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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 [×3], C3, C4 [×11], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×2], Dic3 [×4], C12 [×5], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×4], Dic5 [×3], C20 [×4], 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, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×4], Dic15 [×2], Dic15, C60, C2×C30, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C5×C4⋊C4, C2×Dic10, C4×Dic6, C15⋊Q8 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, Dic53Q8, Dic3×Dic5 [×2], C30.Q8, C12×Dic5, C5×Dic3⋊C4, C30.4Q8, C2×C15⋊Q8, Dic55Dic6
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, 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
(1 405 191 234 116 335 390 64 18 366)(2 367 19 65 391 336 117 235 192 406)(3 407 181 236 118 325 392 66 20 368)(4 369 21 67 393 326 119 237 182 408)(5 397 183 238 120 327 394 68 22 370)(6 371 23 69 395 328 109 239 184 398)(7 399 185 240 110 329 396 70 24 372)(8 361 13 71 385 330 111 229 186 400)(9 401 187 230 112 331 386 72 14 362)(10 363 15 61 387 332 113 231 188 402)(11 403 189 232 114 333 388 62 16 364)(12 365 17 63 389 334 115 233 190 404)(25 258 249 288 437 293 272 88 418 54)(26 55 419 89 273 294 438 277 250 259)(27 260 251 278 439 295 274 90 420 56)(28 57 409 91 275 296 440 279 252 261)(29 262 241 280 441 297 276 92 410 58)(30 59 411 93 265 298 442 281 242 263)(31 264 243 282 443 299 266 94 412 60)(32 49 413 95 267 300 444 283 244 253)(33 254 245 284 433 289 268 96 414 50)(34 51 415 85 269 290 434 285 246 255)(35 256 247 286 435 291 270 86 416 52)(36 53 417 87 271 292 436 287 248 257)(37 74 317 158 150 341 99 143 423 450)(38 451 424 144 100 342 151 159 318 75)(39 76 319 160 152 343 101 133 425 452)(40 453 426 134 102 344 153 161 320 77)(41 78 321 162 154 345 103 135 427 454)(42 455 428 136 104 346 155 163 322 79)(43 80 323 164 156 347 105 137 429 456)(44 445 430 138 106 348 145 165 324 81)(45 82 313 166 146 337 107 139 431 446)(46 447 432 140 108 338 147 167 314 83)(47 84 315 168 148 339 97 141 421 448)(48 449 422 142 98 340 149 157 316 73)(121 302 212 469 382 198 351 458 224 171)(122 172 225 459 352 199 383 470 213 303)(123 304 214 471 384 200 353 460 226 173)(124 174 227 461 354 201 373 472 215 305)(125 306 216 473 374 202 355 462 228 175)(126 176 217 463 356 203 375 474 205 307)(127 308 206 475 376 204 357 464 218 177)(128 178 219 465 358 193 377 476 207 309)(129 310 208 477 378 194 359 466 220 179)(130 180 221 467 360 195 379 478 209 311)(131 312 210 479 380 196 349 468 222 169)(132 170 223 457 350 197 381 480 211 301)
(1 458 335 212)(2 459 336 213)(3 460 325 214)(4 461 326 215)(5 462 327 216)(6 463 328 205)(7 464 329 206)(8 465 330 207)(9 466 331 208)(10 467 332 209)(11 468 333 210)(12 457 334 211)(13 178 229 377)(14 179 230 378)(15 180 231 379)(16 169 232 380)(17 170 233 381)(18 171 234 382)(19 172 235 383)(20 173 236 384)(21 174 237 373)(22 175 238 374)(23 176 239 375)(24 177 240 376)(25 429 293 164)(26 430 294 165)(27 431 295 166)(28 432 296 167)(29 421 297 168)(30 422 298 157)(31 423 299 158)(32 424 300 159)(33 425 289 160)(34 426 290 161)(35 427 291 162)(36 428 292 163)(37 282 341 412)(38 283 342 413)(39 284 343 414)(40 285 344 415)(41 286 345 416)(42 287 346 417)(43 288 347 418)(44 277 348 419)(45 278 337 420)(46 279 338 409)(47 280 339 410)(48 281 340 411)(49 451 444 151)(50 452 433 152)(51 453 434 153)(52 454 435 154)(53 455 436 155)(54 456 437 156)(55 445 438 145)(56 446 439 146)(57 447 440 147)(58 448 441 148)(59 449 442 149)(60 450 443 150)(61 130 188 195)(62 131 189 196)(63 132 190 197)(64 121 191 198)(65 122 192 199)(66 123 181 200)(67 124 182 201)(68 125 183 202)(69 126 184 203)(70 127 185 204)(71 128 186 193)(72 129 187 194)(73 242 98 93)(74 243 99 94)(75 244 100 95)(76 245 101 96)(77 246 102 85)(78 247 103 86)(79 248 104 87)(80 249 105 88)(81 250 106 89)(82 251 107 90)(83 252 108 91)(84 241 97 92)(109 474 371 217)(110 475 372 218)(111 476 361 219)(112 477 362 220)(113 478 363 221)(114 479 364 222)(115 480 365 223)(116 469 366 224)(117 470 367 225)(118 471 368 226)(119 472 369 227)(120 473 370 228)(133 268 319 254)(134 269 320 255)(135 270 321 256)(136 271 322 257)(137 272 323 258)(138 273 324 259)(139 274 313 260)(140 275 314 261)(141 276 315 262)(142 265 316 263)(143 266 317 264)(144 267 318 253)(301 404 350 389)(302 405 351 390)(303 406 352 391)(304 407 353 392)(305 408 354 393)(306 397 355 394)(307 398 356 395)(308 399 357 396)(309 400 358 385)(310 401 359 386)(311 402 360 387)(312 403 349 388)
(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)
(1 314 7 320)(2 313 8 319)(3 324 9 318)(4 323 10 317)(5 322 11 316)(6 321 12 315)(13 152 19 146)(14 151 20 145)(15 150 21 156)(16 149 22 155)(17 148 23 154)(18 147 24 153)(25 221 31 227)(26 220 32 226)(27 219 33 225)(28 218 34 224)(29 217 35 223)(30 228 36 222)(37 182 43 188)(38 181 44 187)(39 192 45 186)(40 191 46 185)(41 190 47 184)(42 189 48 183)(49 173 55 179)(50 172 56 178)(51 171 57 177)(52 170 58 176)(53 169 59 175)(54 180 60 174)(61 341 67 347)(62 340 68 346)(63 339 69 345)(64 338 70 344)(65 337 71 343)(66 348 72 342)(73 397 79 403)(74 408 80 402)(75 407 81 401)(76 406 82 400)(77 405 83 399)(78 404 84 398)(85 302 91 308)(86 301 92 307)(87 312 93 306)(88 311 94 305)(89 310 95 304)(90 309 96 303)(97 395 103 389)(98 394 104 388)(99 393 105 387)(100 392 106 386)(101 391 107 385)(102 390 108 396)(109 427 115 421)(110 426 116 432)(111 425 117 431)(112 424 118 430)(113 423 119 429)(114 422 120 428)(121 409 127 415)(122 420 128 414)(123 419 129 413)(124 418 130 412)(125 417 131 411)(126 416 132 410)(133 336 139 330)(134 335 140 329)(135 334 141 328)(136 333 142 327)(137 332 143 326)(138 331 144 325)(157 370 163 364)(158 369 164 363)(159 368 165 362)(160 367 166 361)(161 366 167 372)(162 365 168 371)(193 284 199 278)(194 283 200 277)(195 282 201 288)(196 281 202 287)(197 280 203 286)(198 279 204 285)(205 270 211 276)(206 269 212 275)(207 268 213 274)(208 267 214 273)(209 266 215 272)(210 265 216 271)(229 452 235 446)(230 451 236 445)(231 450 237 456)(232 449 238 455)(233 448 239 454)(234 447 240 453)(241 356 247 350)(242 355 248 349)(243 354 249 360)(244 353 250 359)(245 352 251 358)(246 351 252 357)(253 460 259 466)(254 459 260 465)(255 458 261 464)(256 457 262 463)(257 468 263 462)(258 467 264 461)(289 470 295 476)(290 469 296 475)(291 480 297 474)(292 479 298 473)(293 478 299 472)(294 477 300 471)(373 437 379 443)(374 436 380 442)(375 435 381 441)(376 434 382 440)(377 433 383 439)(378 444 384 438)

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

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

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

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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