direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic40, C15⋊5Q32, C80.1C6, C48.2D5, C240.2C2, C6.15D40, C30.29D8, C60.171D4, C12.41D20, C24.72D10, Dic20.1C6, C120.85C22, C16.(C3×D5), C5⋊1(C3×Q32), C8.15(C6×D5), C2.5(C3×D40), C4.3(C3×D20), C10.3(C3×D8), C40.16(C2×C6), C20.26(C3×D4), (C3×Dic20).2C2, SmallGroup(480,79)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic40
G = < a,b,c | a3=b80=1, c2=b40, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic40
►Regular action on 480 points
Generators in S480
(1 470 356)(2 471 357)(3 472 358)(4 473 359)(5 474 360)(6 475 361)(7 476 362)(8 477 363)(9 478 364)(10 479 365)(11 480 366)(12 401 367)(13 402 368)(14 403 369)(15 404 370)(16 405 371)(17 406 372)(18 407 373)(19 408 374)(20 409 375)(21 410 376)(22 411 377)(23 412 378)(24 413 379)(25 414 380)(26 415 381)(27 416 382)(28 417 383)(29 418 384)(30 419 385)(31 420 386)(32 421 387)(33 422 388)(34 423 389)(35 424 390)(36 425 391)(37 426 392)(38 427 393)(39 428 394)(40 429 395)(41 430 396)(42 431 397)(43 432 398)(44 433 399)(45 434 400)(46 435 321)(47 436 322)(48 437 323)(49 438 324)(50 439 325)(51 440 326)(52 441 327)(53 442 328)(54 443 329)(55 444 330)(56 445 331)(57 446 332)(58 447 333)(59 448 334)(60 449 335)(61 450 336)(62 451 337)(63 452 338)(64 453 339)(65 454 340)(66 455 341)(67 456 342)(68 457 343)(69 458 344)(70 459 345)(71 460 346)(72 461 347)(73 462 348)(74 463 349)(75 464 350)(76 465 351)(77 466 352)(78 467 353)(79 468 354)(80 469 355)(81 234 285)(82 235 286)(83 236 287)(84 237 288)(85 238 289)(86 239 290)(87 240 291)(88 161 292)(89 162 293)(90 163 294)(91 164 295)(92 165 296)(93 166 297)(94 167 298)(95 168 299)(96 169 300)(97 170 301)(98 171 302)(99 172 303)(100 173 304)(101 174 305)(102 175 306)(103 176 307)(104 177 308)(105 178 309)(106 179 310)(107 180 311)(108 181 312)(109 182 313)(110 183 314)(111 184 315)(112 185 316)(113 186 317)(114 187 318)(115 188 319)(116 189 320)(117 190 241)(118 191 242)(119 192 243)(120 193 244)(121 194 245)(122 195 246)(123 196 247)(124 197 248)(125 198 249)(126 199 250)(127 200 251)(128 201 252)(129 202 253)(130 203 254)(131 204 255)(132 205 256)(133 206 257)(134 207 258)(135 208 259)(136 209 260)(137 210 261)(138 211 262)(139 212 263)(140 213 264)(141 214 265)(142 215 266)(143 216 267)(144 217 268)(145 218 269)(146 219 270)(147 220 271)(148 221 272)(149 222 273)(150 223 274)(151 224 275)(152 225 276)(153 226 277)(154 227 278)(155 228 279)(156 229 280)(157 230 281)(158 231 282)(159 232 283)(160 233 284)
(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 81 41 121)(2 160 42 120)(3 159 43 119)(4 158 44 118)(5 157 45 117)(6 156 46 116)(7 155 47 115)(8 154 48 114)(9 153 49 113)(10 152 50 112)(11 151 51 111)(12 150 52 110)(13 149 53 109)(14 148 54 108)(15 147 55 107)(16 146 56 106)(17 145 57 105)(18 144 58 104)(19 143 59 103)(20 142 60 102)(21 141 61 101)(22 140 62 100)(23 139 63 99)(24 138 64 98)(25 137 65 97)(26 136 66 96)(27 135 67 95)(28 134 68 94)(29 133 69 93)(30 132 70 92)(31 131 71 91)(32 130 72 90)(33 129 73 89)(34 128 74 88)(35 127 75 87)(36 126 76 86)(37 125 77 85)(38 124 78 84)(39 123 79 83)(40 122 80 82)(161 423 201 463)(162 422 202 462)(163 421 203 461)(164 420 204 460)(165 419 205 459)(166 418 206 458)(167 417 207 457)(168 416 208 456)(169 415 209 455)(170 414 210 454)(171 413 211 453)(172 412 212 452)(173 411 213 451)(174 410 214 450)(175 409 215 449)(176 408 216 448)(177 407 217 447)(178 406 218 446)(179 405 219 445)(180 404 220 444)(181 403 221 443)(182 402 222 442)(183 401 223 441)(184 480 224 440)(185 479 225 439)(186 478 226 438)(187 477 227 437)(188 476 228 436)(189 475 229 435)(190 474 230 434)(191 473 231 433)(192 472 232 432)(193 471 233 431)(194 470 234 430)(195 469 235 429)(196 468 236 428)(197 467 237 427)(198 466 238 426)(199 465 239 425)(200 464 240 424)(241 360 281 400)(242 359 282 399)(243 358 283 398)(244 357 284 397)(245 356 285 396)(246 355 286 395)(247 354 287 394)(248 353 288 393)(249 352 289 392)(250 351 290 391)(251 350 291 390)(252 349 292 389)(253 348 293 388)(254 347 294 387)(255 346 295 386)(256 345 296 385)(257 344 297 384)(258 343 298 383)(259 342 299 382)(260 341 300 381)(261 340 301 380)(262 339 302 379)(263 338 303 378)(264 337 304 377)(265 336 305 376)(266 335 306 375)(267 334 307 374)(268 333 308 373)(269 332 309 372)(270 331 310 371)(271 330 311 370)(272 329 312 369)(273 328 313 368)(274 327 314 367)(275 326 315 366)(276 325 316 365)(277 324 317 364)(278 323 318 363)(279 322 319 362)(280 321 320 361)
G:=sub<Sym(480)| (1,470,356)(2,471,357)(3,472,358)(4,473,359)(5,474,360)(6,475,361)(7,476,362)(8,477,363)(9,478,364)(10,479,365)(11,480,366)(12,401,367)(13,402,368)(14,403,369)(15,404,370)(16,405,371)(17,406,372)(18,407,373)(19,408,374)(20,409,375)(21,410,376)(22,411,377)(23,412,378)(24,413,379)(25,414,380)(26,415,381)(27,416,382)(28,417,383)(29,418,384)(30,419,385)(31,420,386)(32,421,387)(33,422,388)(34,423,389)(35,424,390)(36,425,391)(37,426,392)(38,427,393)(39,428,394)(40,429,395)(41,430,396)(42,431,397)(43,432,398)(44,433,399)(45,434,400)(46,435,321)(47,436,322)(48,437,323)(49,438,324)(50,439,325)(51,440,326)(52,441,327)(53,442,328)(54,443,329)(55,444,330)(56,445,331)(57,446,332)(58,447,333)(59,448,334)(60,449,335)(61,450,336)(62,451,337)(63,452,338)(64,453,339)(65,454,340)(66,455,341)(67,456,342)(68,457,343)(69,458,344)(70,459,345)(71,460,346)(72,461,347)(73,462,348)(74,463,349)(75,464,350)(76,465,351)(77,466,352)(78,467,353)(79,468,354)(80,469,355)(81,234,285)(82,235,286)(83,236,287)(84,237,288)(85,238,289)(86,239,290)(87,240,291)(88,161,292)(89,162,293)(90,163,294)(91,164,295)(92,165,296)(93,166,297)(94,167,298)(95,168,299)(96,169,300)(97,170,301)(98,171,302)(99,172,303)(100,173,304)(101,174,305)(102,175,306)(103,176,307)(104,177,308)(105,178,309)(106,179,310)(107,180,311)(108,181,312)(109,182,313)(110,183,314)(111,184,315)(112,185,316)(113,186,317)(114,187,318)(115,188,319)(116,189,320)(117,190,241)(118,191,242)(119,192,243)(120,193,244)(121,194,245)(122,195,246)(123,196,247)(124,197,248)(125,198,249)(126,199,250)(127,200,251)(128,201,252)(129,202,253)(130,203,254)(131,204,255)(132,205,256)(133,206,257)(134,207,258)(135,208,259)(136,209,260)(137,210,261)(138,211,262)(139,212,263)(140,213,264)(141,214,265)(142,215,266)(143,216,267)(144,217,268)(145,218,269)(146,219,270)(147,220,271)(148,221,272)(149,222,273)(150,223,274)(151,224,275)(152,225,276)(153,226,277)(154,227,278)(155,228,279)(156,229,280)(157,230,281)(158,231,282)(159,232,283)(160,233,284), (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), (1,81,41,121)(2,160,42,120)(3,159,43,119)(4,158,44,118)(5,157,45,117)(6,156,46,116)(7,155,47,115)(8,154,48,114)(9,153,49,113)(10,152,50,112)(11,151,51,111)(12,150,52,110)(13,149,53,109)(14,148,54,108)(15,147,55,107)(16,146,56,106)(17,145,57,105)(18,144,58,104)(19,143,59,103)(20,142,60,102)(21,141,61,101)(22,140,62,100)(23,139,63,99)(24,138,64,98)(25,137,65,97)(26,136,66,96)(27,135,67,95)(28,134,68,94)(29,133,69,93)(30,132,70,92)(31,131,71,91)(32,130,72,90)(33,129,73,89)(34,128,74,88)(35,127,75,87)(36,126,76,86)(37,125,77,85)(38,124,78,84)(39,123,79,83)(40,122,80,82)(161,423,201,463)(162,422,202,462)(163,421,203,461)(164,420,204,460)(165,419,205,459)(166,418,206,458)(167,417,207,457)(168,416,208,456)(169,415,209,455)(170,414,210,454)(171,413,211,453)(172,412,212,452)(173,411,213,451)(174,410,214,450)(175,409,215,449)(176,408,216,448)(177,407,217,447)(178,406,218,446)(179,405,219,445)(180,404,220,444)(181,403,221,443)(182,402,222,442)(183,401,223,441)(184,480,224,440)(185,479,225,439)(186,478,226,438)(187,477,227,437)(188,476,228,436)(189,475,229,435)(190,474,230,434)(191,473,231,433)(192,472,232,432)(193,471,233,431)(194,470,234,430)(195,469,235,429)(196,468,236,428)(197,467,237,427)(198,466,238,426)(199,465,239,425)(200,464,240,424)(241,360,281,400)(242,359,282,399)(243,358,283,398)(244,357,284,397)(245,356,285,396)(246,355,286,395)(247,354,287,394)(248,353,288,393)(249,352,289,392)(250,351,290,391)(251,350,291,390)(252,349,292,389)(253,348,293,388)(254,347,294,387)(255,346,295,386)(256,345,296,385)(257,344,297,384)(258,343,298,383)(259,342,299,382)(260,341,300,381)(261,340,301,380)(262,339,302,379)(263,338,303,378)(264,337,304,377)(265,336,305,376)(266,335,306,375)(267,334,307,374)(268,333,308,373)(269,332,309,372)(270,331,310,371)(271,330,311,370)(272,329,312,369)(273,328,313,368)(274,327,314,367)(275,326,315,366)(276,325,316,365)(277,324,317,364)(278,323,318,363)(279,322,319,362)(280,321,320,361)>;
G:=Group( (1,470,356)(2,471,357)(3,472,358)(4,473,359)(5,474,360)(6,475,361)(7,476,362)(8,477,363)(9,478,364)(10,479,365)(11,480,366)(12,401,367)(13,402,368)(14,403,369)(15,404,370)(16,405,371)(17,406,372)(18,407,373)(19,408,374)(20,409,375)(21,410,376)(22,411,377)(23,412,378)(24,413,379)(25,414,380)(26,415,381)(27,416,382)(28,417,383)(29,418,384)(30,419,385)(31,420,386)(32,421,387)(33,422,388)(34,423,389)(35,424,390)(36,425,391)(37,426,392)(38,427,393)(39,428,394)(40,429,395)(41,430,396)(42,431,397)(43,432,398)(44,433,399)(45,434,400)(46,435,321)(47,436,322)(48,437,323)(49,438,324)(50,439,325)(51,440,326)(52,441,327)(53,442,328)(54,443,329)(55,444,330)(56,445,331)(57,446,332)(58,447,333)(59,448,334)(60,449,335)(61,450,336)(62,451,337)(63,452,338)(64,453,339)(65,454,340)(66,455,341)(67,456,342)(68,457,343)(69,458,344)(70,459,345)(71,460,346)(72,461,347)(73,462,348)(74,463,349)(75,464,350)(76,465,351)(77,466,352)(78,467,353)(79,468,354)(80,469,355)(81,234,285)(82,235,286)(83,236,287)(84,237,288)(85,238,289)(86,239,290)(87,240,291)(88,161,292)(89,162,293)(90,163,294)(91,164,295)(92,165,296)(93,166,297)(94,167,298)(95,168,299)(96,169,300)(97,170,301)(98,171,302)(99,172,303)(100,173,304)(101,174,305)(102,175,306)(103,176,307)(104,177,308)(105,178,309)(106,179,310)(107,180,311)(108,181,312)(109,182,313)(110,183,314)(111,184,315)(112,185,316)(113,186,317)(114,187,318)(115,188,319)(116,189,320)(117,190,241)(118,191,242)(119,192,243)(120,193,244)(121,194,245)(122,195,246)(123,196,247)(124,197,248)(125,198,249)(126,199,250)(127,200,251)(128,201,252)(129,202,253)(130,203,254)(131,204,255)(132,205,256)(133,206,257)(134,207,258)(135,208,259)(136,209,260)(137,210,261)(138,211,262)(139,212,263)(140,213,264)(141,214,265)(142,215,266)(143,216,267)(144,217,268)(145,218,269)(146,219,270)(147,220,271)(148,221,272)(149,222,273)(150,223,274)(151,224,275)(152,225,276)(153,226,277)(154,227,278)(155,228,279)(156,229,280)(157,230,281)(158,231,282)(159,232,283)(160,233,284), (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), (1,81,41,121)(2,160,42,120)(3,159,43,119)(4,158,44,118)(5,157,45,117)(6,156,46,116)(7,155,47,115)(8,154,48,114)(9,153,49,113)(10,152,50,112)(11,151,51,111)(12,150,52,110)(13,149,53,109)(14,148,54,108)(15,147,55,107)(16,146,56,106)(17,145,57,105)(18,144,58,104)(19,143,59,103)(20,142,60,102)(21,141,61,101)(22,140,62,100)(23,139,63,99)(24,138,64,98)(25,137,65,97)(26,136,66,96)(27,135,67,95)(28,134,68,94)(29,133,69,93)(30,132,70,92)(31,131,71,91)(32,130,72,90)(33,129,73,89)(34,128,74,88)(35,127,75,87)(36,126,76,86)(37,125,77,85)(38,124,78,84)(39,123,79,83)(40,122,80,82)(161,423,201,463)(162,422,202,462)(163,421,203,461)(164,420,204,460)(165,419,205,459)(166,418,206,458)(167,417,207,457)(168,416,208,456)(169,415,209,455)(170,414,210,454)(171,413,211,453)(172,412,212,452)(173,411,213,451)(174,410,214,450)(175,409,215,449)(176,408,216,448)(177,407,217,447)(178,406,218,446)(179,405,219,445)(180,404,220,444)(181,403,221,443)(182,402,222,442)(183,401,223,441)(184,480,224,440)(185,479,225,439)(186,478,226,438)(187,477,227,437)(188,476,228,436)(189,475,229,435)(190,474,230,434)(191,473,231,433)(192,472,232,432)(193,471,233,431)(194,470,234,430)(195,469,235,429)(196,468,236,428)(197,467,237,427)(198,466,238,426)(199,465,239,425)(200,464,240,424)(241,360,281,400)(242,359,282,399)(243,358,283,398)(244,357,284,397)(245,356,285,396)(246,355,286,395)(247,354,287,394)(248,353,288,393)(249,352,289,392)(250,351,290,391)(251,350,291,390)(252,349,292,389)(253,348,293,388)(254,347,294,387)(255,346,295,386)(256,345,296,385)(257,344,297,384)(258,343,298,383)(259,342,299,382)(260,341,300,381)(261,340,301,380)(262,339,302,379)(263,338,303,378)(264,337,304,377)(265,336,305,376)(266,335,306,375)(267,334,307,374)(268,333,308,373)(269,332,309,372)(270,331,310,371)(271,330,311,370)(272,329,312,369)(273,328,313,368)(274,327,314,367)(275,326,315,366)(276,325,316,365)(277,324,317,364)(278,323,318,363)(279,322,319,362)(280,321,320,361) );
G=PermutationGroup([[(1,470,356),(2,471,357),(3,472,358),(4,473,359),(5,474,360),(6,475,361),(7,476,362),(8,477,363),(9,478,364),(10,479,365),(11,480,366),(12,401,367),(13,402,368),(14,403,369),(15,404,370),(16,405,371),(17,406,372),(18,407,373),(19,408,374),(20,409,375),(21,410,376),(22,411,377),(23,412,378),(24,413,379),(25,414,380),(26,415,381),(27,416,382),(28,417,383),(29,418,384),(30,419,385),(31,420,386),(32,421,387),(33,422,388),(34,423,389),(35,424,390),(36,425,391),(37,426,392),(38,427,393),(39,428,394),(40,429,395),(41,430,396),(42,431,397),(43,432,398),(44,433,399),(45,434,400),(46,435,321),(47,436,322),(48,437,323),(49,438,324),(50,439,325),(51,440,326),(52,441,327),(53,442,328),(54,443,329),(55,444,330),(56,445,331),(57,446,332),(58,447,333),(59,448,334),(60,449,335),(61,450,336),(62,451,337),(63,452,338),(64,453,339),(65,454,340),(66,455,341),(67,456,342),(68,457,343),(69,458,344),(70,459,345),(71,460,346),(72,461,347),(73,462,348),(74,463,349),(75,464,350),(76,465,351),(77,466,352),(78,467,353),(79,468,354),(80,469,355),(81,234,285),(82,235,286),(83,236,287),(84,237,288),(85,238,289),(86,239,290),(87,240,291),(88,161,292),(89,162,293),(90,163,294),(91,164,295),(92,165,296),(93,166,297),(94,167,298),(95,168,299),(96,169,300),(97,170,301),(98,171,302),(99,172,303),(100,173,304),(101,174,305),(102,175,306),(103,176,307),(104,177,308),(105,178,309),(106,179,310),(107,180,311),(108,181,312),(109,182,313),(110,183,314),(111,184,315),(112,185,316),(113,186,317),(114,187,318),(115,188,319),(116,189,320),(117,190,241),(118,191,242),(119,192,243),(120,193,244),(121,194,245),(122,195,246),(123,196,247),(124,197,248),(125,198,249),(126,199,250),(127,200,251),(128,201,252),(129,202,253),(130,203,254),(131,204,255),(132,205,256),(133,206,257),(134,207,258),(135,208,259),(136,209,260),(137,210,261),(138,211,262),(139,212,263),(140,213,264),(141,214,265),(142,215,266),(143,216,267),(144,217,268),(145,218,269),(146,219,270),(147,220,271),(148,221,272),(149,222,273),(150,223,274),(151,224,275),(152,225,276),(153,226,277),(154,227,278),(155,228,279),(156,229,280),(157,230,281),(158,231,282),(159,232,283),(160,233,284)], [(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)], [(1,81,41,121),(2,160,42,120),(3,159,43,119),(4,158,44,118),(5,157,45,117),(6,156,46,116),(7,155,47,115),(8,154,48,114),(9,153,49,113),(10,152,50,112),(11,151,51,111),(12,150,52,110),(13,149,53,109),(14,148,54,108),(15,147,55,107),(16,146,56,106),(17,145,57,105),(18,144,58,104),(19,143,59,103),(20,142,60,102),(21,141,61,101),(22,140,62,100),(23,139,63,99),(24,138,64,98),(25,137,65,97),(26,136,66,96),(27,135,67,95),(28,134,68,94),(29,133,69,93),(30,132,70,92),(31,131,71,91),(32,130,72,90),(33,129,73,89),(34,128,74,88),(35,127,75,87),(36,126,76,86),(37,125,77,85),(38,124,78,84),(39,123,79,83),(40,122,80,82),(161,423,201,463),(162,422,202,462),(163,421,203,461),(164,420,204,460),(165,419,205,459),(166,418,206,458),(167,417,207,457),(168,416,208,456),(169,415,209,455),(170,414,210,454),(171,413,211,453),(172,412,212,452),(173,411,213,451),(174,410,214,450),(175,409,215,449),(176,408,216,448),(177,407,217,447),(178,406,218,446),(179,405,219,445),(180,404,220,444),(181,403,221,443),(182,402,222,442),(183,401,223,441),(184,480,224,440),(185,479,225,439),(186,478,226,438),(187,477,227,437),(188,476,228,436),(189,475,229,435),(190,474,230,434),(191,473,231,433),(192,472,232,432),(193,471,233,431),(194,470,234,430),(195,469,235,429),(196,468,236,428),(197,467,237,427),(198,466,238,426),(199,465,239,425),(200,464,240,424),(241,360,281,400),(242,359,282,399),(243,358,283,398),(244,357,284,397),(245,356,285,396),(246,355,286,395),(247,354,287,394),(248,353,288,393),(249,352,289,392),(250,351,290,391),(251,350,291,390),(252,349,292,389),(253,348,293,388),(254,347,294,387),(255,346,295,386),(256,345,296,385),(257,344,297,384),(258,343,298,383),(259,342,299,382),(260,341,300,381),(261,340,301,380),(262,339,302,379),(263,338,303,378),(264,337,304,377),(265,336,305,376),(266,335,306,375),(267,334,307,374),(268,333,308,373),(269,332,309,372),(270,331,310,371),(271,330,311,370),(272,329,312,369),(273,328,313,368),(274,327,314,367),(275,326,315,366),(276,325,316,365),(277,324,317,364),(278,323,318,363),(279,322,319,362),(280,321,320,361)]])
129 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 1 | 1 | 2 | 40 | 40 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 40 | 40 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
129 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D5 | D8 | D10 | C3×D4 | C3×D5 | Q32 | D20 | C3×D8 | C6×D5 | D40 | C3×Q32 | C3×D20 | Dic40 | C3×D40 | C3×Dic40 |
| kernel | C3×Dic40 | C240 | C3×Dic20 | Dic40 | C80 | Dic20 | C60 | C48 | C30 | C24 | C20 | C16 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C3×Dic40 ►in GL2(𝔽241) generated by
| 15 | 0 |
| 0 | 15 |
| 49 | 84 |
| 103 | 152 |
| 109 | 37 |
| 226 | 132 |
G:=sub<GL(2,GF(241))| [15,0,0,15],[49,103,84,152],[109,226,37,132] >;
C3×Dic40 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{40} % in TeX
G:=Group("C3xDic40"); // GroupNames label
G:=SmallGroup(480,79);
// by ID
G=gap.SmallGroup(480,79);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,336,197,260,1011,192,2524,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^3=b^80=1,c^2=b^40,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export