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G = Dic5.2Dic6order 480 = 25·3·5

2nd non-split extension by Dic5 of Dic6 acting via Dic6/Dic3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5.2Dic6, C6.25(Q8×D5), (C2×C12).4D10, C4⋊Dic3.4D5, C30.12(C2×Q8), C2.9(D5×Dic6), C30.6(C4○D4), (C2×C20).218D6, (C3×Dic5).2Q8, C10.7(C2×Dic6), C53(C4.Dic6), C152(C42.C2), C6.61(C4○D20), C6.3(D42D5), (C2×C30).25C23, (C2×Dic5).87D6, (C2×Dic3).2D10, C10.D4.8S3, C6.Dic10.5C2, C30.Q8.7C2, C2.7(D12⋊D5), (C2×C60).311C22, C33(Dic5.Q8), C30.4Q8.10C2, C10.36(D42S3), C10.21(Q83S3), (Dic3×Dic5).12C2, C2.8(Dic5.D6), (C6×Dic5).10C22, (C2×Dic15).32C22, (C10×Dic3).11C22, (C2×C4).24(S3×D5), C22.118(C2×S3×D5), (C5×C4⋊Dic3).16C2, (C2×C6).37(C22×D5), (C2×C10).37(C22×S3), (C3×C10.D4).10C2, SmallGroup(480,411)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic5.2Dic6
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic5.2Dic6
C15C2×C30 — Dic5.2Dic6
C1C22C2×C4

Generators and relations for Dic5.2Dic6
 G = < a,b,c,d | a10=c12=1, b2=a5, d2=a5c6, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c-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 [×2], C4⋊Dic3, C4⋊Dic3 [×2], 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], C6.Dic10, C3×C10.D4, C5×C4⋊Dic3, C30.4Q8, Dic5.2Dic6
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, D12⋊D5, Dic5.D6, Dic5.2Dic6

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

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

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

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

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×Dic6D12⋊D5Dic5.D6
kernelDic5.2Dic6Dic3×Dic5C30.Q8C6.Dic10C3×C10.D4C5×C4⋊Dic3C30.4Q8C10.D4C3×Dic5C4⋊Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5C6C10C10C2×C4C6C6C22C2C2C2
# reps11211111222144248112222444

Matrix representation of Dic5.2Dic6 in GL6(𝔽61)

010000
60430000
0060000
0006000
000010
000001
,
27250000
27340000
00394700
00392200
0000600
0000060
,
6000000
0600000
0015900
0006000
0000537
0000260
,
6000000
0600000
0050000
0005000
00001611
00002145

G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,43,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,27,0,0,0,0,25,34,0,0,0,0,0,0,39,39,0,0,0,0,47,22,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,59,60,0,0,0,0,0,0,53,26,0,0,0,0,7,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,16,21,0,0,0,0,11,45] >;

Dic5.2Dic6 in GAP, Magma, Sage, TeX

{\rm Dic}_5._2{\rm Dic}_6
% in TeX

G:=Group("Dic5.2Dic6");
// GroupNames label

G:=SmallGroup(480,411);
// by ID

G=gap.SmallGroup(480,411);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,422,303,142,1356,18822]);
// Polycyclic

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

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