direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13×Dic9, C9⋊C52, C117⋊5C4, C18.C26, C78.5S3, C26.2D9, C234.3C2, C39.3Dic3, C2.(C13×D9), C6.1(S3×C13), C3.(Dic3×C13), SmallGroup(468,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C13×Dic9 |
Generators and relations for C13×Dic9
G = < a,b,c | a13=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C13×Dic9
►Regular action on 468 points
Generators in S468
(1 231 209 184 166 152 130 119 94 87 69 40 33)(2 232 210 185 167 153 131 120 95 88 70 41 34)(3 233 211 186 168 154 132 121 96 89 71 42 35)(4 234 212 187 169 155 133 122 97 90 72 43 36)(5 217 213 188 170 156 134 123 98 73 55 44 19)(6 218 214 189 171 157 135 124 99 74 56 45 20)(7 219 215 190 172 158 136 125 100 75 57 46 21)(8 220 216 191 173 159 137 126 101 76 58 47 22)(9 221 199 192 174 160 138 109 102 77 59 48 23)(10 222 200 193 175 161 139 110 103 78 60 49 24)(11 223 201 194 176 162 140 111 104 79 61 50 25)(12 224 202 195 177 145 141 112 105 80 62 51 26)(13 225 203 196 178 146 142 113 106 81 63 52 27)(14 226 204 197 179 147 143 114 107 82 64 53 28)(15 227 205 198 180 148 144 115 108 83 65 54 29)(16 228 206 181 163 149 127 116 91 84 66 37 30)(17 229 207 182 164 150 128 117 92 85 67 38 31)(18 230 208 183 165 151 129 118 93 86 68 39 32)(235 462 439 426 407 391 367 343 332 318 295 276 264)(236 463 440 427 408 392 368 344 333 319 296 277 265)(237 464 441 428 409 393 369 345 334 320 297 278 266)(238 465 442 429 410 394 370 346 335 321 298 279 267)(239 466 443 430 411 395 371 347 336 322 299 280 268)(240 467 444 431 412 396 372 348 337 323 300 281 269)(241 468 445 432 413 379 373 349 338 324 301 282 270)(242 451 446 415 414 380 374 350 339 307 302 283 253)(243 452 447 416 397 381 375 351 340 308 303 284 254)(244 453 448 417 398 382 376 352 341 309 304 285 255)(245 454 449 418 399 383 377 353 342 310 305 286 256)(246 455 450 419 400 384 378 354 325 311 306 287 257)(247 456 433 420 401 385 361 355 326 312 289 288 258)(248 457 434 421 402 386 362 356 327 313 290 271 259)(249 458 435 422 403 387 363 357 328 314 291 272 260)(250 459 436 423 404 388 364 358 329 315 292 273 261)(251 460 437 424 405 389 365 359 330 316 293 274 262)(252 461 438 425 406 390 366 360 331 317 294 275 263)
(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)
(1 250 10 241)(2 249 11 240)(3 248 12 239)(4 247 13 238)(5 246 14 237)(6 245 15 236)(7 244 16 235)(8 243 17 252)(9 242 18 251)(19 257 28 266)(20 256 29 265)(21 255 30 264)(22 254 31 263)(23 253 32 262)(24 270 33 261)(25 269 34 260)(26 268 35 259)(27 267 36 258)(37 276 46 285)(38 275 47 284)(39 274 48 283)(40 273 49 282)(41 272 50 281)(42 271 51 280)(43 288 52 279)(44 287 53 278)(45 286 54 277)(55 306 64 297)(56 305 65 296)(57 304 66 295)(58 303 67 294)(59 302 68 293)(60 301 69 292)(61 300 70 291)(62 299 71 290)(63 298 72 289)(73 311 82 320)(74 310 83 319)(75 309 84 318)(76 308 85 317)(77 307 86 316)(78 324 87 315)(79 323 88 314)(80 322 89 313)(81 321 90 312)(91 332 100 341)(92 331 101 340)(93 330 102 339)(94 329 103 338)(95 328 104 337)(96 327 105 336)(97 326 106 335)(98 325 107 334)(99 342 108 333)(109 350 118 359)(110 349 119 358)(111 348 120 357)(112 347 121 356)(113 346 122 355)(114 345 123 354)(115 344 124 353)(116 343 125 352)(117 360 126 351)(127 367 136 376)(128 366 137 375)(129 365 138 374)(130 364 139 373)(131 363 140 372)(132 362 141 371)(133 361 142 370)(134 378 143 369)(135 377 144 368)(145 395 154 386)(146 394 155 385)(147 393 156 384)(148 392 157 383)(149 391 158 382)(150 390 159 381)(151 389 160 380)(152 388 161 379)(153 387 162 396)(163 407 172 398)(164 406 173 397)(165 405 174 414)(166 404 175 413)(167 403 176 412)(168 402 177 411)(169 401 178 410)(170 400 179 409)(171 399 180 408)(181 426 190 417)(182 425 191 416)(183 424 192 415)(184 423 193 432)(185 422 194 431)(186 421 195 430)(187 420 196 429)(188 419 197 428)(189 418 198 427)(199 446 208 437)(200 445 209 436)(201 444 210 435)(202 443 211 434)(203 442 212 433)(204 441 213 450)(205 440 214 449)(206 439 215 448)(207 438 216 447)(217 455 226 464)(218 454 227 463)(219 453 228 462)(220 452 229 461)(221 451 230 460)(222 468 231 459)(223 467 232 458)(224 466 233 457)(225 465 234 456)
G:=sub<Sym(468)| (1,231,209,184,166,152,130,119,94,87,69,40,33)(2,232,210,185,167,153,131,120,95,88,70,41,34)(3,233,211,186,168,154,132,121,96,89,71,42,35)(4,234,212,187,169,155,133,122,97,90,72,43,36)(5,217,213,188,170,156,134,123,98,73,55,44,19)(6,218,214,189,171,157,135,124,99,74,56,45,20)(7,219,215,190,172,158,136,125,100,75,57,46,21)(8,220,216,191,173,159,137,126,101,76,58,47,22)(9,221,199,192,174,160,138,109,102,77,59,48,23)(10,222,200,193,175,161,139,110,103,78,60,49,24)(11,223,201,194,176,162,140,111,104,79,61,50,25)(12,224,202,195,177,145,141,112,105,80,62,51,26)(13,225,203,196,178,146,142,113,106,81,63,52,27)(14,226,204,197,179,147,143,114,107,82,64,53,28)(15,227,205,198,180,148,144,115,108,83,65,54,29)(16,228,206,181,163,149,127,116,91,84,66,37,30)(17,229,207,182,164,150,128,117,92,85,67,38,31)(18,230,208,183,165,151,129,118,93,86,68,39,32)(235,462,439,426,407,391,367,343,332,318,295,276,264)(236,463,440,427,408,392,368,344,333,319,296,277,265)(237,464,441,428,409,393,369,345,334,320,297,278,266)(238,465,442,429,410,394,370,346,335,321,298,279,267)(239,466,443,430,411,395,371,347,336,322,299,280,268)(240,467,444,431,412,396,372,348,337,323,300,281,269)(241,468,445,432,413,379,373,349,338,324,301,282,270)(242,451,446,415,414,380,374,350,339,307,302,283,253)(243,452,447,416,397,381,375,351,340,308,303,284,254)(244,453,448,417,398,382,376,352,341,309,304,285,255)(245,454,449,418,399,383,377,353,342,310,305,286,256)(246,455,450,419,400,384,378,354,325,311,306,287,257)(247,456,433,420,401,385,361,355,326,312,289,288,258)(248,457,434,421,402,386,362,356,327,313,290,271,259)(249,458,435,422,403,387,363,357,328,314,291,272,260)(250,459,436,423,404,388,364,358,329,315,292,273,261)(251,460,437,424,405,389,365,359,330,316,293,274,262)(252,461,438,425,406,390,366,360,331,317,294,275,263), (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), (1,250,10,241)(2,249,11,240)(3,248,12,239)(4,247,13,238)(5,246,14,237)(6,245,15,236)(7,244,16,235)(8,243,17,252)(9,242,18,251)(19,257,28,266)(20,256,29,265)(21,255,30,264)(22,254,31,263)(23,253,32,262)(24,270,33,261)(25,269,34,260)(26,268,35,259)(27,267,36,258)(37,276,46,285)(38,275,47,284)(39,274,48,283)(40,273,49,282)(41,272,50,281)(42,271,51,280)(43,288,52,279)(44,287,53,278)(45,286,54,277)(55,306,64,297)(56,305,65,296)(57,304,66,295)(58,303,67,294)(59,302,68,293)(60,301,69,292)(61,300,70,291)(62,299,71,290)(63,298,72,289)(73,311,82,320)(74,310,83,319)(75,309,84,318)(76,308,85,317)(77,307,86,316)(78,324,87,315)(79,323,88,314)(80,322,89,313)(81,321,90,312)(91,332,100,341)(92,331,101,340)(93,330,102,339)(94,329,103,338)(95,328,104,337)(96,327,105,336)(97,326,106,335)(98,325,107,334)(99,342,108,333)(109,350,118,359)(110,349,119,358)(111,348,120,357)(112,347,121,356)(113,346,122,355)(114,345,123,354)(115,344,124,353)(116,343,125,352)(117,360,126,351)(127,367,136,376)(128,366,137,375)(129,365,138,374)(130,364,139,373)(131,363,140,372)(132,362,141,371)(133,361,142,370)(134,378,143,369)(135,377,144,368)(145,395,154,386)(146,394,155,385)(147,393,156,384)(148,392,157,383)(149,391,158,382)(150,390,159,381)(151,389,160,380)(152,388,161,379)(153,387,162,396)(163,407,172,398)(164,406,173,397)(165,405,174,414)(166,404,175,413)(167,403,176,412)(168,402,177,411)(169,401,178,410)(170,400,179,409)(171,399,180,408)(181,426,190,417)(182,425,191,416)(183,424,192,415)(184,423,193,432)(185,422,194,431)(186,421,195,430)(187,420,196,429)(188,419,197,428)(189,418,198,427)(199,446,208,437)(200,445,209,436)(201,444,210,435)(202,443,211,434)(203,442,212,433)(204,441,213,450)(205,440,214,449)(206,439,215,448)(207,438,216,447)(217,455,226,464)(218,454,227,463)(219,453,228,462)(220,452,229,461)(221,451,230,460)(222,468,231,459)(223,467,232,458)(224,466,233,457)(225,465,234,456)>;
G:=Group( (1,231,209,184,166,152,130,119,94,87,69,40,33)(2,232,210,185,167,153,131,120,95,88,70,41,34)(3,233,211,186,168,154,132,121,96,89,71,42,35)(4,234,212,187,169,155,133,122,97,90,72,43,36)(5,217,213,188,170,156,134,123,98,73,55,44,19)(6,218,214,189,171,157,135,124,99,74,56,45,20)(7,219,215,190,172,158,136,125,100,75,57,46,21)(8,220,216,191,173,159,137,126,101,76,58,47,22)(9,221,199,192,174,160,138,109,102,77,59,48,23)(10,222,200,193,175,161,139,110,103,78,60,49,24)(11,223,201,194,176,162,140,111,104,79,61,50,25)(12,224,202,195,177,145,141,112,105,80,62,51,26)(13,225,203,196,178,146,142,113,106,81,63,52,27)(14,226,204,197,179,147,143,114,107,82,64,53,28)(15,227,205,198,180,148,144,115,108,83,65,54,29)(16,228,206,181,163,149,127,116,91,84,66,37,30)(17,229,207,182,164,150,128,117,92,85,67,38,31)(18,230,208,183,165,151,129,118,93,86,68,39,32)(235,462,439,426,407,391,367,343,332,318,295,276,264)(236,463,440,427,408,392,368,344,333,319,296,277,265)(237,464,441,428,409,393,369,345,334,320,297,278,266)(238,465,442,429,410,394,370,346,335,321,298,279,267)(239,466,443,430,411,395,371,347,336,322,299,280,268)(240,467,444,431,412,396,372,348,337,323,300,281,269)(241,468,445,432,413,379,373,349,338,324,301,282,270)(242,451,446,415,414,380,374,350,339,307,302,283,253)(243,452,447,416,397,381,375,351,340,308,303,284,254)(244,453,448,417,398,382,376,352,341,309,304,285,255)(245,454,449,418,399,383,377,353,342,310,305,286,256)(246,455,450,419,400,384,378,354,325,311,306,287,257)(247,456,433,420,401,385,361,355,326,312,289,288,258)(248,457,434,421,402,386,362,356,327,313,290,271,259)(249,458,435,422,403,387,363,357,328,314,291,272,260)(250,459,436,423,404,388,364,358,329,315,292,273,261)(251,460,437,424,405,389,365,359,330,316,293,274,262)(252,461,438,425,406,390,366,360,331,317,294,275,263), (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), (1,250,10,241)(2,249,11,240)(3,248,12,239)(4,247,13,238)(5,246,14,237)(6,245,15,236)(7,244,16,235)(8,243,17,252)(9,242,18,251)(19,257,28,266)(20,256,29,265)(21,255,30,264)(22,254,31,263)(23,253,32,262)(24,270,33,261)(25,269,34,260)(26,268,35,259)(27,267,36,258)(37,276,46,285)(38,275,47,284)(39,274,48,283)(40,273,49,282)(41,272,50,281)(42,271,51,280)(43,288,52,279)(44,287,53,278)(45,286,54,277)(55,306,64,297)(56,305,65,296)(57,304,66,295)(58,303,67,294)(59,302,68,293)(60,301,69,292)(61,300,70,291)(62,299,71,290)(63,298,72,289)(73,311,82,320)(74,310,83,319)(75,309,84,318)(76,308,85,317)(77,307,86,316)(78,324,87,315)(79,323,88,314)(80,322,89,313)(81,321,90,312)(91,332,100,341)(92,331,101,340)(93,330,102,339)(94,329,103,338)(95,328,104,337)(96,327,105,336)(97,326,106,335)(98,325,107,334)(99,342,108,333)(109,350,118,359)(110,349,119,358)(111,348,120,357)(112,347,121,356)(113,346,122,355)(114,345,123,354)(115,344,124,353)(116,343,125,352)(117,360,126,351)(127,367,136,376)(128,366,137,375)(129,365,138,374)(130,364,139,373)(131,363,140,372)(132,362,141,371)(133,361,142,370)(134,378,143,369)(135,377,144,368)(145,395,154,386)(146,394,155,385)(147,393,156,384)(148,392,157,383)(149,391,158,382)(150,390,159,381)(151,389,160,380)(152,388,161,379)(153,387,162,396)(163,407,172,398)(164,406,173,397)(165,405,174,414)(166,404,175,413)(167,403,176,412)(168,402,177,411)(169,401,178,410)(170,400,179,409)(171,399,180,408)(181,426,190,417)(182,425,191,416)(183,424,192,415)(184,423,193,432)(185,422,194,431)(186,421,195,430)(187,420,196,429)(188,419,197,428)(189,418,198,427)(199,446,208,437)(200,445,209,436)(201,444,210,435)(202,443,211,434)(203,442,212,433)(204,441,213,450)(205,440,214,449)(206,439,215,448)(207,438,216,447)(217,455,226,464)(218,454,227,463)(219,453,228,462)(220,452,229,461)(221,451,230,460)(222,468,231,459)(223,467,232,458)(224,466,233,457)(225,465,234,456) );
G=PermutationGroup([[(1,231,209,184,166,152,130,119,94,87,69,40,33),(2,232,210,185,167,153,131,120,95,88,70,41,34),(3,233,211,186,168,154,132,121,96,89,71,42,35),(4,234,212,187,169,155,133,122,97,90,72,43,36),(5,217,213,188,170,156,134,123,98,73,55,44,19),(6,218,214,189,171,157,135,124,99,74,56,45,20),(7,219,215,190,172,158,136,125,100,75,57,46,21),(8,220,216,191,173,159,137,126,101,76,58,47,22),(9,221,199,192,174,160,138,109,102,77,59,48,23),(10,222,200,193,175,161,139,110,103,78,60,49,24),(11,223,201,194,176,162,140,111,104,79,61,50,25),(12,224,202,195,177,145,141,112,105,80,62,51,26),(13,225,203,196,178,146,142,113,106,81,63,52,27),(14,226,204,197,179,147,143,114,107,82,64,53,28),(15,227,205,198,180,148,144,115,108,83,65,54,29),(16,228,206,181,163,149,127,116,91,84,66,37,30),(17,229,207,182,164,150,128,117,92,85,67,38,31),(18,230,208,183,165,151,129,118,93,86,68,39,32),(235,462,439,426,407,391,367,343,332,318,295,276,264),(236,463,440,427,408,392,368,344,333,319,296,277,265),(237,464,441,428,409,393,369,345,334,320,297,278,266),(238,465,442,429,410,394,370,346,335,321,298,279,267),(239,466,443,430,411,395,371,347,336,322,299,280,268),(240,467,444,431,412,396,372,348,337,323,300,281,269),(241,468,445,432,413,379,373,349,338,324,301,282,270),(242,451,446,415,414,380,374,350,339,307,302,283,253),(243,452,447,416,397,381,375,351,340,308,303,284,254),(244,453,448,417,398,382,376,352,341,309,304,285,255),(245,454,449,418,399,383,377,353,342,310,305,286,256),(246,455,450,419,400,384,378,354,325,311,306,287,257),(247,456,433,420,401,385,361,355,326,312,289,288,258),(248,457,434,421,402,386,362,356,327,313,290,271,259),(249,458,435,422,403,387,363,357,328,314,291,272,260),(250,459,436,423,404,388,364,358,329,315,292,273,261),(251,460,437,424,405,389,365,359,330,316,293,274,262),(252,461,438,425,406,390,366,360,331,317,294,275,263)], [(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)], [(1,250,10,241),(2,249,11,240),(3,248,12,239),(4,247,13,238),(5,246,14,237),(6,245,15,236),(7,244,16,235),(8,243,17,252),(9,242,18,251),(19,257,28,266),(20,256,29,265),(21,255,30,264),(22,254,31,263),(23,253,32,262),(24,270,33,261),(25,269,34,260),(26,268,35,259),(27,267,36,258),(37,276,46,285),(38,275,47,284),(39,274,48,283),(40,273,49,282),(41,272,50,281),(42,271,51,280),(43,288,52,279),(44,287,53,278),(45,286,54,277),(55,306,64,297),(56,305,65,296),(57,304,66,295),(58,303,67,294),(59,302,68,293),(60,301,69,292),(61,300,70,291),(62,299,71,290),(63,298,72,289),(73,311,82,320),(74,310,83,319),(75,309,84,318),(76,308,85,317),(77,307,86,316),(78,324,87,315),(79,323,88,314),(80,322,89,313),(81,321,90,312),(91,332,100,341),(92,331,101,340),(93,330,102,339),(94,329,103,338),(95,328,104,337),(96,327,105,336),(97,326,106,335),(98,325,107,334),(99,342,108,333),(109,350,118,359),(110,349,119,358),(111,348,120,357),(112,347,121,356),(113,346,122,355),(114,345,123,354),(115,344,124,353),(116,343,125,352),(117,360,126,351),(127,367,136,376),(128,366,137,375),(129,365,138,374),(130,364,139,373),(131,363,140,372),(132,362,141,371),(133,361,142,370),(134,378,143,369),(135,377,144,368),(145,395,154,386),(146,394,155,385),(147,393,156,384),(148,392,157,383),(149,391,158,382),(150,390,159,381),(151,389,160,380),(152,388,161,379),(153,387,162,396),(163,407,172,398),(164,406,173,397),(165,405,174,414),(166,404,175,413),(167,403,176,412),(168,402,177,411),(169,401,178,410),(170,400,179,409),(171,399,180,408),(181,426,190,417),(182,425,191,416),(183,424,192,415),(184,423,193,432),(185,422,194,431),(186,421,195,430),(187,420,196,429),(188,419,197,428),(189,418,198,427),(199,446,208,437),(200,445,209,436),(201,444,210,435),(202,443,211,434),(203,442,212,433),(204,441,213,450),(205,440,214,449),(206,439,215,448),(207,438,216,447),(217,455,226,464),(218,454,227,463),(219,453,228,462),(220,452,229,461),(221,451,230,460),(222,468,231,459),(223,467,232,458),(224,466,233,457),(225,465,234,456)]])
156 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 9A | 9B | 9C | 13A | ··· | 13L | 18A | 18B | 18C | 26A | ··· | 26L | 39A | ··· | 39L | 52A | ··· | 52X | 78A | ··· | 78L | 117A | ··· | 117AJ | 234A | ··· | 234AJ |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 9 | 9 | 9 | 13 | ··· | 13 | 18 | 18 | 18 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 117 | ··· | 117 | 234 | ··· | 234 |
| size | 1 | 1 | 2 | 9 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C4 | C13 | C26 | C52 | S3 | Dic3 | D9 | Dic9 | S3×C13 | Dic3×C13 | C13×D9 | C13×Dic9 |
| kernel | C13×Dic9 | C234 | C117 | Dic9 | C18 | C9 | C78 | C39 | C26 | C13 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 12 | 12 | 24 | 1 | 1 | 3 | 3 | 12 | 12 | 36 | 36 |
Matrix representation of C13×Dic9 ►in GL3(𝔽937) generated by
| 1 | 0 | 0 |
| 0 | 676 | 0 |
| 0 | 0 | 676 |
| 936 | 0 | 0 |
| 0 | 465 | 262 |
| 0 | 675 | 203 |
| 196 | 0 | 0 |
| 0 | 501 | 889 |
| 0 | 388 | 436 |
G:=sub<GL(3,GF(937))| [1,0,0,0,676,0,0,0,676],[936,0,0,0,465,675,0,262,203],[196,0,0,0,501,388,0,889,436] >;
C13×Dic9 in GAP, Magma, Sage, TeX
C_{13}\times {\rm Dic}_9 % in TeX
G:=Group("C13xDic9"); // GroupNames label
G:=SmallGroup(468,3);
// by ID
G=gap.SmallGroup(468,3);
# by ID
G:=PCGroup([5,-2,-13,-2,-3,-3,130,5203,138,7804]);
// Polycyclic
G:=Group<a,b,c|a^13=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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