metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊2Q32, C6.9D40, C3⋊2Dic40, C60.27D4, C12.4D20, C30.11D8, C40.43D6, C24.11D10, C120.5C22, Dic60.2C2, Dic20.1S3, C3⋊C16.D5, C8.10(S3×D5), C5⋊1(C3⋊Q32), C10.4(D4⋊S3), C2.7(C3⋊D40), C4.4(C3⋊D20), C20.52(C3⋊D4), (C3×Dic20).1C2, (C5×C3⋊C16).1C2, SmallGroup(480,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊Dic40
G = < a,b,c | a3=b80=1, c2=b40, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3⋊Dic40
►Regular action on 480 points
Generators in S480
(1 166 430)(2 431 167)(3 168 432)(4 433 169)(5 170 434)(6 435 171)(7 172 436)(8 437 173)(9 174 438)(10 439 175)(11 176 440)(12 441 177)(13 178 442)(14 443 179)(15 180 444)(16 445 181)(17 182 446)(18 447 183)(19 184 448)(20 449 185)(21 186 450)(22 451 187)(23 188 452)(24 453 189)(25 190 454)(26 455 191)(27 192 456)(28 457 193)(29 194 458)(30 459 195)(31 196 460)(32 461 197)(33 198 462)(34 463 199)(35 200 464)(36 465 201)(37 202 466)(38 467 203)(39 204 468)(40 469 205)(41 206 470)(42 471 207)(43 208 472)(44 473 209)(45 210 474)(46 475 211)(47 212 476)(48 477 213)(49 214 478)(50 479 215)(51 216 480)(52 401 217)(53 218 402)(54 403 219)(55 220 404)(56 405 221)(57 222 406)(58 407 223)(59 224 408)(60 409 225)(61 226 410)(62 411 227)(63 228 412)(64 413 229)(65 230 414)(66 415 231)(67 232 416)(68 417 233)(69 234 418)(70 419 235)(71 236 420)(72 421 237)(73 238 422)(74 423 239)(75 240 424)(76 425 161)(77 162 426)(78 427 163)(79 164 428)(80 429 165)(81 291 349)(82 350 292)(83 293 351)(84 352 294)(85 295 353)(86 354 296)(87 297 355)(88 356 298)(89 299 357)(90 358 300)(91 301 359)(92 360 302)(93 303 361)(94 362 304)(95 305 363)(96 364 306)(97 307 365)(98 366 308)(99 309 367)(100 368 310)(101 311 369)(102 370 312)(103 313 371)(104 372 314)(105 315 373)(106 374 316)(107 317 375)(108 376 318)(109 319 377)(110 378 320)(111 241 379)(112 380 242)(113 243 381)(114 382 244)(115 245 383)(116 384 246)(117 247 385)(118 386 248)(119 249 387)(120 388 250)(121 251 389)(122 390 252)(123 253 391)(124 392 254)(125 255 393)(126 394 256)(127 257 395)(128 396 258)(129 259 397)(130 398 260)(131 261 399)(132 400 262)(133 263 321)(134 322 264)(135 265 323)(136 324 266)(137 267 325)(138 326 268)(139 269 327)(140 328 270)(141 271 329)(142 330 272)(143 273 331)(144 332 274)(145 275 333)(146 334 276)(147 277 335)(148 336 278)(149 279 337)(150 338 280)(151 281 339)(152 340 282)(153 283 341)(154 342 284)(155 285 343)(156 344 286)(157 287 345)(158 346 288)(159 289 347)(160 348 290)
(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)
(1 368 41 328)(2 367 42 327)(3 366 43 326)(4 365 44 325)(5 364 45 324)(6 363 46 323)(7 362 47 322)(8 361 48 321)(9 360 49 400)(10 359 50 399)(11 358 51 398)(12 357 52 397)(13 356 53 396)(14 355 54 395)(15 354 55 394)(16 353 56 393)(17 352 57 392)(18 351 58 391)(19 350 59 390)(20 349 60 389)(21 348 61 388)(22 347 62 387)(23 346 63 386)(24 345 64 385)(25 344 65 384)(26 343 66 383)(27 342 67 382)(28 341 68 381)(29 340 69 380)(30 339 70 379)(31 338 71 378)(32 337 72 377)(33 336 73 376)(34 335 74 375)(35 334 75 374)(36 333 76 373)(37 332 77 372)(38 331 78 371)(39 330 79 370)(40 329 80 369)(81 409 121 449)(82 408 122 448)(83 407 123 447)(84 406 124 446)(85 405 125 445)(86 404 126 444)(87 403 127 443)(88 402 128 442)(89 401 129 441)(90 480 130 440)(91 479 131 439)(92 478 132 438)(93 477 133 437)(94 476 134 436)(95 475 135 435)(96 474 136 434)(97 473 137 433)(98 472 138 432)(99 471 139 431)(100 470 140 430)(101 469 141 429)(102 468 142 428)(103 467 143 427)(104 466 144 426)(105 465 145 425)(106 464 146 424)(107 463 147 423)(108 462 148 422)(109 461 149 421)(110 460 150 420)(111 459 151 419)(112 458 152 418)(113 457 153 417)(114 456 154 416)(115 455 155 415)(116 454 156 414)(117 453 157 413)(118 452 158 412)(119 451 159 411)(120 450 160 410)(161 315 201 275)(162 314 202 274)(163 313 203 273)(164 312 204 272)(165 311 205 271)(166 310 206 270)(167 309 207 269)(168 308 208 268)(169 307 209 267)(170 306 210 266)(171 305 211 265)(172 304 212 264)(173 303 213 263)(174 302 214 262)(175 301 215 261)(176 300 216 260)(177 299 217 259)(178 298 218 258)(179 297 219 257)(180 296 220 256)(181 295 221 255)(182 294 222 254)(183 293 223 253)(184 292 224 252)(185 291 225 251)(186 290 226 250)(187 289 227 249)(188 288 228 248)(189 287 229 247)(190 286 230 246)(191 285 231 245)(192 284 232 244)(193 283 233 243)(194 282 234 242)(195 281 235 241)(196 280 236 320)(197 279 237 319)(198 278 238 318)(199 277 239 317)(200 276 240 316)
G:=sub<Sym(480)| (1,166,430)(2,431,167)(3,168,432)(4,433,169)(5,170,434)(6,435,171)(7,172,436)(8,437,173)(9,174,438)(10,439,175)(11,176,440)(12,441,177)(13,178,442)(14,443,179)(15,180,444)(16,445,181)(17,182,446)(18,447,183)(19,184,448)(20,449,185)(21,186,450)(22,451,187)(23,188,452)(24,453,189)(25,190,454)(26,455,191)(27,192,456)(28,457,193)(29,194,458)(30,459,195)(31,196,460)(32,461,197)(33,198,462)(34,463,199)(35,200,464)(36,465,201)(37,202,466)(38,467,203)(39,204,468)(40,469,205)(41,206,470)(42,471,207)(43,208,472)(44,473,209)(45,210,474)(46,475,211)(47,212,476)(48,477,213)(49,214,478)(50,479,215)(51,216,480)(52,401,217)(53,218,402)(54,403,219)(55,220,404)(56,405,221)(57,222,406)(58,407,223)(59,224,408)(60,409,225)(61,226,410)(62,411,227)(63,228,412)(64,413,229)(65,230,414)(66,415,231)(67,232,416)(68,417,233)(69,234,418)(70,419,235)(71,236,420)(72,421,237)(73,238,422)(74,423,239)(75,240,424)(76,425,161)(77,162,426)(78,427,163)(79,164,428)(80,429,165)(81,291,349)(82,350,292)(83,293,351)(84,352,294)(85,295,353)(86,354,296)(87,297,355)(88,356,298)(89,299,357)(90,358,300)(91,301,359)(92,360,302)(93,303,361)(94,362,304)(95,305,363)(96,364,306)(97,307,365)(98,366,308)(99,309,367)(100,368,310)(101,311,369)(102,370,312)(103,313,371)(104,372,314)(105,315,373)(106,374,316)(107,317,375)(108,376,318)(109,319,377)(110,378,320)(111,241,379)(112,380,242)(113,243,381)(114,382,244)(115,245,383)(116,384,246)(117,247,385)(118,386,248)(119,249,387)(120,388,250)(121,251,389)(122,390,252)(123,253,391)(124,392,254)(125,255,393)(126,394,256)(127,257,395)(128,396,258)(129,259,397)(130,398,260)(131,261,399)(132,400,262)(133,263,321)(134,322,264)(135,265,323)(136,324,266)(137,267,325)(138,326,268)(139,269,327)(140,328,270)(141,271,329)(142,330,272)(143,273,331)(144,332,274)(145,275,333)(146,334,276)(147,277,335)(148,336,278)(149,279,337)(150,338,280)(151,281,339)(152,340,282)(153,283,341)(154,342,284)(155,285,343)(156,344,286)(157,287,345)(158,346,288)(159,289,347)(160,348,290), (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), (1,368,41,328)(2,367,42,327)(3,366,43,326)(4,365,44,325)(5,364,45,324)(6,363,46,323)(7,362,47,322)(8,361,48,321)(9,360,49,400)(10,359,50,399)(11,358,51,398)(12,357,52,397)(13,356,53,396)(14,355,54,395)(15,354,55,394)(16,353,56,393)(17,352,57,392)(18,351,58,391)(19,350,59,390)(20,349,60,389)(21,348,61,388)(22,347,62,387)(23,346,63,386)(24,345,64,385)(25,344,65,384)(26,343,66,383)(27,342,67,382)(28,341,68,381)(29,340,69,380)(30,339,70,379)(31,338,71,378)(32,337,72,377)(33,336,73,376)(34,335,74,375)(35,334,75,374)(36,333,76,373)(37,332,77,372)(38,331,78,371)(39,330,79,370)(40,329,80,369)(81,409,121,449)(82,408,122,448)(83,407,123,447)(84,406,124,446)(85,405,125,445)(86,404,126,444)(87,403,127,443)(88,402,128,442)(89,401,129,441)(90,480,130,440)(91,479,131,439)(92,478,132,438)(93,477,133,437)(94,476,134,436)(95,475,135,435)(96,474,136,434)(97,473,137,433)(98,472,138,432)(99,471,139,431)(100,470,140,430)(101,469,141,429)(102,468,142,428)(103,467,143,427)(104,466,144,426)(105,465,145,425)(106,464,146,424)(107,463,147,423)(108,462,148,422)(109,461,149,421)(110,460,150,420)(111,459,151,419)(112,458,152,418)(113,457,153,417)(114,456,154,416)(115,455,155,415)(116,454,156,414)(117,453,157,413)(118,452,158,412)(119,451,159,411)(120,450,160,410)(161,315,201,275)(162,314,202,274)(163,313,203,273)(164,312,204,272)(165,311,205,271)(166,310,206,270)(167,309,207,269)(168,308,208,268)(169,307,209,267)(170,306,210,266)(171,305,211,265)(172,304,212,264)(173,303,213,263)(174,302,214,262)(175,301,215,261)(176,300,216,260)(177,299,217,259)(178,298,218,258)(179,297,219,257)(180,296,220,256)(181,295,221,255)(182,294,222,254)(183,293,223,253)(184,292,224,252)(185,291,225,251)(186,290,226,250)(187,289,227,249)(188,288,228,248)(189,287,229,247)(190,286,230,246)(191,285,231,245)(192,284,232,244)(193,283,233,243)(194,282,234,242)(195,281,235,241)(196,280,236,320)(197,279,237,319)(198,278,238,318)(199,277,239,317)(200,276,240,316)>;
G:=Group( (1,166,430)(2,431,167)(3,168,432)(4,433,169)(5,170,434)(6,435,171)(7,172,436)(8,437,173)(9,174,438)(10,439,175)(11,176,440)(12,441,177)(13,178,442)(14,443,179)(15,180,444)(16,445,181)(17,182,446)(18,447,183)(19,184,448)(20,449,185)(21,186,450)(22,451,187)(23,188,452)(24,453,189)(25,190,454)(26,455,191)(27,192,456)(28,457,193)(29,194,458)(30,459,195)(31,196,460)(32,461,197)(33,198,462)(34,463,199)(35,200,464)(36,465,201)(37,202,466)(38,467,203)(39,204,468)(40,469,205)(41,206,470)(42,471,207)(43,208,472)(44,473,209)(45,210,474)(46,475,211)(47,212,476)(48,477,213)(49,214,478)(50,479,215)(51,216,480)(52,401,217)(53,218,402)(54,403,219)(55,220,404)(56,405,221)(57,222,406)(58,407,223)(59,224,408)(60,409,225)(61,226,410)(62,411,227)(63,228,412)(64,413,229)(65,230,414)(66,415,231)(67,232,416)(68,417,233)(69,234,418)(70,419,235)(71,236,420)(72,421,237)(73,238,422)(74,423,239)(75,240,424)(76,425,161)(77,162,426)(78,427,163)(79,164,428)(80,429,165)(81,291,349)(82,350,292)(83,293,351)(84,352,294)(85,295,353)(86,354,296)(87,297,355)(88,356,298)(89,299,357)(90,358,300)(91,301,359)(92,360,302)(93,303,361)(94,362,304)(95,305,363)(96,364,306)(97,307,365)(98,366,308)(99,309,367)(100,368,310)(101,311,369)(102,370,312)(103,313,371)(104,372,314)(105,315,373)(106,374,316)(107,317,375)(108,376,318)(109,319,377)(110,378,320)(111,241,379)(112,380,242)(113,243,381)(114,382,244)(115,245,383)(116,384,246)(117,247,385)(118,386,248)(119,249,387)(120,388,250)(121,251,389)(122,390,252)(123,253,391)(124,392,254)(125,255,393)(126,394,256)(127,257,395)(128,396,258)(129,259,397)(130,398,260)(131,261,399)(132,400,262)(133,263,321)(134,322,264)(135,265,323)(136,324,266)(137,267,325)(138,326,268)(139,269,327)(140,328,270)(141,271,329)(142,330,272)(143,273,331)(144,332,274)(145,275,333)(146,334,276)(147,277,335)(148,336,278)(149,279,337)(150,338,280)(151,281,339)(152,340,282)(153,283,341)(154,342,284)(155,285,343)(156,344,286)(157,287,345)(158,346,288)(159,289,347)(160,348,290), (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), (1,368,41,328)(2,367,42,327)(3,366,43,326)(4,365,44,325)(5,364,45,324)(6,363,46,323)(7,362,47,322)(8,361,48,321)(9,360,49,400)(10,359,50,399)(11,358,51,398)(12,357,52,397)(13,356,53,396)(14,355,54,395)(15,354,55,394)(16,353,56,393)(17,352,57,392)(18,351,58,391)(19,350,59,390)(20,349,60,389)(21,348,61,388)(22,347,62,387)(23,346,63,386)(24,345,64,385)(25,344,65,384)(26,343,66,383)(27,342,67,382)(28,341,68,381)(29,340,69,380)(30,339,70,379)(31,338,71,378)(32,337,72,377)(33,336,73,376)(34,335,74,375)(35,334,75,374)(36,333,76,373)(37,332,77,372)(38,331,78,371)(39,330,79,370)(40,329,80,369)(81,409,121,449)(82,408,122,448)(83,407,123,447)(84,406,124,446)(85,405,125,445)(86,404,126,444)(87,403,127,443)(88,402,128,442)(89,401,129,441)(90,480,130,440)(91,479,131,439)(92,478,132,438)(93,477,133,437)(94,476,134,436)(95,475,135,435)(96,474,136,434)(97,473,137,433)(98,472,138,432)(99,471,139,431)(100,470,140,430)(101,469,141,429)(102,468,142,428)(103,467,143,427)(104,466,144,426)(105,465,145,425)(106,464,146,424)(107,463,147,423)(108,462,148,422)(109,461,149,421)(110,460,150,420)(111,459,151,419)(112,458,152,418)(113,457,153,417)(114,456,154,416)(115,455,155,415)(116,454,156,414)(117,453,157,413)(118,452,158,412)(119,451,159,411)(120,450,160,410)(161,315,201,275)(162,314,202,274)(163,313,203,273)(164,312,204,272)(165,311,205,271)(166,310,206,270)(167,309,207,269)(168,308,208,268)(169,307,209,267)(170,306,210,266)(171,305,211,265)(172,304,212,264)(173,303,213,263)(174,302,214,262)(175,301,215,261)(176,300,216,260)(177,299,217,259)(178,298,218,258)(179,297,219,257)(180,296,220,256)(181,295,221,255)(182,294,222,254)(183,293,223,253)(184,292,224,252)(185,291,225,251)(186,290,226,250)(187,289,227,249)(188,288,228,248)(189,287,229,247)(190,286,230,246)(191,285,231,245)(192,284,232,244)(193,283,233,243)(194,282,234,242)(195,281,235,241)(196,280,236,320)(197,279,237,319)(198,278,238,318)(199,277,239,317)(200,276,240,316) );
G=PermutationGroup([[(1,166,430),(2,431,167),(3,168,432),(4,433,169),(5,170,434),(6,435,171),(7,172,436),(8,437,173),(9,174,438),(10,439,175),(11,176,440),(12,441,177),(13,178,442),(14,443,179),(15,180,444),(16,445,181),(17,182,446),(18,447,183),(19,184,448),(20,449,185),(21,186,450),(22,451,187),(23,188,452),(24,453,189),(25,190,454),(26,455,191),(27,192,456),(28,457,193),(29,194,458),(30,459,195),(31,196,460),(32,461,197),(33,198,462),(34,463,199),(35,200,464),(36,465,201),(37,202,466),(38,467,203),(39,204,468),(40,469,205),(41,206,470),(42,471,207),(43,208,472),(44,473,209),(45,210,474),(46,475,211),(47,212,476),(48,477,213),(49,214,478),(50,479,215),(51,216,480),(52,401,217),(53,218,402),(54,403,219),(55,220,404),(56,405,221),(57,222,406),(58,407,223),(59,224,408),(60,409,225),(61,226,410),(62,411,227),(63,228,412),(64,413,229),(65,230,414),(66,415,231),(67,232,416),(68,417,233),(69,234,418),(70,419,235),(71,236,420),(72,421,237),(73,238,422),(74,423,239),(75,240,424),(76,425,161),(77,162,426),(78,427,163),(79,164,428),(80,429,165),(81,291,349),(82,350,292),(83,293,351),(84,352,294),(85,295,353),(86,354,296),(87,297,355),(88,356,298),(89,299,357),(90,358,300),(91,301,359),(92,360,302),(93,303,361),(94,362,304),(95,305,363),(96,364,306),(97,307,365),(98,366,308),(99,309,367),(100,368,310),(101,311,369),(102,370,312),(103,313,371),(104,372,314),(105,315,373),(106,374,316),(107,317,375),(108,376,318),(109,319,377),(110,378,320),(111,241,379),(112,380,242),(113,243,381),(114,382,244),(115,245,383),(116,384,246),(117,247,385),(118,386,248),(119,249,387),(120,388,250),(121,251,389),(122,390,252),(123,253,391),(124,392,254),(125,255,393),(126,394,256),(127,257,395),(128,396,258),(129,259,397),(130,398,260),(131,261,399),(132,400,262),(133,263,321),(134,322,264),(135,265,323),(136,324,266),(137,267,325),(138,326,268),(139,269,327),(140,328,270),(141,271,329),(142,330,272),(143,273,331),(144,332,274),(145,275,333),(146,334,276),(147,277,335),(148,336,278),(149,279,337),(150,338,280),(151,281,339),(152,340,282),(153,283,341),(154,342,284),(155,285,343),(156,344,286),(157,287,345),(158,346,288),(159,289,347),(160,348,290)], [(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)], [(1,368,41,328),(2,367,42,327),(3,366,43,326),(4,365,44,325),(5,364,45,324),(6,363,46,323),(7,362,47,322),(8,361,48,321),(9,360,49,400),(10,359,50,399),(11,358,51,398),(12,357,52,397),(13,356,53,396),(14,355,54,395),(15,354,55,394),(16,353,56,393),(17,352,57,392),(18,351,58,391),(19,350,59,390),(20,349,60,389),(21,348,61,388),(22,347,62,387),(23,346,63,386),(24,345,64,385),(25,344,65,384),(26,343,66,383),(27,342,67,382),(28,341,68,381),(29,340,69,380),(30,339,70,379),(31,338,71,378),(32,337,72,377),(33,336,73,376),(34,335,74,375),(35,334,75,374),(36,333,76,373),(37,332,77,372),(38,331,78,371),(39,330,79,370),(40,329,80,369),(81,409,121,449),(82,408,122,448),(83,407,123,447),(84,406,124,446),(85,405,125,445),(86,404,126,444),(87,403,127,443),(88,402,128,442),(89,401,129,441),(90,480,130,440),(91,479,131,439),(92,478,132,438),(93,477,133,437),(94,476,134,436),(95,475,135,435),(96,474,136,434),(97,473,137,433),(98,472,138,432),(99,471,139,431),(100,470,140,430),(101,469,141,429),(102,468,142,428),(103,467,143,427),(104,466,144,426),(105,465,145,425),(106,464,146,424),(107,463,147,423),(108,462,148,422),(109,461,149,421),(110,460,150,420),(111,459,151,419),(112,458,152,418),(113,457,153,417),(114,456,154,416),(115,455,155,415),(116,454,156,414),(117,453,157,413),(118,452,158,412),(119,451,159,411),(120,450,160,410),(161,315,201,275),(162,314,202,274),(163,313,203,273),(164,312,204,272),(165,311,205,271),(166,310,206,270),(167,309,207,269),(168,308,208,268),(169,307,209,267),(170,306,210,266),(171,305,211,265),(172,304,212,264),(173,303,213,263),(174,302,214,262),(175,301,215,261),(176,300,216,260),(177,299,217,259),(178,298,218,258),(179,297,219,257),(180,296,220,256),(181,295,221,255),(182,294,222,254),(183,293,223,253),(184,292,224,252),(185,291,225,251),(186,290,226,250),(187,289,227,249),(188,288,228,248),(189,287,229,247),(190,286,230,246),(191,285,231,245),(192,284,232,244),(193,283,233,243),(194,282,234,242),(195,281,235,241),(196,280,236,320),(197,279,237,319),(198,278,238,318),(199,277,239,317),(200,276,240,316)]])
66 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 40A | ··· | 40H | 60A | 60B | 60C | 60D | 80A | ··· | 80P | 120A | ··· | 120H |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 80 | ··· | 80 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 2 | 40 | 120 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 40 | 40 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | C3⋊D4 | Q32 | D20 | D40 | Dic40 | D4⋊S3 | S3×D5 | C3⋊Q32 | C3⋊D20 | C3⋊D40 | C3⋊Dic40 |
| kernel | C3⋊Dic40 | C5×C3⋊C16 | C3×Dic20 | Dic60 | Dic20 | C60 | C3⋊C16 | C40 | C30 | C24 | C20 | C15 | C12 | C6 | C3 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 16 | 1 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C3⋊Dic40 ►in GL4(𝔽241) generated by
| 240 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 171 | 171 | 0 | 0 |
| 101 | 70 | 0 | 0 |
| 0 | 0 | 192 | 98 |
| 0 | 0 | 143 | 14 |
| 171 | 140 | 0 | 0 |
| 101 | 70 | 0 | 0 |
| 0 | 0 | 154 | 205 |
| 0 | 0 | 63 | 87 |
G:=sub<GL(4,GF(241))| [240,240,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[171,101,0,0,171,70,0,0,0,0,192,143,0,0,98,14],[171,101,0,0,140,70,0,0,0,0,154,63,0,0,205,87] >;
C3⋊Dic40 in GAP, Magma, Sage, TeX
C_3\rtimes {\rm Dic}_{40} % in TeX
G:=Group("C3:Dic40"); // GroupNames label
G:=SmallGroup(480,23);
// by ID
G=gap.SmallGroup(480,23);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,85,92,254,142,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^3=b^80=1,c^2=b^40,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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