metacyclic, supersoluble, monomial, Z-group
Aliases: C13⋊2C36, C26.C18, Dic13⋊C9, C78.2C6, C39.2C12, C13⋊C9⋊2C4, C2.(C13⋊C18), C6.2(C13⋊C6), C3.(C26.C6), (C3×Dic13).C3, (C2×C13⋊C9).C2, SmallGroup(468,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13⋊2C36 |
Generators and relations for C13⋊2C36
G = < a,b | a13=b36=1, bab-1=a4 >
Smallest permutation representation of C13⋊2C36
►Regular action on 468 points
Generators in S468
(1 346 312 287 154 456 113 104 252 405 216 64 387)(2 155 217 388 288 105 65 313 114 181 347 457 406)(3 253 115 407 389 314 458 218 66 348 156 106 182)(4 390 67 183 408 219 107 116 459 157 254 315 349)(5 409 460 350 184 117 316 68 108 255 391 220 158)(6 185 73 159 351 69 221 461 317 392 410 118 256)(7 352 318 257 160 462 119 74 222 411 186 70 393)(8 161 223 394 258 75 71 319 120 187 353 463 412)(9 259 121 413 395 320 464 224 72 354 162 76 188)(10 396 37 189 414 225 77 122 465 163 260 321 355)(11 415 466 356 190 123 322 38 78 261 361 226 164)(12 191 79 165 357 39 227 467 323 362 416 124 262)(13 358 324 263 166 468 125 80 228 417 192 40 363)(14 167 229 364 264 81 41 289 126 193 359 433 418)(15 265 127 419 365 290 434 230 42 360 168 82 194)(16 366 43 195 420 231 83 128 435 169 266 291 325)(17 421 436 326 196 129 292 44 84 267 367 232 170)(18 197 85 171 327 45 233 437 293 368 422 130 268)(19 328 294 269 172 438 131 86 234 423 198 46 369)(20 173 235 370 270 87 47 295 132 199 329 439 424)(21 271 133 425 371 296 440 236 48 330 174 88 200)(22 372 49 201 426 237 89 134 441 175 272 297 331)(23 427 442 332 202 135 298 50 90 273 373 238 176)(24 203 91 177 333 51 239 443 299 374 428 136 274)(25 334 300 275 178 444 137 92 240 429 204 52 375)(26 179 241 376 276 93 53 301 138 205 335 445 430)(27 277 139 431 377 302 446 242 54 336 180 94 206)(28 378 55 207 432 243 95 140 447 145 278 303 337)(29 397 448 338 208 141 304 56 96 279 379 244 146)(30 209 97 147 339 57 245 449 305 380 398 142 280)(31 340 306 281 148 450 143 98 246 399 210 58 381)(32 149 247 382 282 99 59 307 144 211 341 451 400)(33 283 109 401 383 308 452 248 60 342 150 100 212)(34 384 61 213 402 249 101 110 453 151 284 309 343)(35 403 454 344 214 111 310 62 102 285 385 250 152)(36 215 103 153 345 63 251 455 311 386 404 112 286)
(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)
G:=sub<Sym(468)| (1,346,312,287,154,456,113,104,252,405,216,64,387)(2,155,217,388,288,105,65,313,114,181,347,457,406)(3,253,115,407,389,314,458,218,66,348,156,106,182)(4,390,67,183,408,219,107,116,459,157,254,315,349)(5,409,460,350,184,117,316,68,108,255,391,220,158)(6,185,73,159,351,69,221,461,317,392,410,118,256)(7,352,318,257,160,462,119,74,222,411,186,70,393)(8,161,223,394,258,75,71,319,120,187,353,463,412)(9,259,121,413,395,320,464,224,72,354,162,76,188)(10,396,37,189,414,225,77,122,465,163,260,321,355)(11,415,466,356,190,123,322,38,78,261,361,226,164)(12,191,79,165,357,39,227,467,323,362,416,124,262)(13,358,324,263,166,468,125,80,228,417,192,40,363)(14,167,229,364,264,81,41,289,126,193,359,433,418)(15,265,127,419,365,290,434,230,42,360,168,82,194)(16,366,43,195,420,231,83,128,435,169,266,291,325)(17,421,436,326,196,129,292,44,84,267,367,232,170)(18,197,85,171,327,45,233,437,293,368,422,130,268)(19,328,294,269,172,438,131,86,234,423,198,46,369)(20,173,235,370,270,87,47,295,132,199,329,439,424)(21,271,133,425,371,296,440,236,48,330,174,88,200)(22,372,49,201,426,237,89,134,441,175,272,297,331)(23,427,442,332,202,135,298,50,90,273,373,238,176)(24,203,91,177,333,51,239,443,299,374,428,136,274)(25,334,300,275,178,444,137,92,240,429,204,52,375)(26,179,241,376,276,93,53,301,138,205,335,445,430)(27,277,139,431,377,302,446,242,54,336,180,94,206)(28,378,55,207,432,243,95,140,447,145,278,303,337)(29,397,448,338,208,141,304,56,96,279,379,244,146)(30,209,97,147,339,57,245,449,305,380,398,142,280)(31,340,306,281,148,450,143,98,246,399,210,58,381)(32,149,247,382,282,99,59,307,144,211,341,451,400)(33,283,109,401,383,308,452,248,60,342,150,100,212)(34,384,61,213,402,249,101,110,453,151,284,309,343)(35,403,454,344,214,111,310,62,102,285,385,250,152)(36,215,103,153,345,63,251,455,311,386,404,112,286), (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)>;
G:=Group( (1,346,312,287,154,456,113,104,252,405,216,64,387)(2,155,217,388,288,105,65,313,114,181,347,457,406)(3,253,115,407,389,314,458,218,66,348,156,106,182)(4,390,67,183,408,219,107,116,459,157,254,315,349)(5,409,460,350,184,117,316,68,108,255,391,220,158)(6,185,73,159,351,69,221,461,317,392,410,118,256)(7,352,318,257,160,462,119,74,222,411,186,70,393)(8,161,223,394,258,75,71,319,120,187,353,463,412)(9,259,121,413,395,320,464,224,72,354,162,76,188)(10,396,37,189,414,225,77,122,465,163,260,321,355)(11,415,466,356,190,123,322,38,78,261,361,226,164)(12,191,79,165,357,39,227,467,323,362,416,124,262)(13,358,324,263,166,468,125,80,228,417,192,40,363)(14,167,229,364,264,81,41,289,126,193,359,433,418)(15,265,127,419,365,290,434,230,42,360,168,82,194)(16,366,43,195,420,231,83,128,435,169,266,291,325)(17,421,436,326,196,129,292,44,84,267,367,232,170)(18,197,85,171,327,45,233,437,293,368,422,130,268)(19,328,294,269,172,438,131,86,234,423,198,46,369)(20,173,235,370,270,87,47,295,132,199,329,439,424)(21,271,133,425,371,296,440,236,48,330,174,88,200)(22,372,49,201,426,237,89,134,441,175,272,297,331)(23,427,442,332,202,135,298,50,90,273,373,238,176)(24,203,91,177,333,51,239,443,299,374,428,136,274)(25,334,300,275,178,444,137,92,240,429,204,52,375)(26,179,241,376,276,93,53,301,138,205,335,445,430)(27,277,139,431,377,302,446,242,54,336,180,94,206)(28,378,55,207,432,243,95,140,447,145,278,303,337)(29,397,448,338,208,141,304,56,96,279,379,244,146)(30,209,97,147,339,57,245,449,305,380,398,142,280)(31,340,306,281,148,450,143,98,246,399,210,58,381)(32,149,247,382,282,99,59,307,144,211,341,451,400)(33,283,109,401,383,308,452,248,60,342,150,100,212)(34,384,61,213,402,249,101,110,453,151,284,309,343)(35,403,454,344,214,111,310,62,102,285,385,250,152)(36,215,103,153,345,63,251,455,311,386,404,112,286), (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) );
G=PermutationGroup([[(1,346,312,287,154,456,113,104,252,405,216,64,387),(2,155,217,388,288,105,65,313,114,181,347,457,406),(3,253,115,407,389,314,458,218,66,348,156,106,182),(4,390,67,183,408,219,107,116,459,157,254,315,349),(5,409,460,350,184,117,316,68,108,255,391,220,158),(6,185,73,159,351,69,221,461,317,392,410,118,256),(7,352,318,257,160,462,119,74,222,411,186,70,393),(8,161,223,394,258,75,71,319,120,187,353,463,412),(9,259,121,413,395,320,464,224,72,354,162,76,188),(10,396,37,189,414,225,77,122,465,163,260,321,355),(11,415,466,356,190,123,322,38,78,261,361,226,164),(12,191,79,165,357,39,227,467,323,362,416,124,262),(13,358,324,263,166,468,125,80,228,417,192,40,363),(14,167,229,364,264,81,41,289,126,193,359,433,418),(15,265,127,419,365,290,434,230,42,360,168,82,194),(16,366,43,195,420,231,83,128,435,169,266,291,325),(17,421,436,326,196,129,292,44,84,267,367,232,170),(18,197,85,171,327,45,233,437,293,368,422,130,268),(19,328,294,269,172,438,131,86,234,423,198,46,369),(20,173,235,370,270,87,47,295,132,199,329,439,424),(21,271,133,425,371,296,440,236,48,330,174,88,200),(22,372,49,201,426,237,89,134,441,175,272,297,331),(23,427,442,332,202,135,298,50,90,273,373,238,176),(24,203,91,177,333,51,239,443,299,374,428,136,274),(25,334,300,275,178,444,137,92,240,429,204,52,375),(26,179,241,376,276,93,53,301,138,205,335,445,430),(27,277,139,431,377,302,446,242,54,336,180,94,206),(28,378,55,207,432,243,95,140,447,145,278,303,337),(29,397,448,338,208,141,304,56,96,279,379,244,146),(30,209,97,147,339,57,245,449,305,380,398,142,280),(31,340,306,281,148,450,143,98,246,399,210,58,381),(32,149,247,382,282,99,59,307,144,211,341,451,400),(33,283,109,401,383,308,452,248,60,342,150,100,212),(34,384,61,213,402,249,101,110,453,151,284,309,343),(35,403,454,344,214,111,310,62,102,285,385,250,152),(36,215,103,153,345,63,251,455,311,386,404,112,286)], [(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)]])
48 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 13A | 13B | 18A | ··· | 18F | 26A | 26B | 36A | ··· | 36L | 39A | 39B | 39C | 39D | 78A | 78B | 78C | 78D |
| 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 | 39 | 39 | 78 | 78 | 78 | 78 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 1 | 1 | 13 | ··· | 13 | 13 | 13 | 13 | 13 | 6 | 6 | 13 | ··· | 13 | 6 | 6 | 13 | ··· | 13 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | |||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | C13⋊C6 | C26.C6 | C13⋊C18 | C13⋊2C36 |
| kernel | C13⋊2C36 | C2×C13⋊C9 | C3×Dic13 | C13⋊C9 | C78 | Dic13 | C39 | C26 | C13 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 2 | 2 | 4 | 4 |
Matrix representation of C13⋊2C36 ►in GL7(𝔽937)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 936 | 1 | 0 | 0 | 0 | 0 |
| 0 | 936 | 0 | 1 | 0 | 0 | 0 |
| 0 | 936 | 0 | 0 | 1 | 0 | 0 |
| 0 | 936 | 0 | 0 | 0 | 1 | 0 |
| 0 | 936 | 0 | 0 | 0 | 0 | 1 |
| 0 | 660 | 277 | 659 | 278 | 660 | 276 |
| 674 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 528 | 245 | 144 | 279 | 787 | 661 |
| 0 | 795 | 125 | 382 | 227 | 761 | 343 |
| 0 | 159 | 629 | 220 | 424 | 757 | 864 |
| 0 | 58 | 349 | 923 | 620 | 302 | 94 |
| 0 | 639 | 834 | 624 | 830 | 696 | 105 |
| 0 | 358 | 100 | 456 | 674 | 12 | 622 |
G:=sub<GL(7,GF(937))| [1,0,0,0,0,0,0,0,936,936,936,936,936,660,0,1,0,0,0,0,277,0,0,1,0,0,0,659,0,0,0,1,0,0,278,0,0,0,0,1,0,660,0,0,0,0,0,1,276],[674,0,0,0,0,0,0,0,528,795,159,58,639,358,0,245,125,629,349,834,100,0,144,382,220,923,624,456,0,279,227,424,620,830,674,0,787,761,757,302,696,12,0,661,343,864,94,105,622] >;
C13⋊2C36 in GAP, Magma, Sage, TeX
C_{13}\rtimes_2C_{36} % in TeX
G:=Group("C13:2C36"); // GroupNames label
G:=SmallGroup(468,1);
// by ID
G=gap.SmallGroup(468,1);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-13,30,66,10804,1359]);
// Polycyclic
G:=Group<a,b|a^13=b^36=1,b*a*b^-1=a^4>;
// generators/relations
Export