metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.79D4, C20.8D12, C30.3Q16, C30.9SD16, Dic6⋊3Dic5, 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, C15⋊5(Q8⋊C4), (C2×C12).53D10, C3⋊2(Q8⋊Dic5), C5⋊4(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×C15⋊3C8).12C2, (C2×C6).46(C5⋊D4), (C2×C10).46(C3⋊D4), SmallGroup(480,48)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, Q8, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C20, C2×C10, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C30, Q8⋊C4, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C5×Q8, C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C5×Dic3, C3×Dic5, C60, C2×C30, C2×C5⋊2C8, C4⋊Dic5, Q8×C10, C6.SD16, C15⋊3C8, C6×Dic5, C5×Dic6, C5×Dic6, C10×Dic3, C2×C60, Q8⋊Dic5, C3×C4⋊Dic5, C2×C15⋊3C8, C10×Dic6, Dic6⋊Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, SD16, Q16, Dic5, D10, C4×S3, D12, C3⋊D4, Q8⋊C4, C2×Dic5, C5⋊D4, 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
►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)
(1 333 42 471 360 225 135 177 377 142 419 69)(2 332 43 470 341 224 136 176 378 141 420 68)(3 331 44 469 342 223 137 175 379 160 401 67)(4 330 45 468 343 222 138 174 380 159 402 66)(5 329 46 467 344 221 139 173 361 158 403 65)(6 328 47 466 345 240 140 172 362 157 404 64)(7 327 48 465 346 239 121 171 363 156 405 63)(8 326 49 464 347 238 122 170 364 155 406 62)(9 325 50 463 348 237 123 169 365 154 407 61)(10 324 51 462 349 236 124 168 366 153 408 80)(11 323 52 461 350 235 125 167 367 152 409 79)(12 322 53 480 351 234 126 166 368 151 410 78)(13 321 54 479 352 233 127 165 369 150 411 77)(14 340 55 478 353 232 128 164 370 149 412 76)(15 339 56 477 354 231 129 163 371 148 413 75)(16 338 57 476 355 230 130 162 372 147 414 74)(17 337 58 475 356 229 131 161 373 146 415 73)(18 336 59 474 357 228 132 180 374 145 416 72)(19 335 60 473 358 227 133 179 375 144 417 71)(20 334 41 472 359 226 134 178 376 143 418 70)(21 398 242 120 286 206 317 98 182 428 275 450)(22 397 243 119 287 205 318 97 183 427 276 449)(23 396 244 118 288 204 319 96 184 426 277 448)(24 395 245 117 289 203 320 95 185 425 278 447)(25 394 246 116 290 202 301 94 186 424 279 446)(26 393 247 115 291 201 302 93 187 423 280 445)(27 392 248 114 292 220 303 92 188 422 261 444)(28 391 249 113 293 219 304 91 189 421 262 443)(29 390 250 112 294 218 305 90 190 440 263 442)(30 389 251 111 295 217 306 89 191 439 264 441)(31 388 252 110 296 216 307 88 192 438 265 460)(32 387 253 109 297 215 308 87 193 437 266 459)(33 386 254 108 298 214 309 86 194 436 267 458)(34 385 255 107 299 213 310 85 195 435 268 457)(35 384 256 106 300 212 311 84 196 434 269 456)(36 383 257 105 281 211 312 83 197 433 270 455)(37 382 258 104 282 210 313 82 198 432 271 454)(38 381 259 103 283 209 314 81 199 431 272 453)(39 400 260 102 284 208 315 100 200 430 273 452)(40 399 241 101 285 207 316 99 181 429 274 451)
(1 439 6 424 11 429 16 434)(2 428 7 433 12 438 17 423)(3 437 8 422 13 427 18 432)(4 426 9 431 14 436 19 421)(5 435 10 440 15 425 20 430)(21 151 26 156 31 141 36 146)(22 160 27 145 32 150 37 155)(23 149 28 154 33 159 38 144)(24 158 29 143 34 148 39 153)(25 147 30 152 35 157 40 142)(41 100 46 85 51 90 56 95)(42 89 47 94 52 99 57 84)(43 98 48 83 53 88 58 93)(44 87 49 92 54 97 59 82)(45 96 50 81 55 86 60 91)(61 267 66 272 71 277 76 262)(62 276 67 261 72 266 77 271)(63 265 68 270 73 275 78 280)(64 274 69 279 74 264 79 269)(65 263 70 268 75 273 80 278)(101 130 106 135 111 140 116 125)(102 139 107 124 112 129 117 134)(103 128 108 133 113 138 118 123)(104 137 109 122 114 127 119 132)(105 126 110 131 115 136 120 121)(161 242 166 247 171 252 176 257)(162 251 167 256 172 241 177 246)(163 260 168 245 173 250 178 255)(164 249 169 254 174 259 179 244)(165 258 170 243 175 248 180 253)(181 333 186 338 191 323 196 328)(182 322 187 327 192 332 197 337)(183 331 188 336 193 321 198 326)(184 340 189 325 194 330 199 335)(185 329 190 334 195 339 200 324)(201 341 206 346 211 351 216 356)(202 350 207 355 212 360 217 345)(203 359 208 344 213 349 218 354)(204 348 209 353 214 358 219 343)(205 357 210 342 215 347 220 352)(221 294 226 299 231 284 236 289)(222 283 227 288 232 293 237 298)(223 292 228 297 233 282 238 287)(224 281 229 286 234 291 239 296)(225 290 230 295 235 300 240 285)(301 476 306 461 311 466 316 471)(302 465 307 470 312 475 317 480)(303 474 308 479 313 464 318 469)(304 463 309 468 314 473 319 478)(305 472 310 477 315 462 320 467)(361 385 366 390 371 395 376 400)(362 394 367 399 372 384 377 389)(363 383 368 388 373 393 378 398)(364 392 369 397 374 382 379 387)(365 381 370 386 375 391 380 396)(401 459 406 444 411 449 416 454)(402 448 407 453 412 458 417 443)(403 457 408 442 413 447 418 452)(404 446 409 451 414 456 419 441)(405 455 410 460 415 445 420 450)
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), (1,333,42,471,360,225,135,177,377,142,419,69)(2,332,43,470,341,224,136,176,378,141,420,68)(3,331,44,469,342,223,137,175,379,160,401,67)(4,330,45,468,343,222,138,174,380,159,402,66)(5,329,46,467,344,221,139,173,361,158,403,65)(6,328,47,466,345,240,140,172,362,157,404,64)(7,327,48,465,346,239,121,171,363,156,405,63)(8,326,49,464,347,238,122,170,364,155,406,62)(9,325,50,463,348,237,123,169,365,154,407,61)(10,324,51,462,349,236,124,168,366,153,408,80)(11,323,52,461,350,235,125,167,367,152,409,79)(12,322,53,480,351,234,126,166,368,151,410,78)(13,321,54,479,352,233,127,165,369,150,411,77)(14,340,55,478,353,232,128,164,370,149,412,76)(15,339,56,477,354,231,129,163,371,148,413,75)(16,338,57,476,355,230,130,162,372,147,414,74)(17,337,58,475,356,229,131,161,373,146,415,73)(18,336,59,474,357,228,132,180,374,145,416,72)(19,335,60,473,358,227,133,179,375,144,417,71)(20,334,41,472,359,226,134,178,376,143,418,70)(21,398,242,120,286,206,317,98,182,428,275,450)(22,397,243,119,287,205,318,97,183,427,276,449)(23,396,244,118,288,204,319,96,184,426,277,448)(24,395,245,117,289,203,320,95,185,425,278,447)(25,394,246,116,290,202,301,94,186,424,279,446)(26,393,247,115,291,201,302,93,187,423,280,445)(27,392,248,114,292,220,303,92,188,422,261,444)(28,391,249,113,293,219,304,91,189,421,262,443)(29,390,250,112,294,218,305,90,190,440,263,442)(30,389,251,111,295,217,306,89,191,439,264,441)(31,388,252,110,296,216,307,88,192,438,265,460)(32,387,253,109,297,215,308,87,193,437,266,459)(33,386,254,108,298,214,309,86,194,436,267,458)(34,385,255,107,299,213,310,85,195,435,268,457)(35,384,256,106,300,212,311,84,196,434,269,456)(36,383,257,105,281,211,312,83,197,433,270,455)(37,382,258,104,282,210,313,82,198,432,271,454)(38,381,259,103,283,209,314,81,199,431,272,453)(39,400,260,102,284,208,315,100,200,430,273,452)(40,399,241,101,285,207,316,99,181,429,274,451), (1,439,6,424,11,429,16,434)(2,428,7,433,12,438,17,423)(3,437,8,422,13,427,18,432)(4,426,9,431,14,436,19,421)(5,435,10,440,15,425,20,430)(21,151,26,156,31,141,36,146)(22,160,27,145,32,150,37,155)(23,149,28,154,33,159,38,144)(24,158,29,143,34,148,39,153)(25,147,30,152,35,157,40,142)(41,100,46,85,51,90,56,95)(42,89,47,94,52,99,57,84)(43,98,48,83,53,88,58,93)(44,87,49,92,54,97,59,82)(45,96,50,81,55,86,60,91)(61,267,66,272,71,277,76,262)(62,276,67,261,72,266,77,271)(63,265,68,270,73,275,78,280)(64,274,69,279,74,264,79,269)(65,263,70,268,75,273,80,278)(101,130,106,135,111,140,116,125)(102,139,107,124,112,129,117,134)(103,128,108,133,113,138,118,123)(104,137,109,122,114,127,119,132)(105,126,110,131,115,136,120,121)(161,242,166,247,171,252,176,257)(162,251,167,256,172,241,177,246)(163,260,168,245,173,250,178,255)(164,249,169,254,174,259,179,244)(165,258,170,243,175,248,180,253)(181,333,186,338,191,323,196,328)(182,322,187,327,192,332,197,337)(183,331,188,336,193,321,198,326)(184,340,189,325,194,330,199,335)(185,329,190,334,195,339,200,324)(201,341,206,346,211,351,216,356)(202,350,207,355,212,360,217,345)(203,359,208,344,213,349,218,354)(204,348,209,353,214,358,219,343)(205,357,210,342,215,347,220,352)(221,294,226,299,231,284,236,289)(222,283,227,288,232,293,237,298)(223,292,228,297,233,282,238,287)(224,281,229,286,234,291,239,296)(225,290,230,295,235,300,240,285)(301,476,306,461,311,466,316,471)(302,465,307,470,312,475,317,480)(303,474,308,479,313,464,318,469)(304,463,309,468,314,473,319,478)(305,472,310,477,315,462,320,467)(361,385,366,390,371,395,376,400)(362,394,367,399,372,384,377,389)(363,383,368,388,373,393,378,398)(364,392,369,397,374,382,379,387)(365,381,370,386,375,391,380,396)(401,459,406,444,411,449,416,454)(402,448,407,453,412,458,417,443)(403,457,408,442,413,447,418,452)(404,446,409,451,414,456,419,441)(405,455,410,460,415,445,420,450)>;
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), (1,333,42,471,360,225,135,177,377,142,419,69)(2,332,43,470,341,224,136,176,378,141,420,68)(3,331,44,469,342,223,137,175,379,160,401,67)(4,330,45,468,343,222,138,174,380,159,402,66)(5,329,46,467,344,221,139,173,361,158,403,65)(6,328,47,466,345,240,140,172,362,157,404,64)(7,327,48,465,346,239,121,171,363,156,405,63)(8,326,49,464,347,238,122,170,364,155,406,62)(9,325,50,463,348,237,123,169,365,154,407,61)(10,324,51,462,349,236,124,168,366,153,408,80)(11,323,52,461,350,235,125,167,367,152,409,79)(12,322,53,480,351,234,126,166,368,151,410,78)(13,321,54,479,352,233,127,165,369,150,411,77)(14,340,55,478,353,232,128,164,370,149,412,76)(15,339,56,477,354,231,129,163,371,148,413,75)(16,338,57,476,355,230,130,162,372,147,414,74)(17,337,58,475,356,229,131,161,373,146,415,73)(18,336,59,474,357,228,132,180,374,145,416,72)(19,335,60,473,358,227,133,179,375,144,417,71)(20,334,41,472,359,226,134,178,376,143,418,70)(21,398,242,120,286,206,317,98,182,428,275,450)(22,397,243,119,287,205,318,97,183,427,276,449)(23,396,244,118,288,204,319,96,184,426,277,448)(24,395,245,117,289,203,320,95,185,425,278,447)(25,394,246,116,290,202,301,94,186,424,279,446)(26,393,247,115,291,201,302,93,187,423,280,445)(27,392,248,114,292,220,303,92,188,422,261,444)(28,391,249,113,293,219,304,91,189,421,262,443)(29,390,250,112,294,218,305,90,190,440,263,442)(30,389,251,111,295,217,306,89,191,439,264,441)(31,388,252,110,296,216,307,88,192,438,265,460)(32,387,253,109,297,215,308,87,193,437,266,459)(33,386,254,108,298,214,309,86,194,436,267,458)(34,385,255,107,299,213,310,85,195,435,268,457)(35,384,256,106,300,212,311,84,196,434,269,456)(36,383,257,105,281,211,312,83,197,433,270,455)(37,382,258,104,282,210,313,82,198,432,271,454)(38,381,259,103,283,209,314,81,199,431,272,453)(39,400,260,102,284,208,315,100,200,430,273,452)(40,399,241,101,285,207,316,99,181,429,274,451), (1,439,6,424,11,429,16,434)(2,428,7,433,12,438,17,423)(3,437,8,422,13,427,18,432)(4,426,9,431,14,436,19,421)(5,435,10,440,15,425,20,430)(21,151,26,156,31,141,36,146)(22,160,27,145,32,150,37,155)(23,149,28,154,33,159,38,144)(24,158,29,143,34,148,39,153)(25,147,30,152,35,157,40,142)(41,100,46,85,51,90,56,95)(42,89,47,94,52,99,57,84)(43,98,48,83,53,88,58,93)(44,87,49,92,54,97,59,82)(45,96,50,81,55,86,60,91)(61,267,66,272,71,277,76,262)(62,276,67,261,72,266,77,271)(63,265,68,270,73,275,78,280)(64,274,69,279,74,264,79,269)(65,263,70,268,75,273,80,278)(101,130,106,135,111,140,116,125)(102,139,107,124,112,129,117,134)(103,128,108,133,113,138,118,123)(104,137,109,122,114,127,119,132)(105,126,110,131,115,136,120,121)(161,242,166,247,171,252,176,257)(162,251,167,256,172,241,177,246)(163,260,168,245,173,250,178,255)(164,249,169,254,174,259,179,244)(165,258,170,243,175,248,180,253)(181,333,186,338,191,323,196,328)(182,322,187,327,192,332,197,337)(183,331,188,336,193,321,198,326)(184,340,189,325,194,330,199,335)(185,329,190,334,195,339,200,324)(201,341,206,346,211,351,216,356)(202,350,207,355,212,360,217,345)(203,359,208,344,213,349,218,354)(204,348,209,353,214,358,219,343)(205,357,210,342,215,347,220,352)(221,294,226,299,231,284,236,289)(222,283,227,288,232,293,237,298)(223,292,228,297,233,282,238,287)(224,281,229,286,234,291,239,296)(225,290,230,295,235,300,240,285)(301,476,306,461,311,466,316,471)(302,465,307,470,312,475,317,480)(303,474,308,479,313,464,318,469)(304,463,309,468,314,473,319,478)(305,472,310,477,315,462,320,467)(361,385,366,390,371,395,376,400)(362,394,367,399,372,384,377,389)(363,383,368,388,373,393,378,398)(364,392,369,397,374,382,379,387)(365,381,370,386,375,391,380,396)(401,459,406,444,411,449,416,454)(402,448,407,453,412,458,417,443)(403,457,408,442,413,447,418,452)(404,446,409,451,414,456,419,441)(405,455,410,460,415,445,420,450) );
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)], [(1,333,42,471,360,225,135,177,377,142,419,69),(2,332,43,470,341,224,136,176,378,141,420,68),(3,331,44,469,342,223,137,175,379,160,401,67),(4,330,45,468,343,222,138,174,380,159,402,66),(5,329,46,467,344,221,139,173,361,158,403,65),(6,328,47,466,345,240,140,172,362,157,404,64),(7,327,48,465,346,239,121,171,363,156,405,63),(8,326,49,464,347,238,122,170,364,155,406,62),(9,325,50,463,348,237,123,169,365,154,407,61),(10,324,51,462,349,236,124,168,366,153,408,80),(11,323,52,461,350,235,125,167,367,152,409,79),(12,322,53,480,351,234,126,166,368,151,410,78),(13,321,54,479,352,233,127,165,369,150,411,77),(14,340,55,478,353,232,128,164,370,149,412,76),(15,339,56,477,354,231,129,163,371,148,413,75),(16,338,57,476,355,230,130,162,372,147,414,74),(17,337,58,475,356,229,131,161,373,146,415,73),(18,336,59,474,357,228,132,180,374,145,416,72),(19,335,60,473,358,227,133,179,375,144,417,71),(20,334,41,472,359,226,134,178,376,143,418,70),(21,398,242,120,286,206,317,98,182,428,275,450),(22,397,243,119,287,205,318,97,183,427,276,449),(23,396,244,118,288,204,319,96,184,426,277,448),(24,395,245,117,289,203,320,95,185,425,278,447),(25,394,246,116,290,202,301,94,186,424,279,446),(26,393,247,115,291,201,302,93,187,423,280,445),(27,392,248,114,292,220,303,92,188,422,261,444),(28,391,249,113,293,219,304,91,189,421,262,443),(29,390,250,112,294,218,305,90,190,440,263,442),(30,389,251,111,295,217,306,89,191,439,264,441),(31,388,252,110,296,216,307,88,192,438,265,460),(32,387,253,109,297,215,308,87,193,437,266,459),(33,386,254,108,298,214,309,86,194,436,267,458),(34,385,255,107,299,213,310,85,195,435,268,457),(35,384,256,106,300,212,311,84,196,434,269,456),(36,383,257,105,281,211,312,83,197,433,270,455),(37,382,258,104,282,210,313,82,198,432,271,454),(38,381,259,103,283,209,314,81,199,431,272,453),(39,400,260,102,284,208,315,100,200,430,273,452),(40,399,241,101,285,207,316,99,181,429,274,451)], [(1,439,6,424,11,429,16,434),(2,428,7,433,12,438,17,423),(3,437,8,422,13,427,18,432),(4,426,9,431,14,436,19,421),(5,435,10,440,15,425,20,430),(21,151,26,156,31,141,36,146),(22,160,27,145,32,150,37,155),(23,149,28,154,33,159,38,144),(24,158,29,143,34,148,39,153),(25,147,30,152,35,157,40,142),(41,100,46,85,51,90,56,95),(42,89,47,94,52,99,57,84),(43,98,48,83,53,88,58,93),(44,87,49,92,54,97,59,82),(45,96,50,81,55,86,60,91),(61,267,66,272,71,277,76,262),(62,276,67,261,72,266,77,271),(63,265,68,270,73,275,78,280),(64,274,69,279,74,264,79,269),(65,263,70,268,75,273,80,278),(101,130,106,135,111,140,116,125),(102,139,107,124,112,129,117,134),(103,128,108,133,113,138,118,123),(104,137,109,122,114,127,119,132),(105,126,110,131,115,136,120,121),(161,242,166,247,171,252,176,257),(162,251,167,256,172,241,177,246),(163,260,168,245,173,250,178,255),(164,249,169,254,174,259,179,244),(165,258,170,243,175,248,180,253),(181,333,186,338,191,323,196,328),(182,322,187,327,192,332,197,337),(183,331,188,336,193,321,198,326),(184,340,189,325,194,330,199,335),(185,329,190,334,195,339,200,324),(201,341,206,346,211,351,216,356),(202,350,207,355,212,360,217,345),(203,359,208,344,213,349,218,354),(204,348,209,353,214,358,219,343),(205,357,210,342,215,347,220,352),(221,294,226,299,231,284,236,289),(222,283,227,288,232,293,237,298),(223,292,228,297,233,282,238,287),(224,281,229,286,234,291,239,296),(225,290,230,295,235,300,240,285),(301,476,306,461,311,466,316,471),(302,465,307,470,312,475,317,480),(303,474,308,479,313,464,318,469),(304,463,309,468,314,473,319,478),(305,472,310,477,315,462,320,467),(361,385,366,390,371,395,376,400),(362,394,367,399,372,384,377,389),(363,383,368,388,373,393,378,398),(364,392,369,397,374,382,379,387),(365,381,370,386,375,391,380,396),(401,459,406,444,411,449,416,454),(402,448,407,453,412,458,417,443),(403,457,408,442,413,447,418,452),(404,446,409,451,414,456,419,441),(405,455,410,460,415,445,420,450)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | + | - | - | + | + | - | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D5 | D6 | SD16 | Q16 | Dic5 | D10 | C4×S3 | D12 | C3⋊D4 | C5⋊D4 | C5⋊D4 | D4.S3 | C3⋊Q16 | S3×D5 | Q8⋊D5 | C5⋊Q16 | S3×Dic5 | C5⋊D12 | C15⋊D4 | C30.D4 | C15⋊Q16 |
| kernel | Dic6⋊Dic5 | C3×C4⋊Dic5 | C2×C15⋊3C8 | C10×Dic6 | C5×Dic6 | C4⋊Dic5 | C60 | C2×C30 | C2×Dic6 | C2×C20 | C30 | C30 | Dic6 | C2×C12 | C20 | C20 | C2×C10 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of Dic6⋊Dic5 ►in GL6(𝔽241)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 17 | 154 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 218 |
| 0 | 0 | 0 | 0 | 42 | 240 |
| 182 | 115 | 0 | 0 | 0 | 0 |
| 173 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 225 | 236 | 0 | 0 |
| 0 | 0 | 0 | 226 | 0 | 0 |
| 0 | 0 | 0 | 0 | 94 | 158 |
| 0 | 0 | 0 | 0 | 150 | 147 |
| 182 | 115 | 0 | 0 | 0 | 0 |
| 173 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 137 | 196 | 0 | 0 |
| 0 | 0 | 69 | 104 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 45 |
| 0 | 0 | 0 | 0 | 75 | 203 |
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