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## G = C3⋊Dic39order 468 = 22·32·13

### The semidirect product of C3 and Dic39 acting via Dic39/C78=C2

Aliases: C3⋊Dic39, C78.3S3, C6.3D39, C393Dic3, C323Dic13, (C3×C39)⋊7C4, C26.(C3⋊S3), C2.(C3⋊D39), (C3×C78).1C2, (C3×C6).2D13, C132(C3⋊Dic3), SmallGroup(468,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C39 — C3⋊Dic39
 Chief series C1 — C13 — C39 — C3×C39 — C3×C78 — C3⋊Dic39
 Lower central C3×C39 — C3⋊Dic39
 Upper central C1 — C2

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

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

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

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

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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