direct product, abelian, monomial
Aliases: C12×C36, SmallGroup(432,200)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C12×C36 |
| C1 — C12×C36 |
| C1 — C12×C36 |
Generators and relations for C12×C36
G = < a,b | a12=b36=1, ab=ba >
Subgroups: 150, all normal (12 characteristic)
C1, C2, C3, C3, C4, C22, C6, C2×C4, C9, C32, C12, C2×C6, C2×C6, C42, C18, C3×C6, C2×C12, C3×C9, C36, C2×C18, C3×C12, C62, C4×C12, C4×C12, C3×C18, C2×C36, C6×C12, C3×C36, C6×C18, C4×C36, C122, C6×C36, C12×C36
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C9, C32, C12, C2×C6, C42, C18, C3×C6, C2×C12, C3×C9, C36, C2×C18, C3×C12, C62, C4×C12, C3×C18, C2×C36, C6×C12, C3×C36, C6×C18, C4×C36, C122, C6×C36, C12×C36
►Regular action on 432 points
Generators in S432
(1 268 392 225 299 347 76 196 410 137 153 48)(2 269 393 226 300 348 77 197 411 138 154 49)(3 270 394 227 301 349 78 198 412 139 155 50)(4 271 395 228 302 350 79 199 413 140 156 51)(5 272 396 229 303 351 80 200 414 141 157 52)(6 273 361 230 304 352 81 201 415 142 158 53)(7 274 362 231 305 353 82 202 416 143 159 54)(8 275 363 232 306 354 83 203 417 144 160 55)(9 276 364 233 307 355 84 204 418 109 161 56)(10 277 365 234 308 356 85 205 419 110 162 57)(11 278 366 235 309 357 86 206 420 111 163 58)(12 279 367 236 310 358 87 207 421 112 164 59)(13 280 368 237 311 359 88 208 422 113 165 60)(14 281 369 238 312 360 89 209 423 114 166 61)(15 282 370 239 313 325 90 210 424 115 167 62)(16 283 371 240 314 326 91 211 425 116 168 63)(17 284 372 241 315 327 92 212 426 117 169 64)(18 285 373 242 316 328 93 213 427 118 170 65)(19 286 374 243 317 329 94 214 428 119 171 66)(20 287 375 244 318 330 95 215 429 120 172 67)(21 288 376 245 319 331 96 216 430 121 173 68)(22 253 377 246 320 332 97 181 431 122 174 69)(23 254 378 247 321 333 98 182 432 123 175 70)(24 255 379 248 322 334 99 183 397 124 176 71)(25 256 380 249 323 335 100 184 398 125 177 72)(26 257 381 250 324 336 101 185 399 126 178 37)(27 258 382 251 289 337 102 186 400 127 179 38)(28 259 383 252 290 338 103 187 401 128 180 39)(29 260 384 217 291 339 104 188 402 129 145 40)(30 261 385 218 292 340 105 189 403 130 146 41)(31 262 386 219 293 341 106 190 404 131 147 42)(32 263 387 220 294 342 107 191 405 132 148 43)(33 264 388 221 295 343 108 192 406 133 149 44)(34 265 389 222 296 344 73 193 407 134 150 45)(35 266 390 223 297 345 74 194 408 135 151 46)(36 267 391 224 298 346 75 195 409 136 152 47)
(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)
G:=sub<Sym(432)| (1,268,392,225,299,347,76,196,410,137,153,48)(2,269,393,226,300,348,77,197,411,138,154,49)(3,270,394,227,301,349,78,198,412,139,155,50)(4,271,395,228,302,350,79,199,413,140,156,51)(5,272,396,229,303,351,80,200,414,141,157,52)(6,273,361,230,304,352,81,201,415,142,158,53)(7,274,362,231,305,353,82,202,416,143,159,54)(8,275,363,232,306,354,83,203,417,144,160,55)(9,276,364,233,307,355,84,204,418,109,161,56)(10,277,365,234,308,356,85,205,419,110,162,57)(11,278,366,235,309,357,86,206,420,111,163,58)(12,279,367,236,310,358,87,207,421,112,164,59)(13,280,368,237,311,359,88,208,422,113,165,60)(14,281,369,238,312,360,89,209,423,114,166,61)(15,282,370,239,313,325,90,210,424,115,167,62)(16,283,371,240,314,326,91,211,425,116,168,63)(17,284,372,241,315,327,92,212,426,117,169,64)(18,285,373,242,316,328,93,213,427,118,170,65)(19,286,374,243,317,329,94,214,428,119,171,66)(20,287,375,244,318,330,95,215,429,120,172,67)(21,288,376,245,319,331,96,216,430,121,173,68)(22,253,377,246,320,332,97,181,431,122,174,69)(23,254,378,247,321,333,98,182,432,123,175,70)(24,255,379,248,322,334,99,183,397,124,176,71)(25,256,380,249,323,335,100,184,398,125,177,72)(26,257,381,250,324,336,101,185,399,126,178,37)(27,258,382,251,289,337,102,186,400,127,179,38)(28,259,383,252,290,338,103,187,401,128,180,39)(29,260,384,217,291,339,104,188,402,129,145,40)(30,261,385,218,292,340,105,189,403,130,146,41)(31,262,386,219,293,341,106,190,404,131,147,42)(32,263,387,220,294,342,107,191,405,132,148,43)(33,264,388,221,295,343,108,192,406,133,149,44)(34,265,389,222,296,344,73,193,407,134,150,45)(35,266,390,223,297,345,74,194,408,135,151,46)(36,267,391,224,298,346,75,195,409,136,152,47), (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)>;
G:=Group( (1,268,392,225,299,347,76,196,410,137,153,48)(2,269,393,226,300,348,77,197,411,138,154,49)(3,270,394,227,301,349,78,198,412,139,155,50)(4,271,395,228,302,350,79,199,413,140,156,51)(5,272,396,229,303,351,80,200,414,141,157,52)(6,273,361,230,304,352,81,201,415,142,158,53)(7,274,362,231,305,353,82,202,416,143,159,54)(8,275,363,232,306,354,83,203,417,144,160,55)(9,276,364,233,307,355,84,204,418,109,161,56)(10,277,365,234,308,356,85,205,419,110,162,57)(11,278,366,235,309,357,86,206,420,111,163,58)(12,279,367,236,310,358,87,207,421,112,164,59)(13,280,368,237,311,359,88,208,422,113,165,60)(14,281,369,238,312,360,89,209,423,114,166,61)(15,282,370,239,313,325,90,210,424,115,167,62)(16,283,371,240,314,326,91,211,425,116,168,63)(17,284,372,241,315,327,92,212,426,117,169,64)(18,285,373,242,316,328,93,213,427,118,170,65)(19,286,374,243,317,329,94,214,428,119,171,66)(20,287,375,244,318,330,95,215,429,120,172,67)(21,288,376,245,319,331,96,216,430,121,173,68)(22,253,377,246,320,332,97,181,431,122,174,69)(23,254,378,247,321,333,98,182,432,123,175,70)(24,255,379,248,322,334,99,183,397,124,176,71)(25,256,380,249,323,335,100,184,398,125,177,72)(26,257,381,250,324,336,101,185,399,126,178,37)(27,258,382,251,289,337,102,186,400,127,179,38)(28,259,383,252,290,338,103,187,401,128,180,39)(29,260,384,217,291,339,104,188,402,129,145,40)(30,261,385,218,292,340,105,189,403,130,146,41)(31,262,386,219,293,341,106,190,404,131,147,42)(32,263,387,220,294,342,107,191,405,132,148,43)(33,264,388,221,295,343,108,192,406,133,149,44)(34,265,389,222,296,344,73,193,407,134,150,45)(35,266,390,223,297,345,74,194,408,135,151,46)(36,267,391,224,298,346,75,195,409,136,152,47), (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) );
G=PermutationGroup([[(1,268,392,225,299,347,76,196,410,137,153,48),(2,269,393,226,300,348,77,197,411,138,154,49),(3,270,394,227,301,349,78,198,412,139,155,50),(4,271,395,228,302,350,79,199,413,140,156,51),(5,272,396,229,303,351,80,200,414,141,157,52),(6,273,361,230,304,352,81,201,415,142,158,53),(7,274,362,231,305,353,82,202,416,143,159,54),(8,275,363,232,306,354,83,203,417,144,160,55),(9,276,364,233,307,355,84,204,418,109,161,56),(10,277,365,234,308,356,85,205,419,110,162,57),(11,278,366,235,309,357,86,206,420,111,163,58),(12,279,367,236,310,358,87,207,421,112,164,59),(13,280,368,237,311,359,88,208,422,113,165,60),(14,281,369,238,312,360,89,209,423,114,166,61),(15,282,370,239,313,325,90,210,424,115,167,62),(16,283,371,240,314,326,91,211,425,116,168,63),(17,284,372,241,315,327,92,212,426,117,169,64),(18,285,373,242,316,328,93,213,427,118,170,65),(19,286,374,243,317,329,94,214,428,119,171,66),(20,287,375,244,318,330,95,215,429,120,172,67),(21,288,376,245,319,331,96,216,430,121,173,68),(22,253,377,246,320,332,97,181,431,122,174,69),(23,254,378,247,321,333,98,182,432,123,175,70),(24,255,379,248,322,334,99,183,397,124,176,71),(25,256,380,249,323,335,100,184,398,125,177,72),(26,257,381,250,324,336,101,185,399,126,178,37),(27,258,382,251,289,337,102,186,400,127,179,38),(28,259,383,252,290,338,103,187,401,128,180,39),(29,260,384,217,291,339,104,188,402,129,145,40),(30,261,385,218,292,340,105,189,403,130,146,41),(31,262,386,219,293,341,106,190,404,131,147,42),(32,263,387,220,294,342,107,191,405,132,148,43),(33,264,388,221,295,343,108,192,406,133,149,44),(34,265,389,222,296,344,73,193,407,134,150,45),(35,266,390,223,297,345,74,194,408,135,151,46),(36,267,391,224,298,346,75,195,409,136,152,47)], [(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)]])
432 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | ··· | 4L | 6A | ··· | 6X | 9A | ··· | 9R | 12A | ··· | 12CR | 18A | ··· | 18BB | 36A | ··· | 36HH |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
432 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C9 | C12 | C12 | C18 | C36 |
| kernel | C12×C36 | C6×C36 | C4×C36 | C122 | C3×C36 | C2×C36 | C6×C12 | C4×C12 | C36 | C3×C12 | C2×C12 | C12 |
| # reps | 1 | 3 | 6 | 2 | 12 | 18 | 6 | 18 | 72 | 24 | 54 | 216 |
Matrix representation of C12×C36 ►in GL3(𝔽37) generated by
| 8 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 29 |
| 21 | 0 | 0 |
| 0 | 23 | 0 |
| 0 | 0 | 27 |
G:=sub<GL(3,GF(37))| [8,0,0,0,6,0,0,0,29],[21,0,0,0,23,0,0,0,27] >;
C12×C36 in GAP, Magma, Sage, TeX
C_{12}\times C_{36} % in TeX
G:=Group("C12xC36"); // GroupNames label
G:=SmallGroup(432,200);
// by ID
G=gap.SmallGroup(432,200);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,512,772]);
// Polycyclic
G:=Group<a,b|a^12=b^36=1,a*b=b*a>;
// generators/relations