direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×Dic13, C117⋊4C4, C13⋊5C36, C78.10C6, C234.2C2, C39.5C12, C26.3C18, C18.2D13, C2.(C9×D13), C6.2(C3×D13), C3.(C3×Dic13), (C3×Dic13).2C3, SmallGroup(468,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C9×Dic13 |
Generators and relations for C9×Dic13
G = < a,b,c | a9=b26=1, c2=b13, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9×Dic13
►Regular action on 468 points
Generators in S468
(1 418 318 414 284 384 241 94 228)(2 419 319 415 285 385 242 95 229)(3 420 320 416 286 386 243 96 230)(4 421 321 391 261 387 244 97 231)(5 422 322 392 262 388 245 98 232)(6 423 323 393 263 389 246 99 233)(7 424 324 394 264 390 247 100 234)(8 425 325 395 265 365 248 101 209)(9 426 326 396 266 366 249 102 210)(10 427 327 397 267 367 250 103 211)(11 428 328 398 268 368 251 104 212)(12 429 329 399 269 369 252 79 213)(13 430 330 400 270 370 253 80 214)(14 431 331 401 271 371 254 81 215)(15 432 332 402 272 372 255 82 216)(16 433 333 403 273 373 256 83 217)(17 434 334 404 274 374 257 84 218)(18 435 335 405 275 375 258 85 219)(19 436 336 406 276 376 259 86 220)(20 437 337 407 277 377 260 87 221)(21 438 338 408 278 378 235 88 222)(22 439 313 409 279 379 236 89 223)(23 440 314 410 280 380 237 90 224)(24 441 315 411 281 381 238 91 225)(25 442 316 412 282 382 239 92 226)(26 417 317 413 283 383 240 93 227)(27 168 342 443 298 133 55 204 128)(28 169 343 444 299 134 56 205 129)(29 170 344 445 300 135 57 206 130)(30 171 345 446 301 136 58 207 105)(31 172 346 447 302 137 59 208 106)(32 173 347 448 303 138 60 183 107)(33 174 348 449 304 139 61 184 108)(34 175 349 450 305 140 62 185 109)(35 176 350 451 306 141 63 186 110)(36 177 351 452 307 142 64 187 111)(37 178 352 453 308 143 65 188 112)(38 179 353 454 309 144 66 189 113)(39 180 354 455 310 145 67 190 114)(40 181 355 456 311 146 68 191 115)(41 182 356 457 312 147 69 192 116)(42 157 357 458 287 148 70 193 117)(43 158 358 459 288 149 71 194 118)(44 159 359 460 289 150 72 195 119)(45 160 360 461 290 151 73 196 120)(46 161 361 462 291 152 74 197 121)(47 162 362 463 292 153 75 198 122)(48 163 363 464 293 154 76 199 123)(49 164 364 465 294 155 77 200 124)(50 165 339 466 295 156 78 201 125)(51 166 340 467 296 131 53 202 126)(52 167 341 468 297 132 54 203 127)
(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)
(1 293 14 306)(2 292 15 305)(3 291 16 304)(4 290 17 303)(5 289 18 302)(6 288 19 301)(7 287 20 300)(8 312 21 299)(9 311 22 298)(10 310 23 297)(11 309 24 296)(12 308 25 295)(13 307 26 294)(27 366 40 379)(28 365 41 378)(29 390 42 377)(30 389 43 376)(31 388 44 375)(32 387 45 374)(33 386 46 373)(34 385 47 372)(35 384 48 371)(36 383 49 370)(37 382 50 369)(38 381 51 368)(39 380 52 367)(53 328 66 315)(54 327 67 314)(55 326 68 313)(56 325 69 338)(57 324 70 337)(58 323 71 336)(59 322 72 335)(60 321 73 334)(61 320 74 333)(62 319 75 332)(63 318 76 331)(64 317 77 330)(65 316 78 329)(79 352 92 339)(80 351 93 364)(81 350 94 363)(82 349 95 362)(83 348 96 361)(84 347 97 360)(85 346 98 359)(86 345 99 358)(87 344 100 357)(88 343 101 356)(89 342 102 355)(90 341 103 354)(91 340 104 353)(105 263 118 276)(106 262 119 275)(107 261 120 274)(108 286 121 273)(109 285 122 272)(110 284 123 271)(111 283 124 270)(112 282 125 269)(113 281 126 268)(114 280 127 267)(115 279 128 266)(116 278 129 265)(117 277 130 264)(131 428 144 441)(132 427 145 440)(133 426 146 439)(134 425 147 438)(135 424 148 437)(136 423 149 436)(137 422 150 435)(138 421 151 434)(139 420 152 433)(140 419 153 432)(141 418 154 431)(142 417 155 430)(143 442 156 429)(157 260 170 247)(158 259 171 246)(159 258 172 245)(160 257 173 244)(161 256 174 243)(162 255 175 242)(163 254 176 241)(164 253 177 240)(165 252 178 239)(166 251 179 238)(167 250 180 237)(168 249 181 236)(169 248 182 235)(183 391 196 404)(184 416 197 403)(185 415 198 402)(186 414 199 401)(187 413 200 400)(188 412 201 399)(189 411 202 398)(190 410 203 397)(191 409 204 396)(192 408 205 395)(193 407 206 394)(194 406 207 393)(195 405 208 392)(209 457 222 444)(210 456 223 443)(211 455 224 468)(212 454 225 467)(213 453 226 466)(214 452 227 465)(215 451 228 464)(216 450 229 463)(217 449 230 462)(218 448 231 461)(219 447 232 460)(220 446 233 459)(221 445 234 458)
G:=sub<Sym(468)| (1,418,318,414,284,384,241,94,228)(2,419,319,415,285,385,242,95,229)(3,420,320,416,286,386,243,96,230)(4,421,321,391,261,387,244,97,231)(5,422,322,392,262,388,245,98,232)(6,423,323,393,263,389,246,99,233)(7,424,324,394,264,390,247,100,234)(8,425,325,395,265,365,248,101,209)(9,426,326,396,266,366,249,102,210)(10,427,327,397,267,367,250,103,211)(11,428,328,398,268,368,251,104,212)(12,429,329,399,269,369,252,79,213)(13,430,330,400,270,370,253,80,214)(14,431,331,401,271,371,254,81,215)(15,432,332,402,272,372,255,82,216)(16,433,333,403,273,373,256,83,217)(17,434,334,404,274,374,257,84,218)(18,435,335,405,275,375,258,85,219)(19,436,336,406,276,376,259,86,220)(20,437,337,407,277,377,260,87,221)(21,438,338,408,278,378,235,88,222)(22,439,313,409,279,379,236,89,223)(23,440,314,410,280,380,237,90,224)(24,441,315,411,281,381,238,91,225)(25,442,316,412,282,382,239,92,226)(26,417,317,413,283,383,240,93,227)(27,168,342,443,298,133,55,204,128)(28,169,343,444,299,134,56,205,129)(29,170,344,445,300,135,57,206,130)(30,171,345,446,301,136,58,207,105)(31,172,346,447,302,137,59,208,106)(32,173,347,448,303,138,60,183,107)(33,174,348,449,304,139,61,184,108)(34,175,349,450,305,140,62,185,109)(35,176,350,451,306,141,63,186,110)(36,177,351,452,307,142,64,187,111)(37,178,352,453,308,143,65,188,112)(38,179,353,454,309,144,66,189,113)(39,180,354,455,310,145,67,190,114)(40,181,355,456,311,146,68,191,115)(41,182,356,457,312,147,69,192,116)(42,157,357,458,287,148,70,193,117)(43,158,358,459,288,149,71,194,118)(44,159,359,460,289,150,72,195,119)(45,160,360,461,290,151,73,196,120)(46,161,361,462,291,152,74,197,121)(47,162,362,463,292,153,75,198,122)(48,163,363,464,293,154,76,199,123)(49,164,364,465,294,155,77,200,124)(50,165,339,466,295,156,78,201,125)(51,166,340,467,296,131,53,202,126)(52,167,341,468,297,132,54,203,127), (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), (1,293,14,306)(2,292,15,305)(3,291,16,304)(4,290,17,303)(5,289,18,302)(6,288,19,301)(7,287,20,300)(8,312,21,299)(9,311,22,298)(10,310,23,297)(11,309,24,296)(12,308,25,295)(13,307,26,294)(27,366,40,379)(28,365,41,378)(29,390,42,377)(30,389,43,376)(31,388,44,375)(32,387,45,374)(33,386,46,373)(34,385,47,372)(35,384,48,371)(36,383,49,370)(37,382,50,369)(38,381,51,368)(39,380,52,367)(53,328,66,315)(54,327,67,314)(55,326,68,313)(56,325,69,338)(57,324,70,337)(58,323,71,336)(59,322,72,335)(60,321,73,334)(61,320,74,333)(62,319,75,332)(63,318,76,331)(64,317,77,330)(65,316,78,329)(79,352,92,339)(80,351,93,364)(81,350,94,363)(82,349,95,362)(83,348,96,361)(84,347,97,360)(85,346,98,359)(86,345,99,358)(87,344,100,357)(88,343,101,356)(89,342,102,355)(90,341,103,354)(91,340,104,353)(105,263,118,276)(106,262,119,275)(107,261,120,274)(108,286,121,273)(109,285,122,272)(110,284,123,271)(111,283,124,270)(112,282,125,269)(113,281,126,268)(114,280,127,267)(115,279,128,266)(116,278,129,265)(117,277,130,264)(131,428,144,441)(132,427,145,440)(133,426,146,439)(134,425,147,438)(135,424,148,437)(136,423,149,436)(137,422,150,435)(138,421,151,434)(139,420,152,433)(140,419,153,432)(141,418,154,431)(142,417,155,430)(143,442,156,429)(157,260,170,247)(158,259,171,246)(159,258,172,245)(160,257,173,244)(161,256,174,243)(162,255,175,242)(163,254,176,241)(164,253,177,240)(165,252,178,239)(166,251,179,238)(167,250,180,237)(168,249,181,236)(169,248,182,235)(183,391,196,404)(184,416,197,403)(185,415,198,402)(186,414,199,401)(187,413,200,400)(188,412,201,399)(189,411,202,398)(190,410,203,397)(191,409,204,396)(192,408,205,395)(193,407,206,394)(194,406,207,393)(195,405,208,392)(209,457,222,444)(210,456,223,443)(211,455,224,468)(212,454,225,467)(213,453,226,466)(214,452,227,465)(215,451,228,464)(216,450,229,463)(217,449,230,462)(218,448,231,461)(219,447,232,460)(220,446,233,459)(221,445,234,458)>;
G:=Group( (1,418,318,414,284,384,241,94,228)(2,419,319,415,285,385,242,95,229)(3,420,320,416,286,386,243,96,230)(4,421,321,391,261,387,244,97,231)(5,422,322,392,262,388,245,98,232)(6,423,323,393,263,389,246,99,233)(7,424,324,394,264,390,247,100,234)(8,425,325,395,265,365,248,101,209)(9,426,326,396,266,366,249,102,210)(10,427,327,397,267,367,250,103,211)(11,428,328,398,268,368,251,104,212)(12,429,329,399,269,369,252,79,213)(13,430,330,400,270,370,253,80,214)(14,431,331,401,271,371,254,81,215)(15,432,332,402,272,372,255,82,216)(16,433,333,403,273,373,256,83,217)(17,434,334,404,274,374,257,84,218)(18,435,335,405,275,375,258,85,219)(19,436,336,406,276,376,259,86,220)(20,437,337,407,277,377,260,87,221)(21,438,338,408,278,378,235,88,222)(22,439,313,409,279,379,236,89,223)(23,440,314,410,280,380,237,90,224)(24,441,315,411,281,381,238,91,225)(25,442,316,412,282,382,239,92,226)(26,417,317,413,283,383,240,93,227)(27,168,342,443,298,133,55,204,128)(28,169,343,444,299,134,56,205,129)(29,170,344,445,300,135,57,206,130)(30,171,345,446,301,136,58,207,105)(31,172,346,447,302,137,59,208,106)(32,173,347,448,303,138,60,183,107)(33,174,348,449,304,139,61,184,108)(34,175,349,450,305,140,62,185,109)(35,176,350,451,306,141,63,186,110)(36,177,351,452,307,142,64,187,111)(37,178,352,453,308,143,65,188,112)(38,179,353,454,309,144,66,189,113)(39,180,354,455,310,145,67,190,114)(40,181,355,456,311,146,68,191,115)(41,182,356,457,312,147,69,192,116)(42,157,357,458,287,148,70,193,117)(43,158,358,459,288,149,71,194,118)(44,159,359,460,289,150,72,195,119)(45,160,360,461,290,151,73,196,120)(46,161,361,462,291,152,74,197,121)(47,162,362,463,292,153,75,198,122)(48,163,363,464,293,154,76,199,123)(49,164,364,465,294,155,77,200,124)(50,165,339,466,295,156,78,201,125)(51,166,340,467,296,131,53,202,126)(52,167,341,468,297,132,54,203,127), (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), (1,293,14,306)(2,292,15,305)(3,291,16,304)(4,290,17,303)(5,289,18,302)(6,288,19,301)(7,287,20,300)(8,312,21,299)(9,311,22,298)(10,310,23,297)(11,309,24,296)(12,308,25,295)(13,307,26,294)(27,366,40,379)(28,365,41,378)(29,390,42,377)(30,389,43,376)(31,388,44,375)(32,387,45,374)(33,386,46,373)(34,385,47,372)(35,384,48,371)(36,383,49,370)(37,382,50,369)(38,381,51,368)(39,380,52,367)(53,328,66,315)(54,327,67,314)(55,326,68,313)(56,325,69,338)(57,324,70,337)(58,323,71,336)(59,322,72,335)(60,321,73,334)(61,320,74,333)(62,319,75,332)(63,318,76,331)(64,317,77,330)(65,316,78,329)(79,352,92,339)(80,351,93,364)(81,350,94,363)(82,349,95,362)(83,348,96,361)(84,347,97,360)(85,346,98,359)(86,345,99,358)(87,344,100,357)(88,343,101,356)(89,342,102,355)(90,341,103,354)(91,340,104,353)(105,263,118,276)(106,262,119,275)(107,261,120,274)(108,286,121,273)(109,285,122,272)(110,284,123,271)(111,283,124,270)(112,282,125,269)(113,281,126,268)(114,280,127,267)(115,279,128,266)(116,278,129,265)(117,277,130,264)(131,428,144,441)(132,427,145,440)(133,426,146,439)(134,425,147,438)(135,424,148,437)(136,423,149,436)(137,422,150,435)(138,421,151,434)(139,420,152,433)(140,419,153,432)(141,418,154,431)(142,417,155,430)(143,442,156,429)(157,260,170,247)(158,259,171,246)(159,258,172,245)(160,257,173,244)(161,256,174,243)(162,255,175,242)(163,254,176,241)(164,253,177,240)(165,252,178,239)(166,251,179,238)(167,250,180,237)(168,249,181,236)(169,248,182,235)(183,391,196,404)(184,416,197,403)(185,415,198,402)(186,414,199,401)(187,413,200,400)(188,412,201,399)(189,411,202,398)(190,410,203,397)(191,409,204,396)(192,408,205,395)(193,407,206,394)(194,406,207,393)(195,405,208,392)(209,457,222,444)(210,456,223,443)(211,455,224,468)(212,454,225,467)(213,453,226,466)(214,452,227,465)(215,451,228,464)(216,450,229,463)(217,449,230,462)(218,448,231,461)(219,447,232,460)(220,446,233,459)(221,445,234,458) );
G=PermutationGroup([[(1,418,318,414,284,384,241,94,228),(2,419,319,415,285,385,242,95,229),(3,420,320,416,286,386,243,96,230),(4,421,321,391,261,387,244,97,231),(5,422,322,392,262,388,245,98,232),(6,423,323,393,263,389,246,99,233),(7,424,324,394,264,390,247,100,234),(8,425,325,395,265,365,248,101,209),(9,426,326,396,266,366,249,102,210),(10,427,327,397,267,367,250,103,211),(11,428,328,398,268,368,251,104,212),(12,429,329,399,269,369,252,79,213),(13,430,330,400,270,370,253,80,214),(14,431,331,401,271,371,254,81,215),(15,432,332,402,272,372,255,82,216),(16,433,333,403,273,373,256,83,217),(17,434,334,404,274,374,257,84,218),(18,435,335,405,275,375,258,85,219),(19,436,336,406,276,376,259,86,220),(20,437,337,407,277,377,260,87,221),(21,438,338,408,278,378,235,88,222),(22,439,313,409,279,379,236,89,223),(23,440,314,410,280,380,237,90,224),(24,441,315,411,281,381,238,91,225),(25,442,316,412,282,382,239,92,226),(26,417,317,413,283,383,240,93,227),(27,168,342,443,298,133,55,204,128),(28,169,343,444,299,134,56,205,129),(29,170,344,445,300,135,57,206,130),(30,171,345,446,301,136,58,207,105),(31,172,346,447,302,137,59,208,106),(32,173,347,448,303,138,60,183,107),(33,174,348,449,304,139,61,184,108),(34,175,349,450,305,140,62,185,109),(35,176,350,451,306,141,63,186,110),(36,177,351,452,307,142,64,187,111),(37,178,352,453,308,143,65,188,112),(38,179,353,454,309,144,66,189,113),(39,180,354,455,310,145,67,190,114),(40,181,355,456,311,146,68,191,115),(41,182,356,457,312,147,69,192,116),(42,157,357,458,287,148,70,193,117),(43,158,358,459,288,149,71,194,118),(44,159,359,460,289,150,72,195,119),(45,160,360,461,290,151,73,196,120),(46,161,361,462,291,152,74,197,121),(47,162,362,463,292,153,75,198,122),(48,163,363,464,293,154,76,199,123),(49,164,364,465,294,155,77,200,124),(50,165,339,466,295,156,78,201,125),(51,166,340,467,296,131,53,202,126),(52,167,341,468,297,132,54,203,127)], [(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)], [(1,293,14,306),(2,292,15,305),(3,291,16,304),(4,290,17,303),(5,289,18,302),(6,288,19,301),(7,287,20,300),(8,312,21,299),(9,311,22,298),(10,310,23,297),(11,309,24,296),(12,308,25,295),(13,307,26,294),(27,366,40,379),(28,365,41,378),(29,390,42,377),(30,389,43,376),(31,388,44,375),(32,387,45,374),(33,386,46,373),(34,385,47,372),(35,384,48,371),(36,383,49,370),(37,382,50,369),(38,381,51,368),(39,380,52,367),(53,328,66,315),(54,327,67,314),(55,326,68,313),(56,325,69,338),(57,324,70,337),(58,323,71,336),(59,322,72,335),(60,321,73,334),(61,320,74,333),(62,319,75,332),(63,318,76,331),(64,317,77,330),(65,316,78,329),(79,352,92,339),(80,351,93,364),(81,350,94,363),(82,349,95,362),(83,348,96,361),(84,347,97,360),(85,346,98,359),(86,345,99,358),(87,344,100,357),(88,343,101,356),(89,342,102,355),(90,341,103,354),(91,340,104,353),(105,263,118,276),(106,262,119,275),(107,261,120,274),(108,286,121,273),(109,285,122,272),(110,284,123,271),(111,283,124,270),(112,282,125,269),(113,281,126,268),(114,280,127,267),(115,279,128,266),(116,278,129,265),(117,277,130,264),(131,428,144,441),(132,427,145,440),(133,426,146,439),(134,425,147,438),(135,424,148,437),(136,423,149,436),(137,422,150,435),(138,421,151,434),(139,420,152,433),(140,419,153,432),(141,418,154,431),(142,417,155,430),(143,442,156,429),(157,260,170,247),(158,259,171,246),(159,258,172,245),(160,257,173,244),(161,256,174,243),(162,255,175,242),(163,254,176,241),(164,253,177,240),(165,252,178,239),(166,251,179,238),(167,250,180,237),(168,249,181,236),(169,248,182,235),(183,391,196,404),(184,416,197,403),(185,415,198,402),(186,414,199,401),(187,413,200,400),(188,412,201,399),(189,411,202,398),(190,410,203,397),(191,409,204,396),(192,408,205,395),(193,407,206,394),(194,406,207,393),(195,405,208,392),(209,457,222,444),(210,456,223,443),(211,455,224,468),(212,454,225,467),(213,453,226,466),(214,452,227,465),(215,451,228,464),(216,450,229,463),(217,449,230,462),(218,448,231,461),(219,447,232,460),(220,446,233,459),(221,445,234,458)]])
144 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 13A | ··· | 13F | 18A | ··· | 18F | 26A | ··· | 26F | 36A | ··· | 36L | 39A | ··· | 39L | 78A | ··· | 78L | 117A | ··· | 117AJ | 234A | ··· | 234AJ |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 13 | ··· | 13 | 18 | ··· | 18 | 26 | ··· | 26 | 36 | ··· | 36 | 39 | ··· | 39 | 78 | ··· | 78 | 117 | ··· | 117 | 234 | ··· | 234 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 1 | 1 | 1 | ··· | 1 | 13 | 13 | 13 | 13 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | D13 | Dic13 | C3×D13 | C3×Dic13 | C9×D13 | C9×Dic13 |
| kernel | C9×Dic13 | C234 | C3×Dic13 | C117 | C78 | Dic13 | C39 | C26 | C13 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 6 | 6 | 12 | 12 | 36 | 36 |
Matrix representation of C9×Dic13 ►in GL3(𝔽937) generated by
| 1 | 0 | 0 |
| 0 | 924 | 0 |
| 0 | 0 | 924 |
| 936 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 936 | 150 |
| 741 | 0 | 0 |
| 0 | 207 | 367 |
| 0 | 496 | 730 |
G:=sub<GL(3,GF(937))| [1,0,0,0,924,0,0,0,924],[936,0,0,0,0,936,0,1,150],[741,0,0,0,207,496,0,367,730] >;
C9×Dic13 in GAP, Magma, Sage, TeX
C_9\times {\rm Dic}_{13} % in TeX
G:=Group("C9xDic13"); // GroupNames label
G:=SmallGroup(468,4);
// by ID
G=gap.SmallGroup(468,4);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-13,30,66,10804]);
// Polycyclic
G:=Group<a,b,c|a^9=b^26=1,c^2=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export