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## G = C20×Dic6order 480 = 25·3·5

### Direct product of C20 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C20×Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — C10×Dic6 — C20×Dic6
 Lower central C3 — C6 — C20×Dic6
 Upper central C1 — C2×C20 — C4×C20

Generators and relations for C20×Dic6
G = < a,b,c | a20=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 244 in 140 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C42, C42, C4⋊C4, C2×Q8, C20, C20, C2×C10, Dic6, C2×Dic3, C2×C12, C30, C4×Q8, C2×C20, C2×C20, C5×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3, C5×Dic3, C60, C60, C2×C30, C4×C20, C4×C20, C5×C4⋊C4, Q8×C10, C4×Dic6, C5×Dic6, C10×Dic3, C2×C60, Q8×C20, Dic3×C20, C5×Dic3⋊C4, C5×C4⋊Dic3, C4×C60, C10×Dic6, C20×Dic6
Quotients:

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

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 12A ··· 12L 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AF 20AG ··· 20BL 30A ··· 30L 60A ··· 60AV order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 5 5 5 5 6 6 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 1 1 1 1 2 2 2 2 6 ··· 6 1 1 1 1 2 2 2 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C10 C20 S3 Q8 D6 C4○D4 Dic6 C4×S3 C5×S3 C5×Q8 C4○D12 S3×C10 C5×C4○D4 C5×Dic6 S3×C20 C5×C4○D12 kernel C20×Dic6 Dic3×C20 C5×Dic3⋊C4 C5×C4⋊Dic3 C4×C60 C10×Dic6 C5×Dic6 C4×Dic6 C4×Dic3 Dic3⋊C4 C4⋊Dic3 C4×C12 C2×Dic6 Dic6 C4×C20 C60 C2×C20 C30 C20 C20 C42 C12 C10 C2×C4 C6 C4 C4 C2 # reps 1 2 2 1 1 1 8 4 8 8 4 4 4 32 1 2 3 2 4 4 4 8 4 12 8 16 16 16

Matrix representation of C20×Dic6 in GL3(𝔽61) generated by

 11 0 0 0 58 0 0 0 58
,
 1 0 0 0 46 23 0 38 23
,
 1 0 0 0 0 50 0 50 0
G:=sub<GL(3,GF(61))| [11,0,0,0,58,0,0,0,58],[1,0,0,0,46,38,0,23,23],[1,0,0,0,0,50,0,50,0] >;

C20×Dic6 in GAP, Magma, Sage, TeX

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

G:=Group("C20xDic6");
// GroupNames label

G:=SmallGroup(480,747);
// by ID

G=gap.SmallGroup(480,747);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,1149,568,226,15686]);
// Polycyclic

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

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