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G = Dic5.7Dic6order 480 = 25·3·5

1st non-split extension by Dic5 of Dic6 acting via Dic6/C12=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5.7Dic6, C6.30(Q8×D5), C30.27(C2×Q8), (C2×C20).186D6, Dic3⋊C4.1D5, (C3×Dic5).7Q8, C2.14(D5×Dic6), C51(C12.6Q8), C157(C42.C2), C6.29(C4○D20), (C2×C12).260D10, (C2×C30).68C23, (C4×Dic5).11S3, C10.12(C2×Dic6), C10.32(C4○D12), C30.107(C4○D4), C6.40(D42D5), (C2×C60).383C22, (C2×Dic3).21D10, (C2×Dic5).167D6, (C12×Dic5).23C2, C31(Dic5.Q8), C30.4Q8.13C2, Dic155C4.10C2, C6.Dic10.10C2, C30.Q8.11C2, C2.14(Dic3.D10), C2.21(D6.D10), (C2×Dic15).62C22, (C6×Dic5).189C22, (C10×Dic3).40C22, (C2×C4).123(S3×D5), C22.154(C2×S3×D5), (C2×C6).80(C22×D5), (C5×Dic3⋊C4).13C2, (C2×C10).80(C22×S3), SmallGroup(480,454)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic5.7Dic6
C1C5C15C30C2×C30C6×Dic5Dic155C4 — Dic5.7Dic6
C15C2×C30 — Dic5.7Dic6
C1C22C2×C4

Generators and relations for Dic5.7Dic6
 G = < a,b,c,d | a10=c12=1, b2=a5, d2=c6, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a5b, 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], Dic3⋊C4, Dic3⋊C4 [×3], C4⋊Dic3 [×2], C4×C12, C5×Dic3 [×2], C3×Dic5 [×2], C3×Dic5, Dic15 [×2], C60, C2×C30, C4×Dic5, C10.D4 [×4], C4⋊Dic5, C5×C4⋊C4, C12.6Q8, C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, Dic5.Q8, C30.Q8 [×2], Dic155C4, C6.Dic10, C12×Dic5, C5×Dic3⋊C4, C30.4Q8, Dic5.7Dic6
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, C4○D12 [×2], S3×D5, C4○D20, D42D5, Q8×D5, C12.6Q8, C2×S3×D5, Dic5.Q8, D5×Dic6, D6.D10, Dic3.D10, Dic5.7Dic6

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

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

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

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

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A···10F12A12B12C12D12E···12L15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444445566610···101212121212···1215152020202020···2030···3060···60
size11112221010101012126060222222···2222210···1044444412···124···44···4

66 irreducible representations

dim1111111222222222224444444
type++++++++-+++++-+--+-
imageC1C2C2C2C2C2C2S3Q8D5D6D6C4○D4D10D10Dic6C4○D12C4○D20S3×D5D42D5Q8×D5C2×S3×D5D5×Dic6D6.D10Dic3.D10
kernelDic5.7Dic6C30.Q8Dic155C4C6.Dic10C12×Dic5C5×Dic3⋊C4C30.4Q8C4×Dic5C3×Dic5Dic3⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5C10C6C2×C4C6C6C22C2C2C2
# reps1211111122214424882222444

Matrix representation of Dic5.7Dic6 in GL6(𝔽61)

17600000
4510000
001000
000100
0000600
0000060
,
58520000
3530000
001000
000100
00002315
00004638
,
33380000
42280000
0092000
0025200
00005252
0000943
,
14440000
33470000
00475100
00381400
00003019
00004931

G:=sub<GL(6,GF(61))| [17,45,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[58,35,0,0,0,0,52,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,46,0,0,0,0,15,38],[33,42,0,0,0,0,38,28,0,0,0,0,0,0,9,2,0,0,0,0,20,52,0,0,0,0,0,0,52,9,0,0,0,0,52,43],[14,33,0,0,0,0,44,47,0,0,0,0,0,0,47,38,0,0,0,0,51,14,0,0,0,0,0,0,30,49,0,0,0,0,19,31] >;

Dic5.7Dic6 in GAP, Magma, Sage, TeX

{\rm Dic}_5._7{\rm Dic}_6
% in TeX

G:=Group("Dic5.7Dic6");
// GroupNames label

G:=SmallGroup(480,454);
// by ID

G=gap.SmallGroup(480,454);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,219,58,1356,18822]);
// Polycyclic

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

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