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G = C3×C135order 405 = 34·5

Abelian group of type [3,135]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C135, SmallGroup(405,5)

Series: Derived Chief Lower central Upper central

C1 — C3×C135
C1C3C9C45C135 — C3×C135
C1 — C3×C135
C1 — C3×C135

Generators and relations for C3×C135
 G = < a,b | a3=b135=1, ab=ba >


Smallest permutation representation of C3×C135
Regular action on 405 points
Generators in S405
(1 251 362)(2 252 363)(3 253 364)(4 254 365)(5 255 366)(6 256 367)(7 257 368)(8 258 369)(9 259 370)(10 260 371)(11 261 372)(12 262 373)(13 263 374)(14 264 375)(15 265 376)(16 266 377)(17 267 378)(18 268 379)(19 269 380)(20 270 381)(21 136 382)(22 137 383)(23 138 384)(24 139 385)(25 140 386)(26 141 387)(27 142 388)(28 143 389)(29 144 390)(30 145 391)(31 146 392)(32 147 393)(33 148 394)(34 149 395)(35 150 396)(36 151 397)(37 152 398)(38 153 399)(39 154 400)(40 155 401)(41 156 402)(42 157 403)(43 158 404)(44 159 405)(45 160 271)(46 161 272)(47 162 273)(48 163 274)(49 164 275)(50 165 276)(51 166 277)(52 167 278)(53 168 279)(54 169 280)(55 170 281)(56 171 282)(57 172 283)(58 173 284)(59 174 285)(60 175 286)(61 176 287)(62 177 288)(63 178 289)(64 179 290)(65 180 291)(66 181 292)(67 182 293)(68 183 294)(69 184 295)(70 185 296)(71 186 297)(72 187 298)(73 188 299)(74 189 300)(75 190 301)(76 191 302)(77 192 303)(78 193 304)(79 194 305)(80 195 306)(81 196 307)(82 197 308)(83 198 309)(84 199 310)(85 200 311)(86 201 312)(87 202 313)(88 203 314)(89 204 315)(90 205 316)(91 206 317)(92 207 318)(93 208 319)(94 209 320)(95 210 321)(96 211 322)(97 212 323)(98 213 324)(99 214 325)(100 215 326)(101 216 327)(102 217 328)(103 218 329)(104 219 330)(105 220 331)(106 221 332)(107 222 333)(108 223 334)(109 224 335)(110 225 336)(111 226 337)(112 227 338)(113 228 339)(114 229 340)(115 230 341)(116 231 342)(117 232 343)(118 233 344)(119 234 345)(120 235 346)(121 236 347)(122 237 348)(123 238 349)(124 239 350)(125 240 351)(126 241 352)(127 242 353)(128 243 354)(129 244 355)(130 245 356)(131 246 357)(132 247 358)(133 248 359)(134 249 360)(135 250 361)
(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)

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

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

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

405 conjugacy classes

class 1 3A···3H5A5B5C5D9A···9R15A···15AF27A···27BB45A···45BT135A···135HH
order13···355559···915···1527···2745···45135···135
size11···111111···11···11···11···11···1

405 irreducible representations

dim111111111111
type+
imageC1C3C3C5C9C9C15C15C27C45C45C135
kernelC3×C135C135C3×C45C3×C27C45C3×C15C27C3×C9C15C9C32C3
# reps1624126248544824216

Matrix representation of C3×C135 in GL2(𝔽271) generated by

10
0242
,
500
0160
G:=sub<GL(2,GF(271))| [1,0,0,242],[50,0,0,160] >;

C3×C135 in GAP, Magma, Sage, TeX

C_3\times C_{135}
% in TeX

G:=Group("C3xC135");
// GroupNames label

G:=SmallGroup(405,5);
// by ID

G=gap.SmallGroup(405,5);
# by ID

G:=PCGroup([5,-3,-3,-5,-3,-3,225,78]);
// Polycyclic

G:=Group<a,b|a^3=b^135=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C135 in TeX

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