 38 458 450 96 389 57)(2 58 390 85 451 459 39 378 256 442)(3 443 257 379 40 460 452 86 391 59)(4 60 392 87 453 461 41 380 258 444)(5 433 259 381 42 462 454 88 393 49)(6 50 394 89 455 463 43 382 260 434)(7 435 261 383 44 464 456 90 395 51)(8 52 396 91 445 465 45 384 262 436)(9 437 263 373 46 466 446 92 385 53)(10 54 386 93 447 467 47 374 264 438)(11 439 253 375 48 468 448 94 387 55)(12 56 388 95 449 457 37 376 254 440)(13 337 325 132 304 372 479 419 104 400)(14 401 105 420 480 361 305 121 326 338)(15 339 327 122 306 362 469 409 106 402)(16 403 107 410 470 363 307 123 328 340)(17 341 329 124 308 364 471 411 108 404)(18 405 97 412 472 365 309 125 330 342)(19 343 331 126 310 366 473 413 98 406)(20 407 99 414 474 367 311 127 332 344)(21 345 333 128 312 368 475 415 100 408)(22 397 101 416 476 369 301 129 334 346)(23 347 335 130 302 370 477 417 102 398)(24 399 103 418 478 371 303 131 336 348)(25 133 145 316 204 240 119 350 228 289)(26 290 217 351 120 229 193 317 146 134)(27 135 147 318 194 230 109 352 218 291)(28 292 219 353 110 231 195 319 148 136)(29 137 149 320 196 232 111 354 220 293)(30 294 221 355 112 233 197 321 150 138)(31 139 151 322 198 234 113 356 222 295)(32 296 223 357 114 235 199 323 152 140)(33 141 153 324 200 236 115 358 224 297)(34 298 225 359 116 237 201 313 154 142)(35 143 155 314 202 238 117 360 226 299)(36 300 227 349 118 239 203 315 156 144)(61 179 246 74 422 213 167 182 268 286)(62 287 269 183 168 214 423 75 247 180)(63 169 248 76 424 215 157 184 270 288)(64 277 271 185 158 216 425 77 249 170)(65 171 250 78 426 205 159 186 272 278)(66 279 273 187 160 206 427 79 251 172)(67 173 252 80 428 207 161 188 274 280)(68 281 275 189 162 208 429 81 241 174)(69 175 242 82 430 209 163 190 276 282)(70 283 265 191 164 210 431 83 243 176)(71 177 244 84 432 211 165 192 266 284)(72 285 267 181 166 212 421 73 245 178)
(1 287 458 423)(2 288 459 424)(3 277 460 425)(4 278 461 426)(5 279 462 427)(6 280 463 428)(7 281 464 429)(8 282 465 430)(9 283 466 431)(10 284 467 432)(11 285 468 421)(12 286 457 422)(13 110 372 136)(14 111 361 137)(15 112 362 138)(16 113 363 139)(17 114 364 140)(18 115 365 141)(19 116 366 142)(20 117 367 143)(21 118 368 144)(22 119 369 133)(23 120 370 134)(24 109 371 135)(25 397 240 301)(26 398 229 302)(27 399 230 303)(28 400 231 304)(29 401 232 305)(30 402 233 306)(31 403 234 307)(32 404 235 308)(33 405 236 309)(34 406 237 310)(35 407 238 311)(36 408 239 312)(37 74 56 268)(38 75 57 269)(39 76 58 270)(40 77 59 271)(41 78 60 272)(42 79 49 273)(43 80 50 274)(44 81 51 275)(45 82 52 276)(46 83 53 265)(47 84 54 266)(48 73 55 267)(61 449 213 440)(62 450 214 441)(63 451 215 442)(64 452 216 443)(65 453 205 444)(66 454 206 433)(67 455 207 434)(68 456 208 435)(69 445 209 436)(70 446 210 437)(71 447 211 438)(72 448 212 439)(85 157 256 169)(86 158 257 170)(87 159 258 171)(88 160 259 172)(89 161 260 173)(90 162 261 174)(91 163 262 175)(92 164 263 176)(93 165 264 177)(94 166 253 178)(95 167 254 179)(96 168 255 180)(97 200 125 297)(98 201 126 298)(99 202 127 299)(100 203 128 300)(101 204 129 289)(102 193 130 290)(103 194 131 291)(104 195 132 292)(105 196 121 293)(106 197 122 294)(107 198 123 295)(108 199 124 296)(145 346 350 476)(146 347 351 477)(147 348 352 478)(148 337 353 479)(149 338 354 480)(150 339 355 469)(151 340 356 470)(152 341 357 471)(153 342 358 472)(154 343 359 473)(155 344 360 474)(156 345 349 475)(181 375 245 387)(182 376 246 388)(183 377 247 389)(184 378 248 390)(185 379 249 391)(186 380 250 392)(187 381 251 393)(188 382 252 394)(189 383 241 395)(190 384 242 396)(191 373 243 385)(192 374 244 386)(217 417 317 335)(218 418 318 336)(219 419 319 325)(220 420 320 326)(221 409 321 327)(222 410 322 328)(223 411 323 329)(224 412 324 330)(225 413 313 331)(226 414 314 332)(227 415 315 333)(228 416 316 334)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200 201 202 203 204)(205 206 207 208 209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224 225 226 227 228)(229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252)(253 254 255 256 257 258 259 260 261 262 263 264)(265 266 267 268 269 270 271 272 273 274 275 276)(277 278 279 280 281 282 283 284 285 286 287 288)(289 290 291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310 311 312)(313 314 315 316 317 318 319 320 321 322 323 324)(325 326 327 328 329 330 331 332 333 334 335 336)(337 338 339 340 341 342 343 344 345 346 347 348)(349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372)(373 374 375 376 377 378 379 380 381 382 383 384)(385 386 387 388 389 390 391 392 393 394 395 396)(397 398 399 400 401 402 403 404 405 406 407 408)(409 410 411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430 431 432)(433 434 435 436 437 438 439 440 441 442 443 444)(445 446 447 448 449 450 451 452 453 454 455 456)(457 458 459 460 461 462 463 464 465 466 467 468)(469 470 471 472 473 474 475 476 477 478 479 480)
(1 370 7 364)(2 369 8 363)(3 368 9 362)(4 367 10 361)(5 366 11 372)(6 365 12 371)(13 462 19 468)(14 461 20 467)(15 460 21 466)(16 459 22 465)(17 458 23 464)(18 457 24 463)(25 276 31 270)(26 275 32 269)(27 274 33 268)(28 273 34 267)(29 272 35 266)(30 271 36 265)(37 399 43 405)(38 398 44 404)(39 397 45 403)(40 408 46 402)(41 407 47 401)(42 406 48 400)(49 310 55 304)(50 309 56 303)(51 308 57 302)(52 307 58 301)(53 306 59 312)(54 305 60 311)(61 147 67 153)(62 146 68 152)(63 145 69 151)(64 156 70 150)(65 155 71 149)(66 154 72 148)(73 231 79 237)(74 230 80 236)(75 229 81 235)(76 240 82 234)(77 239 83 233)(78 238 84 232)(85 334 91 328)(86 333 92 327)(87 332 93 326)(88 331 94 325)(89 330 95 336)(90 329 96 335)(97 376 103 382)(98 375 104 381)(99 374 105 380)(100 373 106 379)(101 384 107 378)(102 383 108 377)(109 428 115 422)(110 427 116 421)(111 426 117 432)(112 425 118 431)(113 424 119 430)(114 423 120 429)(121 392 127 386)(122 391 128 385)(123 390 129 396)(124 389 130 395)(125 388 131 394)(126 387 132 393)(133 282 139 288)(134 281 140 287)(135 280 141 286)(136 279 142 285)(137 278 143 284)(138 277 144 283)(157 228 163 222)(158 227 164 221)(159 226 165 220)(160 225 166 219)(161 224 167 218)(162 223 168 217)(169 316 175 322)(170 315 176 321)(171 314 177 320)(172 313 178 319)(173 324 179 318)(174 323 180 317)(181 292 187 298)(182 291 188 297)(183 290 189 296)(184 289 190 295)(185 300 191 294)(186 299 192 293)(193 241 199 247)(194 252 200 246)(195 251 201 245)(196 250 202 244)(197 249 203 243)(198 248 204 242)(205 360 211 354)(206 359 212 353)(207 358 213 352)(208 357 214 351)(209 356 215 350)(210 355 216 349)(253 419 259 413)(254 418 260 412)(255 417 261 411)(256 416 262 410)(257 415 263 409)(258 414 264 420)(337 454 343 448)(338 453 344 447)(339 452 345 446)(340 451 346 445)(341 450 347 456)(342 449 348 455)(433 473 439 479)(434 472 440 478)(435 471 441 477)(436 470 442 476)(437 469 443 475)(438 480 444 474)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 12E ··· 12L 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 12 ··· 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 5 5 5 5 6 6 6 6 10 10 30 30 30 30 2 2 2 2 2 2 ··· 2 2 2 2 2 10 ··· 10 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + - + - - + - image C1 C2 C2 C2 C2 C2 C2 C4 S3 Q8 D5 D6 D6 C4○D4 D10 D10 Dic6 C4×S3 C4×D5 C4○D12 S3×D5 D4⋊2D5 Q8×D5 C2×S3×D5 D5×Dic6 C4×S3×D5 Dic3.D10 kernel Dic5⋊5Dic6 Dic3×Dic5 C30.Q8 C12×Dic5 C5×Dic3⋊C4 C30.4Q8 C2×C15⋊Q8 C15⋊Q8 C4×Dic5 C3×Dic5 Dic3⋊C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 Dic5 Dic5 Dic3 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 2 1 1 1 1 1 8 1 2 2 2 1 2 4 2 4 4 8 4 2 2 2 2 4 4 4

Matrix representation of Dic55Dic6 in GL4(𝔽61) generated by

 0 60 0 0 1 44 0 0 0 0 1 0 0 0 0 1
,
 14 22 0 0 16 47 0 0 0 0 1 0 0 0 0 1
,
 32 59 0 0 54 29 0 0 0 0 15 38 0 0 23 38
,
 60 0 0 0 0 60 0 0 0 0 50 0 0 0 11 11
G:=sub<GL(4,GF(61))| [0,1,0,0,60,44,0,0,0,0,1,0,0,0,0,1],[14,16,0,0,22,47,0,0,0,0,1,0,0,0,0,1],[32,54,0,0,59,29,0,0,0,0,15,23,0,0,38,38],[60,0,0,0,0,60,0,0,0,0,50,11,0,0,0,11] >;

Dic55Dic6 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_5{\rm Dic}_6
% in TeX

G:=Group("Dic5:5Dic6");
// GroupNames label

G:=SmallGroup(480,399);
// by ID

G=gap.SmallGroup(480,399);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,135,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=c^6,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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