direct product, abelian, monomial, 3-elementary
Aliases: C3×C138, SmallGroup(414,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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
(1 320 242)(2 321 243)(3 322 244)(4 323 245)(5 324 246)(6 325 247)(7 326 248)(8 327 249)(9 328 250)(10 329 251)(11 330 252)(12 331 253)(13 332 254)(14 333 255)(15 334 256)(16 335 257)(17 336 258)(18 337 259)(19 338 260)(20 339 261)(21 340 262)(22 341 263)(23 342 264)(24 343 265)(25 344 266)(26 345 267)(27 346 268)(28 347 269)(29 348 270)(30 349 271)(31 350 272)(32 351 273)(33 352 274)(34 353 275)(35 354 276)(36 355 139)(37 356 140)(38 357 141)(39 358 142)(40 359 143)(41 360 144)(42 361 145)(43 362 146)(44 363 147)(45 364 148)(46 365 149)(47 366 150)(48 367 151)(49 368 152)(50 369 153)(51 370 154)(52 371 155)(53 372 156)(54 373 157)(55 374 158)(56 375 159)(57 376 160)(58 377 161)(59 378 162)(60 379 163)(61 380 164)(62 381 165)(63 382 166)(64 383 167)(65 384 168)(66 385 169)(67 386 170)(68 387 171)(69 388 172)(70 389 173)(71 390 174)(72 391 175)(73 392 176)(74 393 177)(75 394 178)(76 395 179)(77 396 180)(78 397 181)(79 398 182)(80 399 183)(81 400 184)(82 401 185)(83 402 186)(84 403 187)(85 404 188)(86 405 189)(87 406 190)(88 407 191)(89 408 192)(90 409 193)(91 410 194)(92 411 195)(93 412 196)(94 413 197)(95 414 198)(96 277 199)(97 278 200)(98 279 201)(99 280 202)(100 281 203)(101 282 204)(102 283 205)(103 284 206)(104 285 207)(105 286 208)(106 287 209)(107 288 210)(108 289 211)(109 290 212)(110 291 213)(111 292 214)(112 293 215)(113 294 216)(114 295 217)(115 296 218)(116 297 219)(117 298 220)(118 299 221)(119 300 222)(120 301 223)(121 302 224)(122 303 225)(123 304 226)(124 305 227)(125 306 228)(126 307 229)(127 308 230)(128 309 231)(129 310 232)(130 311 233)(131 312 234)(132 313 235)(133 314 236)(134 315 237)(135 316 238)(136 317 239)(137 318 240)(138 319 241)
(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)| (1,320,242)(2,321,243)(3,322,244)(4,323,245)(5,324,246)(6,325,247)(7,326,248)(8,327,249)(9,328,250)(10,329,251)(11,330,252)(12,331,253)(13,332,254)(14,333,255)(15,334,256)(16,335,257)(17,336,258)(18,337,259)(19,338,260)(20,339,261)(21,340,262)(22,341,263)(23,342,264)(24,343,265)(25,344,266)(26,345,267)(27,346,268)(28,347,269)(29,348,270)(30,349,271)(31,350,272)(32,351,273)(33,352,274)(34,353,275)(35,354,276)(36,355,139)(37,356,140)(38,357,141)(39,358,142)(40,359,143)(41,360,144)(42,361,145)(43,362,146)(44,363,147)(45,364,148)(46,365,149)(47,366,150)(48,367,151)(49,368,152)(50,369,153)(51,370,154)(52,371,155)(53,372,156)(54,373,157)(55,374,158)(56,375,159)(57,376,160)(58,377,161)(59,378,162)(60,379,163)(61,380,164)(62,381,165)(63,382,166)(64,383,167)(65,384,168)(66,385,169)(67,386,170)(68,387,171)(69,388,172)(70,389,173)(71,390,174)(72,391,175)(73,392,176)(74,393,177)(75,394,178)(76,395,179)(77,396,180)(78,397,181)(79,398,182)(80,399,183)(81,400,184)(82,401,185)(83,402,186)(84,403,187)(85,404,188)(86,405,189)(87,406,190)(88,407,191)(89,408,192)(90,409,193)(91,410,194)(92,411,195)(93,412,196)(94,413,197)(95,414,198)(96,277,199)(97,278,200)(98,279,201)(99,280,202)(100,281,203)(101,282,204)(102,283,205)(103,284,206)(104,285,207)(105,286,208)(106,287,209)(107,288,210)(108,289,211)(109,290,212)(110,291,213)(111,292,214)(112,293,215)(113,294,216)(114,295,217)(115,296,218)(116,297,219)(117,298,220)(118,299,221)(119,300,222)(120,301,223)(121,302,224)(122,303,225)(123,304,226)(124,305,227)(125,306,228)(126,307,229)(127,308,230)(128,309,231)(129,310,232)(130,311,233)(131,312,234)(132,313,235)(133,314,236)(134,315,237)(135,316,238)(136,317,239)(137,318,240)(138,319,241), (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( (1,320,242)(2,321,243)(3,322,244)(4,323,245)(5,324,246)(6,325,247)(7,326,248)(8,327,249)(9,328,250)(10,329,251)(11,330,252)(12,331,253)(13,332,254)(14,333,255)(15,334,256)(16,335,257)(17,336,258)(18,337,259)(19,338,260)(20,339,261)(21,340,262)(22,341,263)(23,342,264)(24,343,265)(25,344,266)(26,345,267)(27,346,268)(28,347,269)(29,348,270)(30,349,271)(31,350,272)(32,351,273)(33,352,274)(34,353,275)(35,354,276)(36,355,139)(37,356,140)(38,357,141)(39,358,142)(40,359,143)(41,360,144)(42,361,145)(43,362,146)(44,363,147)(45,364,148)(46,365,149)(47,366,150)(48,367,151)(49,368,152)(50,369,153)(51,370,154)(52,371,155)(53,372,156)(54,373,157)(55,374,158)(56,375,159)(57,376,160)(58,377,161)(59,378,162)(60,379,163)(61,380,164)(62,381,165)(63,382,166)(64,383,167)(65,384,168)(66,385,169)(67,386,170)(68,387,171)(69,388,172)(70,389,173)(71,390,174)(72,391,175)(73,392,176)(74,393,177)(75,394,178)(76,395,179)(77,396,180)(78,397,181)(79,398,182)(80,399,183)(81,400,184)(82,401,185)(83,402,186)(84,403,187)(85,404,188)(86,405,189)(87,406,190)(88,407,191)(89,408,192)(90,409,193)(91,410,194)(92,411,195)(93,412,196)(94,413,197)(95,414,198)(96,277,199)(97,278,200)(98,279,201)(99,280,202)(100,281,203)(101,282,204)(102,283,205)(103,284,206)(104,285,207)(105,286,208)(106,287,209)(107,288,210)(108,289,211)(109,290,212)(110,291,213)(111,292,214)(112,293,215)(113,294,216)(114,295,217)(115,296,218)(116,297,219)(117,298,220)(118,299,221)(119,300,222)(120,301,223)(121,302,224)(122,303,225)(123,304,226)(124,305,227)(125,306,228)(126,307,229)(127,308,230)(128,309,231)(129,310,232)(130,311,233)(131,312,234)(132,313,235)(133,314,236)(134,315,237)(135,316,238)(136,317,239)(137,318,240)(138,319,241), (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([[(1,320,242),(2,321,243),(3,322,244),(4,323,245),(5,324,246),(6,325,247),(7,326,248),(8,327,249),(9,328,250),(10,329,251),(11,330,252),(12,331,253),(13,332,254),(14,333,255),(15,334,256),(16,335,257),(17,336,258),(18,337,259),(19,338,260),(20,339,261),(21,340,262),(22,341,263),(23,342,264),(24,343,265),(25,344,266),(26,345,267),(27,346,268),(28,347,269),(29,348,270),(30,349,271),(31,350,272),(32,351,273),(33,352,274),(34,353,275),(35,354,276),(36,355,139),(37,356,140),(38,357,141),(39,358,142),(40,359,143),(41,360,144),(42,361,145),(43,362,146),(44,363,147),(45,364,148),(46,365,149),(47,366,150),(48,367,151),(49,368,152),(50,369,153),(51,370,154),(52,371,155),(53,372,156),(54,373,157),(55,374,158),(56,375,159),(57,376,160),(58,377,161),(59,378,162),(60,379,163),(61,380,164),(62,381,165),(63,382,166),(64,383,167),(65,384,168),(66,385,169),(67,386,170),(68,387,171),(69,388,172),(70,389,173),(71,390,174),(72,391,175),(73,392,176),(74,393,177),(75,394,178),(76,395,179),(77,396,180),(78,397,181),(79,398,182),(80,399,183),(81,400,184),(82,401,185),(83,402,186),(84,403,187),(85,404,188),(86,405,189),(87,406,190),(88,407,191),(89,408,192),(90,409,193),(91,410,194),(92,411,195),(93,412,196),(94,413,197),(95,414,198),(96,277,199),(97,278,200),(98,279,201),(99,280,202),(100,281,203),(101,282,204),(102,283,205),(103,284,206),(104,285,207),(105,286,208),(106,287,209),(107,288,210),(108,289,211),(109,290,212),(110,291,213),(111,292,214),(112,293,215),(113,294,216),(114,295,217),(115,296,218),(116,297,219),(117,298,220),(118,299,221),(119,300,222),(120,301,223),(121,302,224),(122,303,225),(123,304,226),(124,305,227),(125,306,228),(126,307,229),(127,308,230),(128,309,231),(129,310,232),(130,311,233),(131,312,234),(132,313,235),(133,314,236),(134,315,237),(135,316,238),(136,317,239),(137,318,240),(138,319,241)], [(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 | ··· | 3H | 6A | ··· | 6H | 23A | ··· | 23V | 46A | ··· | 46V | 69A | ··· | 69FT | 138A | ··· | 138FT |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 | 138 | ··· | 138 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
414 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C23 | C46 | C69 | C138 |
| kernel | C3×C138 | C3×C69 | C138 | C69 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 8 | 22 | 22 | 176 | 176 |
Matrix representation of C3×C138 ►in GL2(𝔽139) generated by
| 96 | 0 |
| 0 | 96 |
| 61 | 0 |
| 0 | 62 |
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