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G = C20.Dic6order 480 = 25·3·5

8th non-split extension by C20 of Dic6 acting via Dic6/C6=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.9Q8, C20.8Dic6, C12.12Dic10, C4.6(C15⋊Q8), C4⋊Dic3.6D5, C30.54(C2×Q8), (C2×C20).121D6, (C4×Dic5).4S3, C605C4.25C2, C31(C4.Dic10), C52(C12.6Q8), C10.9(C4○D12), C30.50(C4○D4), (C2×C12).301D10, (C2×C30).78C23, (C12×Dic5).4C2, C10.22(C2×Dic6), C6.21(C2×Dic10), C1512(C42.C2), C6.23(D42D5), (C2×C60).145C22, C6.11(Q82D5), (C2×Dic5).170D6, (C2×Dic3).27D10, C6.Dic10.14C2, Dic155C4.15C2, C2.14(C12.28D10), C2.12(D125D5), (C2×Dic15).67C22, (C6×Dic5).192C22, (C10×Dic3).46C22, C2.7(C2×C15⋊Q8), (C2×C4).158(S3×D5), (C5×C4⋊Dic3).5C2, C22.163(C2×S3×D5), (C2×C6).90(C22×D5), (C2×C10).90(C22×S3), SmallGroup(480,464)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C20.Dic6
C1C5C15C30C2×C30C6×Dic5Dic155C4 — C20.Dic6
C15C2×C30 — C20.Dic6
C1C22C2×C4

Generators and relations for C20.Dic6
 G = < a,b,c | a20=b12=1, c2=b6, bab-1=a9, cac-1=a11, cbc-1=a10b-1 >

Subgroups: 460 in 112 conjugacy classes, 52 normal (34 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×6], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], C10 [×3], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4 [×6], Dic5 [×4], C20 [×2], C20 [×2], 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 [×4], C4⋊Dic3, C4⋊Dic3, C4×C12, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], C2×C30, C4×Dic5, C10.D4 [×2], C4⋊Dic5 [×3], C5×C4⋊C4, C12.6Q8, C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, C4.Dic10, Dic155C4 [×2], C6.Dic10 [×2], C12×Dic5, C5×C4⋊Dic3, C605C4, C20.Dic6
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, Dic10 [×2], C22×D5, C2×Dic6, C4○D12 [×2], S3×D5, C2×Dic10, D42D5, Q82D5, C12.6Q8, C15⋊Q8 [×2], C2×S3×D5, C4.Dic10, D125D5, C12.28D10, C2×C15⋊Q8, C20.Dic6

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

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

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

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

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

dim111111222222222224444444
type+++++++-+++++--+-+-+-+
imageC1C2C2C2C2C2S3Q8D5D6D6C4○D4D10D10Dic6Dic10C4○D12S3×D5D42D5Q82D5C15⋊Q8C2×S3×D5D125D5C12.28D10
kernelC20.Dic6Dic155C4C6.Dic10C12×Dic5C5×C4⋊Dic3C605C4C4×Dic5C60C4⋊Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12C20C12C10C2×C4C6C6C4C22C2C2
# reps122111122214424882224244

Matrix representation of C20.Dic6 in GL6(𝔽61)

100000
010000
00436000
001000
00003513
00002326
,
53280000
3700000
006900
00235500
0000500
0000050
,
1490000
51600000
0036400
00572500
00002341
00005738

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,1,0,0,0,0,60,0,0,0,0,0,0,0,35,23,0,0,0,0,13,26],[53,37,0,0,0,0,28,0,0,0,0,0,0,0,6,23,0,0,0,0,9,55,0,0,0,0,0,0,50,0,0,0,0,0,0,50],[1,51,0,0,0,0,49,60,0,0,0,0,0,0,36,57,0,0,0,0,4,25,0,0,0,0,0,0,23,57,0,0,0,0,41,38] >;

C20.Dic6 in GAP, Magma, Sage, TeX

C_{20}.{\rm Dic}_6
% in TeX

G:=Group("C20.Dic6");
// GroupNames label

G:=SmallGroup(480,464);
// by ID

G=gap.SmallGroup(480,464);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,64,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^20=b^12=1,c^2=b^6,b*a*b^-1=a^9,c*a*c^-1=a^11,c*b*c^-1=a^10*b^-1>;
// generators/relations

׿
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