direct product, metabelian, supersoluble, monomial, A-group
Aliases: C13×C3⋊Dic3, C78.7S3, C32⋊3C52, C39⋊5Dic3, (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
| C32 — C13×C3⋊Dic3 |
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)
(1 135 393)(2 136 394)(3 137 395)(4 138 396)(5 139 397)(6 140 398)(7 141 399)(8 142 400)(9 143 401)(10 131 402)(11 132 403)(12 133 391)(13 134 392)(14 199 291)(15 200 292)(16 201 293)(17 202 294)(18 203 295)(19 204 296)(20 205 297)(21 206 298)(22 207 299)(23 208 287)(24 196 288)(25 197 289)(26 198 290)(27 212 304)(28 213 305)(29 214 306)(30 215 307)(31 216 308)(32 217 309)(33 218 310)(34 219 311)(35 220 312)(36 221 300)(37 209 301)(38 210 302)(39 211 303)(40 346 328)(41 347 329)(42 348 330)(43 349 331)(44 350 332)(45 351 333)(46 339 334)(47 340 335)(48 341 336)(49 342 337)(50 343 338)(51 344 326)(52 345 327)(53 381 84)(54 382 85)(55 383 86)(56 384 87)(57 385 88)(58 386 89)(59 387 90)(60 388 91)(61 389 79)(62 390 80)(63 378 81)(64 379 82)(65 380 83)(66 466 110)(67 467 111)(68 468 112)(69 456 113)(70 457 114)(71 458 115)(72 459 116)(73 460 117)(74 461 105)(75 462 106)(76 463 107)(77 464 108)(78 465 109)(92 257 277)(93 258 278)(94 259 279)(95 260 280)(96 248 281)(97 249 282)(98 250 283)(99 251 284)(100 252 285)(101 253 286)(102 254 274)(103 255 275)(104 256 276)(118 441 164)(119 442 165)(120 430 166)(121 431 167)(122 432 168)(123 433 169)(124 434 157)(125 435 158)(126 436 159)(127 437 160)(128 438 161)(129 439 162)(130 440 163)(144 454 177)(145 455 178)(146 443 179)(147 444 180)(148 445 181)(149 446 182)(150 447 170)(151 448 171)(152 449 172)(153 450 173)(154 451 174)(155 452 175)(156 453 176)(183 355 368)(184 356 369)(185 357 370)(186 358 371)(187 359 372)(188 360 373)(189 361 374)(190 362 375)(191 363 376)(192 364 377)(193 352 365)(194 353 366)(195 354 367)(222 410 423)(223 411 424)(224 412 425)(225 413 426)(226 414 427)(227 415 428)(228 416 429)(229 404 417)(230 405 418)(231 406 419)(232 407 420)(233 408 421)(234 409 422)(235 269 320)(236 270 321)(237 271 322)(238 272 323)(239 273 324)(240 261 325)(241 262 313)(242 263 314)(243 264 315)(244 265 316)(245 266 317)(246 267 318)(247 268 319)
(1 187 426 268 248 46)(2 188 427 269 249 47)(3 189 428 270 250 48)(4 190 429 271 251 49)(5 191 417 272 252 50)(6 192 418 273 253 51)(7 193 419 261 254 52)(8 194 420 262 255 40)(9 195 421 263 256 41)(10 183 422 264 257 42)(11 184 423 265 258 43)(12 185 424 266 259 44)(13 186 425 267 260 45)(14 436 443 75 34 55)(15 437 444 76 35 56)(16 438 445 77 36 57)(17 439 446 78 37 58)(18 440 447 66 38 59)(19 441 448 67 39 60)(20 442 449 68 27 61)(21 430 450 69 28 62)(22 431 451 70 29 63)(23 432 452 71 30 64)(24 433 453 72 31 65)(25 434 454 73 32 53)(26 435 455 74 33 54)(79 297 119 152 112 304)(80 298 120 153 113 305)(81 299 121 154 114 306)(82 287 122 155 115 307)(83 288 123 156 116 308)(84 289 124 144 117 309)(85 290 125 145 105 310)(86 291 126 146 106 311)(87 292 127 147 107 312)(88 293 128 148 108 300)(89 294 129 149 109 301)(90 295 130 150 110 302)(91 296 118 151 111 303)(92 330 402 368 409 243)(93 331 403 369 410 244)(94 332 391 370 411 245)(95 333 392 371 412 246)(96 334 393 372 413 247)(97 335 394 373 414 235)(98 336 395 374 415 236)(99 337 396 375 416 237)(100 338 397 376 404 238)(101 326 398 377 405 239)(102 327 399 365 406 240)(103 328 400 366 407 241)(104 329 401 367 408 242)(131 355 234 315 277 348)(132 356 222 316 278 349)(133 357 223 317 279 350)(134 358 224 318 280 351)(135 359 225 319 281 339)(136 360 226 320 282 340)(137 361 227 321 283 341)(138 362 228 322 284 342)(139 363 229 323 285 343)(140 364 230 324 286 344)(141 352 231 325 274 345)(142 353 232 313 275 346)(143 354 233 314 276 347)(157 177 460 217 381 197)(158 178 461 218 382 198)(159 179 462 219 383 199)(160 180 463 220 384 200)(161 181 464 221 385 201)(162 182 465 209 386 202)(163 170 466 210 387 203)(164 171 467 211 388 204)(165 172 468 212 389 205)(166 173 456 213 390 206)(167 174 457 214 378 207)(168 175 458 215 379 208)(169 176 459 216 380 196)
(1 156 268 83)(2 144 269 84)(3 145 270 85)(4 146 271 86)(5 147 272 87)(6 148 273 88)(7 149 261 89)(8 150 262 90)(9 151 263 91)(10 152 264 79)(11 153 265 80)(12 154 266 81)(13 155 267 82)(14 337 75 416)(15 338 76 404)(16 326 77 405)(17 327 78 406)(18 328 66 407)(19 329 67 408)(20 330 68 409)(21 331 69 410)(22 332 70 411)(23 333 71 412)(24 334 72 413)(25 335 73 414)(26 336 74 415)(27 368 442 92)(28 369 430 93)(29 370 431 94)(30 371 432 95)(31 372 433 96)(32 373 434 97)(33 374 435 98)(34 375 436 99)(35 376 437 100)(36 377 438 101)(37 365 439 102)(38 366 440 103)(39 367 441 104)(40 110 420 295)(41 111 421 296)(42 112 422 297)(43 113 423 298)(44 114 424 299)(45 115 425 287)(46 116 426 288)(47 117 427 289)(48 105 428 290)(49 106 429 291)(50 107 417 292)(51 108 418 293)(52 109 419 294)(53 394 454 235)(54 395 455 236)(55 396 443 237)(56 397 444 238)(57 398 445 239)(58 399 446 240)(59 400 447 241)(60 401 448 242)(61 402 449 243)(62 403 450 244)(63 391 451 245)(64 392 452 246)(65 393 453 247)(118 256 303 195)(119 257 304 183)(120 258 305 184)(121 259 306 185)(122 260 307 186)(123 248 308 187)(124 249 309 188)(125 250 310 189)(126 251 311 190)(127 252 312 191)(128 253 300 192)(129 254 301 193)(130 255 302 194)(131 172 315 389)(132 173 316 390)(133 174 317 378)(134 175 318 379)(135 176 319 380)(136 177 320 381)(137 178 321 382)(138 179 322 383)(139 180 323 384)(140 181 324 385)(141 182 325 386)(142 170 313 387)(143 171 314 388)(157 282 217 360)(158 283 218 361)(159 284 219 362)(160 285 220 363)(161 286 221 364)(162 274 209 352)(163 275 210 353)(164 276 211 354)(165 277 212 355)(166 278 213 356)(167 279 214 357)(168 280 215 358)(169 281 216 359)(196 339 459 225)(197 340 460 226)(198 341 461 227)(199 342 462 228)(200 343 463 229)(201 344 464 230)(202 345 465 231)(203 346 466 232)(204 347 467 233)(205 348 468 234)(206 349 456 222)(207 350 457 223)(208 351 458 224)
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), (1,135,393)(2,136,394)(3,137,395)(4,138,396)(5,139,397)(6,140,398)(7,141,399)(8,142,400)(9,143,401)(10,131,402)(11,132,403)(12,133,391)(13,134,392)(14,199,291)(15,200,292)(16,201,293)(17,202,294)(18,203,295)(19,204,296)(20,205,297)(21,206,298)(22,207,299)(23,208,287)(24,196,288)(25,197,289)(26,198,290)(27,212,304)(28,213,305)(29,214,306)(30,215,307)(31,216,308)(32,217,309)(33,218,310)(34,219,311)(35,220,312)(36,221,300)(37,209,301)(38,210,302)(39,211,303)(40,346,328)(41,347,329)(42,348,330)(43,349,331)(44,350,332)(45,351,333)(46,339,334)(47,340,335)(48,341,336)(49,342,337)(50,343,338)(51,344,326)(52,345,327)(53,381,84)(54,382,85)(55,383,86)(56,384,87)(57,385,88)(58,386,89)(59,387,90)(60,388,91)(61,389,79)(62,390,80)(63,378,81)(64,379,82)(65,380,83)(66,466,110)(67,467,111)(68,468,112)(69,456,113)(70,457,114)(71,458,115)(72,459,116)(73,460,117)(74,461,105)(75,462,106)(76,463,107)(77,464,108)(78,465,109)(92,257,277)(93,258,278)(94,259,279)(95,260,280)(96,248,281)(97,249,282)(98,250,283)(99,251,284)(100,252,285)(101,253,286)(102,254,274)(103,255,275)(104,256,276)(118,441,164)(119,442,165)(120,430,166)(121,431,167)(122,432,168)(123,433,169)(124,434,157)(125,435,158)(126,436,159)(127,437,160)(128,438,161)(129,439,162)(130,440,163)(144,454,177)(145,455,178)(146,443,179)(147,444,180)(148,445,181)(149,446,182)(150,447,170)(151,448,171)(152,449,172)(153,450,173)(154,451,174)(155,452,175)(156,453,176)(183,355,368)(184,356,369)(185,357,370)(186,358,371)(187,359,372)(188,360,373)(189,361,374)(190,362,375)(191,363,376)(192,364,377)(193,352,365)(194,353,366)(195,354,367)(222,410,423)(223,411,424)(224,412,425)(225,413,426)(226,414,427)(227,415,428)(228,416,429)(229,404,417)(230,405,418)(231,406,419)(232,407,420)(233,408,421)(234,409,422)(235,269,320)(236,270,321)(237,271,322)(238,272,323)(239,273,324)(240,261,325)(241,262,313)(242,263,314)(243,264,315)(244,265,316)(245,266,317)(246,267,318)(247,268,319), (1,187,426,268,248,46)(2,188,427,269,249,47)(3,189,428,270,250,48)(4,190,429,271,251,49)(5,191,417,272,252,50)(6,192,418,273,253,51)(7,193,419,261,254,52)(8,194,420,262,255,40)(9,195,421,263,256,41)(10,183,422,264,257,42)(11,184,423,265,258,43)(12,185,424,266,259,44)(13,186,425,267,260,45)(14,436,443,75,34,55)(15,437,444,76,35,56)(16,438,445,77,36,57)(17,439,446,78,37,58)(18,440,447,66,38,59)(19,441,448,67,39,60)(20,442,449,68,27,61)(21,430,450,69,28,62)(22,431,451,70,29,63)(23,432,452,71,30,64)(24,433,453,72,31,65)(25,434,454,73,32,53)(26,435,455,74,33,54)(79,297,119,152,112,304)(80,298,120,153,113,305)(81,299,121,154,114,306)(82,287,122,155,115,307)(83,288,123,156,116,308)(84,289,124,144,117,309)(85,290,125,145,105,310)(86,291,126,146,106,311)(87,292,127,147,107,312)(88,293,128,148,108,300)(89,294,129,149,109,301)(90,295,130,150,110,302)(91,296,118,151,111,303)(92,330,402,368,409,243)(93,331,403,369,410,244)(94,332,391,370,411,245)(95,333,392,371,412,246)(96,334,393,372,413,247)(97,335,394,373,414,235)(98,336,395,374,415,236)(99,337,396,375,416,237)(100,338,397,376,404,238)(101,326,398,377,405,239)(102,327,399,365,406,240)(103,328,400,366,407,241)(104,329,401,367,408,242)(131,355,234,315,277,348)(132,356,222,316,278,349)(133,357,223,317,279,350)(134,358,224,318,280,351)(135,359,225,319,281,339)(136,360,226,320,282,340)(137,361,227,321,283,341)(138,362,228,322,284,342)(139,363,229,323,285,343)(140,364,230,324,286,344)(141,352,231,325,274,345)(142,353,232,313,275,346)(143,354,233,314,276,347)(157,177,460,217,381,197)(158,178,461,218,382,198)(159,179,462,219,383,199)(160,180,463,220,384,200)(161,181,464,221,385,201)(162,182,465,209,386,202)(163,170,466,210,387,203)(164,171,467,211,388,204)(165,172,468,212,389,205)(166,173,456,213,390,206)(167,174,457,214,378,207)(168,175,458,215,379,208)(169,176,459,216,380,196), (1,156,268,83)(2,144,269,84)(3,145,270,85)(4,146,271,86)(5,147,272,87)(6,148,273,88)(7,149,261,89)(8,150,262,90)(9,151,263,91)(10,152,264,79)(11,153,265,80)(12,154,266,81)(13,155,267,82)(14,337,75,416)(15,338,76,404)(16,326,77,405)(17,327,78,406)(18,328,66,407)(19,329,67,408)(20,330,68,409)(21,331,69,410)(22,332,70,411)(23,333,71,412)(24,334,72,413)(25,335,73,414)(26,336,74,415)(27,368,442,92)(28,369,430,93)(29,370,431,94)(30,371,432,95)(31,372,433,96)(32,373,434,97)(33,374,435,98)(34,375,436,99)(35,376,437,100)(36,377,438,101)(37,365,439,102)(38,366,440,103)(39,367,441,104)(40,110,420,295)(41,111,421,296)(42,112,422,297)(43,113,423,298)(44,114,424,299)(45,115,425,287)(46,116,426,288)(47,117,427,289)(48,105,428,290)(49,106,429,291)(50,107,417,292)(51,108,418,293)(52,109,419,294)(53,394,454,235)(54,395,455,236)(55,396,443,237)(56,397,444,238)(57,398,445,239)(58,399,446,240)(59,400,447,241)(60,401,448,242)(61,402,449,243)(62,403,450,244)(63,391,451,245)(64,392,452,246)(65,393,453,247)(118,256,303,195)(119,257,304,183)(120,258,305,184)(121,259,306,185)(122,260,307,186)(123,248,308,187)(124,249,309,188)(125,250,310,189)(126,251,311,190)(127,252,312,191)(128,253,300,192)(129,254,301,193)(130,255,302,194)(131,172,315,389)(132,173,316,390)(133,174,317,378)(134,175,318,379)(135,176,319,380)(136,177,320,381)(137,178,321,382)(138,179,322,383)(139,180,323,384)(140,181,324,385)(141,182,325,386)(142,170,313,387)(143,171,314,388)(157,282,217,360)(158,283,218,361)(159,284,219,362)(160,285,220,363)(161,286,221,364)(162,274,209,352)(163,275,210,353)(164,276,211,354)(165,277,212,355)(166,278,213,356)(167,279,214,357)(168,280,215,358)(169,281,216,359)(196,339,459,225)(197,340,460,226)(198,341,461,227)(199,342,462,228)(200,343,463,229)(201,344,464,230)(202,345,465,231)(203,346,466,232)(204,347,467,233)(205,348,468,234)(206,349,456,222)(207,350,457,223)(208,351,458,224)>;
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), (1,135,393)(2,136,394)(3,137,395)(4,138,396)(5,139,397)(6,140,398)(7,141,399)(8,142,400)(9,143,401)(10,131,402)(11,132,403)(12,133,391)(13,134,392)(14,199,291)(15,200,292)(16,201,293)(17,202,294)(18,203,295)(19,204,296)(20,205,297)(21,206,298)(22,207,299)(23,208,287)(24,196,288)(25,197,289)(26,198,290)(27,212,304)(28,213,305)(29,214,306)(30,215,307)(31,216,308)(32,217,309)(33,218,310)(34,219,311)(35,220,312)(36,221,300)(37,209,301)(38,210,302)(39,211,303)(40,346,328)(41,347,329)(42,348,330)(43,349,331)(44,350,332)(45,351,333)(46,339,334)(47,340,335)(48,341,336)(49,342,337)(50,343,338)(51,344,326)(52,345,327)(53,381,84)(54,382,85)(55,383,86)(56,384,87)(57,385,88)(58,386,89)(59,387,90)(60,388,91)(61,389,79)(62,390,80)(63,378,81)(64,379,82)(65,380,83)(66,466,110)(67,467,111)(68,468,112)(69,456,113)(70,457,114)(71,458,115)(72,459,116)(73,460,117)(74,461,105)(75,462,106)(76,463,107)(77,464,108)(78,465,109)(92,257,277)(93,258,278)(94,259,279)(95,260,280)(96,248,281)(97,249,282)(98,250,283)(99,251,284)(100,252,285)(101,253,286)(102,254,274)(103,255,275)(104,256,276)(118,441,164)(119,442,165)(120,430,166)(121,431,167)(122,432,168)(123,433,169)(124,434,157)(125,435,158)(126,436,159)(127,437,160)(128,438,161)(129,439,162)(130,440,163)(144,454,177)(145,455,178)(146,443,179)(147,444,180)(148,445,181)(149,446,182)(150,447,170)(151,448,171)(152,449,172)(153,450,173)(154,451,174)(155,452,175)(156,453,176)(183,355,368)(184,356,369)(185,357,370)(186,358,371)(187,359,372)(188,360,373)(189,361,374)(190,362,375)(191,363,376)(192,364,377)(193,352,365)(194,353,366)(195,354,367)(222,410,423)(223,411,424)(224,412,425)(225,413,426)(226,414,427)(227,415,428)(228,416,429)(229,404,417)(230,405,418)(231,406,419)(232,407,420)(233,408,421)(234,409,422)(235,269,320)(236,270,321)(237,271,322)(238,272,323)(239,273,324)(240,261,325)(241,262,313)(242,263,314)(243,264,315)(244,265,316)(245,266,317)(246,267,318)(247,268,319), (1,187,426,268,248,46)(2,188,427,269,249,47)(3,189,428,270,250,48)(4,190,429,271,251,49)(5,191,417,272,252,50)(6,192,418,273,253,51)(7,193,419,261,254,52)(8,194,420,262,255,40)(9,195,421,263,256,41)(10,183,422,264,257,42)(11,184,423,265,258,43)(12,185,424,266,259,44)(13,186,425,267,260,45)(14,436,443,75,34,55)(15,437,444,76,35,56)(16,438,445,77,36,57)(17,439,446,78,37,58)(18,440,447,66,38,59)(19,441,448,67,39,60)(20,442,449,68,27,61)(21,430,450,69,28,62)(22,431,451,70,29,63)(23,432,452,71,30,64)(24,433,453,72,31,65)(25,434,454,73,32,53)(26,435,455,74,33,54)(79,297,119,152,112,304)(80,298,120,153,113,305)(81,299,121,154,114,306)(82,287,122,155,115,307)(83,288,123,156,116,308)(84,289,124,144,117,309)(85,290,125,145,105,310)(86,291,126,146,106,311)(87,292,127,147,107,312)(88,293,128,148,108,300)(89,294,129,149,109,301)(90,295,130,150,110,302)(91,296,118,151,111,303)(92,330,402,368,409,243)(93,331,403,369,410,244)(94,332,391,370,411,245)(95,333,392,371,412,246)(96,334,393,372,413,247)(97,335,394,373,414,235)(98,336,395,374,415,236)(99,337,396,375,416,237)(100,338,397,376,404,238)(101,326,398,377,405,239)(102,327,399,365,406,240)(103,328,400,366,407,241)(104,329,401,367,408,242)(131,355,234,315,277,348)(132,356,222,316,278,349)(133,357,223,317,279,350)(134,358,224,318,280,351)(135,359,225,319,281,339)(136,360,226,320,282,340)(137,361,227,321,283,341)(138,362,228,322,284,342)(139,363,229,323,285,343)(140,364,230,324,286,344)(141,352,231,325,274,345)(142,353,232,313,275,346)(143,354,233,314,276,347)(157,177,460,217,381,197)(158,178,461,218,382,198)(159,179,462,219,383,199)(160,180,463,220,384,200)(161,181,464,221,385,201)(162,182,465,209,386,202)(163,170,466,210,387,203)(164,171,467,211,388,204)(165,172,468,212,389,205)(166,173,456,213,390,206)(167,174,457,214,378,207)(168,175,458,215,379,208)(169,176,459,216,380,196), (1,156,268,83)(2,144,269,84)(3,145,270,85)(4,146,271,86)(5,147,272,87)(6,148,273,88)(7,149,261,89)(8,150,262,90)(9,151,263,91)(10,152,264,79)(11,153,265,80)(12,154,266,81)(13,155,267,82)(14,337,75,416)(15,338,76,404)(16,326,77,405)(17,327,78,406)(18,328,66,407)(19,329,67,408)(20,330,68,409)(21,331,69,410)(22,332,70,411)(23,333,71,412)(24,334,72,413)(25,335,73,414)(26,336,74,415)(27,368,442,92)(28,369,430,93)(29,370,431,94)(30,371,432,95)(31,372,433,96)(32,373,434,97)(33,374,435,98)(34,375,436,99)(35,376,437,100)(36,377,438,101)(37,365,439,102)(38,366,440,103)(39,367,441,104)(40,110,420,295)(41,111,421,296)(42,112,422,297)(43,113,423,298)(44,114,424,299)(45,115,425,287)(46,116,426,288)(47,117,427,289)(48,105,428,290)(49,106,429,291)(50,107,417,292)(51,108,418,293)(52,109,419,294)(53,394,454,235)(54,395,455,236)(55,396,443,237)(56,397,444,238)(57,398,445,239)(58,399,446,240)(59,400,447,241)(60,401,448,242)(61,402,449,243)(62,403,450,244)(63,391,451,245)(64,392,452,246)(65,393,453,247)(118,256,303,195)(119,257,304,183)(120,258,305,184)(121,259,306,185)(122,260,307,186)(123,248,308,187)(124,249,309,188)(125,250,310,189)(126,251,311,190)(127,252,312,191)(128,253,300,192)(129,254,301,193)(130,255,302,194)(131,172,315,389)(132,173,316,390)(133,174,317,378)(134,175,318,379)(135,176,319,380)(136,177,320,381)(137,178,321,382)(138,179,322,383)(139,180,323,384)(140,181,324,385)(141,182,325,386)(142,170,313,387)(143,171,314,388)(157,282,217,360)(158,283,218,361)(159,284,219,362)(160,285,220,363)(161,286,221,364)(162,274,209,352)(163,275,210,353)(164,276,211,354)(165,277,212,355)(166,278,213,356)(167,279,214,357)(168,280,215,358)(169,281,216,359)(196,339,459,225)(197,340,460,226)(198,341,461,227)(199,342,462,228)(200,343,463,229)(201,344,464,230)(202,345,465,231)(203,346,466,232)(204,347,467,233)(205,348,468,234)(206,349,456,222)(207,350,457,223)(208,351,458,224) );
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)], [(1,135,393),(2,136,394),(3,137,395),(4,138,396),(5,139,397),(6,140,398),(7,141,399),(8,142,400),(9,143,401),(10,131,402),(11,132,403),(12,133,391),(13,134,392),(14,199,291),(15,200,292),(16,201,293),(17,202,294),(18,203,295),(19,204,296),(20,205,297),(21,206,298),(22,207,299),(23,208,287),(24,196,288),(25,197,289),(26,198,290),(27,212,304),(28,213,305),(29,214,306),(30,215,307),(31,216,308),(32,217,309),(33,218,310),(34,219,311),(35,220,312),(36,221,300),(37,209,301),(38,210,302),(39,211,303),(40,346,328),(41,347,329),(42,348,330),(43,349,331),(44,350,332),(45,351,333),(46,339,334),(47,340,335),(48,341,336),(49,342,337),(50,343,338),(51,344,326),(52,345,327),(53,381,84),(54,382,85),(55,383,86),(56,384,87),(57,385,88),(58,386,89),(59,387,90),(60,388,91),(61,389,79),(62,390,80),(63,378,81),(64,379,82),(65,380,83),(66,466,110),(67,467,111),(68,468,112),(69,456,113),(70,457,114),(71,458,115),(72,459,116),(73,460,117),(74,461,105),(75,462,106),(76,463,107),(77,464,108),(78,465,109),(92,257,277),(93,258,278),(94,259,279),(95,260,280),(96,248,281),(97,249,282),(98,250,283),(99,251,284),(100,252,285),(101,253,286),(102,254,274),(103,255,275),(104,256,276),(118,441,164),(119,442,165),(120,430,166),(121,431,167),(122,432,168),(123,433,169),(124,434,157),(125,435,158),(126,436,159),(127,437,160),(128,438,161),(129,439,162),(130,440,163),(144,454,177),(145,455,178),(146,443,179),(147,444,180),(148,445,181),(149,446,182),(150,447,170),(151,448,171),(152,449,172),(153,450,173),(154,451,174),(155,452,175),(156,453,176),(183,355,368),(184,356,369),(185,357,370),(186,358,371),(187,359,372),(188,360,373),(189,361,374),(190,362,375),(191,363,376),(192,364,377),(193,352,365),(194,353,366),(195,354,367),(222,410,423),(223,411,424),(224,412,425),(225,413,426),(226,414,427),(227,415,428),(228,416,429),(229,404,417),(230,405,418),(231,406,419),(232,407,420),(233,408,421),(234,409,422),(235,269,320),(236,270,321),(237,271,322),(238,272,323),(239,273,324),(240,261,325),(241,262,313),(242,263,314),(243,264,315),(244,265,316),(245,266,317),(246,267,318),(247,268,319)], [(1,187,426,268,248,46),(2,188,427,269,249,47),(3,189,428,270,250,48),(4,190,429,271,251,49),(5,191,417,272,252,50),(6,192,418,273,253,51),(7,193,419,261,254,52),(8,194,420,262,255,40),(9,195,421,263,256,41),(10,183,422,264,257,42),(11,184,423,265,258,43),(12,185,424,266,259,44),(13,186,425,267,260,45),(14,436,443,75,34,55),(15,437,444,76,35,56),(16,438,445,77,36,57),(17,439,446,78,37,58),(18,440,447,66,38,59),(19,441,448,67,39,60),(20,442,449,68,27,61),(21,430,450,69,28,62),(22,431,451,70,29,63),(23,432,452,71,30,64),(24,433,453,72,31,65),(25,434,454,73,32,53),(26,435,455,74,33,54),(79,297,119,152,112,304),(80,298,120,153,113,305),(81,299,121,154,114,306),(82,287,122,155,115,307),(83,288,123,156,116,308),(84,289,124,144,117,309),(85,290,125,145,105,310),(86,291,126,146,106,311),(87,292,127,147,107,312),(88,293,128,148,108,300),(89,294,129,149,109,301),(90,295,130,150,110,302),(91,296,118,151,111,303),(92,330,402,368,409,243),(93,331,403,369,410,244),(94,332,391,370,411,245),(95,333,392,371,412,246),(96,334,393,372,413,247),(97,335,394,373,414,235),(98,336,395,374,415,236),(99,337,396,375,416,237),(100,338,397,376,404,238),(101,326,398,377,405,239),(102,327,399,365,406,240),(103,328,400,366,407,241),(104,329,401,367,408,242),(131,355,234,315,277,348),(132,356,222,316,278,349),(133,357,223,317,279,350),(134,358,224,318,280,351),(135,359,225,319,281,339),(136,360,226,320,282,340),(137,361,227,321,283,341),(138,362,228,322,284,342),(139,363,229,323,285,343),(140,364,230,324,286,344),(141,352,231,325,274,345),(142,353,232,313,275,346),(143,354,233,314,276,347),(157,177,460,217,381,197),(158,178,461,218,382,198),(159,179,462,219,383,199),(160,180,463,220,384,200),(161,181,464,221,385,201),(162,182,465,209,386,202),(163,170,466,210,387,203),(164,171,467,211,388,204),(165,172,468,212,389,205),(166,173,456,213,390,206),(167,174,457,214,378,207),(168,175,458,215,379,208),(169,176,459,216,380,196)], [(1,156,268,83),(2,144,269,84),(3,145,270,85),(4,146,271,86),(5,147,272,87),(6,148,273,88),(7,149,261,89),(8,150,262,90),(9,151,263,91),(10,152,264,79),(11,153,265,80),(12,154,266,81),(13,155,267,82),(14,337,75,416),(15,338,76,404),(16,326,77,405),(17,327,78,406),(18,328,66,407),(19,329,67,408),(20,330,68,409),(21,331,69,410),(22,332,70,411),(23,333,71,412),(24,334,72,413),(25,335,73,414),(26,336,74,415),(27,368,442,92),(28,369,430,93),(29,370,431,94),(30,371,432,95),(31,372,433,96),(32,373,434,97),(33,374,435,98),(34,375,436,99),(35,376,437,100),(36,377,438,101),(37,365,439,102),(38,366,440,103),(39,367,441,104),(40,110,420,295),(41,111,421,296),(42,112,422,297),(43,113,423,298),(44,114,424,299),(45,115,425,287),(46,116,426,288),(47,117,427,289),(48,105,428,290),(49,106,429,291),(50,107,417,292),(51,108,418,293),(52,109,419,294),(53,394,454,235),(54,395,455,236),(55,396,443,237),(56,397,444,238),(57,398,445,239),(58,399,446,240),(59,400,447,241),(60,401,448,242),(61,402,449,243),(62,403,450,244),(63,391,451,245),(64,392,452,246),(65,393,453,247),(118,256,303,195),(119,257,304,183),(120,258,305,184),(121,259,306,185),(122,260,307,186),(123,248,308,187),(124,249,309,188),(125,250,310,189),(126,251,311,190),(127,252,312,191),(128,253,300,192),(129,254,301,193),(130,255,302,194),(131,172,315,389),(132,173,316,390),(133,174,317,378),(134,175,318,379),(135,176,319,380),(136,177,320,381),(137,178,321,382),(138,179,322,383),(139,180,323,384),(140,181,324,385),(141,182,325,386),(142,170,313,387),(143,171,314,388),(157,282,217,360),(158,283,218,361),(159,284,219,362),(160,285,220,363),(161,286,221,364),(162,274,209,352),(163,275,210,353),(164,276,211,354),(165,277,212,355),(166,278,213,356),(167,279,214,357),(168,280,215,358),(169,281,216,359),(196,339,459,225),(197,340,460,226),(198,341,461,227),(199,342,462,228),(200,343,463,229),(201,344,464,230),(202,345,465,231),(203,346,466,232),(204,347,467,233),(205,348,468,234),(206,349,456,222),(207,350,457,223),(208,351,458,224)]])
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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