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## G = Dic5.1Dic6order 480 = 25·3·5

### 1st non-split extension by Dic5 of Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic5.1Dic6
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic5.1Dic6
 Lower central C15 — C2×C30 — Dic5.1Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic5.1Dic6
G = < a,b,c,d | a10=c12=1, b2=a5, d2=c6, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=a5c-1 >

Subgroups: 460 in 112 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C2×C4, C10, Dic3, C12, C2×C6, C15, C42, C4⋊C4, Dic5, Dic5, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C5×C4⋊C4, C4.Dic6, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, Dic5.Q8, Dic3×Dic5, C30.Q8, Dic155C4, C3×C10.D4, C5×Dic3⋊C4, C605C4, Dic5.1Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, Dic6, C22×S3, C42.C2, C22×D5, C2×Dic6, D42S3, Q83S3, S3×D5, C4○D20, D42D5, Q8×D5, C4.Dic6, C2×S3×D5, Dic5.Q8, D5×Dic6, D60⋊C2, C30.C23, Dic5.1Dic6

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + - - + + - - + - + - image C1 C2 C2 C2 C2 C2 C2 S3 Q8 D5 D6 D6 C4○D4 D10 D10 Dic6 C4○D20 D4⋊2S3 Q8⋊3S3 S3×D5 D4⋊2D5 Q8×D5 C2×S3×D5 D5×Dic6 D60⋊C2 C30.C23 kernel Dic5.1Dic6 Dic3×Dic5 C30.Q8 Dic15⋊5C4 C3×C10.D4 C5×Dic3⋊C4 C60⋊5C4 C10.D4 C3×Dic5 Dic3⋊C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 Dic5 C6 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 2 1 1 1 1 1 2 2 2 1 4 4 2 4 8 1 1 2 2 2 2 4 4 4

Matrix representation of Dic5.1Dic6 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 0 16 44
,
 40 21 0 0 0 0 8 21 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 47 0 0 0 0 32 49
,
 50 49 0 0 0 0 0 11 0 0 0 0 0 0 46 23 0 0 0 0 38 23 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 50 0 0 0 0 0 0 50 0 0 0 0 0 0 41 49 0 0 0 0 8 20 0 0 0 0 0 0 60 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,16,0,0,0,0,1,44],[40,8,0,0,0,0,21,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,32,0,0,0,0,47,49],[50,0,0,0,0,0,49,11,0,0,0,0,0,0,46,38,0,0,0,0,23,23,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,41,8,0,0,0,0,49,20,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

Dic5.1Dic6 in GAP, Magma, Sage, TeX

{\rm Dic}_5._1{\rm Dic}_6
% in TeX

G:=Group("Dic5.1Dic6");
// GroupNames label

G:=SmallGroup(480,410);
// by ID

G=gap.SmallGroup(480,410);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,254,219,142,1356,18822]);
// Polycyclic

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

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