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G = Dic6⋊Dic5order 480 = 25·3·5

3rd semidirect product of Dic6 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.79D4, C20.8D12, C30.3Q16, C30.9SD16, Dic63Dic5, C20.36(C4×S3), (C2×C20).51D6, (C2×C30).19D4, C6.5(Q8⋊D5), C4⋊Dic5.8S3, C4.7(S3×Dic5), C60.123(C2×C4), (C5×Dic6)⋊10C4, C155(Q8⋊C4), (C2×C12).53D10, C32(Q8⋊Dic5), C54(C6.SD16), (C2×Dic6).6D5, C12.2(C2×Dic5), C6.5(C5⋊Q16), C10.43(D6⋊C4), C4.22(C5⋊D12), C12.15(C5⋊D4), C2.7(D6⋊Dic5), C10.5(D4.S3), (C10×Dic6).6C2, C10.5(C3⋊Q16), C2.2(C15⋊Q16), C6.6(C23.D5), C30.55(C22⋊C4), (C2×C60).184C22, C2.2(C30.D4), C22.15(C15⋊D4), (C2×C4).187(S3×D5), (C3×C4⋊Dic5).7C2, (C2×C153C8).12C2, (C2×C6).46(C5⋊D4), (C2×C10).46(C3⋊D4), SmallGroup(480,48)

Series: Derived Chief Lower central Upper central

C1C60 — Dic6⋊Dic5
C1C5C15C30C2×C30C2×C60C3×C4⋊Dic5 — Dic6⋊Dic5
C15C30C60 — Dic6⋊Dic5
C1C22C2×C4

Generators and relations for Dic6⋊Dic5
 G = < a,b,c | a20=b12=1, c2=a5, bab-1=a-1, cac-1=a9, cbc-1=a15b-1 >

Subgroups: 300 in 84 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C10 [×3], Dic3 [×2], C12 [×2], C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20 [×2], C20 [×2], C2×C10, C3⋊C8, Dic6 [×2], Dic6, C2×Dic3, C2×C12, C2×C12, C30 [×3], Q8⋊C4, C52C8, C2×Dic5, C2×C20, C2×C20, C5×Q8 [×3], C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C5×Dic3 [×2], C3×Dic5, C60 [×2], C2×C30, C2×C52C8, C4⋊Dic5, Q8×C10, C6.SD16, C153C8, C6×Dic5, C5×Dic6 [×2], C5×Dic6, C10×Dic3, C2×C60, Q8⋊Dic5, C3×C4⋊Dic5, C2×C153C8, C10×Dic6, Dic6⋊Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, SD16, Q16, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, Q8⋊C4, C2×Dic5, C5⋊D4 [×2], D6⋊C4, D4.S3, C3⋊Q16, S3×D5, Q8⋊D5, C5⋊Q16, C23.D5, C6.SD16, S3×Dic5, C15⋊D4, C5⋊D12, Q8⋊Dic5, C30.D4, C15⋊Q16, D6⋊Dic5, Dic6⋊Dic5

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

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

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

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

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444455666888810···1012121212121215152020202020···2030···3060···60
size11112221212202022222303030302···2442020202044444412···124···44···4

60 irreducible representations

dim11111222222222222224444444444
type+++++++++--++--++--+-
imageC1C2C2C2C4S3D4D4D5D6SD16Q16Dic5D10C4×S3D12C3⋊D4C5⋊D4C5⋊D4D4.S3C3⋊Q16S3×D5Q8⋊D5C5⋊Q16S3×Dic5C5⋊D12C15⋊D4C30.D4C15⋊Q16
kernelDic6⋊Dic5C3×C4⋊Dic5C2×C153C8C10×Dic6C5×Dic6C4⋊Dic5C60C2×C30C2×Dic6C2×C20C30C30Dic6C2×C12C20C20C2×C10C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps11114111212242222441122222244

Matrix representation of Dic6⋊Dic5 in GL6(𝔽241)

3600000
171540000
00240000
00024000
00001218
000042240
,
1821150000
173590000
0022523600
00022600
000094158
0000150147
,
1821150000
173590000
0013719600
006910400
0000045
000075203

G:=sub<GL(6,GF(241))| [36,17,0,0,0,0,0,154,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,42,0,0,0,0,218,240],[182,173,0,0,0,0,115,59,0,0,0,0,0,0,225,0,0,0,0,0,236,226,0,0,0,0,0,0,94,150,0,0,0,0,158,147],[182,173,0,0,0,0,115,59,0,0,0,0,0,0,137,69,0,0,0,0,196,104,0,0,0,0,0,0,0,75,0,0,0,0,45,203] >;

Dic6⋊Dic5 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,48);
// by ID

G=gap.SmallGroup(480,48);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^20=b^12=1,c^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^15*b^-1>;
// generators/relations

׿
×
𝔽