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## G = C3×C132order 396 = 22·32·11

### Abelian group of type [3,132]

Aliases: C3×C132, SmallGroup(396,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C132
 Chief series C1 — C2 — C22 — C66 — C3×C66 — C3×C132
 Lower central C1 — C3×C132
 Upper central 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
(1 134 268)(2 135 269)(3 136 270)(4 137 271)(5 138 272)(6 139 273)(7 140 274)(8 141 275)(9 142 276)(10 143 277)(11 144 278)(12 145 279)(13 146 280)(14 147 281)(15 148 282)(16 149 283)(17 150 284)(18 151 285)(19 152 286)(20 153 287)(21 154 288)(22 155 289)(23 156 290)(24 157 291)(25 158 292)(26 159 293)(27 160 294)(28 161 295)(29 162 296)(30 163 297)(31 164 298)(32 165 299)(33 166 300)(34 167 301)(35 168 302)(36 169 303)(37 170 304)(38 171 305)(39 172 306)(40 173 307)(41 174 308)(42 175 309)(43 176 310)(44 177 311)(45 178 312)(46 179 313)(47 180 314)(48 181 315)(49 182 316)(50 183 317)(51 184 318)(52 185 319)(53 186 320)(54 187 321)(55 188 322)(56 189 323)(57 190 324)(58 191 325)(59 192 326)(60 193 327)(61 194 328)(62 195 329)(63 196 330)(64 197 331)(65 198 332)(66 199 333)(67 200 334)(68 201 335)(69 202 336)(70 203 337)(71 204 338)(72 205 339)(73 206 340)(74 207 341)(75 208 342)(76 209 343)(77 210 344)(78 211 345)(79 212 346)(80 213 347)(81 214 348)(82 215 349)(83 216 350)(84 217 351)(85 218 352)(86 219 353)(87 220 354)(88 221 355)(89 222 356)(90 223 357)(91 224 358)(92 225 359)(93 226 360)(94 227 361)(95 228 362)(96 229 363)(97 230 364)(98 231 365)(99 232 366)(100 233 367)(101 234 368)(102 235 369)(103 236 370)(104 237 371)(105 238 372)(106 239 373)(107 240 374)(108 241 375)(109 242 376)(110 243 377)(111 244 378)(112 245 379)(113 246 380)(114 247 381)(115 248 382)(116 249 383)(117 250 384)(118 251 385)(119 252 386)(120 253 387)(121 254 388)(122 255 389)(123 256 390)(124 257 391)(125 258 392)(126 259 393)(127 260 394)(128 261 395)(129 262 396)(130 263 265)(131 264 266)(132 133 267)
(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)| (1,134,268)(2,135,269)(3,136,270)(4,137,271)(5,138,272)(6,139,273)(7,140,274)(8,141,275)(9,142,276)(10,143,277)(11,144,278)(12,145,279)(13,146,280)(14,147,281)(15,148,282)(16,149,283)(17,150,284)(18,151,285)(19,152,286)(20,153,287)(21,154,288)(22,155,289)(23,156,290)(24,157,291)(25,158,292)(26,159,293)(27,160,294)(28,161,295)(29,162,296)(30,163,297)(31,164,298)(32,165,299)(33,166,300)(34,167,301)(35,168,302)(36,169,303)(37,170,304)(38,171,305)(39,172,306)(40,173,307)(41,174,308)(42,175,309)(43,176,310)(44,177,311)(45,178,312)(46,179,313)(47,180,314)(48,181,315)(49,182,316)(50,183,317)(51,184,318)(52,185,319)(53,186,320)(54,187,321)(55,188,322)(56,189,323)(57,190,324)(58,191,325)(59,192,326)(60,193,327)(61,194,328)(62,195,329)(63,196,330)(64,197,331)(65,198,332)(66,199,333)(67,200,334)(68,201,335)(69,202,336)(70,203,337)(71,204,338)(72,205,339)(73,206,340)(74,207,341)(75,208,342)(76,209,343)(77,210,344)(78,211,345)(79,212,346)(80,213,347)(81,214,348)(82,215,349)(83,216,350)(84,217,351)(85,218,352)(86,219,353)(87,220,354)(88,221,355)(89,222,356)(90,223,357)(91,224,358)(92,225,359)(93,226,360)(94,227,361)(95,228,362)(96,229,363)(97,230,364)(98,231,365)(99,232,366)(100,233,367)(101,234,368)(102,235,369)(103,236,370)(104,237,371)(105,238,372)(106,239,373)(107,240,374)(108,241,375)(109,242,376)(110,243,377)(111,244,378)(112,245,379)(113,246,380)(114,247,381)(115,248,382)(116,249,383)(117,250,384)(118,251,385)(119,252,386)(120,253,387)(121,254,388)(122,255,389)(123,256,390)(124,257,391)(125,258,392)(126,259,393)(127,260,394)(128,261,395)(129,262,396)(130,263,265)(131,264,266)(132,133,267), (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( (1,134,268)(2,135,269)(3,136,270)(4,137,271)(5,138,272)(6,139,273)(7,140,274)(8,141,275)(9,142,276)(10,143,277)(11,144,278)(12,145,279)(13,146,280)(14,147,281)(15,148,282)(16,149,283)(17,150,284)(18,151,285)(19,152,286)(20,153,287)(21,154,288)(22,155,289)(23,156,290)(24,157,291)(25,158,292)(26,159,293)(27,160,294)(28,161,295)(29,162,296)(30,163,297)(31,164,298)(32,165,299)(33,166,300)(34,167,301)(35,168,302)(36,169,303)(37,170,304)(38,171,305)(39,172,306)(40,173,307)(41,174,308)(42,175,309)(43,176,310)(44,177,311)(45,178,312)(46,179,313)(47,180,314)(48,181,315)(49,182,316)(50,183,317)(51,184,318)(52,185,319)(53,186,320)(54,187,321)(55,188,322)(56,189,323)(57,190,324)(58,191,325)(59,192,326)(60,193,327)(61,194,328)(62,195,329)(63,196,330)(64,197,331)(65,198,332)(66,199,333)(67,200,334)(68,201,335)(69,202,336)(70,203,337)(71,204,338)(72,205,339)(73,206,340)(74,207,341)(75,208,342)(76,209,343)(77,210,344)(78,211,345)(79,212,346)(80,213,347)(81,214,348)(82,215,349)(83,216,350)(84,217,351)(85,218,352)(86,219,353)(87,220,354)(88,221,355)(89,222,356)(90,223,357)(91,224,358)(92,225,359)(93,226,360)(94,227,361)(95,228,362)(96,229,363)(97,230,364)(98,231,365)(99,232,366)(100,233,367)(101,234,368)(102,235,369)(103,236,370)(104,237,371)(105,238,372)(106,239,373)(107,240,374)(108,241,375)(109,242,376)(110,243,377)(111,244,378)(112,245,379)(113,246,380)(114,247,381)(115,248,382)(116,249,383)(117,250,384)(118,251,385)(119,252,386)(120,253,387)(121,254,388)(122,255,389)(123,256,390)(124,257,391)(125,258,392)(126,259,393)(127,260,394)(128,261,395)(129,262,396)(130,263,265)(131,264,266)(132,133,267), (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([(1,134,268),(2,135,269),(3,136,270),(4,137,271),(5,138,272),(6,139,273),(7,140,274),(8,141,275),(9,142,276),(10,143,277),(11,144,278),(12,145,279),(13,146,280),(14,147,281),(15,148,282),(16,149,283),(17,150,284),(18,151,285),(19,152,286),(20,153,287),(21,154,288),(22,155,289),(23,156,290),(24,157,291),(25,158,292),(26,159,293),(27,160,294),(28,161,295),(29,162,296),(30,163,297),(31,164,298),(32,165,299),(33,166,300),(34,167,301),(35,168,302),(36,169,303),(37,170,304),(38,171,305),(39,172,306),(40,173,307),(41,174,308),(42,175,309),(43,176,310),(44,177,311),(45,178,312),(46,179,313),(47,180,314),(48,181,315),(49,182,316),(50,183,317),(51,184,318),(52,185,319),(53,186,320),(54,187,321),(55,188,322),(56,189,323),(57,190,324),(58,191,325),(59,192,326),(60,193,327),(61,194,328),(62,195,329),(63,196,330),(64,197,331),(65,198,332),(66,199,333),(67,200,334),(68,201,335),(69,202,336),(70,203,337),(71,204,338),(72,205,339),(73,206,340),(74,207,341),(75,208,342),(76,209,343),(77,210,344),(78,211,345),(79,212,346),(80,213,347),(81,214,348),(82,215,349),(83,216,350),(84,217,351),(85,218,352),(86,219,353),(87,220,354),(88,221,355),(89,222,356),(90,223,357),(91,224,358),(92,225,359),(93,226,360),(94,227,361),(95,228,362),(96,229,363),(97,230,364),(98,231,365),(99,232,366),(100,233,367),(101,234,368),(102,235,369),(103,236,370),(104,237,371),(105,238,372),(106,239,373),(107,240,374),(108,241,375),(109,242,376),(110,243,377),(111,244,378),(112,245,379),(113,246,380),(114,247,381),(115,248,382),(116,249,383),(117,250,384),(118,251,385),(119,252,386),(120,253,387),(121,254,388),(122,255,389),(123,256,390),(124,257,391),(125,258,392),(126,259,393),(127,260,394),(128,261,395),(129,262,396),(130,263,265),(131,264,266),(132,133,267)], [(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

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