metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊2Q8, C20⋊4Dic6, C12⋊1Dic10, Dic3.5D20, Dic3⋊4Dic10, C4⋊1(C15⋊Q8), C5⋊1(C12⋊Q8), C15⋊8(C4⋊Q8), (C5×Dic3)⋊7Q8, C3⋊1(C20⋊2Q8), 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, C60⋊5C4.20C2, (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
Generators and relations for C20⋊4Dic6
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, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4⋊Dic5, C4⋊Dic5, C4×C20, C2×Dic10, C12⋊Q8, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C20⋊2Q8, C6.Dic10, C3×C4⋊Dic5, Dic3×C20, C60⋊5C4, C2×C15⋊Q8, C20⋊4Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, D10, Dic6, C22×S3, C4⋊Q8, Dic10, D20, C22×D5, C2×Dic6, S3×D4, S3×Q8, S3×D5, C2×Dic10, C2×D20, C12⋊Q8, C15⋊Q8, C2×S3×D5, C20⋊2Q8, S3×Dic10, S3×D20, C2×C15⋊Q8, C20⋊4Dic6
►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 65 271 207 297 190 459 58 355 155 470 32)(2 64 272 206 298 189 460 57 356 154 471 31)(3 63 273 205 299 188 441 56 357 153 472 30)(4 62 274 204 300 187 442 55 358 152 473 29)(5 61 275 203 281 186 443 54 359 151 474 28)(6 80 276 202 282 185 444 53 360 150 475 27)(7 79 277 201 283 184 445 52 341 149 476 26)(8 78 278 220 284 183 446 51 342 148 477 25)(9 77 279 219 285 182 447 50 343 147 478 24)(10 76 280 218 286 181 448 49 344 146 479 23)(11 75 261 217 287 200 449 48 345 145 480 22)(12 74 262 216 288 199 450 47 346 144 461 21)(13 73 263 215 289 198 451 46 347 143 462 40)(14 72 264 214 290 197 452 45 348 142 463 39)(15 71 265 213 291 196 453 44 349 141 464 38)(16 70 266 212 292 195 454 43 350 160 465 37)(17 69 267 211 293 194 455 42 351 159 466 36)(18 68 268 210 294 193 456 41 352 158 467 35)(19 67 269 209 295 192 457 60 353 157 468 34)(20 66 270 208 296 191 458 59 354 156 469 33)(81 334 231 165 248 131 393 373 401 433 104 302)(82 333 232 164 249 130 394 372 402 432 105 301)(83 332 233 163 250 129 395 371 403 431 106 320)(84 331 234 162 251 128 396 370 404 430 107 319)(85 330 235 161 252 127 397 369 405 429 108 318)(86 329 236 180 253 126 398 368 406 428 109 317)(87 328 237 179 254 125 399 367 407 427 110 316)(88 327 238 178 255 124 400 366 408 426 111 315)(89 326 239 177 256 123 381 365 409 425 112 314)(90 325 240 176 257 122 382 364 410 424 113 313)(91 324 221 175 258 121 383 363 411 423 114 312)(92 323 222 174 259 140 384 362 412 422 115 311)(93 322 223 173 260 139 385 361 413 421 116 310)(94 321 224 172 241 138 386 380 414 440 117 309)(95 340 225 171 242 137 387 379 415 439 118 308)(96 339 226 170 243 136 388 378 416 438 119 307)(97 338 227 169 244 135 389 377 417 437 120 306)(98 337 228 168 245 134 390 376 418 436 101 305)(99 336 229 167 246 133 391 375 419 435 102 304)(100 335 230 166 247 132 392 374 420 434 103 303)
(1 106 459 250)(2 107 460 251)(3 108 441 252)(4 109 442 253)(5 110 443 254)(6 111 444 255)(7 112 445 256)(8 113 446 257)(9 114 447 258)(10 115 448 259)(11 116 449 260)(12 117 450 241)(13 118 451 242)(14 119 452 243)(15 120 453 244)(16 101 454 245)(17 102 455 246)(18 103 456 247)(19 104 457 248)(20 105 458 249)(21 309 199 138)(22 310 200 139)(23 311 181 140)(24 312 182 121)(25 313 183 122)(26 314 184 123)(27 315 185 124)(28 316 186 125)(29 317 187 126)(30 318 188 127)(31 319 189 128)(32 320 190 129)(33 301 191 130)(34 302 192 131)(35 303 193 132)(36 304 194 133)(37 305 195 134)(38 306 196 135)(39 307 197 136)(40 308 198 137)(41 166 68 434)(42 167 69 435)(43 168 70 436)(44 169 71 437)(45 170 72 438)(46 171 73 439)(47 172 74 440)(48 173 75 421)(49 174 76 422)(50 175 77 423)(51 176 78 424)(52 177 79 425)(53 178 80 426)(54 179 61 427)(55 180 62 428)(56 161 63 429)(57 162 64 430)(58 163 65 431)(59 164 66 432)(60 165 67 433)(81 295 393 468)(82 296 394 469)(83 297 395 470)(84 298 396 471)(85 299 397 472)(86 300 398 473)(87 281 399 474)(88 282 400 475)(89 283 381 476)(90 284 382 477)(91 285 383 478)(92 286 384 479)(93 287 385 480)(94 288 386 461)(95 289 387 462)(96 290 388 463)(97 291 389 464)(98 292 390 465)(99 293 391 466)(100 294 392 467)(141 338 213 377)(142 339 214 378)(143 340 215 379)(144 321 216 380)(145 322 217 361)(146 323 218 362)(147 324 219 363)(148 325 220 364)(149 326 201 365)(150 327 202 366)(151 328 203 367)(152 329 204 368)(153 330 205 369)(154 331 206 370)(155 332 207 371)(156 333 208 372)(157 334 209 373)(158 335 210 374)(159 336 211 375)(160 337 212 376)(221 279 411 343)(222 280 412 344)(223 261 413 345)(224 262 414 346)(225 263 415 347)(226 264 416 348)(227 265 417 349)(228 266 418 350)(229 267 419 351)(230 268 420 352)(231 269 401 353)(232 270 402 354)(233 271 403 355)(234 272 404 356)(235 273 405 357)(236 274 406 358)(237 275 407 359)(238 276 408 360)(239 277 409 341)(240 278 410 342)
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), (1,65,271,207,297,190,459,58,355,155,470,32)(2,64,272,206,298,189,460,57,356,154,471,31)(3,63,273,205,299,188,441,56,357,153,472,30)(4,62,274,204,300,187,442,55,358,152,473,29)(5,61,275,203,281,186,443,54,359,151,474,28)(6,80,276,202,282,185,444,53,360,150,475,27)(7,79,277,201,283,184,445,52,341,149,476,26)(8,78,278,220,284,183,446,51,342,148,477,25)(9,77,279,219,285,182,447,50,343,147,478,24)(10,76,280,218,286,181,448,49,344,146,479,23)(11,75,261,217,287,200,449,48,345,145,480,22)(12,74,262,216,288,199,450,47,346,144,461,21)(13,73,263,215,289,198,451,46,347,143,462,40)(14,72,264,214,290,197,452,45,348,142,463,39)(15,71,265,213,291,196,453,44,349,141,464,38)(16,70,266,212,292,195,454,43,350,160,465,37)(17,69,267,211,293,194,455,42,351,159,466,36)(18,68,268,210,294,193,456,41,352,158,467,35)(19,67,269,209,295,192,457,60,353,157,468,34)(20,66,270,208,296,191,458,59,354,156,469,33)(81,334,231,165,248,131,393,373,401,433,104,302)(82,333,232,164,249,130,394,372,402,432,105,301)(83,332,233,163,250,129,395,371,403,431,106,320)(84,331,234,162,251,128,396,370,404,430,107,319)(85,330,235,161,252,127,397,369,405,429,108,318)(86,329,236,180,253,126,398,368,406,428,109,317)(87,328,237,179,254,125,399,367,407,427,110,316)(88,327,238,178,255,124,400,366,408,426,111,315)(89,326,239,177,256,123,381,365,409,425,112,314)(90,325,240,176,257,122,382,364,410,424,113,313)(91,324,221,175,258,121,383,363,411,423,114,312)(92,323,222,174,259,140,384,362,412,422,115,311)(93,322,223,173,260,139,385,361,413,421,116,310)(94,321,224,172,241,138,386,380,414,440,117,309)(95,340,225,171,242,137,387,379,415,439,118,308)(96,339,226,170,243,136,388,378,416,438,119,307)(97,338,227,169,244,135,389,377,417,437,120,306)(98,337,228,168,245,134,390,376,418,436,101,305)(99,336,229,167,246,133,391,375,419,435,102,304)(100,335,230,166,247,132,392,374,420,434,103,303), (1,106,459,250)(2,107,460,251)(3,108,441,252)(4,109,442,253)(5,110,443,254)(6,111,444,255)(7,112,445,256)(8,113,446,257)(9,114,447,258)(10,115,448,259)(11,116,449,260)(12,117,450,241)(13,118,451,242)(14,119,452,243)(15,120,453,244)(16,101,454,245)(17,102,455,246)(18,103,456,247)(19,104,457,248)(20,105,458,249)(21,309,199,138)(22,310,200,139)(23,311,181,140)(24,312,182,121)(25,313,183,122)(26,314,184,123)(27,315,185,124)(28,316,186,125)(29,317,187,126)(30,318,188,127)(31,319,189,128)(32,320,190,129)(33,301,191,130)(34,302,192,131)(35,303,193,132)(36,304,194,133)(37,305,195,134)(38,306,196,135)(39,307,197,136)(40,308,198,137)(41,166,68,434)(42,167,69,435)(43,168,70,436)(44,169,71,437)(45,170,72,438)(46,171,73,439)(47,172,74,440)(48,173,75,421)(49,174,76,422)(50,175,77,423)(51,176,78,424)(52,177,79,425)(53,178,80,426)(54,179,61,427)(55,180,62,428)(56,161,63,429)(57,162,64,430)(58,163,65,431)(59,164,66,432)(60,165,67,433)(81,295,393,468)(82,296,394,469)(83,297,395,470)(84,298,396,471)(85,299,397,472)(86,300,398,473)(87,281,399,474)(88,282,400,475)(89,283,381,476)(90,284,382,477)(91,285,383,478)(92,286,384,479)(93,287,385,480)(94,288,386,461)(95,289,387,462)(96,290,388,463)(97,291,389,464)(98,292,390,465)(99,293,391,466)(100,294,392,467)(141,338,213,377)(142,339,214,378)(143,340,215,379)(144,321,216,380)(145,322,217,361)(146,323,218,362)(147,324,219,363)(148,325,220,364)(149,326,201,365)(150,327,202,366)(151,328,203,367)(152,329,204,368)(153,330,205,369)(154,331,206,370)(155,332,207,371)(156,333,208,372)(157,334,209,373)(158,335,210,374)(159,336,211,375)(160,337,212,376)(221,279,411,343)(222,280,412,344)(223,261,413,345)(224,262,414,346)(225,263,415,347)(226,264,416,348)(227,265,417,349)(228,266,418,350)(229,267,419,351)(230,268,420,352)(231,269,401,353)(232,270,402,354)(233,271,403,355)(234,272,404,356)(235,273,405,357)(236,274,406,358)(237,275,407,359)(238,276,408,360)(239,277,409,341)(240,278,410,342)>;
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), (1,65,271,207,297,190,459,58,355,155,470,32)(2,64,272,206,298,189,460,57,356,154,471,31)(3,63,273,205,299,188,441,56,357,153,472,30)(4,62,274,204,300,187,442,55,358,152,473,29)(5,61,275,203,281,186,443,54,359,151,474,28)(6,80,276,202,282,185,444,53,360,150,475,27)(7,79,277,201,283,184,445,52,341,149,476,26)(8,78,278,220,284,183,446,51,342,148,477,25)(9,77,279,219,285,182,447,50,343,147,478,24)(10,76,280,218,286,181,448,49,344,146,479,23)(11,75,261,217,287,200,449,48,345,145,480,22)(12,74,262,216,288,199,450,47,346,144,461,21)(13,73,263,215,289,198,451,46,347,143,462,40)(14,72,264,214,290,197,452,45,348,142,463,39)(15,71,265,213,291,196,453,44,349,141,464,38)(16,70,266,212,292,195,454,43,350,160,465,37)(17,69,267,211,293,194,455,42,351,159,466,36)(18,68,268,210,294,193,456,41,352,158,467,35)(19,67,269,209,295,192,457,60,353,157,468,34)(20,66,270,208,296,191,458,59,354,156,469,33)(81,334,231,165,248,131,393,373,401,433,104,302)(82,333,232,164,249,130,394,372,402,432,105,301)(83,332,233,163,250,129,395,371,403,431,106,320)(84,331,234,162,251,128,396,370,404,430,107,319)(85,330,235,161,252,127,397,369,405,429,108,318)(86,329,236,180,253,126,398,368,406,428,109,317)(87,328,237,179,254,125,399,367,407,427,110,316)(88,327,238,178,255,124,400,366,408,426,111,315)(89,326,239,177,256,123,381,365,409,425,112,314)(90,325,240,176,257,122,382,364,410,424,113,313)(91,324,221,175,258,121,383,363,411,423,114,312)(92,323,222,174,259,140,384,362,412,422,115,311)(93,322,223,173,260,139,385,361,413,421,116,310)(94,321,224,172,241,138,386,380,414,440,117,309)(95,340,225,171,242,137,387,379,415,439,118,308)(96,339,226,170,243,136,388,378,416,438,119,307)(97,338,227,169,244,135,389,377,417,437,120,306)(98,337,228,168,245,134,390,376,418,436,101,305)(99,336,229,167,246,133,391,375,419,435,102,304)(100,335,230,166,247,132,392,374,420,434,103,303), (1,106,459,250)(2,107,460,251)(3,108,441,252)(4,109,442,253)(5,110,443,254)(6,111,444,255)(7,112,445,256)(8,113,446,257)(9,114,447,258)(10,115,448,259)(11,116,449,260)(12,117,450,241)(13,118,451,242)(14,119,452,243)(15,120,453,244)(16,101,454,245)(17,102,455,246)(18,103,456,247)(19,104,457,248)(20,105,458,249)(21,309,199,138)(22,310,200,139)(23,311,181,140)(24,312,182,121)(25,313,183,122)(26,314,184,123)(27,315,185,124)(28,316,186,125)(29,317,187,126)(30,318,188,127)(31,319,189,128)(32,320,190,129)(33,301,191,130)(34,302,192,131)(35,303,193,132)(36,304,194,133)(37,305,195,134)(38,306,196,135)(39,307,197,136)(40,308,198,137)(41,166,68,434)(42,167,69,435)(43,168,70,436)(44,169,71,437)(45,170,72,438)(46,171,73,439)(47,172,74,440)(48,173,75,421)(49,174,76,422)(50,175,77,423)(51,176,78,424)(52,177,79,425)(53,178,80,426)(54,179,61,427)(55,180,62,428)(56,161,63,429)(57,162,64,430)(58,163,65,431)(59,164,66,432)(60,165,67,433)(81,295,393,468)(82,296,394,469)(83,297,395,470)(84,298,396,471)(85,299,397,472)(86,300,398,473)(87,281,399,474)(88,282,400,475)(89,283,381,476)(90,284,382,477)(91,285,383,478)(92,286,384,479)(93,287,385,480)(94,288,386,461)(95,289,387,462)(96,290,388,463)(97,291,389,464)(98,292,390,465)(99,293,391,466)(100,294,392,467)(141,338,213,377)(142,339,214,378)(143,340,215,379)(144,321,216,380)(145,322,217,361)(146,323,218,362)(147,324,219,363)(148,325,220,364)(149,326,201,365)(150,327,202,366)(151,328,203,367)(152,329,204,368)(153,330,205,369)(154,331,206,370)(155,332,207,371)(156,333,208,372)(157,334,209,373)(158,335,210,374)(159,336,211,375)(160,337,212,376)(221,279,411,343)(222,280,412,344)(223,261,413,345)(224,262,414,346)(225,263,415,347)(226,264,416,348)(227,265,417,349)(228,266,418,350)(229,267,419,351)(230,268,420,352)(231,269,401,353)(232,270,402,354)(233,271,403,355)(234,272,404,356)(235,273,405,357)(236,274,406,358)(237,275,407,359)(238,276,408,360)(239,277,409,341)(240,278,410,342) );
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)], [(1,65,271,207,297,190,459,58,355,155,470,32),(2,64,272,206,298,189,460,57,356,154,471,31),(3,63,273,205,299,188,441,56,357,153,472,30),(4,62,274,204,300,187,442,55,358,152,473,29),(5,61,275,203,281,186,443,54,359,151,474,28),(6,80,276,202,282,185,444,53,360,150,475,27),(7,79,277,201,283,184,445,52,341,149,476,26),(8,78,278,220,284,183,446,51,342,148,477,25),(9,77,279,219,285,182,447,50,343,147,478,24),(10,76,280,218,286,181,448,49,344,146,479,23),(11,75,261,217,287,200,449,48,345,145,480,22),(12,74,262,216,288,199,450,47,346,144,461,21),(13,73,263,215,289,198,451,46,347,143,462,40),(14,72,264,214,290,197,452,45,348,142,463,39),(15,71,265,213,291,196,453,44,349,141,464,38),(16,70,266,212,292,195,454,43,350,160,465,37),(17,69,267,211,293,194,455,42,351,159,466,36),(18,68,268,210,294,193,456,41,352,158,467,35),(19,67,269,209,295,192,457,60,353,157,468,34),(20,66,270,208,296,191,458,59,354,156,469,33),(81,334,231,165,248,131,393,373,401,433,104,302),(82,333,232,164,249,130,394,372,402,432,105,301),(83,332,233,163,250,129,395,371,403,431,106,320),(84,331,234,162,251,128,396,370,404,430,107,319),(85,330,235,161,252,127,397,369,405,429,108,318),(86,329,236,180,253,126,398,368,406,428,109,317),(87,328,237,179,254,125,399,367,407,427,110,316),(88,327,238,178,255,124,400,366,408,426,111,315),(89,326,239,177,256,123,381,365,409,425,112,314),(90,325,240,176,257,122,382,364,410,424,113,313),(91,324,221,175,258,121,383,363,411,423,114,312),(92,323,222,174,259,140,384,362,412,422,115,311),(93,322,223,173,260,139,385,361,413,421,116,310),(94,321,224,172,241,138,386,380,414,440,117,309),(95,340,225,171,242,137,387,379,415,439,118,308),(96,339,226,170,243,136,388,378,416,438,119,307),(97,338,227,169,244,135,389,377,417,437,120,306),(98,337,228,168,245,134,390,376,418,436,101,305),(99,336,229,167,246,133,391,375,419,435,102,304),(100,335,230,166,247,132,392,374,420,434,103,303)], [(1,106,459,250),(2,107,460,251),(3,108,441,252),(4,109,442,253),(5,110,443,254),(6,111,444,255),(7,112,445,256),(8,113,446,257),(9,114,447,258),(10,115,448,259),(11,116,449,260),(12,117,450,241),(13,118,451,242),(14,119,452,243),(15,120,453,244),(16,101,454,245),(17,102,455,246),(18,103,456,247),(19,104,457,248),(20,105,458,249),(21,309,199,138),(22,310,200,139),(23,311,181,140),(24,312,182,121),(25,313,183,122),(26,314,184,123),(27,315,185,124),(28,316,186,125),(29,317,187,126),(30,318,188,127),(31,319,189,128),(32,320,190,129),(33,301,191,130),(34,302,192,131),(35,303,193,132),(36,304,194,133),(37,305,195,134),(38,306,196,135),(39,307,197,136),(40,308,198,137),(41,166,68,434),(42,167,69,435),(43,168,70,436),(44,169,71,437),(45,170,72,438),(46,171,73,439),(47,172,74,440),(48,173,75,421),(49,174,76,422),(50,175,77,423),(51,176,78,424),(52,177,79,425),(53,178,80,426),(54,179,61,427),(55,180,62,428),(56,161,63,429),(57,162,64,430),(58,163,65,431),(59,164,66,432),(60,165,67,433),(81,295,393,468),(82,296,394,469),(83,297,395,470),(84,298,396,471),(85,299,397,472),(86,300,398,473),(87,281,399,474),(88,282,400,475),(89,283,381,476),(90,284,382,477),(91,285,383,478),(92,286,384,479),(93,287,385,480),(94,288,386,461),(95,289,387,462),(96,290,388,463),(97,291,389,464),(98,292,390,465),(99,293,391,466),(100,294,392,467),(141,338,213,377),(142,339,214,378),(143,340,215,379),(144,321,216,380),(145,322,217,361),(146,323,218,362),(147,324,219,363),(148,325,220,364),(149,326,201,365),(150,327,202,366),(151,328,203,367),(152,329,204,368),(153,330,205,369),(154,331,206,370),(155,332,207,371),(156,333,208,372),(157,334,209,373),(158,335,210,374),(159,336,211,375),(160,337,212,376),(221,279,411,343),(222,280,412,344),(223,261,413,345),(224,262,414,346),(225,263,415,347),(226,264,416,348),(227,265,417,349),(228,266,418,350),(229,267,419,351),(230,268,420,352),(231,269,401,353),(232,270,402,354),(233,271,403,355),(234,272,404,356),(235,273,405,357),(236,274,406,358),(237,275,407,359),(238,276,408,360),(239,277,409,341),(240,278,410,342)]])
72 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 | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 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 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 20 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | D4 | Q8 | Q8 | D5 | D6 | D6 | D10 | D10 | Dic6 | Dic10 | D20 | Dic10 | S3×D4 | S3×Q8 | S3×D5 | C15⋊Q8 | C2×S3×D5 | S3×Dic10 | S3×D20 |
| kernel | C20⋊4Dic6 | C6.Dic10 | C3×C4⋊Dic5 | Dic3×C20 | C60⋊5C4 | C2×C15⋊Q8 | C4⋊Dic5 | C5×Dic3 | C5×Dic3 | C60 | C4×Dic3 | C2×Dic5 | C2×C20 | C2×Dic3 | C2×C12 | C20 | Dic3 | Dic3 | C12 | C10 | C10 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C20⋊4Dic6 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 41 |
| 0 | 0 | 0 | 0 | 42 | 30 |
| 0 | 30 | 0 | 0 | 0 | 0 |
| 2 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 14 | 0 | 0 |
| 0 | 0 | 28 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 55 |
| 0 | 0 | 0 | 0 | 43 | 47 |
| 29 | 48 | 0 | 0 | 0 | 0 |
| 46 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 41 |
| 0 | 0 | 0 | 0 | 42 | 30 |
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] >;
C20⋊4Dic6 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