Copied to
clipboard

G = C20×Dic6order 480 = 25·3·5

Direct product of C20 and Dic6

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20×Dic6, C6013Q8, C31(Q8×C20), C123(C5×Q8), C1517(C4×Q8), C4.9(S3×C20), (C4×C20).9S3, C6.1(Q8×C10), C20.98(C4×S3), (C4×C60).15C2, (C4×C12).5C10, C42.3(C5×S3), C30.82(C2×Q8), C60.206(C2×C4), C12.19(C2×C20), (C2×C20).425D6, C6.1(C22×C20), C2.1(C10×Dic6), Dic3⋊C4.7C10, C4⋊Dic3.13C10, (C4×Dic3).8C10, Dic3.1(C2×C20), C10.41(C2×Dic6), C30.197(C4○D4), C30.192(C22×C4), (C2×C60).449C22, (C2×C30).388C23, (Dic3×C20).19C2, (C2×Dic6).10C10, (C10×Dic6).20C2, C10.108(C4○D12), (C10×Dic3).211C22, C2.4(S3×C2×C20), C6.1(C5×C4○D4), C10.128(S3×C2×C4), C2.1(C5×C4○D12), C22.8(S3×C2×C10), (C2×C4).75(S3×C10), (C2×C12).71(C2×C10), (C5×C4⋊Dic3).27C2, (C2×C6).9(C22×C10), (C5×Dic3⋊C4).17C2, (C5×Dic3).37(C2×C4), (C2×C10).322(C22×S3), (C2×Dic3).17(C2×C10), SmallGroup(480,747)

Series: Derived Chief Lower central Upper central

C1C6 — C20×Dic6
C1C3C6C2×C6C2×C30C10×Dic3C10×Dic6 — C20×Dic6
C3C6 — C20×Dic6
C1C2×C20C4×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 [×3], C3, C4 [×4], C4 [×7], C22, C5, C6 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×4], C10 [×3], Dic3 [×4], Dic3 [×2], C12 [×4], C12, C2×C6, C15, C42, C42 [×2], C4⋊C4 [×3], C2×Q8, C20 [×4], C20 [×7], C2×C10, Dic6 [×4], C2×Dic3 [×4], C2×C12 [×3], C30 [×3], C4×Q8, C2×C20 [×3], C2×C20 [×4], C5×Q8 [×4], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3 [×4], C5×Dic3 [×2], C60 [×4], C60, C2×C30, C4×C20, C4×C20 [×2], C5×C4⋊C4 [×3], Q8×C10, C4×Dic6, C5×Dic6 [×4], C10×Dic3 [×4], C2×C60 [×3], Q8×C20, Dic3×C20 [×2], C5×Dic3⋊C4 [×2], C5×C4⋊Dic3, C4×C60, C10×Dic6, C20×Dic6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], Q8 [×2], C23, C10 [×7], D6 [×3], C22×C4, C2×Q8, C4○D4, C20 [×4], C2×C10 [×7], Dic6 [×2], C4×S3 [×2], C22×S3, C5×S3, C4×Q8, C2×C20 [×6], C5×Q8 [×2], C22×C10, C2×Dic6, S3×C2×C4, C4○D12, S3×C10 [×3], C22×C20, Q8×C10, C5×C4○D4, C4×Dic6, C5×Dic6 [×2], S3×C20 [×2], S3×C2×C10, Q8×C20, C10×Dic6, S3×C2×C20, C5×C4○D12, 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)
(1 156 359 214 452 334 279 320 90 163 52 379)(2 157 360 215 453 335 280 301 91 164 53 380)(3 158 341 216 454 336 261 302 92 165 54 361)(4 159 342 217 455 337 262 303 93 166 55 362)(5 160 343 218 456 338 263 304 94 167 56 363)(6 141 344 219 457 339 264 305 95 168 57 364)(7 142 345 220 458 340 265 306 96 169 58 365)(8 143 346 201 459 321 266 307 97 170 59 366)(9 144 347 202 460 322 267 308 98 171 60 367)(10 145 348 203 441 323 268 309 99 172 41 368)(11 146 349 204 442 324 269 310 100 173 42 369)(12 147 350 205 443 325 270 311 81 174 43 370)(13 148 351 206 444 326 271 312 82 175 44 371)(14 149 352 207 445 327 272 313 83 176 45 372)(15 150 353 208 446 328 273 314 84 177 46 373)(16 151 354 209 447 329 274 315 85 178 47 374)(17 152 355 210 448 330 275 316 86 179 48 375)(18 153 356 211 449 331 276 317 87 180 49 376)(19 154 357 212 450 332 277 318 88 161 50 377)(20 155 358 213 451 333 278 319 89 162 51 378)(21 244 237 478 126 68 403 423 289 199 391 110)(22 245 238 479 127 69 404 424 290 200 392 111)(23 246 239 480 128 70 405 425 291 181 393 112)(24 247 240 461 129 71 406 426 292 182 394 113)(25 248 221 462 130 72 407 427 293 183 395 114)(26 249 222 463 131 73 408 428 294 184 396 115)(27 250 223 464 132 74 409 429 295 185 397 116)(28 251 224 465 133 75 410 430 296 186 398 117)(29 252 225 466 134 76 411 431 297 187 399 118)(30 253 226 467 135 77 412 432 298 188 400 119)(31 254 227 468 136 78 413 433 299 189 381 120)(32 255 228 469 137 79 414 434 300 190 382 101)(33 256 229 470 138 80 415 435 281 191 383 102)(34 257 230 471 139 61 416 436 282 192 384 103)(35 258 231 472 140 62 417 437 283 193 385 104)(36 259 232 473 121 63 418 438 284 194 386 105)(37 260 233 474 122 64 419 439 285 195 387 106)(38 241 234 475 123 65 420 440 286 196 388 107)(39 242 235 476 124 66 401 421 287 197 389 108)(40 243 236 477 125 67 402 422 288 198 390 109)
(1 65 279 107)(2 66 280 108)(3 67 261 109)(4 68 262 110)(5 69 263 111)(6 70 264 112)(7 71 265 113)(8 72 266 114)(9 73 267 115)(10 74 268 116)(11 75 269 117)(12 76 270 118)(13 77 271 119)(14 78 272 120)(15 79 273 101)(16 80 274 102)(17 61 275 103)(18 62 276 104)(19 63 277 105)(20 64 278 106)(21 362 403 337)(22 363 404 338)(23 364 405 339)(24 365 406 340)(25 366 407 321)(26 367 408 322)(27 368 409 323)(28 369 410 324)(29 370 411 325)(30 371 412 326)(31 372 413 327)(32 373 414 328)(33 374 415 329)(34 375 416 330)(35 376 417 331)(36 377 418 332)(37 378 419 333)(38 379 420 334)(39 380 401 335)(40 361 402 336)(41 429 441 250)(42 430 442 251)(43 431 443 252)(44 432 444 253)(45 433 445 254)(46 434 446 255)(47 435 447 256)(48 436 448 257)(49 437 449 258)(50 438 450 259)(51 439 451 260)(52 440 452 241)(53 421 453 242)(54 422 454 243)(55 423 455 244)(56 424 456 245)(57 425 457 246)(58 426 458 247)(59 427 459 248)(60 428 460 249)(81 187 350 466)(82 188 351 467)(83 189 352 468)(84 190 353 469)(85 191 354 470)(86 192 355 471)(87 193 356 472)(88 194 357 473)(89 195 358 474)(90 196 359 475)(91 197 360 476)(92 198 341 477)(93 199 342 478)(94 200 343 479)(95 181 344 480)(96 182 345 461)(97 183 346 462)(98 184 347 463)(99 185 348 464)(100 186 349 465)(121 318 386 154)(122 319 387 155)(123 320 388 156)(124 301 389 157)(125 302 390 158)(126 303 391 159)(127 304 392 160)(128 305 393 141)(129 306 394 142)(130 307 395 143)(131 308 396 144)(132 309 397 145)(133 310 398 146)(134 311 399 147)(135 312 400 148)(136 313 381 149)(137 314 382 150)(138 315 383 151)(139 316 384 152)(140 317 385 153)(161 284 212 232)(162 285 213 233)(163 286 214 234)(164 287 215 235)(165 288 216 236)(166 289 217 237)(167 290 218 238)(168 291 219 239)(169 292 220 240)(170 293 201 221)(171 294 202 222)(172 295 203 223)(173 296 204 224)(174 297 205 225)(175 298 206 226)(176 299 207 227)(177 300 208 228)(178 281 209 229)(179 282 210 230)(180 283 211 231)

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

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

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

180 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I···4P5A5B5C5D6A6B6C10A···10L12A···12L15A15B15C15D20A···20P20Q···20AF20AG···20BL30A···30L60A···60AV
order12223444444444···4555566610···1012···121515151520···2020···2020···2030···3060···60
size11112111122226···611112221···12···222221···12···26···62···22···2

180 irreducible representations

dim1111111111111122222222222222
type+++++++-+-
imageC1C2C2C2C2C2C4C5C10C10C10C10C10C20S3Q8D6C4○D4Dic6C4×S3C5×S3C5×Q8C4○D12S3×C10C5×C4○D4C5×Dic6S3×C20C5×C4○D12
kernelC20×Dic6Dic3×C20C5×Dic3⋊C4C5×C4⋊Dic3C4×C60C10×Dic6C5×Dic6C4×Dic6C4×Dic3Dic3⋊C4C4⋊Dic3C4×C12C2×Dic6Dic6C4×C20C60C2×C20C30C20C20C42C12C10C2×C4C6C4C4C2
# reps122111848844432123244484128161616

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

1100
0580
0058
,
100
04623
03823
,
100
0050
0500
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

׿
×
𝔽