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G = C12×C36order 432 = 24·33

Abelian group of type [12,36]

direct product, abelian, monomial

Aliases: C12×C36, SmallGroup(432,200)

Series: Derived Chief Lower central Upper central

C1 — C12×C36
C1C3C6C2×C6C62C6×C18C6×C36 — 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 [×3], C3, C3 [×3], C4 [×6], C22, C6 [×12], C2×C4 [×3], C9 [×3], C32, C12 [×24], C2×C6, C2×C6 [×3], C42, C18 [×9], C3×C6 [×3], C2×C12 [×12], C3×C9, C36 [×18], C2×C18 [×3], C3×C12 [×6], C62, C4×C12, C4×C12 [×3], C3×C18 [×3], C2×C36 [×9], C6×C12 [×3], C3×C36 [×6], C6×C18, C4×C36 [×3], C122, C6×C36 [×3], C12×C36
Quotients: C1, C2 [×3], C3 [×4], C4 [×6], C22, C6 [×12], C2×C4 [×3], C9 [×3], C32, C12 [×24], C2×C6 [×4], C42, C18 [×9], C3×C6 [×3], C2×C12 [×12], C3×C9, C36 [×18], C2×C18 [×3], C3×C12 [×6], C62, C4×C12 [×4], C3×C18 [×3], C2×C36 [×9], C6×C12 [×3], C3×C36 [×6], C6×C18, C4×C36 [×3], C122, C6×C36 [×3], C12×C36

Smallest permutation representation of C12×C36
Regular action on 432 points
Generators in S432
(1 316 367 238 59 80 215 355 118 413 148 284)(2 317 368 239 60 81 216 356 119 414 149 285)(3 318 369 240 61 82 181 357 120 415 150 286)(4 319 370 241 62 83 182 358 121 416 151 287)(5 320 371 242 63 84 183 359 122 417 152 288)(6 321 372 243 64 85 184 360 123 418 153 253)(7 322 373 244 65 86 185 325 124 419 154 254)(8 323 374 245 66 87 186 326 125 420 155 255)(9 324 375 246 67 88 187 327 126 421 156 256)(10 289 376 247 68 89 188 328 127 422 157 257)(11 290 377 248 69 90 189 329 128 423 158 258)(12 291 378 249 70 91 190 330 129 424 159 259)(13 292 379 250 71 92 191 331 130 425 160 260)(14 293 380 251 72 93 192 332 131 426 161 261)(15 294 381 252 37 94 193 333 132 427 162 262)(16 295 382 217 38 95 194 334 133 428 163 263)(17 296 383 218 39 96 195 335 134 429 164 264)(18 297 384 219 40 97 196 336 135 430 165 265)(19 298 385 220 41 98 197 337 136 431 166 266)(20 299 386 221 42 99 198 338 137 432 167 267)(21 300 387 222 43 100 199 339 138 397 168 268)(22 301 388 223 44 101 200 340 139 398 169 269)(23 302 389 224 45 102 201 341 140 399 170 270)(24 303 390 225 46 103 202 342 141 400 171 271)(25 304 391 226 47 104 203 343 142 401 172 272)(26 305 392 227 48 105 204 344 143 402 173 273)(27 306 393 228 49 106 205 345 144 403 174 274)(28 307 394 229 50 107 206 346 109 404 175 275)(29 308 395 230 51 108 207 347 110 405 176 276)(30 309 396 231 52 73 208 348 111 406 177 277)(31 310 361 232 53 74 209 349 112 407 178 278)(32 311 362 233 54 75 210 350 113 408 179 279)(33 312 363 234 55 76 211 351 114 409 180 280)(34 313 364 235 56 77 212 352 115 410 145 281)(35 314 365 236 57 78 213 353 116 411 146 282)(36 315 366 237 58 79 214 354 117 412 147 283)
(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)| (1,316,367,238,59,80,215,355,118,413,148,284)(2,317,368,239,60,81,216,356,119,414,149,285)(3,318,369,240,61,82,181,357,120,415,150,286)(4,319,370,241,62,83,182,358,121,416,151,287)(5,320,371,242,63,84,183,359,122,417,152,288)(6,321,372,243,64,85,184,360,123,418,153,253)(7,322,373,244,65,86,185,325,124,419,154,254)(8,323,374,245,66,87,186,326,125,420,155,255)(9,324,375,246,67,88,187,327,126,421,156,256)(10,289,376,247,68,89,188,328,127,422,157,257)(11,290,377,248,69,90,189,329,128,423,158,258)(12,291,378,249,70,91,190,330,129,424,159,259)(13,292,379,250,71,92,191,331,130,425,160,260)(14,293,380,251,72,93,192,332,131,426,161,261)(15,294,381,252,37,94,193,333,132,427,162,262)(16,295,382,217,38,95,194,334,133,428,163,263)(17,296,383,218,39,96,195,335,134,429,164,264)(18,297,384,219,40,97,196,336,135,430,165,265)(19,298,385,220,41,98,197,337,136,431,166,266)(20,299,386,221,42,99,198,338,137,432,167,267)(21,300,387,222,43,100,199,339,138,397,168,268)(22,301,388,223,44,101,200,340,139,398,169,269)(23,302,389,224,45,102,201,341,140,399,170,270)(24,303,390,225,46,103,202,342,141,400,171,271)(25,304,391,226,47,104,203,343,142,401,172,272)(26,305,392,227,48,105,204,344,143,402,173,273)(27,306,393,228,49,106,205,345,144,403,174,274)(28,307,394,229,50,107,206,346,109,404,175,275)(29,308,395,230,51,108,207,347,110,405,176,276)(30,309,396,231,52,73,208,348,111,406,177,277)(31,310,361,232,53,74,209,349,112,407,178,278)(32,311,362,233,54,75,210,350,113,408,179,279)(33,312,363,234,55,76,211,351,114,409,180,280)(34,313,364,235,56,77,212,352,115,410,145,281)(35,314,365,236,57,78,213,353,116,411,146,282)(36,315,366,237,58,79,214,354,117,412,147,283), (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( (1,316,367,238,59,80,215,355,118,413,148,284)(2,317,368,239,60,81,216,356,119,414,149,285)(3,318,369,240,61,82,181,357,120,415,150,286)(4,319,370,241,62,83,182,358,121,416,151,287)(5,320,371,242,63,84,183,359,122,417,152,288)(6,321,372,243,64,85,184,360,123,418,153,253)(7,322,373,244,65,86,185,325,124,419,154,254)(8,323,374,245,66,87,186,326,125,420,155,255)(9,324,375,246,67,88,187,327,126,421,156,256)(10,289,376,247,68,89,188,328,127,422,157,257)(11,290,377,248,69,90,189,329,128,423,158,258)(12,291,378,249,70,91,190,330,129,424,159,259)(13,292,379,250,71,92,191,331,130,425,160,260)(14,293,380,251,72,93,192,332,131,426,161,261)(15,294,381,252,37,94,193,333,132,427,162,262)(16,295,382,217,38,95,194,334,133,428,163,263)(17,296,383,218,39,96,195,335,134,429,164,264)(18,297,384,219,40,97,196,336,135,430,165,265)(19,298,385,220,41,98,197,337,136,431,166,266)(20,299,386,221,42,99,198,338,137,432,167,267)(21,300,387,222,43,100,199,339,138,397,168,268)(22,301,388,223,44,101,200,340,139,398,169,269)(23,302,389,224,45,102,201,341,140,399,170,270)(24,303,390,225,46,103,202,342,141,400,171,271)(25,304,391,226,47,104,203,343,142,401,172,272)(26,305,392,227,48,105,204,344,143,402,173,273)(27,306,393,228,49,106,205,345,144,403,174,274)(28,307,394,229,50,107,206,346,109,404,175,275)(29,308,395,230,51,108,207,347,110,405,176,276)(30,309,396,231,52,73,208,348,111,406,177,277)(31,310,361,232,53,74,209,349,112,407,178,278)(32,311,362,233,54,75,210,350,113,408,179,279)(33,312,363,234,55,76,211,351,114,409,180,280)(34,313,364,235,56,77,212,352,115,410,145,281)(35,314,365,236,57,78,213,353,116,411,146,282)(36,315,366,237,58,79,214,354,117,412,147,283), (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([(1,316,367,238,59,80,215,355,118,413,148,284),(2,317,368,239,60,81,216,356,119,414,149,285),(3,318,369,240,61,82,181,357,120,415,150,286),(4,319,370,241,62,83,182,358,121,416,151,287),(5,320,371,242,63,84,183,359,122,417,152,288),(6,321,372,243,64,85,184,360,123,418,153,253),(7,322,373,244,65,86,185,325,124,419,154,254),(8,323,374,245,66,87,186,326,125,420,155,255),(9,324,375,246,67,88,187,327,126,421,156,256),(10,289,376,247,68,89,188,328,127,422,157,257),(11,290,377,248,69,90,189,329,128,423,158,258),(12,291,378,249,70,91,190,330,129,424,159,259),(13,292,379,250,71,92,191,331,130,425,160,260),(14,293,380,251,72,93,192,332,131,426,161,261),(15,294,381,252,37,94,193,333,132,427,162,262),(16,295,382,217,38,95,194,334,133,428,163,263),(17,296,383,218,39,96,195,335,134,429,164,264),(18,297,384,219,40,97,196,336,135,430,165,265),(19,298,385,220,41,98,197,337,136,431,166,266),(20,299,386,221,42,99,198,338,137,432,167,267),(21,300,387,222,43,100,199,339,138,397,168,268),(22,301,388,223,44,101,200,340,139,398,169,269),(23,302,389,224,45,102,201,341,140,399,170,270),(24,303,390,225,46,103,202,342,141,400,171,271),(25,304,391,226,47,104,203,343,142,401,172,272),(26,305,392,227,48,105,204,344,143,402,173,273),(27,306,393,228,49,106,205,345,144,403,174,274),(28,307,394,229,50,107,206,346,109,404,175,275),(29,308,395,230,51,108,207,347,110,405,176,276),(30,309,396,231,52,73,208,348,111,406,177,277),(31,310,361,232,53,74,209,349,112,407,178,278),(32,311,362,233,54,75,210,350,113,408,179,279),(33,312,363,234,55,76,211,351,114,409,180,280),(34,313,364,235,56,77,212,352,115,410,145,281),(35,314,365,236,57,78,213,353,116,411,146,282),(36,315,366,237,58,79,214,354,117,412,147,283)], [(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 2A2B2C3A···3H4A···4L6A···6X9A···9R12A···12CR18A···18BB36A···36HH
order12223···34···46···69···912···1218···1836···36
size11111···11···11···11···11···11···11···1

432 irreducible representations

dim111111111111
type++
imageC1C2C3C3C4C6C6C9C12C12C18C36
kernelC12×C36C6×C36C4×C36C122C3×C36C2×C36C6×C12C4×C12C36C3×C12C2×C12C12
# reps13621218618722454216

Matrix representation of C12×C36 in GL3(𝔽37) generated by

800
060
0029
,
2100
0230
0027
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

׿
×
𝔽