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G = C20⋊Dic6order 480 = 25·3·5

2nd semidirect product of C20 and Dic6 acting via Dic6/C6=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C604Q8, C202Dic6, Dic159Q8, C122Dic10, Dic15.32D4, C159(C4⋊Q8), C53(C12⋊Q8), C33(C20⋊Q8), C43(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, Dic155C4.16C2, (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

C1C2×C30 — C20⋊Dic6
C1C5C15C30C2×C30C6×Dic5C2×C15⋊Q8 — 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=a-1, cac-1=a11, cbc-1=b-1 >

Subgroups: 652 in 136 conjugacy classes, 56 normal (32 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×8], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×6], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×6], C20 [×2], C20 [×2], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4⋊Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, C2×Dic6 [×2], C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×4], C60 [×2], C2×C30, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4, C2×Dic10 [×2], C12⋊Q8, C15⋊Q8 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, C20⋊Q8, Dic155C4 [×2], C3×C4⋊Dic5, C5×C4⋊Dic3, C4×Dic15, C2×C15⋊Q8 [×2], C20⋊Dic6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D5, D6 [×3], C2×D4, C2×Q8 [×2], D10 [×3], Dic6 [×2], C22×S3, C4⋊Q8, Dic10 [×2], C22×D5, C2×Dic6, S3×D4, S3×Q8, S3×D5, C2×Dic10, D4×D5, Q8×D5, C12⋊Q8, C15⋊Q8 [×2], C2×S3×D5, C20⋊Q8, D15⋊Q8, C20⋊D6, 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)
(1 409 280 468 439 379 37 314 260 117 127 146)(2 408 261 467 440 378 38 313 241 116 128 145)(3 407 262 466 421 377 39 312 242 115 129 144)(4 406 263 465 422 376 40 311 243 114 130 143)(5 405 264 464 423 375 21 310 244 113 131 142)(6 404 265 463 424 374 22 309 245 112 132 141)(7 403 266 462 425 373 23 308 246 111 133 160)(8 402 267 461 426 372 24 307 247 110 134 159)(9 401 268 480 427 371 25 306 248 109 135 158)(10 420 269 479 428 370 26 305 249 108 136 157)(11 419 270 478 429 369 27 304 250 107 137 156)(12 418 271 477 430 368 28 303 251 106 138 155)(13 417 272 476 431 367 29 302 252 105 139 154)(14 416 273 475 432 366 30 301 253 104 140 153)(15 415 274 474 433 365 31 320 254 103 121 152)(16 414 275 473 434 364 32 319 255 102 122 151)(17 413 276 472 435 363 33 318 256 101 123 150)(18 412 277 471 436 362 34 317 257 120 124 149)(19 411 278 470 437 361 35 316 258 119 125 148)(20 410 279 469 438 380 36 315 259 118 126 147)(41 66 392 347 235 165 332 81 449 294 189 210)(42 65 393 346 236 164 333 100 450 293 190 209)(43 64 394 345 237 163 334 99 451 292 191 208)(44 63 395 344 238 162 335 98 452 291 192 207)(45 62 396 343 239 161 336 97 453 290 193 206)(46 61 397 342 240 180 337 96 454 289 194 205)(47 80 398 341 221 179 338 95 455 288 195 204)(48 79 399 360 222 178 339 94 456 287 196 203)(49 78 400 359 223 177 340 93 457 286 197 202)(50 77 381 358 224 176 321 92 458 285 198 201)(51 76 382 357 225 175 322 91 459 284 199 220)(52 75 383 356 226 174 323 90 460 283 200 219)(53 74 384 355 227 173 324 89 441 282 181 218)(54 73 385 354 228 172 325 88 442 281 182 217)(55 72 386 353 229 171 326 87 443 300 183 216)(56 71 387 352 230 170 327 86 444 299 184 215)(57 70 388 351 231 169 328 85 445 298 185 214)(58 69 389 350 232 168 329 84 446 297 186 213)(59 68 390 349 233 167 330 83 447 296 187 212)(60 67 391 348 234 166 331 82 448 295 188 211)
(1 162 37 207)(2 173 38 218)(3 164 39 209)(4 175 40 220)(5 166 21 211)(6 177 22 202)(7 168 23 213)(8 179 24 204)(9 170 25 215)(10 161 26 206)(11 172 27 217)(12 163 28 208)(13 174 29 219)(14 165 30 210)(15 176 31 201)(16 167 32 212)(17 178 33 203)(18 169 34 214)(19 180 35 205)(20 171 36 216)(41 153 332 366)(42 144 333 377)(43 155 334 368)(44 146 335 379)(45 157 336 370)(46 148 337 361)(47 159 338 372)(48 150 339 363)(49 141 340 374)(50 152 321 365)(51 143 322 376)(52 154 323 367)(53 145 324 378)(54 156 325 369)(55 147 326 380)(56 158 327 371)(57 149 328 362)(58 160 329 373)(59 151 330 364)(60 142 331 375)(61 125 96 437)(62 136 97 428)(63 127 98 439)(64 138 99 430)(65 129 100 421)(66 140 81 432)(67 131 82 423)(68 122 83 434)(69 133 84 425)(70 124 85 436)(71 135 86 427)(72 126 87 438)(73 137 88 429)(74 128 89 440)(75 139 90 431)(76 130 91 422)(77 121 92 433)(78 132 93 424)(79 123 94 435)(80 134 95 426)(101 456 472 399)(102 447 473 390)(103 458 474 381)(104 449 475 392)(105 460 476 383)(106 451 477 394)(107 442 478 385)(108 453 479 396)(109 444 480 387)(110 455 461 398)(111 446 462 389)(112 457 463 400)(113 448 464 391)(114 459 465 382)(115 450 466 393)(116 441 467 384)(117 452 468 395)(118 443 469 386)(119 454 470 397)(120 445 471 388)(181 408 227 313)(182 419 228 304)(183 410 229 315)(184 401 230 306)(185 412 231 317)(186 403 232 308)(187 414 233 319)(188 405 234 310)(189 416 235 301)(190 407 236 312)(191 418 237 303)(192 409 238 314)(193 420 239 305)(194 411 240 316)(195 402 221 307)(196 413 222 318)(197 404 223 309)(198 415 224 320)(199 406 225 311)(200 417 226 302)(241 282 261 355)(242 293 262 346)(243 284 263 357)(244 295 264 348)(245 286 265 359)(246 297 266 350)(247 288 267 341)(248 299 268 352)(249 290 269 343)(250 281 270 354)(251 292 271 345)(252 283 272 356)(253 294 273 347)(254 285 274 358)(255 296 275 349)(256 287 276 360)(257 298 277 351)(258 289 278 342)(259 300 279 353)(260 291 280 344)

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

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

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

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444445566610···1012121212121215152020202020···2030···3060···60
size11112221212202030303030222222···2442020202044444412···124···44···4

60 irreducible representations

dim11111122222222222444444444
type++++++++--+++++--+-++--+
imageC1C2C2C2C2C2S3D4Q8Q8D5D6D6D10D10Dic6Dic10S3×D4S3×Q8S3×D5D4×D5Q8×D5C15⋊Q8C2×S3×D5D15⋊Q8C20⋊D6
kernelC20⋊Dic6Dic155C4C3×C4⋊Dic5C5×C4⋊Dic3C4×Dic15C2×C15⋊Q8C4⋊Dic5Dic15Dic15C60C4⋊Dic3C2×Dic5C2×C20C2×Dic3C2×C12C20C12C10C10C2×C4C6C6C4C22C2C2
# reps12111212222214248112224244

Matrix representation of C20⋊Dic6 in GL6(𝔽61)

010000
60180000
0060000
0006000
00001717
00002644
,
39590000
29220000
0014600
00495900
00004444
00006017
,
36570000
4250000
0032200
00445800
00004444
00006017

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

׿
×
𝔽