metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊4Q8, C20⋊2Dic6, Dic15⋊9Q8, C12⋊2Dic10, Dic15.32D4, C15⋊9(C4⋊Q8), C5⋊3(C12⋊Q8), C3⋊3(C20⋊Q8), C4⋊3(C15⋊Q8), C6.51(D4×D5), C6.18(Q8×D5), C10.52(S3×D4), C30.67(C2×D4), C10.18(S3×Q8), C30.58(C2×Q8), (C2×C20).136D6, C4⋊Dic5.13S3, C4⋊Dic3.13D5, (C2×C12).137D10, C2.19(D15⋊Q8), (C2×Dic5).51D6, C10.25(C2×Dic6), C6.25(C2×Dic10), C2.27(C20⋊D6), (C2×C60).206C22, (C2×C30).160C23, (C4×Dic15).16C2, (C2×Dic3).51D10, Dic15⋊5C4.16C2, (C6×Dic5).96C22, (C10×Dic3).96C22, (C2×Dic15).219C22, (C2×C15⋊Q8).9C2, C2.10(C2×C15⋊Q8), (C2×C4).216(S3×D5), C22.211(C2×S3×D5), (C3×C4⋊Dic5).12C2, (C5×C4⋊Dic3).12C2, (C2×C6).172(C22×D5), (C2×C10).172(C22×S3), SmallGroup(480,546)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊Dic6
G = < a,b,c | a20=b12=1, c2=b6, bab-1=a-1, cac-1=a11, cbc-1=b-1 >
Subgroups: 652 in 136 conjugacy classes, 56 normal (32 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×Dic10, C12⋊Q8, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C20⋊Q8, Dic15⋊5C4, C3×C4⋊Dic5, C5×C4⋊Dic3, C4×Dic15, C2×C15⋊Q8, C20⋊Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, D10, Dic6, C22×S3, C4⋊Q8, Dic10, C22×D5, C2×Dic6, S3×D4, S3×Q8, S3×D5, C2×Dic10, D4×D5, Q8×D5, C12⋊Q8, C15⋊Q8, C2×S3×D5, C20⋊Q8, D15⋊Q8, C20⋊D6, C2×C15⋊Q8, C20⋊Dic6
►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)
(1 126 75 313 202 249 34 143 84 112 267 395)(2 125 76 312 203 248 35 142 85 111 268 394)(3 124 77 311 204 247 36 141 86 110 269 393)(4 123 78 310 205 246 37 160 87 109 270 392)(5 122 79 309 206 245 38 159 88 108 271 391)(6 121 80 308 207 244 39 158 89 107 272 390)(7 140 61 307 208 243 40 157 90 106 273 389)(8 139 62 306 209 242 21 156 91 105 274 388)(9 138 63 305 210 241 22 155 92 104 275 387)(10 137 64 304 211 260 23 154 93 103 276 386)(11 136 65 303 212 259 24 153 94 102 277 385)(12 135 66 302 213 258 25 152 95 101 278 384)(13 134 67 301 214 257 26 151 96 120 279 383)(14 133 68 320 215 256 27 150 97 119 280 382)(15 132 69 319 216 255 28 149 98 118 261 381)(16 131 70 318 217 254 29 148 99 117 262 400)(17 130 71 317 218 253 30 147 100 116 263 399)(18 129 72 316 219 252 31 146 81 115 264 398)(19 128 73 315 220 251 32 145 82 114 265 397)(20 127 74 314 201 250 33 144 83 113 266 396)(41 182 174 370 405 340 351 430 230 441 477 295)(42 181 175 369 406 339 352 429 231 460 478 294)(43 200 176 368 407 338 353 428 232 459 479 293)(44 199 177 367 408 337 354 427 233 458 480 292)(45 198 178 366 409 336 355 426 234 457 461 291)(46 197 179 365 410 335 356 425 235 456 462 290)(47 196 180 364 411 334 357 424 236 455 463 289)(48 195 161 363 412 333 358 423 237 454 464 288)(49 194 162 362 413 332 359 422 238 453 465 287)(50 193 163 361 414 331 360 421 239 452 466 286)(51 192 164 380 415 330 341 440 240 451 467 285)(52 191 165 379 416 329 342 439 221 450 468 284)(53 190 166 378 417 328 343 438 222 449 469 283)(54 189 167 377 418 327 344 437 223 448 470 282)(55 188 168 376 419 326 345 436 224 447 471 281)(56 187 169 375 420 325 346 435 225 446 472 300)(57 186 170 374 401 324 347 434 226 445 473 299)(58 185 171 373 402 323 348 433 227 444 474 298)(59 184 172 372 403 322 349 432 228 443 475 297)(60 183 173 371 404 321 350 431 229 442 476 296)
(1 59 34 349)(2 50 35 360)(3 41 36 351)(4 52 37 342)(5 43 38 353)(6 54 39 344)(7 45 40 355)(8 56 21 346)(9 47 22 357)(10 58 23 348)(11 49 24 359)(12 60 25 350)(13 51 26 341)(14 42 27 352)(15 53 28 343)(16 44 29 354)(17 55 30 345)(18 46 31 356)(19 57 32 347)(20 48 33 358)(61 461 90 409)(62 472 91 420)(63 463 92 411)(64 474 93 402)(65 465 94 413)(66 476 95 404)(67 467 96 415)(68 478 97 406)(69 469 98 417)(70 480 99 408)(71 471 100 419)(72 462 81 410)(73 473 82 401)(74 464 83 412)(75 475 84 403)(76 466 85 414)(77 477 86 405)(78 468 87 416)(79 479 88 407)(80 470 89 418)(101 371 302 442)(102 362 303 453)(103 373 304 444)(104 364 305 455)(105 375 306 446)(106 366 307 457)(107 377 308 448)(108 368 309 459)(109 379 310 450)(110 370 311 441)(111 361 312 452)(112 372 313 443)(113 363 314 454)(114 374 315 445)(115 365 316 456)(116 376 317 447)(117 367 318 458)(118 378 319 449)(119 369 320 460)(120 380 301 451)(121 282 158 327)(122 293 159 338)(123 284 160 329)(124 295 141 340)(125 286 142 331)(126 297 143 322)(127 288 144 333)(128 299 145 324)(129 290 146 335)(130 281 147 326)(131 292 148 337)(132 283 149 328)(133 294 150 339)(134 285 151 330)(135 296 152 321)(136 287 153 332)(137 298 154 323)(138 289 155 334)(139 300 156 325)(140 291 157 336)(161 201 237 266)(162 212 238 277)(163 203 239 268)(164 214 240 279)(165 205 221 270)(166 216 222 261)(167 207 223 272)(168 218 224 263)(169 209 225 274)(170 220 226 265)(171 211 227 276)(172 202 228 267)(173 213 229 278)(174 204 230 269)(175 215 231 280)(176 206 232 271)(177 217 233 262)(178 208 234 273)(179 219 235 264)(180 210 236 275)(181 256 429 382)(182 247 430 393)(183 258 431 384)(184 249 432 395)(185 260 433 386)(186 251 434 397)(187 242 435 388)(188 253 436 399)(189 244 437 390)(190 255 438 381)(191 246 439 392)(192 257 440 383)(193 248 421 394)(194 259 422 385)(195 250 423 396)(196 241 424 387)(197 252 425 398)(198 243 426 389)(199 254 427 400)(200 245 428 391)
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), (1,126,75,313,202,249,34,143,84,112,267,395)(2,125,76,312,203,248,35,142,85,111,268,394)(3,124,77,311,204,247,36,141,86,110,269,393)(4,123,78,310,205,246,37,160,87,109,270,392)(5,122,79,309,206,245,38,159,88,108,271,391)(6,121,80,308,207,244,39,158,89,107,272,390)(7,140,61,307,208,243,40,157,90,106,273,389)(8,139,62,306,209,242,21,156,91,105,274,388)(9,138,63,305,210,241,22,155,92,104,275,387)(10,137,64,304,211,260,23,154,93,103,276,386)(11,136,65,303,212,259,24,153,94,102,277,385)(12,135,66,302,213,258,25,152,95,101,278,384)(13,134,67,301,214,257,26,151,96,120,279,383)(14,133,68,320,215,256,27,150,97,119,280,382)(15,132,69,319,216,255,28,149,98,118,261,381)(16,131,70,318,217,254,29,148,99,117,262,400)(17,130,71,317,218,253,30,147,100,116,263,399)(18,129,72,316,219,252,31,146,81,115,264,398)(19,128,73,315,220,251,32,145,82,114,265,397)(20,127,74,314,201,250,33,144,83,113,266,396)(41,182,174,370,405,340,351,430,230,441,477,295)(42,181,175,369,406,339,352,429,231,460,478,294)(43,200,176,368,407,338,353,428,232,459,479,293)(44,199,177,367,408,337,354,427,233,458,480,292)(45,198,178,366,409,336,355,426,234,457,461,291)(46,197,179,365,410,335,356,425,235,456,462,290)(47,196,180,364,411,334,357,424,236,455,463,289)(48,195,161,363,412,333,358,423,237,454,464,288)(49,194,162,362,413,332,359,422,238,453,465,287)(50,193,163,361,414,331,360,421,239,452,466,286)(51,192,164,380,415,330,341,440,240,451,467,285)(52,191,165,379,416,329,342,439,221,450,468,284)(53,190,166,378,417,328,343,438,222,449,469,283)(54,189,167,377,418,327,344,437,223,448,470,282)(55,188,168,376,419,326,345,436,224,447,471,281)(56,187,169,375,420,325,346,435,225,446,472,300)(57,186,170,374,401,324,347,434,226,445,473,299)(58,185,171,373,402,323,348,433,227,444,474,298)(59,184,172,372,403,322,349,432,228,443,475,297)(60,183,173,371,404,321,350,431,229,442,476,296), (1,59,34,349)(2,50,35,360)(3,41,36,351)(4,52,37,342)(5,43,38,353)(6,54,39,344)(7,45,40,355)(8,56,21,346)(9,47,22,357)(10,58,23,348)(11,49,24,359)(12,60,25,350)(13,51,26,341)(14,42,27,352)(15,53,28,343)(16,44,29,354)(17,55,30,345)(18,46,31,356)(19,57,32,347)(20,48,33,358)(61,461,90,409)(62,472,91,420)(63,463,92,411)(64,474,93,402)(65,465,94,413)(66,476,95,404)(67,467,96,415)(68,478,97,406)(69,469,98,417)(70,480,99,408)(71,471,100,419)(72,462,81,410)(73,473,82,401)(74,464,83,412)(75,475,84,403)(76,466,85,414)(77,477,86,405)(78,468,87,416)(79,479,88,407)(80,470,89,418)(101,371,302,442)(102,362,303,453)(103,373,304,444)(104,364,305,455)(105,375,306,446)(106,366,307,457)(107,377,308,448)(108,368,309,459)(109,379,310,450)(110,370,311,441)(111,361,312,452)(112,372,313,443)(113,363,314,454)(114,374,315,445)(115,365,316,456)(116,376,317,447)(117,367,318,458)(118,378,319,449)(119,369,320,460)(120,380,301,451)(121,282,158,327)(122,293,159,338)(123,284,160,329)(124,295,141,340)(125,286,142,331)(126,297,143,322)(127,288,144,333)(128,299,145,324)(129,290,146,335)(130,281,147,326)(131,292,148,337)(132,283,149,328)(133,294,150,339)(134,285,151,330)(135,296,152,321)(136,287,153,332)(137,298,154,323)(138,289,155,334)(139,300,156,325)(140,291,157,336)(161,201,237,266)(162,212,238,277)(163,203,239,268)(164,214,240,279)(165,205,221,270)(166,216,222,261)(167,207,223,272)(168,218,224,263)(169,209,225,274)(170,220,226,265)(171,211,227,276)(172,202,228,267)(173,213,229,278)(174,204,230,269)(175,215,231,280)(176,206,232,271)(177,217,233,262)(178,208,234,273)(179,219,235,264)(180,210,236,275)(181,256,429,382)(182,247,430,393)(183,258,431,384)(184,249,432,395)(185,260,433,386)(186,251,434,397)(187,242,435,388)(188,253,436,399)(189,244,437,390)(190,255,438,381)(191,246,439,392)(192,257,440,383)(193,248,421,394)(194,259,422,385)(195,250,423,396)(196,241,424,387)(197,252,425,398)(198,243,426,389)(199,254,427,400)(200,245,428,391)>;
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), (1,126,75,313,202,249,34,143,84,112,267,395)(2,125,76,312,203,248,35,142,85,111,268,394)(3,124,77,311,204,247,36,141,86,110,269,393)(4,123,78,310,205,246,37,160,87,109,270,392)(5,122,79,309,206,245,38,159,88,108,271,391)(6,121,80,308,207,244,39,158,89,107,272,390)(7,140,61,307,208,243,40,157,90,106,273,389)(8,139,62,306,209,242,21,156,91,105,274,388)(9,138,63,305,210,241,22,155,92,104,275,387)(10,137,64,304,211,260,23,154,93,103,276,386)(11,136,65,303,212,259,24,153,94,102,277,385)(12,135,66,302,213,258,25,152,95,101,278,384)(13,134,67,301,214,257,26,151,96,120,279,383)(14,133,68,320,215,256,27,150,97,119,280,382)(15,132,69,319,216,255,28,149,98,118,261,381)(16,131,70,318,217,254,29,148,99,117,262,400)(17,130,71,317,218,253,30,147,100,116,263,399)(18,129,72,316,219,252,31,146,81,115,264,398)(19,128,73,315,220,251,32,145,82,114,265,397)(20,127,74,314,201,250,33,144,83,113,266,396)(41,182,174,370,405,340,351,430,230,441,477,295)(42,181,175,369,406,339,352,429,231,460,478,294)(43,200,176,368,407,338,353,428,232,459,479,293)(44,199,177,367,408,337,354,427,233,458,480,292)(45,198,178,366,409,336,355,426,234,457,461,291)(46,197,179,365,410,335,356,425,235,456,462,290)(47,196,180,364,411,334,357,424,236,455,463,289)(48,195,161,363,412,333,358,423,237,454,464,288)(49,194,162,362,413,332,359,422,238,453,465,287)(50,193,163,361,414,331,360,421,239,452,466,286)(51,192,164,380,415,330,341,440,240,451,467,285)(52,191,165,379,416,329,342,439,221,450,468,284)(53,190,166,378,417,328,343,438,222,449,469,283)(54,189,167,377,418,327,344,437,223,448,470,282)(55,188,168,376,419,326,345,436,224,447,471,281)(56,187,169,375,420,325,346,435,225,446,472,300)(57,186,170,374,401,324,347,434,226,445,473,299)(58,185,171,373,402,323,348,433,227,444,474,298)(59,184,172,372,403,322,349,432,228,443,475,297)(60,183,173,371,404,321,350,431,229,442,476,296), (1,59,34,349)(2,50,35,360)(3,41,36,351)(4,52,37,342)(5,43,38,353)(6,54,39,344)(7,45,40,355)(8,56,21,346)(9,47,22,357)(10,58,23,348)(11,49,24,359)(12,60,25,350)(13,51,26,341)(14,42,27,352)(15,53,28,343)(16,44,29,354)(17,55,30,345)(18,46,31,356)(19,57,32,347)(20,48,33,358)(61,461,90,409)(62,472,91,420)(63,463,92,411)(64,474,93,402)(65,465,94,413)(66,476,95,404)(67,467,96,415)(68,478,97,406)(69,469,98,417)(70,480,99,408)(71,471,100,419)(72,462,81,410)(73,473,82,401)(74,464,83,412)(75,475,84,403)(76,466,85,414)(77,477,86,405)(78,468,87,416)(79,479,88,407)(80,470,89,418)(101,371,302,442)(102,362,303,453)(103,373,304,444)(104,364,305,455)(105,375,306,446)(106,366,307,457)(107,377,308,448)(108,368,309,459)(109,379,310,450)(110,370,311,441)(111,361,312,452)(112,372,313,443)(113,363,314,454)(114,374,315,445)(115,365,316,456)(116,376,317,447)(117,367,318,458)(118,378,319,449)(119,369,320,460)(120,380,301,451)(121,282,158,327)(122,293,159,338)(123,284,160,329)(124,295,141,340)(125,286,142,331)(126,297,143,322)(127,288,144,333)(128,299,145,324)(129,290,146,335)(130,281,147,326)(131,292,148,337)(132,283,149,328)(133,294,150,339)(134,285,151,330)(135,296,152,321)(136,287,153,332)(137,298,154,323)(138,289,155,334)(139,300,156,325)(140,291,157,336)(161,201,237,266)(162,212,238,277)(163,203,239,268)(164,214,240,279)(165,205,221,270)(166,216,222,261)(167,207,223,272)(168,218,224,263)(169,209,225,274)(170,220,226,265)(171,211,227,276)(172,202,228,267)(173,213,229,278)(174,204,230,269)(175,215,231,280)(176,206,232,271)(177,217,233,262)(178,208,234,273)(179,219,235,264)(180,210,236,275)(181,256,429,382)(182,247,430,393)(183,258,431,384)(184,249,432,395)(185,260,433,386)(186,251,434,397)(187,242,435,388)(188,253,436,399)(189,244,437,390)(190,255,438,381)(191,246,439,392)(192,257,440,383)(193,248,421,394)(194,259,422,385)(195,250,423,396)(196,241,424,387)(197,252,425,398)(198,243,426,389)(199,254,427,400)(200,245,428,391) );
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)], [(1,126,75,313,202,249,34,143,84,112,267,395),(2,125,76,312,203,248,35,142,85,111,268,394),(3,124,77,311,204,247,36,141,86,110,269,393),(4,123,78,310,205,246,37,160,87,109,270,392),(5,122,79,309,206,245,38,159,88,108,271,391),(6,121,80,308,207,244,39,158,89,107,272,390),(7,140,61,307,208,243,40,157,90,106,273,389),(8,139,62,306,209,242,21,156,91,105,274,388),(9,138,63,305,210,241,22,155,92,104,275,387),(10,137,64,304,211,260,23,154,93,103,276,386),(11,136,65,303,212,259,24,153,94,102,277,385),(12,135,66,302,213,258,25,152,95,101,278,384),(13,134,67,301,214,257,26,151,96,120,279,383),(14,133,68,320,215,256,27,150,97,119,280,382),(15,132,69,319,216,255,28,149,98,118,261,381),(16,131,70,318,217,254,29,148,99,117,262,400),(17,130,71,317,218,253,30,147,100,116,263,399),(18,129,72,316,219,252,31,146,81,115,264,398),(19,128,73,315,220,251,32,145,82,114,265,397),(20,127,74,314,201,250,33,144,83,113,266,396),(41,182,174,370,405,340,351,430,230,441,477,295),(42,181,175,369,406,339,352,429,231,460,478,294),(43,200,176,368,407,338,353,428,232,459,479,293),(44,199,177,367,408,337,354,427,233,458,480,292),(45,198,178,366,409,336,355,426,234,457,461,291),(46,197,179,365,410,335,356,425,235,456,462,290),(47,196,180,364,411,334,357,424,236,455,463,289),(48,195,161,363,412,333,358,423,237,454,464,288),(49,194,162,362,413,332,359,422,238,453,465,287),(50,193,163,361,414,331,360,421,239,452,466,286),(51,192,164,380,415,330,341,440,240,451,467,285),(52,191,165,379,416,329,342,439,221,450,468,284),(53,190,166,378,417,328,343,438,222,449,469,283),(54,189,167,377,418,327,344,437,223,448,470,282),(55,188,168,376,419,326,345,436,224,447,471,281),(56,187,169,375,420,325,346,435,225,446,472,300),(57,186,170,374,401,324,347,434,226,445,473,299),(58,185,171,373,402,323,348,433,227,444,474,298),(59,184,172,372,403,322,349,432,228,443,475,297),(60,183,173,371,404,321,350,431,229,442,476,296)], [(1,59,34,349),(2,50,35,360),(3,41,36,351),(4,52,37,342),(5,43,38,353),(6,54,39,344),(7,45,40,355),(8,56,21,346),(9,47,22,357),(10,58,23,348),(11,49,24,359),(12,60,25,350),(13,51,26,341),(14,42,27,352),(15,53,28,343),(16,44,29,354),(17,55,30,345),(18,46,31,356),(19,57,32,347),(20,48,33,358),(61,461,90,409),(62,472,91,420),(63,463,92,411),(64,474,93,402),(65,465,94,413),(66,476,95,404),(67,467,96,415),(68,478,97,406),(69,469,98,417),(70,480,99,408),(71,471,100,419),(72,462,81,410),(73,473,82,401),(74,464,83,412),(75,475,84,403),(76,466,85,414),(77,477,86,405),(78,468,87,416),(79,479,88,407),(80,470,89,418),(101,371,302,442),(102,362,303,453),(103,373,304,444),(104,364,305,455),(105,375,306,446),(106,366,307,457),(107,377,308,448),(108,368,309,459),(109,379,310,450),(110,370,311,441),(111,361,312,452),(112,372,313,443),(113,363,314,454),(114,374,315,445),(115,365,316,456),(116,376,317,447),(117,367,318,458),(118,378,319,449),(119,369,320,460),(120,380,301,451),(121,282,158,327),(122,293,159,338),(123,284,160,329),(124,295,141,340),(125,286,142,331),(126,297,143,322),(127,288,144,333),(128,299,145,324),(129,290,146,335),(130,281,147,326),(131,292,148,337),(132,283,149,328),(133,294,150,339),(134,285,151,330),(135,296,152,321),(136,287,153,332),(137,298,154,323),(138,289,155,334),(139,300,156,325),(140,291,157,336),(161,201,237,266),(162,212,238,277),(163,203,239,268),(164,214,240,279),(165,205,221,270),(166,216,222,261),(167,207,223,272),(168,218,224,263),(169,209,225,274),(170,220,226,265),(171,211,227,276),(172,202,228,267),(173,213,229,278),(174,204,230,269),(175,215,231,280),(176,206,232,271),(177,217,233,262),(178,208,234,273),(179,219,235,264),(180,210,236,275),(181,256,429,382),(182,247,430,393),(183,258,431,384),(184,249,432,395),(185,260,433,386),(186,251,434,397),(187,242,435,388),(188,253,436,399),(189,244,437,390),(190,255,438,381),(191,246,439,392),(192,257,440,383),(193,248,421,394),(194,259,422,385),(195,250,423,396),(196,241,424,387),(197,252,425,398),(198,243,426,389),(199,254,427,400),(200,245,428,391)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 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 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 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 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | + | + | - | - | + | - | + | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | Q8 | D5 | D6 | D6 | D10 | D10 | Dic6 | Dic10 | S3×D4 | S3×Q8 | S3×D5 | D4×D5 | Q8×D5 | C15⋊Q8 | C2×S3×D5 | D15⋊Q8 | C20⋊D6 |
| kernel | C20⋊Dic6 | Dic15⋊5C4 | C3×C4⋊Dic5 | C5×C4⋊Dic3 | C4×Dic15 | C2×C15⋊Q8 | C4⋊Dic5 | Dic15 | Dic15 | C60 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C2×Dic3 | C2×C12 | C20 | C12 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C20⋊Dic6 ►in GL6(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 17 |
| 0 | 0 | 0 | 0 | 26 | 44 |
| 39 | 59 | 0 | 0 | 0 | 0 |
| 29 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 46 | 0 | 0 |
| 0 | 0 | 49 | 59 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 44 |
| 0 | 0 | 0 | 0 | 60 | 17 |
| 36 | 57 | 0 | 0 | 0 | 0 |
| 4 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 22 | 0 | 0 |
| 0 | 0 | 44 | 58 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 44 |
| 0 | 0 | 0 | 0 | 60 | 17 |
G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,18,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,17,26,0,0,0,0,17,44],[39,29,0,0,0,0,59,22,0,0,0,0,0,0,1,49,0,0,0,0,46,59,0,0,0,0,0,0,44,60,0,0,0,0,44,17],[36,4,0,0,0,0,57,25,0,0,0,0,0,0,3,44,0,0,0,0,22,58,0,0,0,0,0,0,44,60,0,0,0,0,44,17] >;
C20⋊Dic6 in GAP, Magma, Sage, TeX
C_{20}\rtimes {\rm Dic}_6 % in TeX
G:=Group("C20:Dic6"); // GroupNames label
G:=SmallGroup(480,546);
// by ID
G=gap.SmallGroup(480,546);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,64,422,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^20=b^12=1,c^2=b^6,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=b^-1>;
// generators/relations