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## G = Dic5×Dic6order 480 = 25·3·5

### Direct product of Dic5 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic5×Dic6
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic5×Dic6
 Lower central C15 — 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, bd=db, dcd-1=c-1 >

Subgroups: 492 in 140 conjugacy classes, 72 normal (34 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×9], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×4], Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×4], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4×Q8, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3 [×4], C3×Dic5 [×2], C3×Dic5, Dic15 [×2], C60 [×2], C2×C30, C4×Dic5, C4×Dic5 [×2], C4⋊Dic5 [×3], Q8×C10, C4×Dic6, C6×Dic5 [×2], C5×Dic6 [×4], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, Q8×Dic5, Dic3×Dic5 [×2], C6.Dic10 [×2], C12×Dic5, C605C4, C10×Dic6, Dic5×Dic6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D5, D6 [×3], C22×C4, C2×Q8, C4○D4, Dic5 [×4], D10 [×3], Dic6 [×2], C4×S3 [×2], C22×S3, C4×Q8, C2×Dic5 [×6], C22×D5, C2×Dic6, S3×C2×C4, C4○D12, S3×D5, Q8×D5, Q82D5, C22×Dic5, C4×Dic6, S3×Dic5 [×2], C2×S3×D5, Q8×Dic5, D5×Dic6, C12.28D10, C2×S3×Dic5, Dic5×Dic6

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

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

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

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

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 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 C4 S3 Q8 D5 D6 D6 C4○D4 Dic5 D10 D10 Dic6 C4×S3 C4○D12 S3×D5 Q8×D5 Q8⋊2D5 S3×Dic5 C2×S3×D5 D5×Dic6 C12.28D10 kernel Dic5×Dic6 Dic3×Dic5 C6.Dic10 C12×Dic5 C60⋊5C4 C10×Dic6 C5×Dic6 C4×Dic5 C3×Dic5 C2×Dic6 C2×Dic5 C2×C20 C30 Dic6 C2×Dic3 C2×C12 Dic5 C20 C10 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 2 1 1 1 8 1 2 2 2 1 2 8 4 2 4 4 4 2 2 2 4 2 4 4

Matrix representation of Dic5×Dic6 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 1 60 0 0 19 43
,
 60 0 0 0 0 60 0 0 0 0 5 51 0 0 27 56
,
 46 38 0 0 23 23 0 0 0 0 1 0 0 0 0 1
,
 50 0 0 0 50 11 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,1,19,0,0,60,43],[60,0,0,0,0,60,0,0,0,0,5,27,0,0,51,56],[46,23,0,0,38,23,0,0,0,0,1,0,0,0,0,1],[50,50,0,0,0,11,0,0,0,0,1,0,0,0,0,1] >;

Dic5×Dic6 in GAP, Magma, Sage, TeX

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

G:=Group("Dic5xDic6");
// GroupNames label

G:=SmallGroup(480,408);
// by ID

G=gap.SmallGroup(480,408);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,219,100,1356,18822]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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