metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.8D12, C60.79D4, C30.3Q16, C30.9SD16, Dic6⋊3Dic5, C20.36(C4×S3), (C2×C30).19D4, (C2×C20).51D6, C6.5(Q8⋊D5), C4⋊Dic5.8S3, C4.7(S3×Dic5), C60.123(C2×C4), (C5×Dic6)⋊10C4, C15⋊5(Q8⋊C4), (C2×C12).53D10, C3⋊2(Q8⋊Dic5), C5⋊4(C6.Q16), (C2×Dic6).6D5, C12.2(C2×Dic5), C6.5(C5⋊Q16), C10.43(D6⋊C4), C4.22(C5⋊D12), C12.15(C5⋊D4), C2.7(D6⋊Dic5), C10.5(D4.S3), (C10×Dic6).6C2, C2.2(C15⋊Q16), C10.5(C3⋊Q16), C6.6(C23.D5), C30.55(C22⋊C4), (C2×C60).184C22, C2.2(C20.D6), C22.15(C15⋊D4), (C2×C4).187(S3×D5), (C3×C4⋊Dic5).7C2, (C2×C15⋊3C8).12C2, (C2×C6).46(C5⋊D4), (C2×C10).46(C3⋊D4), SmallGroup(480,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.8D12
G = < a,b,c | a20=b12=1, c2=a5, bab-1=a-1, cac-1=a9, cbc-1=a15b-1 >
Subgroups: 300 in 84 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C10 [×3], Dic3 [×2], C12 [×2], C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20 [×2], C20 [×2], C2×C10, C3⋊C8, Dic6 [×2], Dic6, C2×Dic3, C2×C12, C2×C12, C30 [×3], Q8⋊C4, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C5×Q8 [×3], C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C5×Dic3 [×2], C3×Dic5, C60 [×2], C2×C30, C2×C5⋊2C8, C4⋊Dic5, Q8×C10, C6.Q16, C15⋊3C8, C6×Dic5, C5×Dic6 [×2], C5×Dic6, C10×Dic3, C2×C60, Q8⋊Dic5, C3×C4⋊Dic5, C2×C15⋊3C8, C10×Dic6, C20.8D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, SD16, Q16, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, Q8⋊C4, C2×Dic5, C5⋊D4 [×2], D6⋊C4, D4.S3, C3⋊Q16, S3×D5, Q8⋊D5, C5⋊Q16, C23.D5, C6.Q16, S3×Dic5, C15⋊D4, C5⋊D12, Q8⋊Dic5, C20.D6, C15⋊Q16, D6⋊Dic5, C20.8D12
►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)
(1 319 63 235 424 247 464 178 32 287 338 149)(2 318 64 234 425 246 465 177 33 286 339 148)(3 317 65 233 426 245 466 176 34 285 340 147)(4 316 66 232 427 244 467 175 35 284 321 146)(5 315 67 231 428 243 468 174 36 283 322 145)(6 314 68 230 429 242 469 173 37 282 323 144)(7 313 69 229 430 241 470 172 38 281 324 143)(8 312 70 228 431 260 471 171 39 300 325 142)(9 311 71 227 432 259 472 170 40 299 326 141)(10 310 72 226 433 258 473 169 21 298 327 160)(11 309 73 225 434 257 474 168 22 297 328 159)(12 308 74 224 435 256 475 167 23 296 329 158)(13 307 75 223 436 255 476 166 24 295 330 157)(14 306 76 222 437 254 477 165 25 294 331 156)(15 305 77 221 438 253 478 164 26 293 332 155)(16 304 78 240 439 252 479 163 27 292 333 154)(17 303 79 239 440 251 480 162 28 291 334 153)(18 302 80 238 421 250 461 161 29 290 335 152)(19 301 61 237 422 249 462 180 30 289 336 151)(20 320 62 236 423 248 463 179 31 288 337 150)(41 136 182 370 453 93 382 110 342 271 207 405)(42 135 183 369 454 92 383 109 343 270 208 404)(43 134 184 368 455 91 384 108 344 269 209 403)(44 133 185 367 456 90 385 107 345 268 210 402)(45 132 186 366 457 89 386 106 346 267 211 401)(46 131 187 365 458 88 387 105 347 266 212 420)(47 130 188 364 459 87 388 104 348 265 213 419)(48 129 189 363 460 86 389 103 349 264 214 418)(49 128 190 362 441 85 390 102 350 263 215 417)(50 127 191 361 442 84 391 101 351 262 216 416)(51 126 192 380 443 83 392 120 352 261 217 415)(52 125 193 379 444 82 393 119 353 280 218 414)(53 124 194 378 445 81 394 118 354 279 219 413)(54 123 195 377 446 100 395 117 355 278 220 412)(55 122 196 376 447 99 396 116 356 277 201 411)(56 121 197 375 448 98 397 115 357 276 202 410)(57 140 198 374 449 97 398 114 358 275 203 409)(58 139 199 373 450 96 399 113 359 274 204 408)(59 138 200 372 451 95 400 112 360 273 205 407)(60 137 181 371 452 94 381 111 341 272 206 406)
(1 41 6 46 11 51 16 56)(2 50 7 55 12 60 17 45)(3 59 8 44 13 49 18 54)(4 48 9 53 14 58 19 43)(5 57 10 42 15 47 20 52)(21 454 26 459 31 444 36 449)(22 443 27 448 32 453 37 458)(23 452 28 457 33 442 38 447)(24 441 29 446 34 451 39 456)(25 450 30 455 35 460 40 445)(61 209 66 214 71 219 76 204)(62 218 67 203 72 208 77 213)(63 207 68 212 73 217 78 202)(64 216 69 201 74 206 79 211)(65 205 70 210 75 215 80 220)(81 180 86 165 91 170 96 175)(82 169 87 174 92 179 97 164)(83 178 88 163 93 168 98 173)(84 167 89 172 94 177 99 162)(85 176 90 161 95 166 100 171)(101 256 106 241 111 246 116 251)(102 245 107 250 112 255 117 260)(103 254 108 259 113 244 118 249)(104 243 109 248 114 253 119 258)(105 252 110 257 115 242 120 247)(121 144 126 149 131 154 136 159)(122 153 127 158 132 143 137 148)(123 142 128 147 133 152 138 157)(124 151 129 156 134 141 139 146)(125 160 130 145 135 150 140 155)(181 334 186 339 191 324 196 329)(182 323 187 328 192 333 197 338)(183 332 188 337 193 322 198 327)(184 321 189 326 194 331 199 336)(185 330 190 335 195 340 200 325)(221 280 226 265 231 270 236 275)(222 269 227 274 232 279 237 264)(223 278 228 263 233 268 238 273)(224 267 229 272 234 277 239 262)(225 276 230 261 235 266 240 271)(281 371 286 376 291 361 296 366)(282 380 287 365 292 370 297 375)(283 369 288 374 293 379 298 364)(284 378 289 363 294 368 299 373)(285 367 290 372 295 377 300 362)(301 418 306 403 311 408 316 413)(302 407 307 412 312 417 317 402)(303 416 308 401 313 406 318 411)(304 405 309 410 314 415 319 420)(305 414 310 419 315 404 320 409)(341 440 346 425 351 430 356 435)(342 429 347 434 352 439 357 424)(343 438 348 423 353 428 358 433)(344 427 349 432 354 437 359 422)(345 436 350 421 355 426 360 431)(381 480 386 465 391 470 396 475)(382 469 387 474 392 479 397 464)(383 478 388 463 393 468 398 473)(384 467 389 472 394 477 399 462)(385 476 390 461 395 466 400 471)
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), (1,319,63,235,424,247,464,178,32,287,338,149)(2,318,64,234,425,246,465,177,33,286,339,148)(3,317,65,233,426,245,466,176,34,285,340,147)(4,316,66,232,427,244,467,175,35,284,321,146)(5,315,67,231,428,243,468,174,36,283,322,145)(6,314,68,230,429,242,469,173,37,282,323,144)(7,313,69,229,430,241,470,172,38,281,324,143)(8,312,70,228,431,260,471,171,39,300,325,142)(9,311,71,227,432,259,472,170,40,299,326,141)(10,310,72,226,433,258,473,169,21,298,327,160)(11,309,73,225,434,257,474,168,22,297,328,159)(12,308,74,224,435,256,475,167,23,296,329,158)(13,307,75,223,436,255,476,166,24,295,330,157)(14,306,76,222,437,254,477,165,25,294,331,156)(15,305,77,221,438,253,478,164,26,293,332,155)(16,304,78,240,439,252,479,163,27,292,333,154)(17,303,79,239,440,251,480,162,28,291,334,153)(18,302,80,238,421,250,461,161,29,290,335,152)(19,301,61,237,422,249,462,180,30,289,336,151)(20,320,62,236,423,248,463,179,31,288,337,150)(41,136,182,370,453,93,382,110,342,271,207,405)(42,135,183,369,454,92,383,109,343,270,208,404)(43,134,184,368,455,91,384,108,344,269,209,403)(44,133,185,367,456,90,385,107,345,268,210,402)(45,132,186,366,457,89,386,106,346,267,211,401)(46,131,187,365,458,88,387,105,347,266,212,420)(47,130,188,364,459,87,388,104,348,265,213,419)(48,129,189,363,460,86,389,103,349,264,214,418)(49,128,190,362,441,85,390,102,350,263,215,417)(50,127,191,361,442,84,391,101,351,262,216,416)(51,126,192,380,443,83,392,120,352,261,217,415)(52,125,193,379,444,82,393,119,353,280,218,414)(53,124,194,378,445,81,394,118,354,279,219,413)(54,123,195,377,446,100,395,117,355,278,220,412)(55,122,196,376,447,99,396,116,356,277,201,411)(56,121,197,375,448,98,397,115,357,276,202,410)(57,140,198,374,449,97,398,114,358,275,203,409)(58,139,199,373,450,96,399,113,359,274,204,408)(59,138,200,372,451,95,400,112,360,273,205,407)(60,137,181,371,452,94,381,111,341,272,206,406), (1,41,6,46,11,51,16,56)(2,50,7,55,12,60,17,45)(3,59,8,44,13,49,18,54)(4,48,9,53,14,58,19,43)(5,57,10,42,15,47,20,52)(21,454,26,459,31,444,36,449)(22,443,27,448,32,453,37,458)(23,452,28,457,33,442,38,447)(24,441,29,446,34,451,39,456)(25,450,30,455,35,460,40,445)(61,209,66,214,71,219,76,204)(62,218,67,203,72,208,77,213)(63,207,68,212,73,217,78,202)(64,216,69,201,74,206,79,211)(65,205,70,210,75,215,80,220)(81,180,86,165,91,170,96,175)(82,169,87,174,92,179,97,164)(83,178,88,163,93,168,98,173)(84,167,89,172,94,177,99,162)(85,176,90,161,95,166,100,171)(101,256,106,241,111,246,116,251)(102,245,107,250,112,255,117,260)(103,254,108,259,113,244,118,249)(104,243,109,248,114,253,119,258)(105,252,110,257,115,242,120,247)(121,144,126,149,131,154,136,159)(122,153,127,158,132,143,137,148)(123,142,128,147,133,152,138,157)(124,151,129,156,134,141,139,146)(125,160,130,145,135,150,140,155)(181,334,186,339,191,324,196,329)(182,323,187,328,192,333,197,338)(183,332,188,337,193,322,198,327)(184,321,189,326,194,331,199,336)(185,330,190,335,195,340,200,325)(221,280,226,265,231,270,236,275)(222,269,227,274,232,279,237,264)(223,278,228,263,233,268,238,273)(224,267,229,272,234,277,239,262)(225,276,230,261,235,266,240,271)(281,371,286,376,291,361,296,366)(282,380,287,365,292,370,297,375)(283,369,288,374,293,379,298,364)(284,378,289,363,294,368,299,373)(285,367,290,372,295,377,300,362)(301,418,306,403,311,408,316,413)(302,407,307,412,312,417,317,402)(303,416,308,401,313,406,318,411)(304,405,309,410,314,415,319,420)(305,414,310,419,315,404,320,409)(341,440,346,425,351,430,356,435)(342,429,347,434,352,439,357,424)(343,438,348,423,353,428,358,433)(344,427,349,432,354,437,359,422)(345,436,350,421,355,426,360,431)(381,480,386,465,391,470,396,475)(382,469,387,474,392,479,397,464)(383,478,388,463,393,468,398,473)(384,467,389,472,394,477,399,462)(385,476,390,461,395,466,400,471)>;
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), (1,319,63,235,424,247,464,178,32,287,338,149)(2,318,64,234,425,246,465,177,33,286,339,148)(3,317,65,233,426,245,466,176,34,285,340,147)(4,316,66,232,427,244,467,175,35,284,321,146)(5,315,67,231,428,243,468,174,36,283,322,145)(6,314,68,230,429,242,469,173,37,282,323,144)(7,313,69,229,430,241,470,172,38,281,324,143)(8,312,70,228,431,260,471,171,39,300,325,142)(9,311,71,227,432,259,472,170,40,299,326,141)(10,310,72,226,433,258,473,169,21,298,327,160)(11,309,73,225,434,257,474,168,22,297,328,159)(12,308,74,224,435,256,475,167,23,296,329,158)(13,307,75,223,436,255,476,166,24,295,330,157)(14,306,76,222,437,254,477,165,25,294,331,156)(15,305,77,221,438,253,478,164,26,293,332,155)(16,304,78,240,439,252,479,163,27,292,333,154)(17,303,79,239,440,251,480,162,28,291,334,153)(18,302,80,238,421,250,461,161,29,290,335,152)(19,301,61,237,422,249,462,180,30,289,336,151)(20,320,62,236,423,248,463,179,31,288,337,150)(41,136,182,370,453,93,382,110,342,271,207,405)(42,135,183,369,454,92,383,109,343,270,208,404)(43,134,184,368,455,91,384,108,344,269,209,403)(44,133,185,367,456,90,385,107,345,268,210,402)(45,132,186,366,457,89,386,106,346,267,211,401)(46,131,187,365,458,88,387,105,347,266,212,420)(47,130,188,364,459,87,388,104,348,265,213,419)(48,129,189,363,460,86,389,103,349,264,214,418)(49,128,190,362,441,85,390,102,350,263,215,417)(50,127,191,361,442,84,391,101,351,262,216,416)(51,126,192,380,443,83,392,120,352,261,217,415)(52,125,193,379,444,82,393,119,353,280,218,414)(53,124,194,378,445,81,394,118,354,279,219,413)(54,123,195,377,446,100,395,117,355,278,220,412)(55,122,196,376,447,99,396,116,356,277,201,411)(56,121,197,375,448,98,397,115,357,276,202,410)(57,140,198,374,449,97,398,114,358,275,203,409)(58,139,199,373,450,96,399,113,359,274,204,408)(59,138,200,372,451,95,400,112,360,273,205,407)(60,137,181,371,452,94,381,111,341,272,206,406), (1,41,6,46,11,51,16,56)(2,50,7,55,12,60,17,45)(3,59,8,44,13,49,18,54)(4,48,9,53,14,58,19,43)(5,57,10,42,15,47,20,52)(21,454,26,459,31,444,36,449)(22,443,27,448,32,453,37,458)(23,452,28,457,33,442,38,447)(24,441,29,446,34,451,39,456)(25,450,30,455,35,460,40,445)(61,209,66,214,71,219,76,204)(62,218,67,203,72,208,77,213)(63,207,68,212,73,217,78,202)(64,216,69,201,74,206,79,211)(65,205,70,210,75,215,80,220)(81,180,86,165,91,170,96,175)(82,169,87,174,92,179,97,164)(83,178,88,163,93,168,98,173)(84,167,89,172,94,177,99,162)(85,176,90,161,95,166,100,171)(101,256,106,241,111,246,116,251)(102,245,107,250,112,255,117,260)(103,254,108,259,113,244,118,249)(104,243,109,248,114,253,119,258)(105,252,110,257,115,242,120,247)(121,144,126,149,131,154,136,159)(122,153,127,158,132,143,137,148)(123,142,128,147,133,152,138,157)(124,151,129,156,134,141,139,146)(125,160,130,145,135,150,140,155)(181,334,186,339,191,324,196,329)(182,323,187,328,192,333,197,338)(183,332,188,337,193,322,198,327)(184,321,189,326,194,331,199,336)(185,330,190,335,195,340,200,325)(221,280,226,265,231,270,236,275)(222,269,227,274,232,279,237,264)(223,278,228,263,233,268,238,273)(224,267,229,272,234,277,239,262)(225,276,230,261,235,266,240,271)(281,371,286,376,291,361,296,366)(282,380,287,365,292,370,297,375)(283,369,288,374,293,379,298,364)(284,378,289,363,294,368,299,373)(285,367,290,372,295,377,300,362)(301,418,306,403,311,408,316,413)(302,407,307,412,312,417,317,402)(303,416,308,401,313,406,318,411)(304,405,309,410,314,415,319,420)(305,414,310,419,315,404,320,409)(341,440,346,425,351,430,356,435)(342,429,347,434,352,439,357,424)(343,438,348,423,353,428,358,433)(344,427,349,432,354,437,359,422)(345,436,350,421,355,426,360,431)(381,480,386,465,391,470,396,475)(382,469,387,474,392,479,397,464)(383,478,388,463,393,468,398,473)(384,467,389,472,394,477,399,462)(385,476,390,461,395,466,400,471) );
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)], [(1,319,63,235,424,247,464,178,32,287,338,149),(2,318,64,234,425,246,465,177,33,286,339,148),(3,317,65,233,426,245,466,176,34,285,340,147),(4,316,66,232,427,244,467,175,35,284,321,146),(5,315,67,231,428,243,468,174,36,283,322,145),(6,314,68,230,429,242,469,173,37,282,323,144),(7,313,69,229,430,241,470,172,38,281,324,143),(8,312,70,228,431,260,471,171,39,300,325,142),(9,311,71,227,432,259,472,170,40,299,326,141),(10,310,72,226,433,258,473,169,21,298,327,160),(11,309,73,225,434,257,474,168,22,297,328,159),(12,308,74,224,435,256,475,167,23,296,329,158),(13,307,75,223,436,255,476,166,24,295,330,157),(14,306,76,222,437,254,477,165,25,294,331,156),(15,305,77,221,438,253,478,164,26,293,332,155),(16,304,78,240,439,252,479,163,27,292,333,154),(17,303,79,239,440,251,480,162,28,291,334,153),(18,302,80,238,421,250,461,161,29,290,335,152),(19,301,61,237,422,249,462,180,30,289,336,151),(20,320,62,236,423,248,463,179,31,288,337,150),(41,136,182,370,453,93,382,110,342,271,207,405),(42,135,183,369,454,92,383,109,343,270,208,404),(43,134,184,368,455,91,384,108,344,269,209,403),(44,133,185,367,456,90,385,107,345,268,210,402),(45,132,186,366,457,89,386,106,346,267,211,401),(46,131,187,365,458,88,387,105,347,266,212,420),(47,130,188,364,459,87,388,104,348,265,213,419),(48,129,189,363,460,86,389,103,349,264,214,418),(49,128,190,362,441,85,390,102,350,263,215,417),(50,127,191,361,442,84,391,101,351,262,216,416),(51,126,192,380,443,83,392,120,352,261,217,415),(52,125,193,379,444,82,393,119,353,280,218,414),(53,124,194,378,445,81,394,118,354,279,219,413),(54,123,195,377,446,100,395,117,355,278,220,412),(55,122,196,376,447,99,396,116,356,277,201,411),(56,121,197,375,448,98,397,115,357,276,202,410),(57,140,198,374,449,97,398,114,358,275,203,409),(58,139,199,373,450,96,399,113,359,274,204,408),(59,138,200,372,451,95,400,112,360,273,205,407),(60,137,181,371,452,94,381,111,341,272,206,406)], [(1,41,6,46,11,51,16,56),(2,50,7,55,12,60,17,45),(3,59,8,44,13,49,18,54),(4,48,9,53,14,58,19,43),(5,57,10,42,15,47,20,52),(21,454,26,459,31,444,36,449),(22,443,27,448,32,453,37,458),(23,452,28,457,33,442,38,447),(24,441,29,446,34,451,39,456),(25,450,30,455,35,460,40,445),(61,209,66,214,71,219,76,204),(62,218,67,203,72,208,77,213),(63,207,68,212,73,217,78,202),(64,216,69,201,74,206,79,211),(65,205,70,210,75,215,80,220),(81,180,86,165,91,170,96,175),(82,169,87,174,92,179,97,164),(83,178,88,163,93,168,98,173),(84,167,89,172,94,177,99,162),(85,176,90,161,95,166,100,171),(101,256,106,241,111,246,116,251),(102,245,107,250,112,255,117,260),(103,254,108,259,113,244,118,249),(104,243,109,248,114,253,119,258),(105,252,110,257,115,242,120,247),(121,144,126,149,131,154,136,159),(122,153,127,158,132,143,137,148),(123,142,128,147,133,152,138,157),(124,151,129,156,134,141,139,146),(125,160,130,145,135,150,140,155),(181,334,186,339,191,324,196,329),(182,323,187,328,192,333,197,338),(183,332,188,337,193,322,198,327),(184,321,189,326,194,331,199,336),(185,330,190,335,195,340,200,325),(221,280,226,265,231,270,236,275),(222,269,227,274,232,279,237,264),(223,278,228,263,233,268,238,273),(224,267,229,272,234,277,239,262),(225,276,230,261,235,266,240,271),(281,371,286,376,291,361,296,366),(282,380,287,365,292,370,297,375),(283,369,288,374,293,379,298,364),(284,378,289,363,294,368,299,373),(285,367,290,372,295,377,300,362),(301,418,306,403,311,408,316,413),(302,407,307,412,312,417,317,402),(303,416,308,401,313,406,318,411),(304,405,309,410,314,415,319,420),(305,414,310,419,315,404,320,409),(341,440,346,425,351,430,356,435),(342,429,347,434,352,439,357,424),(343,438,348,423,353,428,358,433),(344,427,349,432,354,437,359,422),(345,436,350,421,355,426,360,431),(381,480,386,465,391,470,396,475),(382,469,387,474,392,479,397,464),(383,478,388,463,393,468,398,473),(384,467,389,472,394,477,399,462),(385,476,390,461,395,466,400,471)])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 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 | 12 | 12 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | + | - | - | + | + | - | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D5 | D6 | SD16 | Q16 | Dic5 | D10 | C4×S3 | D12 | C3⋊D4 | C5⋊D4 | C5⋊D4 | D4.S3 | C3⋊Q16 | S3×D5 | Q8⋊D5 | C5⋊Q16 | S3×Dic5 | C5⋊D12 | C15⋊D4 | C20.D6 | C15⋊Q16 |
| kernel | C20.8D12 | C3×C4⋊Dic5 | C2×C15⋊3C8 | C10×Dic6 | C5×Dic6 | C4⋊Dic5 | C60 | C2×C30 | C2×Dic6 | C2×C20 | C30 | C30 | Dic6 | C2×C12 | C20 | C20 | C2×C10 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C20.8D12 ►in GL6(𝔽241)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 17 | 154 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 218 |
| 0 | 0 | 0 | 0 | 42 | 240 |
| 182 | 115 | 0 | 0 | 0 | 0 |
| 173 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 225 | 236 | 0 | 0 |
| 0 | 0 | 0 | 226 | 0 | 0 |
| 0 | 0 | 0 | 0 | 94 | 158 |
| 0 | 0 | 0 | 0 | 150 | 147 |
| 182 | 115 | 0 | 0 | 0 | 0 |
| 173 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 137 | 196 | 0 | 0 |
| 0 | 0 | 69 | 104 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 45 |
| 0 | 0 | 0 | 0 | 75 | 203 |
G:=sub<GL(6,GF(241))| [36,17,0,0,0,0,0,154,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,42,0,0,0,0,218,240],[182,173,0,0,0,0,115,59,0,0,0,0,0,0,225,0,0,0,0,0,236,226,0,0,0,0,0,0,94,150,0,0,0,0,158,147],[182,173,0,0,0,0,115,59,0,0,0,0,0,0,137,69,0,0,0,0,196,104,0,0,0,0,0,0,0,75,0,0,0,0,45,203] >;
C20.8D12 in GAP, Magma, Sage, TeX
C_{20}._8D_{12} % in TeX
G:=Group("C20.8D12"); // GroupNames label
G:=SmallGroup(480,48);
// by ID
G=gap.SmallGroup(480,48);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^20=b^12=1,c^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^15*b^-1>;
// generators/relations