direct product, abelian, monomial, 3-elementary
Aliases: C3×C132, SmallGroup(396,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C132 |
| C1 — C3×C132 |
| C1 — C3×C132 |
Generators and relations for C3×C132
G = < a,b | a3=b132=1, ab=ba >
Smallest permutation representation of C3×C132
►Regular action on 396 points
Generators in S396
(1 148 340)(2 149 341)(3 150 342)(4 151 343)(5 152 344)(6 153 345)(7 154 346)(8 155 347)(9 156 348)(10 157 349)(11 158 350)(12 159 351)(13 160 352)(14 161 353)(15 162 354)(16 163 355)(17 164 356)(18 165 357)(19 166 358)(20 167 359)(21 168 360)(22 169 361)(23 170 362)(24 171 363)(25 172 364)(26 173 365)(27 174 366)(28 175 367)(29 176 368)(30 177 369)(31 178 370)(32 179 371)(33 180 372)(34 181 373)(35 182 374)(36 183 375)(37 184 376)(38 185 377)(39 186 378)(40 187 379)(41 188 380)(42 189 381)(43 190 382)(44 191 383)(45 192 384)(46 193 385)(47 194 386)(48 195 387)(49 196 388)(50 197 389)(51 198 390)(52 199 391)(53 200 392)(54 201 393)(55 202 394)(56 203 395)(57 204 396)(58 205 265)(59 206 266)(60 207 267)(61 208 268)(62 209 269)(63 210 270)(64 211 271)(65 212 272)(66 213 273)(67 214 274)(68 215 275)(69 216 276)(70 217 277)(71 218 278)(72 219 279)(73 220 280)(74 221 281)(75 222 282)(76 223 283)(77 224 284)(78 225 285)(79 226 286)(80 227 287)(81 228 288)(82 229 289)(83 230 290)(84 231 291)(85 232 292)(86 233 293)(87 234 294)(88 235 295)(89 236 296)(90 237 297)(91 238 298)(92 239 299)(93 240 300)(94 241 301)(95 242 302)(96 243 303)(97 244 304)(98 245 305)(99 246 306)(100 247 307)(101 248 308)(102 249 309)(103 250 310)(104 251 311)(105 252 312)(106 253 313)(107 254 314)(108 255 315)(109 256 316)(110 257 317)(111 258 318)(112 259 319)(113 260 320)(114 261 321)(115 262 322)(116 263 323)(117 264 324)(118 133 325)(119 134 326)(120 135 327)(121 136 328)(122 137 329)(123 138 330)(124 139 331)(125 140 332)(126 141 333)(127 142 334)(128 143 335)(129 144 336)(130 145 337)(131 146 338)(132 147 339)
(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)
G:=sub<Sym(396)| (1,148,340)(2,149,341)(3,150,342)(4,151,343)(5,152,344)(6,153,345)(7,154,346)(8,155,347)(9,156,348)(10,157,349)(11,158,350)(12,159,351)(13,160,352)(14,161,353)(15,162,354)(16,163,355)(17,164,356)(18,165,357)(19,166,358)(20,167,359)(21,168,360)(22,169,361)(23,170,362)(24,171,363)(25,172,364)(26,173,365)(27,174,366)(28,175,367)(29,176,368)(30,177,369)(31,178,370)(32,179,371)(33,180,372)(34,181,373)(35,182,374)(36,183,375)(37,184,376)(38,185,377)(39,186,378)(40,187,379)(41,188,380)(42,189,381)(43,190,382)(44,191,383)(45,192,384)(46,193,385)(47,194,386)(48,195,387)(49,196,388)(50,197,389)(51,198,390)(52,199,391)(53,200,392)(54,201,393)(55,202,394)(56,203,395)(57,204,396)(58,205,265)(59,206,266)(60,207,267)(61,208,268)(62,209,269)(63,210,270)(64,211,271)(65,212,272)(66,213,273)(67,214,274)(68,215,275)(69,216,276)(70,217,277)(71,218,278)(72,219,279)(73,220,280)(74,221,281)(75,222,282)(76,223,283)(77,224,284)(78,225,285)(79,226,286)(80,227,287)(81,228,288)(82,229,289)(83,230,290)(84,231,291)(85,232,292)(86,233,293)(87,234,294)(88,235,295)(89,236,296)(90,237,297)(91,238,298)(92,239,299)(93,240,300)(94,241,301)(95,242,302)(96,243,303)(97,244,304)(98,245,305)(99,246,306)(100,247,307)(101,248,308)(102,249,309)(103,250,310)(104,251,311)(105,252,312)(106,253,313)(107,254,314)(108,255,315)(109,256,316)(110,257,317)(111,258,318)(112,259,319)(113,260,320)(114,261,321)(115,262,322)(116,263,323)(117,264,324)(118,133,325)(119,134,326)(120,135,327)(121,136,328)(122,137,329)(123,138,330)(124,139,331)(125,140,332)(126,141,333)(127,142,334)(128,143,335)(129,144,336)(130,145,337)(131,146,338)(132,147,339), (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)>;
G:=Group( (1,148,340)(2,149,341)(3,150,342)(4,151,343)(5,152,344)(6,153,345)(7,154,346)(8,155,347)(9,156,348)(10,157,349)(11,158,350)(12,159,351)(13,160,352)(14,161,353)(15,162,354)(16,163,355)(17,164,356)(18,165,357)(19,166,358)(20,167,359)(21,168,360)(22,169,361)(23,170,362)(24,171,363)(25,172,364)(26,173,365)(27,174,366)(28,175,367)(29,176,368)(30,177,369)(31,178,370)(32,179,371)(33,180,372)(34,181,373)(35,182,374)(36,183,375)(37,184,376)(38,185,377)(39,186,378)(40,187,379)(41,188,380)(42,189,381)(43,190,382)(44,191,383)(45,192,384)(46,193,385)(47,194,386)(48,195,387)(49,196,388)(50,197,389)(51,198,390)(52,199,391)(53,200,392)(54,201,393)(55,202,394)(56,203,395)(57,204,396)(58,205,265)(59,206,266)(60,207,267)(61,208,268)(62,209,269)(63,210,270)(64,211,271)(65,212,272)(66,213,273)(67,214,274)(68,215,275)(69,216,276)(70,217,277)(71,218,278)(72,219,279)(73,220,280)(74,221,281)(75,222,282)(76,223,283)(77,224,284)(78,225,285)(79,226,286)(80,227,287)(81,228,288)(82,229,289)(83,230,290)(84,231,291)(85,232,292)(86,233,293)(87,234,294)(88,235,295)(89,236,296)(90,237,297)(91,238,298)(92,239,299)(93,240,300)(94,241,301)(95,242,302)(96,243,303)(97,244,304)(98,245,305)(99,246,306)(100,247,307)(101,248,308)(102,249,309)(103,250,310)(104,251,311)(105,252,312)(106,253,313)(107,254,314)(108,255,315)(109,256,316)(110,257,317)(111,258,318)(112,259,319)(113,260,320)(114,261,321)(115,262,322)(116,263,323)(117,264,324)(118,133,325)(119,134,326)(120,135,327)(121,136,328)(122,137,329)(123,138,330)(124,139,331)(125,140,332)(126,141,333)(127,142,334)(128,143,335)(129,144,336)(130,145,337)(131,146,338)(132,147,339), (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) );
G=PermutationGroup([[(1,148,340),(2,149,341),(3,150,342),(4,151,343),(5,152,344),(6,153,345),(7,154,346),(8,155,347),(9,156,348),(10,157,349),(11,158,350),(12,159,351),(13,160,352),(14,161,353),(15,162,354),(16,163,355),(17,164,356),(18,165,357),(19,166,358),(20,167,359),(21,168,360),(22,169,361),(23,170,362),(24,171,363),(25,172,364),(26,173,365),(27,174,366),(28,175,367),(29,176,368),(30,177,369),(31,178,370),(32,179,371),(33,180,372),(34,181,373),(35,182,374),(36,183,375),(37,184,376),(38,185,377),(39,186,378),(40,187,379),(41,188,380),(42,189,381),(43,190,382),(44,191,383),(45,192,384),(46,193,385),(47,194,386),(48,195,387),(49,196,388),(50,197,389),(51,198,390),(52,199,391),(53,200,392),(54,201,393),(55,202,394),(56,203,395),(57,204,396),(58,205,265),(59,206,266),(60,207,267),(61,208,268),(62,209,269),(63,210,270),(64,211,271),(65,212,272),(66,213,273),(67,214,274),(68,215,275),(69,216,276),(70,217,277),(71,218,278),(72,219,279),(73,220,280),(74,221,281),(75,222,282),(76,223,283),(77,224,284),(78,225,285),(79,226,286),(80,227,287),(81,228,288),(82,229,289),(83,230,290),(84,231,291),(85,232,292),(86,233,293),(87,234,294),(88,235,295),(89,236,296),(90,237,297),(91,238,298),(92,239,299),(93,240,300),(94,241,301),(95,242,302),(96,243,303),(97,244,304),(98,245,305),(99,246,306),(100,247,307),(101,248,308),(102,249,309),(103,250,310),(104,251,311),(105,252,312),(106,253,313),(107,254,314),(108,255,315),(109,256,316),(110,257,317),(111,258,318),(112,259,319),(113,260,320),(114,261,321),(115,262,322),(116,263,323),(117,264,324),(118,133,325),(119,134,326),(120,135,327),(121,136,328),(122,137,329),(123,138,330),(124,139,331),(125,140,332),(126,141,333),(127,142,334),(128,143,335),(129,144,336),(130,145,337),(131,146,338),(132,147,339)], [(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)]])
396 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 11A | ··· | 11J | 12A | ··· | 12P | 22A | ··· | 22J | 33A | ··· | 33CB | 44A | ··· | 44T | 66A | ··· | 66CB | 132A | ··· | 132FD |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 11 | ··· | 11 | 12 | ··· | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
396 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C11 | C12 | C22 | C33 | C44 | C66 | C132 |
| kernel | C3×C132 | C3×C66 | C132 | C3×C33 | C66 | C3×C12 | C33 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 10 | 16 | 10 | 80 | 20 | 80 | 160 |
Matrix representation of C3×C132 ►in GL2(𝔽397) generated by
| 1 | 0 |
| 0 | 34 |
| 212 | 0 |
| 0 | 389 |
G:=sub<GL(2,GF(397))| [1,0,0,34],[212,0,0,389] >;
C3×C132 in GAP, Magma, Sage, TeX
C_3\times C_{132} % in TeX
G:=Group("C3xC132"); // GroupNames label
G:=SmallGroup(396,16);
// by ID
G=gap.SmallGroup(396,16);
# by ID
G:=PCGroup([5,-2,-3,-3,-11,-2,990]);
// Polycyclic
G:=Group<a,b|a^3=b^132=1,a*b=b*a>;
// generators/relations
Export