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G = C204Dic6order 480 = 25·3·5

1st semidirect product of C20 and Dic6 acting via Dic6/Dic3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C602Q8, C204Dic6, C121Dic10, Dic3.5D20, Dic34Dic10, C41(C15⋊Q8), C51(C12⋊Q8), C158(C4⋊Q8), (C5×Dic3)⋊7Q8, C31(C202Q8), C2.27(S3×D20), C30.66(C2×D4), C6.25(C2×D20), C10.24(S3×D4), C4⋊Dic5.7S3, C30.57(C2×Q8), C10.37(S3×Q8), (C2×C20).304D6, (C4×Dic3).5D5, C605C4.20C2, (C2×C12).136D10, (C2×Dic5).50D6, (Dic3×C20).5C2, (C5×Dic3).27D4, C2.19(S3×Dic10), C10.24(C2×Dic6), C6.24(C2×Dic10), (C2×C30).159C23, (C2×C60).123C22, C6.Dic10.16C2, (C2×Dic3).160D10, (C6×Dic5).95C22, (C2×Dic15).117C22, (C10×Dic3).194C22, C2.9(C2×C15⋊Q8), (C2×C15⋊Q8).8C2, (C2×C4).114(S3×D5), (C3×C4⋊Dic5).6C2, C22.210(C2×S3×D5), (C2×C6).171(C22×D5), (C2×C10).171(C22×S3), SmallGroup(480,545)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C204Dic6
C1C5C15C30C2×C30C6×Dic5C2×C15⋊Q8 — C204Dic6
C15C2×C30 — C204Dic6
C1C22C2×C4

Generators and relations for C204Dic6
 G = < a,b,c | a20=b12=1, c2=b6, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 652 in 136 conjugacy classes, 60 normal (34 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×8], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×4], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×4], C20 [×2], C20 [×4], 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 [×4], C3×Dic5 [×2], Dic15 [×2], C60 [×2], C2×C30, C4⋊Dic5, C4⋊Dic5 [×3], C4×C20, C2×Dic10 [×2], C12⋊Q8, C15⋊Q8 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, C202Q8, C6.Dic10 [×2], C3×C4⋊Dic5, Dic3×C20, C605C4, C2×C15⋊Q8 [×2], C204Dic6
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 [×4], D20 [×2], C22×D5, C2×Dic6, S3×D4, S3×Q8, S3×D5, C2×Dic10 [×2], C2×D20, C12⋊Q8, C15⋊Q8 [×2], C2×S3×D5, C202Q8, S3×Dic10, S3×D20, C2×C15⋊Q8, C204Dic6

Smallest permutation representation of C204Dic6
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)
(1 301 371 237 420 198 93 388 108 435 446 206)(2 320 372 236 401 197 94 387 109 434 447 205)(3 319 373 235 402 196 95 386 110 433 448 204)(4 318 374 234 403 195 96 385 111 432 449 203)(5 317 375 233 404 194 97 384 112 431 450 202)(6 316 376 232 405 193 98 383 113 430 451 201)(7 315 377 231 406 192 99 382 114 429 452 220)(8 314 378 230 407 191 100 381 115 428 453 219)(9 313 379 229 408 190 81 400 116 427 454 218)(10 312 380 228 409 189 82 399 117 426 455 217)(11 311 361 227 410 188 83 398 118 425 456 216)(12 310 362 226 411 187 84 397 119 424 457 215)(13 309 363 225 412 186 85 396 120 423 458 214)(14 308 364 224 413 185 86 395 101 422 459 213)(15 307 365 223 414 184 87 394 102 421 460 212)(16 306 366 222 415 183 88 393 103 440 441 211)(17 305 367 221 416 182 89 392 104 439 442 210)(18 304 368 240 417 181 90 391 105 438 443 209)(19 303 369 239 418 200 91 390 106 437 444 208)(20 302 370 238 419 199 92 389 107 436 445 207)(21 344 258 42 145 62 273 477 123 334 295 168)(22 343 259 41 146 61 274 476 124 333 296 167)(23 342 260 60 147 80 275 475 125 332 297 166)(24 341 241 59 148 79 276 474 126 331 298 165)(25 360 242 58 149 78 277 473 127 330 299 164)(26 359 243 57 150 77 278 472 128 329 300 163)(27 358 244 56 151 76 279 471 129 328 281 162)(28 357 245 55 152 75 280 470 130 327 282 161)(29 356 246 54 153 74 261 469 131 326 283 180)(30 355 247 53 154 73 262 468 132 325 284 179)(31 354 248 52 155 72 263 467 133 324 285 178)(32 353 249 51 156 71 264 466 134 323 286 177)(33 352 250 50 157 70 265 465 135 322 287 176)(34 351 251 49 158 69 266 464 136 321 288 175)(35 350 252 48 159 68 267 463 137 340 289 174)(36 349 253 47 160 67 268 462 138 339 290 173)(37 348 254 46 141 66 269 461 139 338 291 172)(38 347 255 45 142 65 270 480 140 337 292 171)(39 346 256 44 143 64 271 479 121 336 293 170)(40 345 257 43 144 63 272 478 122 335 294 169)
(1 155 93 285)(2 156 94 286)(3 157 95 287)(4 158 96 288)(5 159 97 289)(6 160 98 290)(7 141 99 291)(8 142 100 292)(9 143 81 293)(10 144 82 294)(11 145 83 295)(12 146 84 296)(13 147 85 297)(14 148 86 298)(15 149 87 299)(16 150 88 300)(17 151 89 281)(18 152 90 282)(19 153 91 283)(20 154 92 284)(21 456 273 410)(22 457 274 411)(23 458 275 412)(24 459 276 413)(25 460 277 414)(26 441 278 415)(27 442 279 416)(28 443 280 417)(29 444 261 418)(30 445 262 419)(31 446 263 420)(32 447 264 401)(33 448 265 402)(34 449 266 403)(35 450 267 404)(36 451 268 405)(37 452 269 406)(38 453 270 407)(39 454 271 408)(40 455 272 409)(41 397 333 310)(42 398 334 311)(43 399 335 312)(44 400 336 313)(45 381 337 314)(46 382 338 315)(47 383 339 316)(48 384 340 317)(49 385 321 318)(50 386 322 319)(51 387 323 320)(52 388 324 301)(53 389 325 302)(54 390 326 303)(55 391 327 304)(56 392 328 305)(57 393 329 306)(58 394 330 307)(59 395 331 308)(60 396 332 309)(61 187 167 215)(62 188 168 216)(63 189 169 217)(64 190 170 218)(65 191 171 219)(66 192 172 220)(67 193 173 201)(68 194 174 202)(69 195 175 203)(70 196 176 204)(71 197 177 205)(72 198 178 206)(73 199 179 207)(74 200 180 208)(75 181 161 209)(76 182 162 210)(77 183 163 211)(78 184 164 212)(79 185 165 213)(80 186 166 214)(101 126 364 241)(102 127 365 242)(103 128 366 243)(104 129 367 244)(105 130 368 245)(106 131 369 246)(107 132 370 247)(108 133 371 248)(109 134 372 249)(110 135 373 250)(111 136 374 251)(112 137 375 252)(113 138 376 253)(114 139 377 254)(115 140 378 255)(116 121 379 256)(117 122 380 257)(118 123 361 258)(119 124 362 259)(120 125 363 260)(221 358 439 471)(222 359 440 472)(223 360 421 473)(224 341 422 474)(225 342 423 475)(226 343 424 476)(227 344 425 477)(228 345 426 478)(229 346 427 479)(230 347 428 480)(231 348 429 461)(232 349 430 462)(233 350 431 463)(234 351 432 464)(235 352 433 465)(236 353 434 466)(237 354 435 467)(238 355 436 468)(239 356 437 469)(240 357 438 470)

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

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

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

72 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A···20H20I···20X30A···30F60A···60H
order1222344444444445566610···10121212121212151520···2020···2030···3060···60
size1111222666620206060222222···24420202020442···26···64···44···4

72 irreducible representations

dim11111122222222222224444444
type++++++++--+++++--+-+-+-+-+
imageC1C2C2C2C2C2S3D4Q8Q8D5D6D6D10D10Dic6Dic10D20Dic10S3×D4S3×Q8S3×D5C15⋊Q8C2×S3×D5S3×Dic10S3×D20
kernelC204Dic6C6.Dic10C3×C4⋊Dic5Dic3×C20C605C4C2×C15⋊Q8C4⋊Dic5C5×Dic3C5×Dic3C60C4×Dic3C2×Dic5C2×C20C2×Dic3C2×C12C20Dic3Dic3C12C10C10C2×C4C4C22C2C2
# reps12111212222214248881124244

Matrix representation of C204Dic6 in GL6(𝔽61)

6000000
0600000
0018100
0060000
00003141
00004230
,
0300000
2530000
0061400
00285500
00001455
00004347
,
29480000
46320000
0060000
0006000
00003141
00004230

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,18,60,0,0,0,0,1,0,0,0,0,0,0,0,31,42,0,0,0,0,41,30],[0,2,0,0,0,0,30,53,0,0,0,0,0,0,6,28,0,0,0,0,14,55,0,0,0,0,0,0,14,43,0,0,0,0,55,47],[29,46,0,0,0,0,48,32,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,31,42,0,0,0,0,41,30] >;

C204Dic6 in GAP, Magma, Sage, TeX

C_{20}\rtimes_4{\rm Dic}_6
% in TeX

G:=Group("C20:4Dic6");
// GroupNames label

G:=SmallGroup(480,545);
// by ID

G=gap.SmallGroup(480,545);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,64,422,100,1356,18822]);
// Polycyclic

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

׿
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