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## G = C2×C3⋊Dic20order 480 = 25·3·5

### Direct product of C2 and C3⋊Dic20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×C3⋊Dic20
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — C3⋊Dic20 — C2×C3⋊Dic20
 Lower central C15 — C30 — C60 — C2×C3⋊Dic20
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C3⋊Dic20
G = < a,b,c,d | a2=b3=c40=1, d2=c20, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 572 in 120 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C12, C2×C6, C15, C2×C8, Q16, C2×Q8, Dic5, C20, C2×C10, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C30, C30, C2×Q16, C40, Dic10, Dic10, C2×Dic5, C2×C20, C2×C3⋊C8, C3⋊Q16, C2×Dic6, C6×Q8, C3×Dic5, Dic15, C60, C2×C30, Dic20, C2×C40, C2×Dic10, C2×Dic10, C2×C3⋊Q16, C5×C3⋊C8, C3×Dic10, C3×Dic10, C6×Dic5, Dic30, Dic30, C2×Dic15, C2×C60, C2×Dic20, C3⋊Dic20, C10×C3⋊C8, C6×Dic10, C2×Dic30, C2×C3⋊Dic20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, Q16, C2×D4, D10, C3⋊D4, C22×S3, C2×Q16, D20, C22×D5, C3⋊Q16, C2×C3⋊D4, S3×D5, Dic20, C2×D20, C2×C3⋊Q16, C3⋊D20, C2×S3×D5, C2×Dic20, C3⋊Dic20, C2×C3⋊D20, C2×C3⋊Dic20

Smallest permutation representation of C2×C3⋊Dic20
Regular action on 480 points
Generators in S480
(1 116)(2 117)(3 118)(4 119)(5 120)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 114)(40 115)(41 448)(42 449)(43 450)(44 451)(45 452)(46 453)(47 454)(48 455)(49 456)(50 457)(51 458)(52 459)(53 460)(54 461)(55 462)(56 463)(57 464)(58 465)(59 466)(60 467)(61 468)(62 469)(63 470)(64 471)(65 472)(66 473)(67 474)(68 475)(69 476)(70 477)(71 478)(72 479)(73 480)(74 441)(75 442)(76 443)(77 444)(78 445)(79 446)(80 447)(121 279)(122 280)(123 241)(124 242)(125 243)(126 244)(127 245)(128 246)(129 247)(130 248)(131 249)(132 250)(133 251)(134 252)(135 253)(136 254)(137 255)(138 256)(139 257)(140 258)(141 259)(142 260)(143 261)(144 262)(145 263)(146 264)(147 265)(148 266)(149 267)(150 268)(151 269)(152 270)(153 271)(154 272)(155 273)(156 274)(157 275)(158 276)(159 277)(160 278)(161 428)(162 429)(163 430)(164 431)(165 432)(166 433)(167 434)(168 435)(169 436)(170 437)(171 438)(172 439)(173 440)(174 401)(175 402)(176 403)(177 404)(178 405)(179 406)(180 407)(181 408)(182 409)(183 410)(184 411)(185 412)(186 413)(187 414)(188 415)(189 416)(190 417)(191 418)(192 419)(193 420)(194 421)(195 422)(196 423)(197 424)(198 425)(199 426)(200 427)(201 320)(202 281)(203 282)(204 283)(205 284)(206 285)(207 286)(208 287)(209 288)(210 289)(211 290)(212 291)(213 292)(214 293)(215 294)(216 295)(217 296)(218 297)(219 298)(220 299)(221 300)(222 301)(223 302)(224 303)(225 304)(226 305)(227 306)(228 307)(229 308)(230 309)(231 310)(232 311)(233 312)(234 313)(235 314)(236 315)(237 316)(238 317)(239 318)(240 319)(321 397)(322 398)(323 399)(324 400)(325 361)(326 362)(327 363)(328 364)(329 365)(330 366)(331 367)(332 368)(333 369)(334 370)(335 371)(336 372)(337 373)(338 374)(339 375)(340 376)(341 377)(342 378)(343 379)(344 380)(345 381)(346 382)(347 383)(348 384)(349 385)(350 386)(351 387)(352 388)(353 389)(354 390)(355 391)(356 392)(357 393)(358 394)(359 395)(360 396)
(1 345 409)(2 410 346)(3 347 411)(4 412 348)(5 349 413)(6 414 350)(7 351 415)(8 416 352)(9 353 417)(10 418 354)(11 355 419)(12 420 356)(13 357 421)(14 422 358)(15 359 423)(16 424 360)(17 321 425)(18 426 322)(19 323 427)(20 428 324)(21 325 429)(22 430 326)(23 327 431)(24 432 328)(25 329 433)(26 434 330)(27 331 435)(28 436 332)(29 333 437)(30 438 334)(31 335 439)(32 440 336)(33 337 401)(34 402 338)(35 339 403)(36 404 340)(37 341 405)(38 406 342)(39 343 407)(40 408 344)(41 156 217)(42 218 157)(43 158 219)(44 220 159)(45 160 221)(46 222 121)(47 122 223)(48 224 123)(49 124 225)(50 226 125)(51 126 227)(52 228 127)(53 128 229)(54 230 129)(55 130 231)(56 232 131)(57 132 233)(58 234 133)(59 134 235)(60 236 135)(61 136 237)(62 238 137)(63 138 239)(64 240 139)(65 140 201)(66 202 141)(67 142 203)(68 204 143)(69 144 205)(70 206 145)(71 146 207)(72 208 147)(73 148 209)(74 210 149)(75 150 211)(76 212 151)(77 152 213)(78 214 153)(79 154 215)(80 216 155)(81 187 386)(82 387 188)(83 189 388)(84 389 190)(85 191 390)(86 391 192)(87 193 392)(88 393 194)(89 195 394)(90 395 196)(91 197 396)(92 397 198)(93 199 398)(94 399 200)(95 161 400)(96 361 162)(97 163 362)(98 363 164)(99 165 364)(100 365 166)(101 167 366)(102 367 168)(103 169 368)(104 369 170)(105 171 370)(106 371 172)(107 173 372)(108 373 174)(109 175 374)(110 375 176)(111 177 376)(112 377 178)(113 179 378)(114 379 180)(115 181 380)(116 381 182)(117 183 382)(118 383 184)(119 185 384)(120 385 186)(241 455 303)(242 304 456)(243 457 305)(244 306 458)(245 459 307)(246 308 460)(247 461 309)(248 310 462)(249 463 311)(250 312 464)(251 465 313)(252 314 466)(253 467 315)(254 316 468)(255 469 317)(256 318 470)(257 471 319)(258 320 472)(259 473 281)(260 282 474)(261 475 283)(262 284 476)(263 477 285)(264 286 478)(265 479 287)(266 288 480)(267 441 289)(268 290 442)(269 443 291)(270 292 444)(271 445 293)(272 294 446)(273 447 295)(274 296 448)(275 449 297)(276 298 450)(277 451 299)(278 300 452)(279 453 301)(280 302 454)
(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 69 21 49)(2 68 22 48)(3 67 23 47)(4 66 24 46)(5 65 25 45)(6 64 26 44)(7 63 27 43)(8 62 28 42)(9 61 29 41)(10 60 30 80)(11 59 31 79)(12 58 32 78)(13 57 33 77)(14 56 34 76)(15 55 35 75)(16 54 36 74)(17 53 37 73)(18 52 38 72)(19 51 39 71)(20 50 40 70)(81 471 101 451)(82 470 102 450)(83 469 103 449)(84 468 104 448)(85 467 105 447)(86 466 106 446)(87 465 107 445)(88 464 108 444)(89 463 109 443)(90 462 110 442)(91 461 111 441)(92 460 112 480)(93 459 113 479)(94 458 114 478)(95 457 115 477)(96 456 116 476)(97 455 117 475)(98 454 118 474)(99 453 119 473)(100 452 120 472)(121 348 141 328)(122 347 142 327)(123 346 143 326)(124 345 144 325)(125 344 145 324)(126 343 146 323)(127 342 147 322)(128 341 148 321)(129 340 149 360)(130 339 150 359)(131 338 151 358)(132 337 152 357)(133 336 153 356)(134 335 154 355)(135 334 155 354)(136 333 156 353)(137 332 157 352)(138 331 158 351)(139 330 159 350)(140 329 160 349)(161 305 181 285)(162 304 182 284)(163 303 183 283)(164 302 184 282)(165 301 185 281)(166 300 186 320)(167 299 187 319)(168 298 188 318)(169 297 189 317)(170 296 190 316)(171 295 191 315)(172 294 192 314)(173 293 193 313)(174 292 194 312)(175 291 195 311)(176 290 196 310)(177 289 197 309)(178 288 198 308)(179 287 199 307)(180 286 200 306)(201 433 221 413)(202 432 222 412)(203 431 223 411)(204 430 224 410)(205 429 225 409)(206 428 226 408)(207 427 227 407)(208 426 228 406)(209 425 229 405)(210 424 230 404)(211 423 231 403)(212 422 232 402)(213 421 233 401)(214 420 234 440)(215 419 235 439)(216 418 236 438)(217 417 237 437)(218 416 238 436)(219 415 239 435)(220 414 240 434)(241 382 261 362)(242 381 262 361)(243 380 263 400)(244 379 264 399)(245 378 265 398)(246 377 266 397)(247 376 267 396)(248 375 268 395)(249 374 269 394)(250 373 270 393)(251 372 271 392)(252 371 272 391)(253 370 273 390)(254 369 274 389)(255 368 275 388)(256 367 276 387)(257 366 277 386)(258 365 278 385)(259 364 279 384)(260 363 280 383)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A ··· 20H 30A ··· 30F 40A ··· 40P 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 12 12 15 15 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 2 2 2 20 20 60 60 2 2 2 2 2 6 6 6 6 2 ··· 2 4 4 20 20 20 20 4 4 2 ··· 2 4 ··· 4 6 ··· 6 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + - + + + + - - + + + + - image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 Q16 D10 D10 C3⋊D4 C3⋊D4 D20 D20 Dic20 C3⋊Q16 S3×D5 C3⋊D20 C2×S3×D5 C3⋊D20 C3⋊Dic20 kernel C2×C3⋊Dic20 C3⋊Dic20 C10×C3⋊C8 C6×Dic10 C2×Dic30 C2×Dic10 C60 C2×C30 C2×C3⋊C8 Dic10 C2×C20 C30 C3⋊C8 C2×C12 C20 C2×C10 C12 C2×C6 C6 C10 C2×C4 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 2 2 1 4 4 2 2 2 4 4 16 2 2 2 2 2 8

Matrix representation of C2×C3⋊Dic20 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 1 240
,
 214 228 0 0 0 0 13 33 0 0 0 0 0 0 14 199 0 0 0 0 42 29 0 0 0 0 0 0 166 62 0 0 0 0 228 75
,
 162 45 0 0 0 0 113 79 0 0 0 0 0 0 133 210 0 0 0 0 42 108 0 0 0 0 0 0 171 140 0 0 0 0 101 70

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[214,13,0,0,0,0,228,33,0,0,0,0,0,0,14,42,0,0,0,0,199,29,0,0,0,0,0,0,166,228,0,0,0,0,62,75],[162,113,0,0,0,0,45,79,0,0,0,0,0,0,133,42,0,0,0,0,210,108,0,0,0,0,0,0,171,101,0,0,0,0,140,70] >;

C2×C3⋊Dic20 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes {\rm Dic}_{20}
% in TeX

G:=Group("C2xC3:Dic20");
// GroupNames label

G:=SmallGroup(480,395);
// by ID

G=gap.SmallGroup(480,395);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,176,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^40=1,d^2=c^20,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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