metabelian, supersoluble, monomial, A-group
Aliases: C3⋊Dic39, C78.3S3, C6.3D39, C39⋊3Dic3, C32⋊3Dic13, (C3×C39)⋊7C4, C26.(C3⋊S3), C2.(C3⋊D39), (C3×C78).1C2, (C3×C6).2D13, C13⋊2(C3⋊Dic3), SmallGroup(468,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C39 — C3⋊Dic39 |
Generators and relations for C3⋊Dic39
G = < a,b,c | a3=b78=1, c2=b39, ab=ba, cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C3⋊Dic39
►Regular action on 468 points
Generators in S468
(1 236 163)(2 237 164)(3 238 165)(4 239 166)(5 240 167)(6 241 168)(7 242 169)(8 243 170)(9 244 171)(10 245 172)(11 246 173)(12 247 174)(13 248 175)(14 249 176)(15 250 177)(16 251 178)(17 252 179)(18 253 180)(19 254 181)(20 255 182)(21 256 183)(22 257 184)(23 258 185)(24 259 186)(25 260 187)(26 261 188)(27 262 189)(28 263 190)(29 264 191)(30 265 192)(31 266 193)(32 267 194)(33 268 195)(34 269 196)(35 270 197)(36 271 198)(37 272 199)(38 273 200)(39 274 201)(40 275 202)(41 276 203)(42 277 204)(43 278 205)(44 279 206)(45 280 207)(46 281 208)(47 282 209)(48 283 210)(49 284 211)(50 285 212)(51 286 213)(52 287 214)(53 288 215)(54 289 216)(55 290 217)(56 291 218)(57 292 219)(58 293 220)(59 294 221)(60 295 222)(61 296 223)(62 297 224)(63 298 225)(64 299 226)(65 300 227)(66 301 228)(67 302 229)(68 303 230)(69 304 231)(70 305 232)(71 306 233)(72 307 234)(73 308 157)(74 309 158)(75 310 159)(76 311 160)(77 312 161)(78 235 162)(79 353 410)(80 354 411)(81 355 412)(82 356 413)(83 357 414)(84 358 415)(85 359 416)(86 360 417)(87 361 418)(88 362 419)(89 363 420)(90 364 421)(91 365 422)(92 366 423)(93 367 424)(94 368 425)(95 369 426)(96 370 427)(97 371 428)(98 372 429)(99 373 430)(100 374 431)(101 375 432)(102 376 433)(103 377 434)(104 378 435)(105 379 436)(106 380 437)(107 381 438)(108 382 439)(109 383 440)(110 384 441)(111 385 442)(112 386 443)(113 387 444)(114 388 445)(115 389 446)(116 390 447)(117 313 448)(118 314 449)(119 315 450)(120 316 451)(121 317 452)(122 318 453)(123 319 454)(124 320 455)(125 321 456)(126 322 457)(127 323 458)(128 324 459)(129 325 460)(130 326 461)(131 327 462)(132 328 463)(133 329 464)(134 330 465)(135 331 466)(136 332 467)(137 333 468)(138 334 391)(139 335 392)(140 336 393)(141 337 394)(142 338 395)(143 339 396)(144 340 397)(145 341 398)(146 342 399)(147 343 400)(148 344 401)(149 345 402)(150 346 403)(151 347 404)(152 348 405)(153 349 406)(154 350 407)(155 351 408)(156 352 409)
(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)
(1 104 40 143)(2 103 41 142)(3 102 42 141)(4 101 43 140)(5 100 44 139)(6 99 45 138)(7 98 46 137)(8 97 47 136)(9 96 48 135)(10 95 49 134)(11 94 50 133)(12 93 51 132)(13 92 52 131)(14 91 53 130)(15 90 54 129)(16 89 55 128)(17 88 56 127)(18 87 57 126)(19 86 58 125)(20 85 59 124)(21 84 60 123)(22 83 61 122)(23 82 62 121)(24 81 63 120)(25 80 64 119)(26 79 65 118)(27 156 66 117)(28 155 67 116)(29 154 68 115)(30 153 69 114)(31 152 70 113)(32 151 71 112)(33 150 72 111)(34 149 73 110)(35 148 74 109)(36 147 75 108)(37 146 76 107)(38 145 77 106)(39 144 78 105)(157 384 196 345)(158 383 197 344)(159 382 198 343)(160 381 199 342)(161 380 200 341)(162 379 201 340)(163 378 202 339)(164 377 203 338)(165 376 204 337)(166 375 205 336)(167 374 206 335)(168 373 207 334)(169 372 208 333)(170 371 209 332)(171 370 210 331)(172 369 211 330)(173 368 212 329)(174 367 213 328)(175 366 214 327)(176 365 215 326)(177 364 216 325)(178 363 217 324)(179 362 218 323)(180 361 219 322)(181 360 220 321)(182 359 221 320)(183 358 222 319)(184 357 223 318)(185 356 224 317)(186 355 225 316)(187 354 226 315)(188 353 227 314)(189 352 228 313)(190 351 229 390)(191 350 230 389)(192 349 231 388)(193 348 232 387)(194 347 233 386)(195 346 234 385)(235 436 274 397)(236 435 275 396)(237 434 276 395)(238 433 277 394)(239 432 278 393)(240 431 279 392)(241 430 280 391)(242 429 281 468)(243 428 282 467)(244 427 283 466)(245 426 284 465)(246 425 285 464)(247 424 286 463)(248 423 287 462)(249 422 288 461)(250 421 289 460)(251 420 290 459)(252 419 291 458)(253 418 292 457)(254 417 293 456)(255 416 294 455)(256 415 295 454)(257 414 296 453)(258 413 297 452)(259 412 298 451)(260 411 299 450)(261 410 300 449)(262 409 301 448)(263 408 302 447)(264 407 303 446)(265 406 304 445)(266 405 305 444)(267 404 306 443)(268 403 307 442)(269 402 308 441)(270 401 309 440)(271 400 310 439)(272 399 311 438)(273 398 312 437)
G:=sub<Sym(468)| (1,236,163)(2,237,164)(3,238,165)(4,239,166)(5,240,167)(6,241,168)(7,242,169)(8,243,170)(9,244,171)(10,245,172)(11,246,173)(12,247,174)(13,248,175)(14,249,176)(15,250,177)(16,251,178)(17,252,179)(18,253,180)(19,254,181)(20,255,182)(21,256,183)(22,257,184)(23,258,185)(24,259,186)(25,260,187)(26,261,188)(27,262,189)(28,263,190)(29,264,191)(30,265,192)(31,266,193)(32,267,194)(33,268,195)(34,269,196)(35,270,197)(36,271,198)(37,272,199)(38,273,200)(39,274,201)(40,275,202)(41,276,203)(42,277,204)(43,278,205)(44,279,206)(45,280,207)(46,281,208)(47,282,209)(48,283,210)(49,284,211)(50,285,212)(51,286,213)(52,287,214)(53,288,215)(54,289,216)(55,290,217)(56,291,218)(57,292,219)(58,293,220)(59,294,221)(60,295,222)(61,296,223)(62,297,224)(63,298,225)(64,299,226)(65,300,227)(66,301,228)(67,302,229)(68,303,230)(69,304,231)(70,305,232)(71,306,233)(72,307,234)(73,308,157)(74,309,158)(75,310,159)(76,311,160)(77,312,161)(78,235,162)(79,353,410)(80,354,411)(81,355,412)(82,356,413)(83,357,414)(84,358,415)(85,359,416)(86,360,417)(87,361,418)(88,362,419)(89,363,420)(90,364,421)(91,365,422)(92,366,423)(93,367,424)(94,368,425)(95,369,426)(96,370,427)(97,371,428)(98,372,429)(99,373,430)(100,374,431)(101,375,432)(102,376,433)(103,377,434)(104,378,435)(105,379,436)(106,380,437)(107,381,438)(108,382,439)(109,383,440)(110,384,441)(111,385,442)(112,386,443)(113,387,444)(114,388,445)(115,389,446)(116,390,447)(117,313,448)(118,314,449)(119,315,450)(120,316,451)(121,317,452)(122,318,453)(123,319,454)(124,320,455)(125,321,456)(126,322,457)(127,323,458)(128,324,459)(129,325,460)(130,326,461)(131,327,462)(132,328,463)(133,329,464)(134,330,465)(135,331,466)(136,332,467)(137,333,468)(138,334,391)(139,335,392)(140,336,393)(141,337,394)(142,338,395)(143,339,396)(144,340,397)(145,341,398)(146,342,399)(147,343,400)(148,344,401)(149,345,402)(150,346,403)(151,347,404)(152,348,405)(153,349,406)(154,350,407)(155,351,408)(156,352,409), (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), (1,104,40,143)(2,103,41,142)(3,102,42,141)(4,101,43,140)(5,100,44,139)(6,99,45,138)(7,98,46,137)(8,97,47,136)(9,96,48,135)(10,95,49,134)(11,94,50,133)(12,93,51,132)(13,92,52,131)(14,91,53,130)(15,90,54,129)(16,89,55,128)(17,88,56,127)(18,87,57,126)(19,86,58,125)(20,85,59,124)(21,84,60,123)(22,83,61,122)(23,82,62,121)(24,81,63,120)(25,80,64,119)(26,79,65,118)(27,156,66,117)(28,155,67,116)(29,154,68,115)(30,153,69,114)(31,152,70,113)(32,151,71,112)(33,150,72,111)(34,149,73,110)(35,148,74,109)(36,147,75,108)(37,146,76,107)(38,145,77,106)(39,144,78,105)(157,384,196,345)(158,383,197,344)(159,382,198,343)(160,381,199,342)(161,380,200,341)(162,379,201,340)(163,378,202,339)(164,377,203,338)(165,376,204,337)(166,375,205,336)(167,374,206,335)(168,373,207,334)(169,372,208,333)(170,371,209,332)(171,370,210,331)(172,369,211,330)(173,368,212,329)(174,367,213,328)(175,366,214,327)(176,365,215,326)(177,364,216,325)(178,363,217,324)(179,362,218,323)(180,361,219,322)(181,360,220,321)(182,359,221,320)(183,358,222,319)(184,357,223,318)(185,356,224,317)(186,355,225,316)(187,354,226,315)(188,353,227,314)(189,352,228,313)(190,351,229,390)(191,350,230,389)(192,349,231,388)(193,348,232,387)(194,347,233,386)(195,346,234,385)(235,436,274,397)(236,435,275,396)(237,434,276,395)(238,433,277,394)(239,432,278,393)(240,431,279,392)(241,430,280,391)(242,429,281,468)(243,428,282,467)(244,427,283,466)(245,426,284,465)(246,425,285,464)(247,424,286,463)(248,423,287,462)(249,422,288,461)(250,421,289,460)(251,420,290,459)(252,419,291,458)(253,418,292,457)(254,417,293,456)(255,416,294,455)(256,415,295,454)(257,414,296,453)(258,413,297,452)(259,412,298,451)(260,411,299,450)(261,410,300,449)(262,409,301,448)(263,408,302,447)(264,407,303,446)(265,406,304,445)(266,405,305,444)(267,404,306,443)(268,403,307,442)(269,402,308,441)(270,401,309,440)(271,400,310,439)(272,399,311,438)(273,398,312,437)>;
G:=Group( (1,236,163)(2,237,164)(3,238,165)(4,239,166)(5,240,167)(6,241,168)(7,242,169)(8,243,170)(9,244,171)(10,245,172)(11,246,173)(12,247,174)(13,248,175)(14,249,176)(15,250,177)(16,251,178)(17,252,179)(18,253,180)(19,254,181)(20,255,182)(21,256,183)(22,257,184)(23,258,185)(24,259,186)(25,260,187)(26,261,188)(27,262,189)(28,263,190)(29,264,191)(30,265,192)(31,266,193)(32,267,194)(33,268,195)(34,269,196)(35,270,197)(36,271,198)(37,272,199)(38,273,200)(39,274,201)(40,275,202)(41,276,203)(42,277,204)(43,278,205)(44,279,206)(45,280,207)(46,281,208)(47,282,209)(48,283,210)(49,284,211)(50,285,212)(51,286,213)(52,287,214)(53,288,215)(54,289,216)(55,290,217)(56,291,218)(57,292,219)(58,293,220)(59,294,221)(60,295,222)(61,296,223)(62,297,224)(63,298,225)(64,299,226)(65,300,227)(66,301,228)(67,302,229)(68,303,230)(69,304,231)(70,305,232)(71,306,233)(72,307,234)(73,308,157)(74,309,158)(75,310,159)(76,311,160)(77,312,161)(78,235,162)(79,353,410)(80,354,411)(81,355,412)(82,356,413)(83,357,414)(84,358,415)(85,359,416)(86,360,417)(87,361,418)(88,362,419)(89,363,420)(90,364,421)(91,365,422)(92,366,423)(93,367,424)(94,368,425)(95,369,426)(96,370,427)(97,371,428)(98,372,429)(99,373,430)(100,374,431)(101,375,432)(102,376,433)(103,377,434)(104,378,435)(105,379,436)(106,380,437)(107,381,438)(108,382,439)(109,383,440)(110,384,441)(111,385,442)(112,386,443)(113,387,444)(114,388,445)(115,389,446)(116,390,447)(117,313,448)(118,314,449)(119,315,450)(120,316,451)(121,317,452)(122,318,453)(123,319,454)(124,320,455)(125,321,456)(126,322,457)(127,323,458)(128,324,459)(129,325,460)(130,326,461)(131,327,462)(132,328,463)(133,329,464)(134,330,465)(135,331,466)(136,332,467)(137,333,468)(138,334,391)(139,335,392)(140,336,393)(141,337,394)(142,338,395)(143,339,396)(144,340,397)(145,341,398)(146,342,399)(147,343,400)(148,344,401)(149,345,402)(150,346,403)(151,347,404)(152,348,405)(153,349,406)(154,350,407)(155,351,408)(156,352,409), (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), (1,104,40,143)(2,103,41,142)(3,102,42,141)(4,101,43,140)(5,100,44,139)(6,99,45,138)(7,98,46,137)(8,97,47,136)(9,96,48,135)(10,95,49,134)(11,94,50,133)(12,93,51,132)(13,92,52,131)(14,91,53,130)(15,90,54,129)(16,89,55,128)(17,88,56,127)(18,87,57,126)(19,86,58,125)(20,85,59,124)(21,84,60,123)(22,83,61,122)(23,82,62,121)(24,81,63,120)(25,80,64,119)(26,79,65,118)(27,156,66,117)(28,155,67,116)(29,154,68,115)(30,153,69,114)(31,152,70,113)(32,151,71,112)(33,150,72,111)(34,149,73,110)(35,148,74,109)(36,147,75,108)(37,146,76,107)(38,145,77,106)(39,144,78,105)(157,384,196,345)(158,383,197,344)(159,382,198,343)(160,381,199,342)(161,380,200,341)(162,379,201,340)(163,378,202,339)(164,377,203,338)(165,376,204,337)(166,375,205,336)(167,374,206,335)(168,373,207,334)(169,372,208,333)(170,371,209,332)(171,370,210,331)(172,369,211,330)(173,368,212,329)(174,367,213,328)(175,366,214,327)(176,365,215,326)(177,364,216,325)(178,363,217,324)(179,362,218,323)(180,361,219,322)(181,360,220,321)(182,359,221,320)(183,358,222,319)(184,357,223,318)(185,356,224,317)(186,355,225,316)(187,354,226,315)(188,353,227,314)(189,352,228,313)(190,351,229,390)(191,350,230,389)(192,349,231,388)(193,348,232,387)(194,347,233,386)(195,346,234,385)(235,436,274,397)(236,435,275,396)(237,434,276,395)(238,433,277,394)(239,432,278,393)(240,431,279,392)(241,430,280,391)(242,429,281,468)(243,428,282,467)(244,427,283,466)(245,426,284,465)(246,425,285,464)(247,424,286,463)(248,423,287,462)(249,422,288,461)(250,421,289,460)(251,420,290,459)(252,419,291,458)(253,418,292,457)(254,417,293,456)(255,416,294,455)(256,415,295,454)(257,414,296,453)(258,413,297,452)(259,412,298,451)(260,411,299,450)(261,410,300,449)(262,409,301,448)(263,408,302,447)(264,407,303,446)(265,406,304,445)(266,405,305,444)(267,404,306,443)(268,403,307,442)(269,402,308,441)(270,401,309,440)(271,400,310,439)(272,399,311,438)(273,398,312,437) );
G=PermutationGroup([[(1,236,163),(2,237,164),(3,238,165),(4,239,166),(5,240,167),(6,241,168),(7,242,169),(8,243,170),(9,244,171),(10,245,172),(11,246,173),(12,247,174),(13,248,175),(14,249,176),(15,250,177),(16,251,178),(17,252,179),(18,253,180),(19,254,181),(20,255,182),(21,256,183),(22,257,184),(23,258,185),(24,259,186),(25,260,187),(26,261,188),(27,262,189),(28,263,190),(29,264,191),(30,265,192),(31,266,193),(32,267,194),(33,268,195),(34,269,196),(35,270,197),(36,271,198),(37,272,199),(38,273,200),(39,274,201),(40,275,202),(41,276,203),(42,277,204),(43,278,205),(44,279,206),(45,280,207),(46,281,208),(47,282,209),(48,283,210),(49,284,211),(50,285,212),(51,286,213),(52,287,214),(53,288,215),(54,289,216),(55,290,217),(56,291,218),(57,292,219),(58,293,220),(59,294,221),(60,295,222),(61,296,223),(62,297,224),(63,298,225),(64,299,226),(65,300,227),(66,301,228),(67,302,229),(68,303,230),(69,304,231),(70,305,232),(71,306,233),(72,307,234),(73,308,157),(74,309,158),(75,310,159),(76,311,160),(77,312,161),(78,235,162),(79,353,410),(80,354,411),(81,355,412),(82,356,413),(83,357,414),(84,358,415),(85,359,416),(86,360,417),(87,361,418),(88,362,419),(89,363,420),(90,364,421),(91,365,422),(92,366,423),(93,367,424),(94,368,425),(95,369,426),(96,370,427),(97,371,428),(98,372,429),(99,373,430),(100,374,431),(101,375,432),(102,376,433),(103,377,434),(104,378,435),(105,379,436),(106,380,437),(107,381,438),(108,382,439),(109,383,440),(110,384,441),(111,385,442),(112,386,443),(113,387,444),(114,388,445),(115,389,446),(116,390,447),(117,313,448),(118,314,449),(119,315,450),(120,316,451),(121,317,452),(122,318,453),(123,319,454),(124,320,455),(125,321,456),(126,322,457),(127,323,458),(128,324,459),(129,325,460),(130,326,461),(131,327,462),(132,328,463),(133,329,464),(134,330,465),(135,331,466),(136,332,467),(137,333,468),(138,334,391),(139,335,392),(140,336,393),(141,337,394),(142,338,395),(143,339,396),(144,340,397),(145,341,398),(146,342,399),(147,343,400),(148,344,401),(149,345,402),(150,346,403),(151,347,404),(152,348,405),(153,349,406),(154,350,407),(155,351,408),(156,352,409)], [(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)], [(1,104,40,143),(2,103,41,142),(3,102,42,141),(4,101,43,140),(5,100,44,139),(6,99,45,138),(7,98,46,137),(8,97,47,136),(9,96,48,135),(10,95,49,134),(11,94,50,133),(12,93,51,132),(13,92,52,131),(14,91,53,130),(15,90,54,129),(16,89,55,128),(17,88,56,127),(18,87,57,126),(19,86,58,125),(20,85,59,124),(21,84,60,123),(22,83,61,122),(23,82,62,121),(24,81,63,120),(25,80,64,119),(26,79,65,118),(27,156,66,117),(28,155,67,116),(29,154,68,115),(30,153,69,114),(31,152,70,113),(32,151,71,112),(33,150,72,111),(34,149,73,110),(35,148,74,109),(36,147,75,108),(37,146,76,107),(38,145,77,106),(39,144,78,105),(157,384,196,345),(158,383,197,344),(159,382,198,343),(160,381,199,342),(161,380,200,341),(162,379,201,340),(163,378,202,339),(164,377,203,338),(165,376,204,337),(166,375,205,336),(167,374,206,335),(168,373,207,334),(169,372,208,333),(170,371,209,332),(171,370,210,331),(172,369,211,330),(173,368,212,329),(174,367,213,328),(175,366,214,327),(176,365,215,326),(177,364,216,325),(178,363,217,324),(179,362,218,323),(180,361,219,322),(181,360,220,321),(182,359,221,320),(183,358,222,319),(184,357,223,318),(185,356,224,317),(186,355,225,316),(187,354,226,315),(188,353,227,314),(189,352,228,313),(190,351,229,390),(191,350,230,389),(192,349,231,388),(193,348,232,387),(194,347,233,386),(195,346,234,385),(235,436,274,397),(236,435,275,396),(237,434,276,395),(238,433,277,394),(239,432,278,393),(240,431,279,392),(241,430,280,391),(242,429,281,468),(243,428,282,467),(244,427,283,466),(245,426,284,465),(246,425,285,464),(247,424,286,463),(248,423,287,462),(249,422,288,461),(250,421,289,460),(251,420,290,459),(252,419,291,458),(253,418,292,457),(254,417,293,456),(255,416,294,455),(256,415,295,454),(257,414,296,453),(258,413,297,452),(259,412,298,451),(260,411,299,450),(261,410,300,449),(262,409,301,448),(263,408,302,447),(264,407,303,446),(265,406,304,445),(266,405,305,444),(267,404,306,443),(268,403,307,442),(269,402,308,441),(270,401,309,440),(271,400,310,439),(272,399,311,438),(273,398,312,437)]])
120 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39AV | 78A | ··· | 78AV |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 117 | 117 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D13 | Dic13 | D39 | Dic39 |
| kernel | C3⋊Dic39 | C3×C78 | C3×C39 | C78 | C39 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 4 | 6 | 6 | 48 | 48 |
Matrix representation of C3⋊Dic39 ►in GL4(𝔽157) generated by
| 7 | 34 | 0 | 0 |
| 123 | 149 | 0 | 0 |
| 0 | 0 | 54 | 111 |
| 0 | 0 | 68 | 102 |
| 34 | 7 | 0 | 0 |
| 150 | 54 | 0 | 0 |
| 0 | 0 | 103 | 156 |
| 0 | 0 | 138 | 145 |
| 152 | 99 | 0 | 0 |
| 152 | 5 | 0 | 0 |
| 0 | 0 | 138 | 58 |
| 0 | 0 | 10 | 19 |
G:=sub<GL(4,GF(157))| [7,123,0,0,34,149,0,0,0,0,54,68,0,0,111,102],[34,150,0,0,7,54,0,0,0,0,103,138,0,0,156,145],[152,152,0,0,99,5,0,0,0,0,138,10,0,0,58,19] >;
C3⋊Dic39 in GAP, Magma, Sage, TeX
C_3\rtimes {\rm Dic}_{39} % in TeX
G:=Group("C3:Dic39"); // GroupNames label
G:=SmallGroup(468,27);
// by ID
G=gap.SmallGroup(468,27);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-13,10,122,483,10804]);
// Polycyclic
G:=Group<a,b,c|a^3=b^78=1,c^2=b^39,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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