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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, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×Dic3, C2×C12, C22×C6, C30, C30, C2×C42, C2×C20, C2×C20, C22×C10, C4×Dic3, C22×Dic3, C22×C12, C5×Dic3, C60, C2×C30, C2×C30, C4×C20, C22×C20, C22×C20, C2×C4×Dic3, C10×Dic3, C2×C60, C22×C30, C2×C4×C20, Dic3×C20, Dic3×C2×C10, C22×C60, Dic3×C2×C20
Quotients:

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

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

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

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

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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