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G = Dic5.1Dic6order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5.1Dic6, (C2×C20).4D6, C6.24(Q8×D5), (C2×C12).3D10, C30.11(C2×Q8), C605C4.4C2, C2.8(D5×Dic6), C6.3(C4○D20), Dic3⋊C4.3D5, (C2×C60).2C22, (C3×Dic5).1Q8, C10.6(C2×Dic6), C151(C42.C2), C52(C4.Dic6), (C2×C30).24C23, (C2×Dic5).86D6, (C2×Dic3).1D10, C10.D4.3S3, Dic155C4.5C2, C30.Q8.6C2, C30.101(C4○D4), C2.7(D60⋊C2), C6.63(D42D5), C10.3(Q83S3), C32(Dic5.Q8), C10.62(D42S3), (C6×Dic5).9C22, (Dic3×Dic5).11C2, C2.8(C30.C23), (C2×Dic15).31C22, (C10×Dic3).10C22, (C2×C4).23(S3×D5), C22.117(C2×S3×D5), (C5×Dic3⋊C4).3C2, (C2×C6).36(C22×D5), (C2×C10).36(C22×S3), (C3×C10.D4).3C2, SmallGroup(480,410)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic5.1Dic6
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic5.1Dic6
C15C2×C30 — Dic5.1Dic6
C1C22C2×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 [×3], C3, C4 [×8], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], C10 [×3], Dic3 [×4], C12 [×4], C2×C6, C15, C42, C4⋊C4 [×6], Dic5 [×2], Dic5 [×3], C20 [×3], C2×C10, C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C42.C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3 [×3], C3×C4⋊C4, C5×Dic3 [×2], C3×Dic5 [×2], C3×Dic5, Dic15 [×2], C60, C2×C30, C4×Dic5, C10.D4, C10.D4 [×3], C4⋊Dic5, C5×C4⋊C4, C4.Dic6, C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, Dic5.Q8, Dic3×Dic5, C30.Q8 [×2], Dic155C4, C3×C10.D4, C5×Dic3⋊C4, C605C4, Dic5.1Dic6
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D5, D6 [×3], C2×Q8, C4○D4 [×2], D10 [×3], Dic6 [×2], 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 82 93 35 131 205 139 171 219 16)(2 83 94 36 132 206 140 172 220 17)(3 84 95 25 121 207 141 173 221 18)(4 73 96 26 122 208 142 174 222 19)(5 74 85 27 123 209 143 175 223 20)(6 75 86 28 124 210 144 176 224 21)(7 76 87 29 125 211 133 177 225 22)(8 77 88 30 126 212 134 178 226 23)(9 78 89 31 127 213 135 179 227 24)(10 79 90 32 128 214 136 180 228 13)(11 80 91 33 129 215 137 169 217 14)(12 81 92 34 130 216 138 170 218 15)(37 115 60 99 366 409 473 422 434 68)(38 116 49 100 367 410 474 423 435 69)(39 117 50 101 368 411 475 424 436 70)(40 118 51 102 369 412 476 425 437 71)(41 119 52 103 370 413 477 426 438 72)(42 120 53 104 371 414 478 427 439 61)(43 109 54 105 372 415 479 428 440 62)(44 110 55 106 361 416 480 429 441 63)(45 111 56 107 362 417 469 430 442 64)(46 112 57 108 363 418 470 431 443 65)(47 113 58 97 364 419 471 432 444 66)(48 114 59 98 365 420 472 421 433 67)(145 294 331 324 242 341 195 164 234 188)(146 295 332 313 243 342 196 165 235 189)(147 296 333 314 244 343 197 166 236 190)(148 297 334 315 245 344 198 167 237 191)(149 298 335 316 246 345 199 168 238 192)(150 299 336 317 247 346 200 157 239 181)(151 300 325 318 248 347 201 158 240 182)(152 289 326 319 249 348 202 159 229 183)(153 290 327 320 250 337 203 160 230 184)(154 291 328 321 251 338 204 161 231 185)(155 292 329 322 252 339 193 162 232 186)(156 293 330 323 241 340 194 163 233 187)(253 302 460 452 375 398 392 272 284 351)(254 303 461 453 376 399 393 273 285 352)(255 304 462 454 377 400 394 274 286 353)(256 305 463 455 378 401 395 275 287 354)(257 306 464 456 379 402 396 276 288 355)(258 307 465 445 380 403 385 265 277 356)(259 308 466 446 381 404 386 266 278 357)(260 309 467 447 382 405 387 267 279 358)(261 310 468 448 383 406 388 268 280 359)(262 311 457 449 384 407 389 269 281 360)(263 312 458 450 373 408 390 270 282 349)(264 301 459 451 374 397 391 271 283 350)
(1 403 205 258)(2 259 206 404)(3 405 207 260)(4 261 208 406)(5 407 209 262)(6 263 210 408)(7 397 211 264)(8 253 212 398)(9 399 213 254)(10 255 214 400)(11 401 215 256)(12 257 216 402)(13 304 128 394)(14 395 129 305)(15 306 130 396)(16 385 131 307)(17 308 132 386)(18 387 121 309)(19 310 122 388)(20 389 123 311)(21 312 124 390)(22 391 125 301)(23 302 126 392)(24 393 127 303)(25 467 221 267)(26 268 222 468)(27 457 223 269)(28 270 224 458)(29 459 225 271)(30 272 226 460)(31 461 227 273)(32 274 228 462)(33 463 217 275)(34 276 218 464)(35 465 219 265)(36 266 220 466)(37 250 409 184)(38 185 410 251)(39 252 411 186)(40 187 412 241)(41 242 413 188)(42 189 414 243)(43 244 415 190)(44 191 416 245)(45 246 417 192)(46 181 418 247)(47 248 419 182)(48 183 420 249)(49 161 423 328)(50 329 424 162)(51 163 425 330)(52 331 426 164)(53 165 427 332)(54 333 428 166)(55 167 429 334)(56 335 430 168)(57 157 431 336)(58 325 432 158)(59 159 421 326)(60 327 422 160)(61 146 371 342)(62 343 372 147)(63 148 361 344)(64 345 362 149)(65 150 363 346)(66 347 364 151)(67 152 365 348)(68 337 366 153)(69 154 367 338)(70 339 368 155)(71 156 369 340)(72 341 370 145)(73 359 142 383)(74 384 143 360)(75 349 144 373)(76 374 133 350)(77 351 134 375)(78 376 135 352)(79 353 136 377)(80 378 137 354)(81 355 138 379)(82 380 139 356)(83 357 140 381)(84 382 141 358)(85 449 175 281)(86 282 176 450)(87 451 177 283)(88 284 178 452)(89 453 179 285)(90 286 180 454)(91 455 169 287)(92 288 170 456)(93 445 171 277)(94 278 172 446)(95 447 173 279)(96 280 174 448)(97 300 444 201)(98 202 433 289)(99 290 434 203)(100 204 435 291)(101 292 436 193)(102 194 437 293)(103 294 438 195)(104 196 439 295)(105 296 440 197)(106 198 441 297)(107 298 442 199)(108 200 443 299)(109 314 479 236)(110 237 480 315)(111 316 469 238)(112 239 470 317)(113 318 471 240)(114 229 472 319)(115 320 473 230)(116 231 474 321)(117 322 475 232)(118 233 476 323)(119 324 477 234)(120 235 478 313)
(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)
(1 329 7 335)(2 161 8 167)(3 327 9 333)(4 159 10 165)(5 325 11 331)(6 157 12 163)(13 196 19 202)(14 294 20 300)(15 194 21 200)(16 292 22 298)(17 204 23 198)(18 290 24 296)(25 337 31 343)(26 152 32 146)(27 347 33 341)(28 150 34 156)(29 345 35 339)(30 148 36 154)(37 285 43 279)(38 452 44 446)(39 283 45 277)(40 450 46 456)(41 281 47 287)(42 448 48 454)(49 398 55 404)(50 264 56 258)(51 408 57 402)(52 262 58 256)(53 406 59 400)(54 260 60 254)(61 468 67 462)(62 267 68 273)(63 466 69 460)(64 265 70 271)(65 464 71 458)(66 275 72 269)(73 229 79 235)(74 318 80 324)(75 239 81 233)(76 316 82 322)(77 237 83 231)(78 314 84 320)(85 248 91 242)(86 181 92 187)(87 246 93 252)(88 191 94 185)(89 244 95 250)(90 189 96 183)(97 305 103 311)(98 394 104 388)(99 303 105 309)(100 392 106 386)(101 301 107 307)(102 390 108 396)(109 358 115 352)(110 381 116 375)(111 356 117 350)(112 379 118 373)(113 354 119 360)(114 377 120 383)(121 203 127 197)(122 289 128 295)(123 201 129 195)(124 299 130 293)(125 199 131 193)(126 297 132 291)(133 238 139 232)(134 315 140 321)(135 236 141 230)(136 313 142 319)(137 234 143 240)(138 323 144 317)(145 223 151 217)(147 221 153 227)(149 219 155 225)(158 215 164 209)(160 213 166 207)(162 211 168 205)(169 188 175 182)(170 241 176 247)(171 186 177 192)(172 251 178 245)(173 184 179 190)(174 249 180 243)(206 328 212 334)(208 326 214 332)(210 336 216 330)(218 340 224 346)(220 338 226 344)(222 348 228 342)(253 429 259 423)(255 427 261 421)(257 425 263 431)(266 367 272 361)(268 365 274 371)(270 363 276 369)(278 410 284 416)(280 420 286 414)(282 418 288 412)(302 441 308 435)(304 439 310 433)(306 437 312 443)(349 470 355 476)(351 480 357 474)(353 478 359 472)(362 465 368 459)(364 463 370 457)(366 461 372 467)(374 469 380 475)(376 479 382 473)(378 477 384 471)(385 436 391 442)(387 434 393 440)(389 444 395 438)(397 430 403 424)(399 428 405 422)(401 426 407 432)(409 453 415 447)(411 451 417 445)(413 449 419 455)

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

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

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

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444445566610···1012121212121215152020202020···2030···3060···60
size1111246610101220303060222222···2442020202044444412···124···44···4

60 irreducible representations

dim11111112222222222444444444
type++++++++-+++++--++--+-+-
imageC1C2C2C2C2C2C2S3Q8D5D6D6C4○D4D10D10Dic6C4○D20D42S3Q83S3S3×D5D42D5Q8×D5C2×S3×D5D5×Dic6D60⋊C2C30.C23
kernelDic5.1Dic6Dic3×Dic5C30.Q8Dic155C4C3×C10.D4C5×Dic3⋊C4C605C4C10.D4C3×Dic5Dic3⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5C6C10C10C2×C4C6C6C22C2C2C2
# reps11211111222144248112222444

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

6000000
0600000
001000
000100
0000601
00001644
,
40210000
8210000
001000
000100
00001247
00003249
,
50490000
0110000
00462300
00382300
000010
000001
,
5000000
0500000
00414900
0082000
0000600
0000060

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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