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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 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C10, C10 [×2], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×4], C20 [×2], C2×C10, C3⋊C8 [×2], Dic6 [×3], C2×Dic3, C2×C12, C2×C12, C3×Q8 [×3], C30, C30 [×2], C2×Q16, C40 [×2], Dic10 [×2], Dic10 [×4], C2×Dic5 [×2], C2×C20, C2×C3⋊C8, C3⋊Q16 [×4], C2×Dic6, C6×Q8, C3×Dic5 [×2], Dic15 [×2], C60 [×2], C2×C30, Dic20 [×4], C2×C40, C2×Dic10, C2×Dic10, C2×C3⋊Q16, C5×C3⋊C8 [×2], C3×Dic10 [×2], C3×Dic10, C6×Dic5, Dic30 [×2], Dic30, C2×Dic15, C2×C60, C2×Dic20, C3⋊Dic20 [×4], C10×C3⋊C8, C6×Dic10, C2×Dic30, C2×C3⋊Dic20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], Q16 [×2], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C2×Q16, D20 [×2], C22×D5, C3⋊Q16 [×2], C2×C3⋊D4, S3×D5, Dic20 [×2], C2×D20, C2×C3⋊Q16, C3⋊D20 [×2], C2×S3×D5, C2×Dic20, C3⋊Dic20 [×2], C2×C3⋊D20, C2×C3⋊Dic20

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

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

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

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

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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