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## G = C20.Dic6order 480 = 25·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C20.Dic6
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic15⋊5C4 — C20.Dic6
 Lower central C15 — C2×C30 — C20.Dic6
 Upper central C1 — C22 — C2×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, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, Dic5, C20, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4×C12, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C12.6Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C4.Dic10, Dic155C4, C6.Dic10, C12×Dic5, C5×C4⋊Dic3, C605C4, C20.Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, Dic6, C22×S3, C42.C2, Dic10, C22×D5, C2×Dic6, C4○D12, S3×D5, C2×Dic10, D42D5, Q82D5, C12.6Q8, C15⋊Q8, 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)
(1 324 168 423 413 228 119 86 213 48 398 476)(2 333 169 432 414 237 120 95 214 57 399 465)(3 322 170 421 415 226 101 84 215 46 400 474)(4 331 171 430 416 235 102 93 216 55 381 463)(5 340 172 439 417 224 103 82 217 44 382 472)(6 329 173 428 418 233 104 91 218 53 383 461)(7 338 174 437 419 222 105 100 219 42 384 470)(8 327 175 426 420 231 106 89 220 51 385 479)(9 336 176 435 401 240 107 98 201 60 386 468)(10 325 177 424 402 229 108 87 202 49 387 477)(11 334 178 433 403 238 109 96 203 58 388 466)(12 323 179 422 404 227 110 85 204 47 389 475)(13 332 180 431 405 236 111 94 205 56 390 464)(14 321 161 440 406 225 112 83 206 45 391 473)(15 330 162 429 407 234 113 92 207 54 392 462)(16 339 163 438 408 223 114 81 208 43 393 471)(17 328 164 427 409 232 115 90 209 52 394 480)(18 337 165 436 410 221 116 99 210 41 395 469)(19 326 166 425 411 230 117 88 211 50 396 478)(20 335 167 434 412 239 118 97 212 59 397 467)(21 121 368 320 264 453 200 253 353 150 297 70)(22 130 369 309 265 442 181 242 354 159 298 79)(23 139 370 318 266 451 182 251 355 148 299 68)(24 128 371 307 267 460 183 260 356 157 300 77)(25 137 372 316 268 449 184 249 357 146 281 66)(26 126 373 305 269 458 185 258 358 155 282 75)(27 135 374 314 270 447 186 247 359 144 283 64)(28 124 375 303 271 456 187 256 360 153 284 73)(29 133 376 312 272 445 188 245 341 142 285 62)(30 122 377 301 273 454 189 254 342 151 286 71)(31 131 378 310 274 443 190 243 343 160 287 80)(32 140 379 319 275 452 191 252 344 149 288 69)(33 129 380 308 276 441 192 241 345 158 289 78)(34 138 361 317 277 450 193 250 346 147 290 67)(35 127 362 306 278 459 194 259 347 156 291 76)(36 136 363 315 279 448 195 248 348 145 292 65)(37 125 364 304 280 457 196 257 349 154 293 74)(38 134 365 313 261 446 197 246 350 143 294 63)(39 123 366 302 262 455 198 255 351 152 295 72)(40 132 367 311 263 444 199 244 352 141 296 61)
(1 264 119 297)(2 275 120 288)(3 266 101 299)(4 277 102 290)(5 268 103 281)(6 279 104 292)(7 270 105 283)(8 261 106 294)(9 272 107 285)(10 263 108 296)(11 274 109 287)(12 265 110 298)(13 276 111 289)(14 267 112 300)(15 278 113 291)(16 269 114 282)(17 280 115 293)(18 271 116 284)(19 262 117 295)(20 273 118 286)(21 398 200 413)(22 389 181 404)(23 400 182 415)(24 391 183 406)(25 382 184 417)(26 393 185 408)(27 384 186 419)(28 395 187 410)(29 386 188 401)(30 397 189 412)(31 388 190 403)(32 399 191 414)(33 390 192 405)(34 381 193 416)(35 392 194 407)(36 383 195 418)(37 394 196 409)(38 385 197 420)(39 396 198 411)(40 387 199 402)(41 246 436 134)(42 257 437 125)(43 248 438 136)(44 259 439 127)(45 250 440 138)(46 241 421 129)(47 252 422 140)(48 243 423 131)(49 254 424 122)(50 245 425 133)(51 256 426 124)(52 247 427 135)(53 258 428 126)(54 249 429 137)(55 260 430 128)(56 251 431 139)(57 242 432 130)(58 253 433 121)(59 244 434 132)(60 255 435 123)(61 467 444 239)(62 478 445 230)(63 469 446 221)(64 480 447 232)(65 471 448 223)(66 462 449 234)(67 473 450 225)(68 464 451 236)(69 475 452 227)(70 466 453 238)(71 477 454 229)(72 468 455 240)(73 479 456 231)(74 470 457 222)(75 461 458 233)(76 472 459 224)(77 463 460 235)(78 474 441 226)(79 465 442 237)(80 476 443 228)(81 145 339 315)(82 156 340 306)(83 147 321 317)(84 158 322 308)(85 149 323 319)(86 160 324 310)(87 151 325 301)(88 142 326 312)(89 153 327 303)(90 144 328 314)(91 155 329 305)(92 146 330 316)(93 157 331 307)(94 148 332 318)(95 159 333 309)(96 150 334 320)(97 141 335 311)(98 152 336 302)(99 143 337 313)(100 154 338 304)(161 371 206 356)(162 362 207 347)(163 373 208 358)(164 364 209 349)(165 375 210 360)(166 366 211 351)(167 377 212 342)(168 368 213 353)(169 379 214 344)(170 370 215 355)(171 361 216 346)(172 372 217 357)(173 363 218 348)(174 374 219 359)(175 365 220 350)(176 376 201 341)(177 367 202 352)(178 378 203 343)(179 369 204 354)(180 380 205 345)

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

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

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

66 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 ··· 12L 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 10 10 10 10 12 12 60 60 2 2 2 2 2 2 ··· 2 2 2 2 2 10 ··· 10 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

66 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 type + + + + + + + - + + + + + - - + - + - + - + image C1 C2 C2 C2 C2 C2 S3 Q8 D5 D6 D6 C4○D4 D10 D10 Dic6 Dic10 C4○D12 S3×D5 D4⋊2D5 Q8⋊2D5 C15⋊Q8 C2×S3×D5 D12⋊5D5 C12.28D10 kernel C20.Dic6 Dic15⋊5C4 C6.Dic10 C12×Dic5 C5×C4⋊Dic3 C60⋊5C4 C4×Dic5 C60 C4⋊Dic3 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C20 C12 C10 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 2 1 1 1 1 2 2 2 1 4 4 2 4 8 8 2 2 2 4 2 4 4

Matrix representation of C20.Dic6 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 43 60 0 0 0 0 1 0 0 0 0 0 0 0 35 13 0 0 0 0 23 26
,
 53 28 0 0 0 0 37 0 0 0 0 0 0 0 6 9 0 0 0 0 23 55 0 0 0 0 0 0 50 0 0 0 0 0 0 50
,
 1 49 0 0 0 0 51 60 0 0 0 0 0 0 36 4 0 0 0 0 57 25 0 0 0 0 0 0 23 41 0 0 0 0 57 38

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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