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## G = C13×C3⋊Dic3order 468 = 22·32·13

### Direct product of C13 and C3⋊Dic3

Aliases: C13×C3⋊Dic3, C78.7S3, C323C52, C395Dic3, (C3×C39)⋊11C4, C3⋊(Dic3×C13), C6.3(S3×C13), (C3×C6).2C26, (C3×C78).5C2, C26.2(C3⋊S3), C2.(C13×C3⋊S3), SmallGroup(468,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C13×C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C78 — C13×C3⋊Dic3
 Lower central C32 — C13×C3⋊Dic3
 Upper central C1 — C26

Generators and relations for C13×C3⋊Dic3
G = < a,b,c,d | a13=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

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

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

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

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

156 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 13A ··· 13L 26A ··· 26L 39A ··· 39AV 52A ··· 52X 78A ··· 78AV order 1 2 3 3 3 3 4 4 6 6 6 6 13 ··· 13 26 ··· 26 39 ··· 39 52 ··· 52 78 ··· 78 size 1 1 2 2 2 2 9 9 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2

156 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C13 C26 C52 S3 Dic3 S3×C13 Dic3×C13 kernel C13×C3⋊Dic3 C3×C78 C3×C39 C3⋊Dic3 C3×C6 C32 C78 C39 C6 C3 # reps 1 1 2 12 12 24 4 4 48 48

Matrix representation of C13×C3⋊Dic3 in GL4(𝔽157) generated by

 130 0 0 0 0 130 0 0 0 0 93 0 0 0 0 93
,
 0 1 0 0 156 156 0 0 0 0 0 1 0 0 156 156
,
 0 1 0 0 156 156 0 0 0 0 1 1 0 0 156 0
,
 68 22 0 0 111 89 0 0 0 0 154 138 0 0 141 3
G:=sub<GL(4,GF(157))| [130,0,0,0,0,130,0,0,0,0,93,0,0,0,0,93],[0,156,0,0,1,156,0,0,0,0,0,156,0,0,1,156],[0,156,0,0,1,156,0,0,0,0,1,156,0,0,1,0],[68,111,0,0,22,89,0,0,0,0,154,141,0,0,138,3] >;

C13×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_{13}\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C13xC3:Dic3");
// GroupNames label

G:=SmallGroup(468,26);
// by ID

G=gap.SmallGroup(468,26);
# by ID

G:=PCGroup([5,-2,-13,-2,-3,-3,130,2083,7804]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^3=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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