direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C32×Dic13, C39⋊10C12, C78.12C6, (C3×C39)⋊9C4, C13⋊5(C3×C12), (C3×C78).3C2, C26.3(C3×C6), C6.4(C3×D13), (C3×C6).3D13, C2.(C32×D13), SmallGroup(468,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C32×Dic13 |
Generators and relations for C32×Dic13
G = < a,b,c,d | a3=b3=c26=1, d2=c13, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of C32×Dic13
►Regular action on 468 points
Generators in S468
(1 228 128)(2 229 129)(3 230 130)(4 231 105)(5 232 106)(6 233 107)(7 234 108)(8 209 109)(9 210 110)(10 211 111)(11 212 112)(12 213 113)(13 214 114)(14 215 115)(15 216 116)(16 217 117)(17 218 118)(18 219 119)(19 220 120)(20 221 121)(21 222 122)(22 223 123)(23 224 124)(24 225 125)(25 226 126)(26 227 127)(27 165 150)(28 166 151)(29 167 152)(30 168 153)(31 169 154)(32 170 155)(33 171 156)(34 172 131)(35 173 132)(36 174 133)(37 175 134)(38 176 135)(39 177 136)(40 178 137)(41 179 138)(42 180 139)(43 181 140)(44 182 141)(45 157 142)(46 158 143)(47 159 144)(48 160 145)(49 161 146)(50 162 147)(51 163 148)(52 164 149)(53 193 90)(54 194 91)(55 195 92)(56 196 93)(57 197 94)(58 198 95)(59 199 96)(60 200 97)(61 201 98)(62 202 99)(63 203 100)(64 204 101)(65 205 102)(66 206 103)(67 207 104)(68 208 79)(69 183 80)(70 184 81)(71 185 82)(72 186 83)(73 187 84)(74 188 85)(75 189 86)(76 190 87)(77 191 88)(78 192 89)(235 443 339)(236 444 340)(237 445 341)(238 446 342)(239 447 343)(240 448 344)(241 449 345)(242 450 346)(243 451 347)(244 452 348)(245 453 349)(246 454 350)(247 455 351)(248 456 352)(249 457 353)(250 458 354)(251 459 355)(252 460 356)(253 461 357)(254 462 358)(255 463 359)(256 464 360)(257 465 361)(258 466 362)(259 467 363)(260 468 364)(261 404 365)(262 405 366)(263 406 367)(264 407 368)(265 408 369)(266 409 370)(267 410 371)(268 411 372)(269 412 373)(270 413 374)(271 414 375)(272 415 376)(273 416 377)(274 391 378)(275 392 379)(276 393 380)(277 394 381)(278 395 382)(279 396 383)(280 397 384)(281 398 385)(282 399 386)(283 400 387)(284 401 388)(285 402 389)(286 403 390)(287 417 326)(288 418 327)(289 419 328)(290 420 329)(291 421 330)(292 422 331)(293 423 332)(294 424 333)(295 425 334)(296 426 335)(297 427 336)(298 428 337)(299 429 338)(300 430 313)(301 431 314)(302 432 315)(303 433 316)(304 434 317)(305 435 318)(306 436 319)(307 437 320)(308 438 321)(309 439 322)(310 440 323)(311 441 324)(312 442 325)
(1 69 27)(2 70 28)(3 71 29)(4 72 30)(5 73 31)(6 74 32)(7 75 33)(8 76 34)(9 77 35)(10 78 36)(11 53 37)(12 54 38)(13 55 39)(14 56 40)(15 57 41)(16 58 42)(17 59 43)(18 60 44)(19 61 45)(20 62 46)(21 63 47)(22 64 48)(23 65 49)(24 66 50)(25 67 51)(26 68 52)(79 149 127)(80 150 128)(81 151 129)(82 152 130)(83 153 105)(84 154 106)(85 155 107)(86 156 108)(87 131 109)(88 132 110)(89 133 111)(90 134 112)(91 135 113)(92 136 114)(93 137 115)(94 138 116)(95 139 117)(96 140 118)(97 141 119)(98 142 120)(99 143 121)(100 144 122)(101 145 123)(102 146 124)(103 147 125)(104 148 126)(157 220 201)(158 221 202)(159 222 203)(160 223 204)(161 224 205)(162 225 206)(163 226 207)(164 227 208)(165 228 183)(166 229 184)(167 230 185)(168 231 186)(169 232 187)(170 233 188)(171 234 189)(172 209 190)(173 210 191)(174 211 192)(175 212 193)(176 213 194)(177 214 195)(178 215 196)(179 216 197)(180 217 198)(181 218 199)(182 219 200)(235 287 261)(236 288 262)(237 289 263)(238 290 264)(239 291 265)(240 292 266)(241 293 267)(242 294 268)(243 295 269)(244 296 270)(245 297 271)(246 298 272)(247 299 273)(248 300 274)(249 301 275)(250 302 276)(251 303 277)(252 304 278)(253 305 279)(254 306 280)(255 307 281)(256 308 282)(257 309 283)(258 310 284)(259 311 285)(260 312 286)(313 378 352)(314 379 353)(315 380 354)(316 381 355)(317 382 356)(318 383 357)(319 384 358)(320 385 359)(321 386 360)(322 387 361)(323 388 362)(324 389 363)(325 390 364)(326 365 339)(327 366 340)(328 367 341)(329 368 342)(330 369 343)(331 370 344)(332 371 345)(333 372 346)(334 373 347)(335 374 348)(336 375 349)(337 376 350)(338 377 351)(391 456 430)(392 457 431)(393 458 432)(394 459 433)(395 460 434)(396 461 435)(397 462 436)(398 463 437)(399 464 438)(400 465 439)(401 466 440)(402 467 441)(403 468 442)(404 443 417)(405 444 418)(406 445 419)(407 446 420)(408 447 421)(409 448 422)(410 449 423)(411 450 424)(412 451 425)(413 452 426)(414 453 427)(415 454 428)(416 455 429)
(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)
(1 235 14 248)(2 260 15 247)(3 259 16 246)(4 258 17 245)(5 257 18 244)(6 256 19 243)(7 255 20 242)(8 254 21 241)(9 253 22 240)(10 252 23 239)(11 251 24 238)(12 250 25 237)(13 249 26 236)(27 261 40 274)(28 286 41 273)(29 285 42 272)(30 284 43 271)(31 283 44 270)(32 282 45 269)(33 281 46 268)(34 280 47 267)(35 279 48 266)(36 278 49 265)(37 277 50 264)(38 276 51 263)(39 275 52 262)(53 303 66 290)(54 302 67 289)(55 301 68 288)(56 300 69 287)(57 299 70 312)(58 298 71 311)(59 297 72 310)(60 296 73 309)(61 295 74 308)(62 294 75 307)(63 293 76 306)(64 292 77 305)(65 291 78 304)(79 327 92 314)(80 326 93 313)(81 325 94 338)(82 324 95 337)(83 323 96 336)(84 322 97 335)(85 321 98 334)(86 320 99 333)(87 319 100 332)(88 318 101 331)(89 317 102 330)(90 316 103 329)(91 315 104 328)(105 362 118 349)(106 361 119 348)(107 360 120 347)(108 359 121 346)(109 358 122 345)(110 357 123 344)(111 356 124 343)(112 355 125 342)(113 354 126 341)(114 353 127 340)(115 352 128 339)(116 351 129 364)(117 350 130 363)(131 384 144 371)(132 383 145 370)(133 382 146 369)(134 381 147 368)(135 380 148 367)(136 379 149 366)(137 378 150 365)(138 377 151 390)(139 376 152 389)(140 375 153 388)(141 374 154 387)(142 373 155 386)(143 372 156 385)(157 412 170 399)(158 411 171 398)(159 410 172 397)(160 409 173 396)(161 408 174 395)(162 407 175 394)(163 406 176 393)(164 405 177 392)(165 404 178 391)(166 403 179 416)(167 402 180 415)(168 401 181 414)(169 400 182 413)(183 417 196 430)(184 442 197 429)(185 441 198 428)(186 440 199 427)(187 439 200 426)(188 438 201 425)(189 437 202 424)(190 436 203 423)(191 435 204 422)(192 434 205 421)(193 433 206 420)(194 432 207 419)(195 431 208 418)(209 462 222 449)(210 461 223 448)(211 460 224 447)(212 459 225 446)(213 458 226 445)(214 457 227 444)(215 456 228 443)(216 455 229 468)(217 454 230 467)(218 453 231 466)(219 452 232 465)(220 451 233 464)(221 450 234 463)
G:=sub<Sym(468)| (1,228,128)(2,229,129)(3,230,130)(4,231,105)(5,232,106)(6,233,107)(7,234,108)(8,209,109)(9,210,110)(10,211,111)(11,212,112)(12,213,113)(13,214,114)(14,215,115)(15,216,116)(16,217,117)(17,218,118)(18,219,119)(19,220,120)(20,221,121)(21,222,122)(22,223,123)(23,224,124)(24,225,125)(25,226,126)(26,227,127)(27,165,150)(28,166,151)(29,167,152)(30,168,153)(31,169,154)(32,170,155)(33,171,156)(34,172,131)(35,173,132)(36,174,133)(37,175,134)(38,176,135)(39,177,136)(40,178,137)(41,179,138)(42,180,139)(43,181,140)(44,182,141)(45,157,142)(46,158,143)(47,159,144)(48,160,145)(49,161,146)(50,162,147)(51,163,148)(52,164,149)(53,193,90)(54,194,91)(55,195,92)(56,196,93)(57,197,94)(58,198,95)(59,199,96)(60,200,97)(61,201,98)(62,202,99)(63,203,100)(64,204,101)(65,205,102)(66,206,103)(67,207,104)(68,208,79)(69,183,80)(70,184,81)(71,185,82)(72,186,83)(73,187,84)(74,188,85)(75,189,86)(76,190,87)(77,191,88)(78,192,89)(235,443,339)(236,444,340)(237,445,341)(238,446,342)(239,447,343)(240,448,344)(241,449,345)(242,450,346)(243,451,347)(244,452,348)(245,453,349)(246,454,350)(247,455,351)(248,456,352)(249,457,353)(250,458,354)(251,459,355)(252,460,356)(253,461,357)(254,462,358)(255,463,359)(256,464,360)(257,465,361)(258,466,362)(259,467,363)(260,468,364)(261,404,365)(262,405,366)(263,406,367)(264,407,368)(265,408,369)(266,409,370)(267,410,371)(268,411,372)(269,412,373)(270,413,374)(271,414,375)(272,415,376)(273,416,377)(274,391,378)(275,392,379)(276,393,380)(277,394,381)(278,395,382)(279,396,383)(280,397,384)(281,398,385)(282,399,386)(283,400,387)(284,401,388)(285,402,389)(286,403,390)(287,417,326)(288,418,327)(289,419,328)(290,420,329)(291,421,330)(292,422,331)(293,423,332)(294,424,333)(295,425,334)(296,426,335)(297,427,336)(298,428,337)(299,429,338)(300,430,313)(301,431,314)(302,432,315)(303,433,316)(304,434,317)(305,435,318)(306,436,319)(307,437,320)(308,438,321)(309,439,322)(310,440,323)(311,441,324)(312,442,325), (1,69,27)(2,70,28)(3,71,29)(4,72,30)(5,73,31)(6,74,32)(7,75,33)(8,76,34)(9,77,35)(10,78,36)(11,53,37)(12,54,38)(13,55,39)(14,56,40)(15,57,41)(16,58,42)(17,59,43)(18,60,44)(19,61,45)(20,62,46)(21,63,47)(22,64,48)(23,65,49)(24,66,50)(25,67,51)(26,68,52)(79,149,127)(80,150,128)(81,151,129)(82,152,130)(83,153,105)(84,154,106)(85,155,107)(86,156,108)(87,131,109)(88,132,110)(89,133,111)(90,134,112)(91,135,113)(92,136,114)(93,137,115)(94,138,116)(95,139,117)(96,140,118)(97,141,119)(98,142,120)(99,143,121)(100,144,122)(101,145,123)(102,146,124)(103,147,125)(104,148,126)(157,220,201)(158,221,202)(159,222,203)(160,223,204)(161,224,205)(162,225,206)(163,226,207)(164,227,208)(165,228,183)(166,229,184)(167,230,185)(168,231,186)(169,232,187)(170,233,188)(171,234,189)(172,209,190)(173,210,191)(174,211,192)(175,212,193)(176,213,194)(177,214,195)(178,215,196)(179,216,197)(180,217,198)(181,218,199)(182,219,200)(235,287,261)(236,288,262)(237,289,263)(238,290,264)(239,291,265)(240,292,266)(241,293,267)(242,294,268)(243,295,269)(244,296,270)(245,297,271)(246,298,272)(247,299,273)(248,300,274)(249,301,275)(250,302,276)(251,303,277)(252,304,278)(253,305,279)(254,306,280)(255,307,281)(256,308,282)(257,309,283)(258,310,284)(259,311,285)(260,312,286)(313,378,352)(314,379,353)(315,380,354)(316,381,355)(317,382,356)(318,383,357)(319,384,358)(320,385,359)(321,386,360)(322,387,361)(323,388,362)(324,389,363)(325,390,364)(326,365,339)(327,366,340)(328,367,341)(329,368,342)(330,369,343)(331,370,344)(332,371,345)(333,372,346)(334,373,347)(335,374,348)(336,375,349)(337,376,350)(338,377,351)(391,456,430)(392,457,431)(393,458,432)(394,459,433)(395,460,434)(396,461,435)(397,462,436)(398,463,437)(399,464,438)(400,465,439)(401,466,440)(402,467,441)(403,468,442)(404,443,417)(405,444,418)(406,445,419)(407,446,420)(408,447,421)(409,448,422)(410,449,423)(411,450,424)(412,451,425)(413,452,426)(414,453,427)(415,454,428)(416,455,429), (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), (1,235,14,248)(2,260,15,247)(3,259,16,246)(4,258,17,245)(5,257,18,244)(6,256,19,243)(7,255,20,242)(8,254,21,241)(9,253,22,240)(10,252,23,239)(11,251,24,238)(12,250,25,237)(13,249,26,236)(27,261,40,274)(28,286,41,273)(29,285,42,272)(30,284,43,271)(31,283,44,270)(32,282,45,269)(33,281,46,268)(34,280,47,267)(35,279,48,266)(36,278,49,265)(37,277,50,264)(38,276,51,263)(39,275,52,262)(53,303,66,290)(54,302,67,289)(55,301,68,288)(56,300,69,287)(57,299,70,312)(58,298,71,311)(59,297,72,310)(60,296,73,309)(61,295,74,308)(62,294,75,307)(63,293,76,306)(64,292,77,305)(65,291,78,304)(79,327,92,314)(80,326,93,313)(81,325,94,338)(82,324,95,337)(83,323,96,336)(84,322,97,335)(85,321,98,334)(86,320,99,333)(87,319,100,332)(88,318,101,331)(89,317,102,330)(90,316,103,329)(91,315,104,328)(105,362,118,349)(106,361,119,348)(107,360,120,347)(108,359,121,346)(109,358,122,345)(110,357,123,344)(111,356,124,343)(112,355,125,342)(113,354,126,341)(114,353,127,340)(115,352,128,339)(116,351,129,364)(117,350,130,363)(131,384,144,371)(132,383,145,370)(133,382,146,369)(134,381,147,368)(135,380,148,367)(136,379,149,366)(137,378,150,365)(138,377,151,390)(139,376,152,389)(140,375,153,388)(141,374,154,387)(142,373,155,386)(143,372,156,385)(157,412,170,399)(158,411,171,398)(159,410,172,397)(160,409,173,396)(161,408,174,395)(162,407,175,394)(163,406,176,393)(164,405,177,392)(165,404,178,391)(166,403,179,416)(167,402,180,415)(168,401,181,414)(169,400,182,413)(183,417,196,430)(184,442,197,429)(185,441,198,428)(186,440,199,427)(187,439,200,426)(188,438,201,425)(189,437,202,424)(190,436,203,423)(191,435,204,422)(192,434,205,421)(193,433,206,420)(194,432,207,419)(195,431,208,418)(209,462,222,449)(210,461,223,448)(211,460,224,447)(212,459,225,446)(213,458,226,445)(214,457,227,444)(215,456,228,443)(216,455,229,468)(217,454,230,467)(218,453,231,466)(219,452,232,465)(220,451,233,464)(221,450,234,463)>;
G:=Group( (1,228,128)(2,229,129)(3,230,130)(4,231,105)(5,232,106)(6,233,107)(7,234,108)(8,209,109)(9,210,110)(10,211,111)(11,212,112)(12,213,113)(13,214,114)(14,215,115)(15,216,116)(16,217,117)(17,218,118)(18,219,119)(19,220,120)(20,221,121)(21,222,122)(22,223,123)(23,224,124)(24,225,125)(25,226,126)(26,227,127)(27,165,150)(28,166,151)(29,167,152)(30,168,153)(31,169,154)(32,170,155)(33,171,156)(34,172,131)(35,173,132)(36,174,133)(37,175,134)(38,176,135)(39,177,136)(40,178,137)(41,179,138)(42,180,139)(43,181,140)(44,182,141)(45,157,142)(46,158,143)(47,159,144)(48,160,145)(49,161,146)(50,162,147)(51,163,148)(52,164,149)(53,193,90)(54,194,91)(55,195,92)(56,196,93)(57,197,94)(58,198,95)(59,199,96)(60,200,97)(61,201,98)(62,202,99)(63,203,100)(64,204,101)(65,205,102)(66,206,103)(67,207,104)(68,208,79)(69,183,80)(70,184,81)(71,185,82)(72,186,83)(73,187,84)(74,188,85)(75,189,86)(76,190,87)(77,191,88)(78,192,89)(235,443,339)(236,444,340)(237,445,341)(238,446,342)(239,447,343)(240,448,344)(241,449,345)(242,450,346)(243,451,347)(244,452,348)(245,453,349)(246,454,350)(247,455,351)(248,456,352)(249,457,353)(250,458,354)(251,459,355)(252,460,356)(253,461,357)(254,462,358)(255,463,359)(256,464,360)(257,465,361)(258,466,362)(259,467,363)(260,468,364)(261,404,365)(262,405,366)(263,406,367)(264,407,368)(265,408,369)(266,409,370)(267,410,371)(268,411,372)(269,412,373)(270,413,374)(271,414,375)(272,415,376)(273,416,377)(274,391,378)(275,392,379)(276,393,380)(277,394,381)(278,395,382)(279,396,383)(280,397,384)(281,398,385)(282,399,386)(283,400,387)(284,401,388)(285,402,389)(286,403,390)(287,417,326)(288,418,327)(289,419,328)(290,420,329)(291,421,330)(292,422,331)(293,423,332)(294,424,333)(295,425,334)(296,426,335)(297,427,336)(298,428,337)(299,429,338)(300,430,313)(301,431,314)(302,432,315)(303,433,316)(304,434,317)(305,435,318)(306,436,319)(307,437,320)(308,438,321)(309,439,322)(310,440,323)(311,441,324)(312,442,325), (1,69,27)(2,70,28)(3,71,29)(4,72,30)(5,73,31)(6,74,32)(7,75,33)(8,76,34)(9,77,35)(10,78,36)(11,53,37)(12,54,38)(13,55,39)(14,56,40)(15,57,41)(16,58,42)(17,59,43)(18,60,44)(19,61,45)(20,62,46)(21,63,47)(22,64,48)(23,65,49)(24,66,50)(25,67,51)(26,68,52)(79,149,127)(80,150,128)(81,151,129)(82,152,130)(83,153,105)(84,154,106)(85,155,107)(86,156,108)(87,131,109)(88,132,110)(89,133,111)(90,134,112)(91,135,113)(92,136,114)(93,137,115)(94,138,116)(95,139,117)(96,140,118)(97,141,119)(98,142,120)(99,143,121)(100,144,122)(101,145,123)(102,146,124)(103,147,125)(104,148,126)(157,220,201)(158,221,202)(159,222,203)(160,223,204)(161,224,205)(162,225,206)(163,226,207)(164,227,208)(165,228,183)(166,229,184)(167,230,185)(168,231,186)(169,232,187)(170,233,188)(171,234,189)(172,209,190)(173,210,191)(174,211,192)(175,212,193)(176,213,194)(177,214,195)(178,215,196)(179,216,197)(180,217,198)(181,218,199)(182,219,200)(235,287,261)(236,288,262)(237,289,263)(238,290,264)(239,291,265)(240,292,266)(241,293,267)(242,294,268)(243,295,269)(244,296,270)(245,297,271)(246,298,272)(247,299,273)(248,300,274)(249,301,275)(250,302,276)(251,303,277)(252,304,278)(253,305,279)(254,306,280)(255,307,281)(256,308,282)(257,309,283)(258,310,284)(259,311,285)(260,312,286)(313,378,352)(314,379,353)(315,380,354)(316,381,355)(317,382,356)(318,383,357)(319,384,358)(320,385,359)(321,386,360)(322,387,361)(323,388,362)(324,389,363)(325,390,364)(326,365,339)(327,366,340)(328,367,341)(329,368,342)(330,369,343)(331,370,344)(332,371,345)(333,372,346)(334,373,347)(335,374,348)(336,375,349)(337,376,350)(338,377,351)(391,456,430)(392,457,431)(393,458,432)(394,459,433)(395,460,434)(396,461,435)(397,462,436)(398,463,437)(399,464,438)(400,465,439)(401,466,440)(402,467,441)(403,468,442)(404,443,417)(405,444,418)(406,445,419)(407,446,420)(408,447,421)(409,448,422)(410,449,423)(411,450,424)(412,451,425)(413,452,426)(414,453,427)(415,454,428)(416,455,429), (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), (1,235,14,248)(2,260,15,247)(3,259,16,246)(4,258,17,245)(5,257,18,244)(6,256,19,243)(7,255,20,242)(8,254,21,241)(9,253,22,240)(10,252,23,239)(11,251,24,238)(12,250,25,237)(13,249,26,236)(27,261,40,274)(28,286,41,273)(29,285,42,272)(30,284,43,271)(31,283,44,270)(32,282,45,269)(33,281,46,268)(34,280,47,267)(35,279,48,266)(36,278,49,265)(37,277,50,264)(38,276,51,263)(39,275,52,262)(53,303,66,290)(54,302,67,289)(55,301,68,288)(56,300,69,287)(57,299,70,312)(58,298,71,311)(59,297,72,310)(60,296,73,309)(61,295,74,308)(62,294,75,307)(63,293,76,306)(64,292,77,305)(65,291,78,304)(79,327,92,314)(80,326,93,313)(81,325,94,338)(82,324,95,337)(83,323,96,336)(84,322,97,335)(85,321,98,334)(86,320,99,333)(87,319,100,332)(88,318,101,331)(89,317,102,330)(90,316,103,329)(91,315,104,328)(105,362,118,349)(106,361,119,348)(107,360,120,347)(108,359,121,346)(109,358,122,345)(110,357,123,344)(111,356,124,343)(112,355,125,342)(113,354,126,341)(114,353,127,340)(115,352,128,339)(116,351,129,364)(117,350,130,363)(131,384,144,371)(132,383,145,370)(133,382,146,369)(134,381,147,368)(135,380,148,367)(136,379,149,366)(137,378,150,365)(138,377,151,390)(139,376,152,389)(140,375,153,388)(141,374,154,387)(142,373,155,386)(143,372,156,385)(157,412,170,399)(158,411,171,398)(159,410,172,397)(160,409,173,396)(161,408,174,395)(162,407,175,394)(163,406,176,393)(164,405,177,392)(165,404,178,391)(166,403,179,416)(167,402,180,415)(168,401,181,414)(169,400,182,413)(183,417,196,430)(184,442,197,429)(185,441,198,428)(186,440,199,427)(187,439,200,426)(188,438,201,425)(189,437,202,424)(190,436,203,423)(191,435,204,422)(192,434,205,421)(193,433,206,420)(194,432,207,419)(195,431,208,418)(209,462,222,449)(210,461,223,448)(211,460,224,447)(212,459,225,446)(213,458,226,445)(214,457,227,444)(215,456,228,443)(216,455,229,468)(217,454,230,467)(218,453,231,466)(219,452,232,465)(220,451,233,464)(221,450,234,463) );
G=PermutationGroup([[(1,228,128),(2,229,129),(3,230,130),(4,231,105),(5,232,106),(6,233,107),(7,234,108),(8,209,109),(9,210,110),(10,211,111),(11,212,112),(12,213,113),(13,214,114),(14,215,115),(15,216,116),(16,217,117),(17,218,118),(18,219,119),(19,220,120),(20,221,121),(21,222,122),(22,223,123),(23,224,124),(24,225,125),(25,226,126),(26,227,127),(27,165,150),(28,166,151),(29,167,152),(30,168,153),(31,169,154),(32,170,155),(33,171,156),(34,172,131),(35,173,132),(36,174,133),(37,175,134),(38,176,135),(39,177,136),(40,178,137),(41,179,138),(42,180,139),(43,181,140),(44,182,141),(45,157,142),(46,158,143),(47,159,144),(48,160,145),(49,161,146),(50,162,147),(51,163,148),(52,164,149),(53,193,90),(54,194,91),(55,195,92),(56,196,93),(57,197,94),(58,198,95),(59,199,96),(60,200,97),(61,201,98),(62,202,99),(63,203,100),(64,204,101),(65,205,102),(66,206,103),(67,207,104),(68,208,79),(69,183,80),(70,184,81),(71,185,82),(72,186,83),(73,187,84),(74,188,85),(75,189,86),(76,190,87),(77,191,88),(78,192,89),(235,443,339),(236,444,340),(237,445,341),(238,446,342),(239,447,343),(240,448,344),(241,449,345),(242,450,346),(243,451,347),(244,452,348),(245,453,349),(246,454,350),(247,455,351),(248,456,352),(249,457,353),(250,458,354),(251,459,355),(252,460,356),(253,461,357),(254,462,358),(255,463,359),(256,464,360),(257,465,361),(258,466,362),(259,467,363),(260,468,364),(261,404,365),(262,405,366),(263,406,367),(264,407,368),(265,408,369),(266,409,370),(267,410,371),(268,411,372),(269,412,373),(270,413,374),(271,414,375),(272,415,376),(273,416,377),(274,391,378),(275,392,379),(276,393,380),(277,394,381),(278,395,382),(279,396,383),(280,397,384),(281,398,385),(282,399,386),(283,400,387),(284,401,388),(285,402,389),(286,403,390),(287,417,326),(288,418,327),(289,419,328),(290,420,329),(291,421,330),(292,422,331),(293,423,332),(294,424,333),(295,425,334),(296,426,335),(297,427,336),(298,428,337),(299,429,338),(300,430,313),(301,431,314),(302,432,315),(303,433,316),(304,434,317),(305,435,318),(306,436,319),(307,437,320),(308,438,321),(309,439,322),(310,440,323),(311,441,324),(312,442,325)], [(1,69,27),(2,70,28),(3,71,29),(4,72,30),(5,73,31),(6,74,32),(7,75,33),(8,76,34),(9,77,35),(10,78,36),(11,53,37),(12,54,38),(13,55,39),(14,56,40),(15,57,41),(16,58,42),(17,59,43),(18,60,44),(19,61,45),(20,62,46),(21,63,47),(22,64,48),(23,65,49),(24,66,50),(25,67,51),(26,68,52),(79,149,127),(80,150,128),(81,151,129),(82,152,130),(83,153,105),(84,154,106),(85,155,107),(86,156,108),(87,131,109),(88,132,110),(89,133,111),(90,134,112),(91,135,113),(92,136,114),(93,137,115),(94,138,116),(95,139,117),(96,140,118),(97,141,119),(98,142,120),(99,143,121),(100,144,122),(101,145,123),(102,146,124),(103,147,125),(104,148,126),(157,220,201),(158,221,202),(159,222,203),(160,223,204),(161,224,205),(162,225,206),(163,226,207),(164,227,208),(165,228,183),(166,229,184),(167,230,185),(168,231,186),(169,232,187),(170,233,188),(171,234,189),(172,209,190),(173,210,191),(174,211,192),(175,212,193),(176,213,194),(177,214,195),(178,215,196),(179,216,197),(180,217,198),(181,218,199),(182,219,200),(235,287,261),(236,288,262),(237,289,263),(238,290,264),(239,291,265),(240,292,266),(241,293,267),(242,294,268),(243,295,269),(244,296,270),(245,297,271),(246,298,272),(247,299,273),(248,300,274),(249,301,275),(250,302,276),(251,303,277),(252,304,278),(253,305,279),(254,306,280),(255,307,281),(256,308,282),(257,309,283),(258,310,284),(259,311,285),(260,312,286),(313,378,352),(314,379,353),(315,380,354),(316,381,355),(317,382,356),(318,383,357),(319,384,358),(320,385,359),(321,386,360),(322,387,361),(323,388,362),(324,389,363),(325,390,364),(326,365,339),(327,366,340),(328,367,341),(329,368,342),(330,369,343),(331,370,344),(332,371,345),(333,372,346),(334,373,347),(335,374,348),(336,375,349),(337,376,350),(338,377,351),(391,456,430),(392,457,431),(393,458,432),(394,459,433),(395,460,434),(396,461,435),(397,462,436),(398,463,437),(399,464,438),(400,465,439),(401,466,440),(402,467,441),(403,468,442),(404,443,417),(405,444,418),(406,445,419),(407,446,420),(408,447,421),(409,448,422),(410,449,423),(411,450,424),(412,451,425),(413,452,426),(414,453,427),(415,454,428),(416,455,429)], [(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)], [(1,235,14,248),(2,260,15,247),(3,259,16,246),(4,258,17,245),(5,257,18,244),(6,256,19,243),(7,255,20,242),(8,254,21,241),(9,253,22,240),(10,252,23,239),(11,251,24,238),(12,250,25,237),(13,249,26,236),(27,261,40,274),(28,286,41,273),(29,285,42,272),(30,284,43,271),(31,283,44,270),(32,282,45,269),(33,281,46,268),(34,280,47,267),(35,279,48,266),(36,278,49,265),(37,277,50,264),(38,276,51,263),(39,275,52,262),(53,303,66,290),(54,302,67,289),(55,301,68,288),(56,300,69,287),(57,299,70,312),(58,298,71,311),(59,297,72,310),(60,296,73,309),(61,295,74,308),(62,294,75,307),(63,293,76,306),(64,292,77,305),(65,291,78,304),(79,327,92,314),(80,326,93,313),(81,325,94,338),(82,324,95,337),(83,323,96,336),(84,322,97,335),(85,321,98,334),(86,320,99,333),(87,319,100,332),(88,318,101,331),(89,317,102,330),(90,316,103,329),(91,315,104,328),(105,362,118,349),(106,361,119,348),(107,360,120,347),(108,359,121,346),(109,358,122,345),(110,357,123,344),(111,356,124,343),(112,355,125,342),(113,354,126,341),(114,353,127,340),(115,352,128,339),(116,351,129,364),(117,350,130,363),(131,384,144,371),(132,383,145,370),(133,382,146,369),(134,381,147,368),(135,380,148,367),(136,379,149,366),(137,378,150,365),(138,377,151,390),(139,376,152,389),(140,375,153,388),(141,374,154,387),(142,373,155,386),(143,372,156,385),(157,412,170,399),(158,411,171,398),(159,410,172,397),(160,409,173,396),(161,408,174,395),(162,407,175,394),(163,406,176,393),(164,405,177,392),(165,404,178,391),(166,403,179,416),(167,402,180,415),(168,401,181,414),(169,400,182,413),(183,417,196,430),(184,442,197,429),(185,441,198,428),(186,440,199,427),(187,439,200,426),(188,438,201,425),(189,437,202,424),(190,436,203,423),(191,435,204,422),(192,434,205,421),(193,433,206,420),(194,432,207,419),(195,431,208,418),(209,462,222,449),(210,461,223,448),(211,460,224,447),(212,459,225,446),(213,458,226,445),(214,457,227,444),(215,456,228,443),(216,455,229,468),(217,454,230,467),(218,453,231,466),(219,452,232,465),(220,451,233,464),(221,450,234,463)]])
144 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 12A | ··· | 12P | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39AV | 78A | ··· | 78AV |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | ··· | 1 | 13 | 13 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D13 | Dic13 | C3×D13 | C3×Dic13 |
| kernel | C32×Dic13 | C3×C78 | C3×Dic13 | C3×C39 | C78 | C39 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 16 | 6 | 6 | 48 | 48 |
Matrix representation of C32×Dic13 ►in GL3(𝔽157) generated by
| 12 | 0 | 0 |
| 0 | 144 | 0 |
| 0 | 0 | 144 |
| 1 | 0 | 0 |
| 0 | 144 | 0 |
| 0 | 0 | 144 |
| 156 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 156 | 124 |
| 28 | 0 | 0 |
| 0 | 94 | 136 |
| 0 | 17 | 63 |
G:=sub<GL(3,GF(157))| [12,0,0,0,144,0,0,0,144],[1,0,0,0,144,0,0,0,144],[156,0,0,0,0,156,0,1,124],[28,0,0,0,94,17,0,136,63] >;
C32×Dic13 in GAP, Magma, Sage, TeX
C_3^2\times {\rm Dic}_{13} % in TeX
G:=Group("C3^2xDic13"); // GroupNames label
G:=SmallGroup(468,23);
// by ID
G=gap.SmallGroup(468,23);
# by ID
G:=PCGroup([5,-2,-3,-3,-2,-13,90,10804]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^26=1,d^2=c^13,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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