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G = C3×C138order 414 = 2·32·23

Abelian group of type [3,138]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C138, SmallGroup(414,10)

Series: Derived Chief Lower central Upper central

C1 — C3×C138
C1C23C69C3×C69 — C3×C138
C1 — C3×C138
C1 — C3×C138

Generators and relations for C3×C138
 G = < a,b | a3=b138=1, ab=ba >


Smallest permutation representation of C3×C138
Regular action on 414 points
Generators in S414
(1 217 364)(2 218 365)(3 219 366)(4 220 367)(5 221 368)(6 222 369)(7 223 370)(8 224 371)(9 225 372)(10 226 373)(11 227 374)(12 228 375)(13 229 376)(14 230 377)(15 231 378)(16 232 379)(17 233 380)(18 234 381)(19 235 382)(20 236 383)(21 237 384)(22 238 385)(23 239 386)(24 240 387)(25 241 388)(26 242 389)(27 243 390)(28 244 391)(29 245 392)(30 246 393)(31 247 394)(32 248 395)(33 249 396)(34 250 397)(35 251 398)(36 252 399)(37 253 400)(38 254 401)(39 255 402)(40 256 403)(41 257 404)(42 258 405)(43 259 406)(44 260 407)(45 261 408)(46 262 409)(47 263 410)(48 264 411)(49 265 412)(50 266 413)(51 267 414)(52 268 277)(53 269 278)(54 270 279)(55 271 280)(56 272 281)(57 273 282)(58 274 283)(59 275 284)(60 276 285)(61 139 286)(62 140 287)(63 141 288)(64 142 289)(65 143 290)(66 144 291)(67 145 292)(68 146 293)(69 147 294)(70 148 295)(71 149 296)(72 150 297)(73 151 298)(74 152 299)(75 153 300)(76 154 301)(77 155 302)(78 156 303)(79 157 304)(80 158 305)(81 159 306)(82 160 307)(83 161 308)(84 162 309)(85 163 310)(86 164 311)(87 165 312)(88 166 313)(89 167 314)(90 168 315)(91 169 316)(92 170 317)(93 171 318)(94 172 319)(95 173 320)(96 174 321)(97 175 322)(98 176 323)(99 177 324)(100 178 325)(101 179 326)(102 180 327)(103 181 328)(104 182 329)(105 183 330)(106 184 331)(107 185 332)(108 186 333)(109 187 334)(110 188 335)(111 189 336)(112 190 337)(113 191 338)(114 192 339)(115 193 340)(116 194 341)(117 195 342)(118 196 343)(119 197 344)(120 198 345)(121 199 346)(122 200 347)(123 201 348)(124 202 349)(125 203 350)(126 204 351)(127 205 352)(128 206 353)(129 207 354)(130 208 355)(131 209 356)(132 210 357)(133 211 358)(134 212 359)(135 213 360)(136 214 361)(137 215 362)(138 216 363)
(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)

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

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

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

414 conjugacy classes

class 1  2 3A···3H6A···6H23A···23V46A···46V69A···69FT138A···138FT
order123···36···623···2346···4669···69138···138
size111···11···11···11···11···11···1

414 irreducible representations

dim11111111
type++
imageC1C2C3C6C23C46C69C138
kernelC3×C138C3×C69C138C69C3×C6C32C6C3
# reps11882222176176

Matrix representation of C3×C138 in GL2(𝔽139) generated by

960
096
,
610
062
G:=sub<GL(2,GF(139))| [96,0,0,96],[61,0,0,62] >;

C3×C138 in GAP, Magma, Sage, TeX

C_3\times C_{138}
% in TeX

G:=Group("C3xC138");
// GroupNames label

G:=SmallGroup(414,10);
// by ID

G=gap.SmallGroup(414,10);
# by ID

G:=PCGroup([4,-2,-3,-3,-23]);
// Polycyclic

G:=Group<a,b|a^3=b^138=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C138 in TeX

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