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G = C132C36order 468 = 22·32·13

The semidirect product of C13 and C36 acting via C36/C6=C6

metacyclic, supersoluble, monomial, Z-group

Aliases: C132C36, C26.C18, Dic13⋊C9, C78.2C6, C39.2C12, C13⋊C92C4, 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

C1C13 — C132C36
C1C13C39C78C2×C13⋊C9 — C132C36
C13 — C132C36
C1C6

Generators and relations for C132C36
 G = < a,b | a13=b36=1, bab-1=a4 >

13C4
13C9
13C12
13C18
13C36

Smallest permutation representation of C132C36
Regular action on 468 points
Generators in S468
(1 422 230 109 347 290 283 463 216 45 79 394 162)(2 348 181 163 110 464 395 231 284 80 423 291 46)(3 111 285 47 164 232 292 182 396 424 349 465 81)(4 165 361 82 48 183 466 286 293 350 112 233 425)(5 49 294 426 83 287 234 362 467 113 166 184 351)(6 84 468 352 427 363 185 295 235 167 50 288 114)(7 428 236 115 353 296 253 433 186 51 85 364 168)(8 354 187 169 116 434 365 237 254 86 429 297 52)(9 117 255 53 170 238 298 188 366 430 355 435 87)(10 171 367 88 54 189 436 256 299 356 118 239 431)(11 55 300 432 89 257 240 368 437 119 172 190 357)(12 90 438 358 397 369 191 301 241 173 56 258 120)(13 398 242 121 359 302 259 439 192 57 91 370 174)(14 360 193 175 122 440 371 243 260 92 399 303 58)(15 123 261 59 176 244 304 194 372 400 325 441 93)(16 177 373 94 60 195 442 262 305 326 124 245 401)(17 61 306 402 95 263 246 374 443 125 178 196 327)(18 96 444 328 403 375 197 307 247 179 62 264 126)(19 404 248 127 329 308 265 445 198 63 97 376 180)(20 330 199 145 128 446 377 249 266 98 405 309 64)(21 129 267 65 146 250 310 200 378 406 331 447 99)(22 147 379 100 66 201 448 268 311 332 130 251 407)(23 67 312 408 101 269 252 380 449 131 148 202 333)(24 102 450 334 409 381 203 313 217 149 68 270 132)(25 410 218 133 335 314 271 451 204 69 103 382 150)(26 336 205 151 134 452 383 219 272 104 411 315 70)(27 135 273 71 152 220 316 206 384 412 337 453 105)(28 153 385 106 72 207 454 274 317 338 136 221 413)(29 37 318 414 107 275 222 386 455 137 154 208 339)(30 108 456 340 415 387 209 319 223 155 38 276 138)(31 416 224 139 341 320 277 457 210 39 73 388 156)(32 342 211 157 140 458 389 225 278 74 417 321 40)(33 141 279 41 158 226 322 212 390 418 343 459 75)(34 159 391 76 42 213 460 280 323 344 142 227 419)(35 43 324 420 77 281 228 392 461 143 160 214 345)(36 78 462 346 421 393 215 289 229 161 44 282 144)
(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)| (1,422,230,109,347,290,283,463,216,45,79,394,162)(2,348,181,163,110,464,395,231,284,80,423,291,46)(3,111,285,47,164,232,292,182,396,424,349,465,81)(4,165,361,82,48,183,466,286,293,350,112,233,425)(5,49,294,426,83,287,234,362,467,113,166,184,351)(6,84,468,352,427,363,185,295,235,167,50,288,114)(7,428,236,115,353,296,253,433,186,51,85,364,168)(8,354,187,169,116,434,365,237,254,86,429,297,52)(9,117,255,53,170,238,298,188,366,430,355,435,87)(10,171,367,88,54,189,436,256,299,356,118,239,431)(11,55,300,432,89,257,240,368,437,119,172,190,357)(12,90,438,358,397,369,191,301,241,173,56,258,120)(13,398,242,121,359,302,259,439,192,57,91,370,174)(14,360,193,175,122,440,371,243,260,92,399,303,58)(15,123,261,59,176,244,304,194,372,400,325,441,93)(16,177,373,94,60,195,442,262,305,326,124,245,401)(17,61,306,402,95,263,246,374,443,125,178,196,327)(18,96,444,328,403,375,197,307,247,179,62,264,126)(19,404,248,127,329,308,265,445,198,63,97,376,180)(20,330,199,145,128,446,377,249,266,98,405,309,64)(21,129,267,65,146,250,310,200,378,406,331,447,99)(22,147,379,100,66,201,448,268,311,332,130,251,407)(23,67,312,408,101,269,252,380,449,131,148,202,333)(24,102,450,334,409,381,203,313,217,149,68,270,132)(25,410,218,133,335,314,271,451,204,69,103,382,150)(26,336,205,151,134,452,383,219,272,104,411,315,70)(27,135,273,71,152,220,316,206,384,412,337,453,105)(28,153,385,106,72,207,454,274,317,338,136,221,413)(29,37,318,414,107,275,222,386,455,137,154,208,339)(30,108,456,340,415,387,209,319,223,155,38,276,138)(31,416,224,139,341,320,277,457,210,39,73,388,156)(32,342,211,157,140,458,389,225,278,74,417,321,40)(33,141,279,41,158,226,322,212,390,418,343,459,75)(34,159,391,76,42,213,460,280,323,344,142,227,419)(35,43,324,420,77,281,228,392,461,143,160,214,345)(36,78,462,346,421,393,215,289,229,161,44,282,144), (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( (1,422,230,109,347,290,283,463,216,45,79,394,162)(2,348,181,163,110,464,395,231,284,80,423,291,46)(3,111,285,47,164,232,292,182,396,424,349,465,81)(4,165,361,82,48,183,466,286,293,350,112,233,425)(5,49,294,426,83,287,234,362,467,113,166,184,351)(6,84,468,352,427,363,185,295,235,167,50,288,114)(7,428,236,115,353,296,253,433,186,51,85,364,168)(8,354,187,169,116,434,365,237,254,86,429,297,52)(9,117,255,53,170,238,298,188,366,430,355,435,87)(10,171,367,88,54,189,436,256,299,356,118,239,431)(11,55,300,432,89,257,240,368,437,119,172,190,357)(12,90,438,358,397,369,191,301,241,173,56,258,120)(13,398,242,121,359,302,259,439,192,57,91,370,174)(14,360,193,175,122,440,371,243,260,92,399,303,58)(15,123,261,59,176,244,304,194,372,400,325,441,93)(16,177,373,94,60,195,442,262,305,326,124,245,401)(17,61,306,402,95,263,246,374,443,125,178,196,327)(18,96,444,328,403,375,197,307,247,179,62,264,126)(19,404,248,127,329,308,265,445,198,63,97,376,180)(20,330,199,145,128,446,377,249,266,98,405,309,64)(21,129,267,65,146,250,310,200,378,406,331,447,99)(22,147,379,100,66,201,448,268,311,332,130,251,407)(23,67,312,408,101,269,252,380,449,131,148,202,333)(24,102,450,334,409,381,203,313,217,149,68,270,132)(25,410,218,133,335,314,271,451,204,69,103,382,150)(26,336,205,151,134,452,383,219,272,104,411,315,70)(27,135,273,71,152,220,316,206,384,412,337,453,105)(28,153,385,106,72,207,454,274,317,338,136,221,413)(29,37,318,414,107,275,222,386,455,137,154,208,339)(30,108,456,340,415,387,209,319,223,155,38,276,138)(31,416,224,139,341,320,277,457,210,39,73,388,156)(32,342,211,157,140,458,389,225,278,74,417,321,40)(33,141,279,41,158,226,322,212,390,418,343,459,75)(34,159,391,76,42,213,460,280,323,344,142,227,419)(35,43,324,420,77,281,228,392,461,143,160,214,345)(36,78,462,346,421,393,215,289,229,161,44,282,144), (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([(1,422,230,109,347,290,283,463,216,45,79,394,162),(2,348,181,163,110,464,395,231,284,80,423,291,46),(3,111,285,47,164,232,292,182,396,424,349,465,81),(4,165,361,82,48,183,466,286,293,350,112,233,425),(5,49,294,426,83,287,234,362,467,113,166,184,351),(6,84,468,352,427,363,185,295,235,167,50,288,114),(7,428,236,115,353,296,253,433,186,51,85,364,168),(8,354,187,169,116,434,365,237,254,86,429,297,52),(9,117,255,53,170,238,298,188,366,430,355,435,87),(10,171,367,88,54,189,436,256,299,356,118,239,431),(11,55,300,432,89,257,240,368,437,119,172,190,357),(12,90,438,358,397,369,191,301,241,173,56,258,120),(13,398,242,121,359,302,259,439,192,57,91,370,174),(14,360,193,175,122,440,371,243,260,92,399,303,58),(15,123,261,59,176,244,304,194,372,400,325,441,93),(16,177,373,94,60,195,442,262,305,326,124,245,401),(17,61,306,402,95,263,246,374,443,125,178,196,327),(18,96,444,328,403,375,197,307,247,179,62,264,126),(19,404,248,127,329,308,265,445,198,63,97,376,180),(20,330,199,145,128,446,377,249,266,98,405,309,64),(21,129,267,65,146,250,310,200,378,406,331,447,99),(22,147,379,100,66,201,448,268,311,332,130,251,407),(23,67,312,408,101,269,252,380,449,131,148,202,333),(24,102,450,334,409,381,203,313,217,149,68,270,132),(25,410,218,133,335,314,271,451,204,69,103,382,150),(26,336,205,151,134,452,383,219,272,104,411,315,70),(27,135,273,71,152,220,316,206,384,412,337,453,105),(28,153,385,106,72,207,454,274,317,338,136,221,413),(29,37,318,414,107,275,222,386,455,137,154,208,339),(30,108,456,340,415,387,209,319,223,155,38,276,138),(31,416,224,139,341,320,277,457,210,39,73,388,156),(32,342,211,157,140,458,389,225,278,74,417,321,40),(33,141,279,41,158,226,322,212,390,418,343,459,75),(34,159,391,76,42,213,460,280,323,344,142,227,419),(35,43,324,420,77,281,228,392,461,143,160,214,345),(36,78,462,346,421,393,215,289,229,161,44,282,144)], [(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 3A3B4A4B6A6B9A···9F12A12B12C12D13A13B18A···18F26A26B36A···36L39A39B39C39D78A78B78C78D
order123344669···912121212131318···18262636···363939393978787878
size111113131113···13131313136613···136613···1366666666

48 irreducible representations

dim1111111116666
type+++-
imageC1C2C3C4C6C9C12C18C36C13⋊C6C26.C6C13⋊C18C132C36
kernelC132C36C2×C13⋊C9C3×Dic13C13⋊C9C78Dic13C39C26C13C6C3C2C1
# reps11222646122244

Matrix representation of C132C36 in GL7(𝔽937)

1000000
093610000
093601000
093600100
093600010
093600001
0660277659278660276
,
674000000
0528245144279787661
0795125382227761343
0159629220424757864
05834992362030294
0639834624830696105
035810045667412622

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] >;

C132C36 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

Subgroup lattice of C132C36 in TeX

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