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## G = Dic3×C2×C20order 480 = 25·3·5

### Direct product of C2×C20 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C2×C20
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — Dic3×C2×C10 — Dic3×C2×C20
 Lower central C3 — Dic3×C2×C20
 Upper central C1 — C22×C20

Generators and relations for Dic3×C2×C20
G = < a,b,c,d | a2=b20=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 324 in 216 conjugacy classes, 162 normal (22 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C4 [×8], C22, C22 [×6], C5, C6, C6 [×6], C2×C4 [×6], C2×C4 [×12], C23, C10, C10 [×6], Dic3 [×8], C12 [×4], C2×C6, C2×C6 [×6], C15, C42 [×4], C22×C4, C22×C4 [×2], C20 [×4], C20 [×8], C2×C10, C2×C10 [×6], C2×Dic3 [×12], C2×C12 [×6], C22×C6, C30, C30 [×6], C2×C42, C2×C20 [×6], C2×C20 [×12], C22×C10, C4×Dic3 [×4], C22×Dic3 [×2], C22×C12, C5×Dic3 [×8], C60 [×4], C2×C30, C2×C30 [×6], C4×C20 [×4], C22×C20, C22×C20 [×2], C2×C4×Dic3, C10×Dic3 [×12], C2×C60 [×6], C22×C30, C2×C4×C20, Dic3×C20 [×4], Dic3×C2×C10 [×2], C22×C60, Dic3×C2×C20
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C5, S3, C2×C4 [×18], C23, C10 [×7], Dic3 [×4], D6 [×3], C42 [×4], C22×C4 [×3], C20 [×12], C2×C10 [×7], C4×S3 [×4], C2×Dic3 [×6], C22×S3, C5×S3, C2×C42, C2×C20 [×18], C22×C10, C4×Dic3 [×4], S3×C2×C4 [×2], C22×Dic3, C5×Dic3 [×4], S3×C10 [×3], C4×C20 [×4], C22×C20 [×3], C2×C4×Dic3, S3×C20 [×4], C10×Dic3 [×6], S3×C2×C10, C2×C4×C20, Dic3×C20 [×4], S3×C2×C20 [×2], Dic3×C2×C10, Dic3×C2×C20

Smallest permutation representation of Dic3×C2×C20
Regular action on 480 points
Generators in S480
(1 258)(2 259)(3 260)(4 241)(5 242)(6 243)(7 244)(8 245)(9 246)(10 247)(11 248)(12 249)(13 250)(14 251)(15 252)(16 253)(17 254)(18 255)(19 256)(20 257)(21 207)(22 208)(23 209)(24 210)(25 211)(26 212)(27 213)(28 214)(29 215)(30 216)(31 217)(32 218)(33 219)(34 220)(35 201)(36 202)(37 203)(38 204)(39 205)(40 206)(41 380)(42 361)(43 362)(44 363)(45 364)(46 365)(47 366)(48 367)(49 368)(50 369)(51 370)(52 371)(53 372)(54 373)(55 374)(56 375)(57 376)(58 377)(59 378)(60 379)(61 465)(62 466)(63 467)(64 468)(65 469)(66 470)(67 471)(68 472)(69 473)(70 474)(71 475)(72 476)(73 477)(74 478)(75 479)(76 480)(77 461)(78 462)(79 463)(80 464)(81 174)(82 175)(83 176)(84 177)(85 178)(86 179)(87 180)(88 161)(89 162)(90 163)(91 164)(92 165)(93 166)(94 167)(95 168)(96 169)(97 170)(98 171)(99 172)(100 173)(101 443)(102 444)(103 445)(104 446)(105 447)(106 448)(107 449)(108 450)(109 451)(110 452)(111 453)(112 454)(113 455)(114 456)(115 457)(116 458)(117 459)(118 460)(119 441)(120 442)(121 345)(122 346)(123 347)(124 348)(125 349)(126 350)(127 351)(128 352)(129 353)(130 354)(131 355)(132 356)(133 357)(134 358)(135 359)(136 360)(137 341)(138 342)(139 343)(140 344)(141 292)(142 293)(143 294)(144 295)(145 296)(146 297)(147 298)(148 299)(149 300)(150 281)(151 282)(152 283)(153 284)(154 285)(155 286)(156 287)(157 288)(158 289)(159 290)(160 291)(181 302)(182 303)(183 304)(184 305)(185 306)(186 307)(187 308)(188 309)(189 310)(190 311)(191 312)(192 313)(193 314)(194 315)(195 316)(196 317)(197 318)(198 319)(199 320)(200 301)(221 394)(222 395)(223 396)(224 397)(225 398)(226 399)(227 400)(228 381)(229 382)(230 383)(231 384)(232 385)(233 386)(234 387)(235 388)(236 389)(237 390)(238 391)(239 392)(240 393)(261 324)(262 325)(263 326)(264 327)(265 328)(266 329)(267 330)(268 331)(269 332)(270 333)(271 334)(272 335)(273 336)(274 337)(275 338)(276 339)(277 340)(278 321)(279 322)(280 323)(401 434)(402 435)(403 436)(404 437)(405 438)(406 439)(407 440)(408 421)(409 422)(410 423)(411 424)(412 425)(413 426)(414 427)(415 428)(416 429)(417 430)(418 431)(419 432)(420 433)
(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)
(1 296 83 454 471 36)(2 297 84 455 472 37)(3 298 85 456 473 38)(4 299 86 457 474 39)(5 300 87 458 475 40)(6 281 88 459 476 21)(7 282 89 460 477 22)(8 283 90 441 478 23)(9 284 91 442 479 24)(10 285 92 443 480 25)(11 286 93 444 461 26)(12 287 94 445 462 27)(13 288 95 446 463 28)(14 289 96 447 464 29)(15 290 97 448 465 30)(16 291 98 449 466 31)(17 292 99 450 467 32)(18 293 100 451 468 33)(19 294 81 452 469 34)(20 295 82 453 470 35)(41 397 184 140 425 329)(42 398 185 121 426 330)(43 399 186 122 427 331)(44 400 187 123 428 332)(45 381 188 124 429 333)(46 382 189 125 430 334)(47 383 190 126 431 335)(48 384 191 127 432 336)(49 385 192 128 433 337)(50 386 193 129 434 338)(51 387 194 130 435 339)(52 388 195 131 436 340)(53 389 196 132 437 321)(54 390 197 133 438 322)(55 391 198 134 439 323)(56 392 199 135 440 324)(57 393 200 136 421 325)(58 394 181 137 422 326)(59 395 182 138 423 327)(60 396 183 139 424 328)(61 216 252 159 170 106)(62 217 253 160 171 107)(63 218 254 141 172 108)(64 219 255 142 173 109)(65 220 256 143 174 110)(66 201 257 144 175 111)(67 202 258 145 176 112)(68 203 259 146 177 113)(69 204 260 147 178 114)(70 205 241 148 179 115)(71 206 242 149 180 116)(72 207 243 150 161 117)(73 208 244 151 162 118)(74 209 245 152 163 119)(75 210 246 153 164 120)(76 211 247 154 165 101)(77 212 248 155 166 102)(78 213 249 156 167 103)(79 214 250 157 168 104)(80 215 251 158 169 105)(221 302 341 409 263 377)(222 303 342 410 264 378)(223 304 343 411 265 379)(224 305 344 412 266 380)(225 306 345 413 267 361)(226 307 346 414 268 362)(227 308 347 415 269 363)(228 309 348 416 270 364)(229 310 349 417 271 365)(230 311 350 418 272 366)(231 312 351 419 273 367)(232 313 352 420 274 368)(233 314 353 401 275 369)(234 315 354 402 276 370)(235 316 355 403 277 371)(236 317 356 404 278 372)(237 318 357 405 279 373)(238 319 358 406 280 374)(239 320 359 407 261 375)(240 301 360 408 262 376)
(1 414 454 226)(2 415 455 227)(3 416 456 228)(4 417 457 229)(5 418 458 230)(6 419 459 231)(7 420 460 232)(8 401 441 233)(9 402 442 234)(10 403 443 235)(11 404 444 236)(12 405 445 237)(13 406 446 238)(14 407 447 239)(15 408 448 240)(16 409 449 221)(17 410 450 222)(18 411 451 223)(19 412 452 224)(20 413 453 225)(21 273 88 312)(22 274 89 313)(23 275 90 314)(24 276 91 315)(25 277 92 316)(26 278 93 317)(27 279 94 318)(28 280 95 319)(29 261 96 320)(30 262 97 301)(31 263 98 302)(32 264 99 303)(33 265 100 304)(34 266 81 305)(35 267 82 306)(36 268 83 307)(37 269 84 308)(38 270 85 309)(39 271 86 310)(40 272 87 311)(41 143 140 65)(42 144 121 66)(43 145 122 67)(44 146 123 68)(45 147 124 69)(46 148 125 70)(47 149 126 71)(48 150 127 72)(49 151 128 73)(50 152 129 74)(51 153 130 75)(52 154 131 76)(53 155 132 77)(54 156 133 78)(55 157 134 79)(56 158 135 80)(57 159 136 61)(58 160 137 62)(59 141 138 63)(60 142 139 64)(101 388 247 436)(102 389 248 437)(103 390 249 438)(104 391 250 439)(105 392 251 440)(106 393 252 421)(107 394 253 422)(108 395 254 423)(109 396 255 424)(110 397 256 425)(111 398 257 426)(112 399 258 427)(113 400 259 428)(114 381 260 429)(115 382 241 430)(116 383 242 431)(117 384 243 432)(118 385 244 433)(119 386 245 434)(120 387 246 435)(161 191 207 336)(162 192 208 337)(163 193 209 338)(164 194 210 339)(165 195 211 340)(166 196 212 321)(167 197 213 322)(168 198 214 323)(169 199 215 324)(170 200 216 325)(171 181 217 326)(172 182 218 327)(173 183 219 328)(174 184 220 329)(175 185 201 330)(176 186 202 331)(177 187 203 332)(178 188 204 333)(179 189 205 334)(180 190 206 335)(281 351 476 367)(282 352 477 368)(283 353 478 369)(284 354 479 370)(285 355 480 371)(286 356 461 372)(287 357 462 373)(288 358 463 374)(289 359 464 375)(290 360 465 376)(291 341 466 377)(292 342 467 378)(293 343 468 379)(294 344 469 380)(295 345 470 361)(296 346 471 362)(297 347 472 363)(298 348 473 364)(299 349 474 365)(300 350 475 366)

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

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

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

240 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 4I ··· 4X 5A 5B 5C 5D 6A ··· 6G 10A ··· 10AB 12A ··· 12H 15A 15B 15C 15D 20A ··· 20AF 20AG ··· 20CR 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 5 5 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 2 1 ··· 1 3 ··· 3 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

240 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 S3 Dic3 D6 D6 C4×S3 C5×S3 C5×Dic3 S3×C10 S3×C10 S3×C20 kernel Dic3×C2×C20 Dic3×C20 Dic3×C2×C10 C22×C60 C10×Dic3 C2×C60 C2×C4×Dic3 C4×Dic3 C22×Dic3 C22×C12 C2×Dic3 C2×C12 C22×C20 C2×C20 C2×C20 C22×C10 C2×C10 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 16 8 4 16 8 4 64 32 1 4 2 1 8 4 16 8 4 32

Matrix representation of Dic3×C2×C20 in GL5(𝔽61)

 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 50 0 0 0 0 0 60 0 0 0 0 0 34 0 0 0 0 0 34
,
 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 60 1 0 0 0 60 0
,
 50 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 25 53 0 0 0 17 36

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,50,0,0,0,0,0,60,0,0,0,0,0,34,0,0,0,0,0,34],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,60,0,0,0,1,0],[50,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,25,17,0,0,0,53,36] >;

Dic3×C2×C20 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_2\times C_{20}
% in TeX

G:=Group("Dic3xC2xC20");
// GroupNames label

G:=SmallGroup(480,801);
// by ID

G=gap.SmallGroup(480,801);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,436,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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