direct product, abelian, monomial, 3-elementary
Aliases: C3×C129, SmallGroup(387,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C129 |
| C1 — C3×C129 |
| C1 — C3×C129 |
Generators and relations for C3×C129
G = < a,b | a3=b129=1, ab=ba >
Smallest permutation representation of C3×C129
►Regular action on 387 points
Generators in S387
(1 281 178)(2 282 179)(3 283 180)(4 284 181)(5 285 182)(6 286 183)(7 287 184)(8 288 185)(9 289 186)(10 290 187)(11 291 188)(12 292 189)(13 293 190)(14 294 191)(15 295 192)(16 296 193)(17 297 194)(18 298 195)(19 299 196)(20 300 197)(21 301 198)(22 302 199)(23 303 200)(24 304 201)(25 305 202)(26 306 203)(27 307 204)(28 308 205)(29 309 206)(30 310 207)(31 311 208)(32 312 209)(33 313 210)(34 314 211)(35 315 212)(36 316 213)(37 317 214)(38 318 215)(39 319 216)(40 320 217)(41 321 218)(42 322 219)(43 323 220)(44 324 221)(45 325 222)(46 326 223)(47 327 224)(48 328 225)(49 329 226)(50 330 227)(51 331 228)(52 332 229)(53 333 230)(54 334 231)(55 335 232)(56 336 233)(57 337 234)(58 338 235)(59 339 236)(60 340 237)(61 341 238)(62 342 239)(63 343 240)(64 344 241)(65 345 242)(66 346 243)(67 347 244)(68 348 245)(69 349 246)(70 350 247)(71 351 248)(72 352 249)(73 353 250)(74 354 251)(75 355 252)(76 356 253)(77 357 254)(78 358 255)(79 359 256)(80 360 257)(81 361 258)(82 362 130)(83 363 131)(84 364 132)(85 365 133)(86 366 134)(87 367 135)(88 368 136)(89 369 137)(90 370 138)(91 371 139)(92 372 140)(93 373 141)(94 374 142)(95 375 143)(96 376 144)(97 377 145)(98 378 146)(99 379 147)(100 380 148)(101 381 149)(102 382 150)(103 383 151)(104 384 152)(105 385 153)(106 386 154)(107 387 155)(108 259 156)(109 260 157)(110 261 158)(111 262 159)(112 263 160)(113 264 161)(114 265 162)(115 266 163)(116 267 164)(117 268 165)(118 269 166)(119 270 167)(120 271 168)(121 272 169)(122 273 170)(123 274 171)(124 275 172)(125 276 173)(126 277 174)(127 278 175)(128 279 176)(129 280 177)
(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)
G:=sub<Sym(387)| (1,281,178)(2,282,179)(3,283,180)(4,284,181)(5,285,182)(6,286,183)(7,287,184)(8,288,185)(9,289,186)(10,290,187)(11,291,188)(12,292,189)(13,293,190)(14,294,191)(15,295,192)(16,296,193)(17,297,194)(18,298,195)(19,299,196)(20,300,197)(21,301,198)(22,302,199)(23,303,200)(24,304,201)(25,305,202)(26,306,203)(27,307,204)(28,308,205)(29,309,206)(30,310,207)(31,311,208)(32,312,209)(33,313,210)(34,314,211)(35,315,212)(36,316,213)(37,317,214)(38,318,215)(39,319,216)(40,320,217)(41,321,218)(42,322,219)(43,323,220)(44,324,221)(45,325,222)(46,326,223)(47,327,224)(48,328,225)(49,329,226)(50,330,227)(51,331,228)(52,332,229)(53,333,230)(54,334,231)(55,335,232)(56,336,233)(57,337,234)(58,338,235)(59,339,236)(60,340,237)(61,341,238)(62,342,239)(63,343,240)(64,344,241)(65,345,242)(66,346,243)(67,347,244)(68,348,245)(69,349,246)(70,350,247)(71,351,248)(72,352,249)(73,353,250)(74,354,251)(75,355,252)(76,356,253)(77,357,254)(78,358,255)(79,359,256)(80,360,257)(81,361,258)(82,362,130)(83,363,131)(84,364,132)(85,365,133)(86,366,134)(87,367,135)(88,368,136)(89,369,137)(90,370,138)(91,371,139)(92,372,140)(93,373,141)(94,374,142)(95,375,143)(96,376,144)(97,377,145)(98,378,146)(99,379,147)(100,380,148)(101,381,149)(102,382,150)(103,383,151)(104,384,152)(105,385,153)(106,386,154)(107,387,155)(108,259,156)(109,260,157)(110,261,158)(111,262,159)(112,263,160)(113,264,161)(114,265,162)(115,266,163)(116,267,164)(117,268,165)(118,269,166)(119,270,167)(120,271,168)(121,272,169)(122,273,170)(123,274,171)(124,275,172)(125,276,173)(126,277,174)(127,278,175)(128,279,176)(129,280,177), (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)>;
G:=Group( (1,281,178)(2,282,179)(3,283,180)(4,284,181)(5,285,182)(6,286,183)(7,287,184)(8,288,185)(9,289,186)(10,290,187)(11,291,188)(12,292,189)(13,293,190)(14,294,191)(15,295,192)(16,296,193)(17,297,194)(18,298,195)(19,299,196)(20,300,197)(21,301,198)(22,302,199)(23,303,200)(24,304,201)(25,305,202)(26,306,203)(27,307,204)(28,308,205)(29,309,206)(30,310,207)(31,311,208)(32,312,209)(33,313,210)(34,314,211)(35,315,212)(36,316,213)(37,317,214)(38,318,215)(39,319,216)(40,320,217)(41,321,218)(42,322,219)(43,323,220)(44,324,221)(45,325,222)(46,326,223)(47,327,224)(48,328,225)(49,329,226)(50,330,227)(51,331,228)(52,332,229)(53,333,230)(54,334,231)(55,335,232)(56,336,233)(57,337,234)(58,338,235)(59,339,236)(60,340,237)(61,341,238)(62,342,239)(63,343,240)(64,344,241)(65,345,242)(66,346,243)(67,347,244)(68,348,245)(69,349,246)(70,350,247)(71,351,248)(72,352,249)(73,353,250)(74,354,251)(75,355,252)(76,356,253)(77,357,254)(78,358,255)(79,359,256)(80,360,257)(81,361,258)(82,362,130)(83,363,131)(84,364,132)(85,365,133)(86,366,134)(87,367,135)(88,368,136)(89,369,137)(90,370,138)(91,371,139)(92,372,140)(93,373,141)(94,374,142)(95,375,143)(96,376,144)(97,377,145)(98,378,146)(99,379,147)(100,380,148)(101,381,149)(102,382,150)(103,383,151)(104,384,152)(105,385,153)(106,386,154)(107,387,155)(108,259,156)(109,260,157)(110,261,158)(111,262,159)(112,263,160)(113,264,161)(114,265,162)(115,266,163)(116,267,164)(117,268,165)(118,269,166)(119,270,167)(120,271,168)(121,272,169)(122,273,170)(123,274,171)(124,275,172)(125,276,173)(126,277,174)(127,278,175)(128,279,176)(129,280,177), (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) );
G=PermutationGroup([[(1,281,178),(2,282,179),(3,283,180),(4,284,181),(5,285,182),(6,286,183),(7,287,184),(8,288,185),(9,289,186),(10,290,187),(11,291,188),(12,292,189),(13,293,190),(14,294,191),(15,295,192),(16,296,193),(17,297,194),(18,298,195),(19,299,196),(20,300,197),(21,301,198),(22,302,199),(23,303,200),(24,304,201),(25,305,202),(26,306,203),(27,307,204),(28,308,205),(29,309,206),(30,310,207),(31,311,208),(32,312,209),(33,313,210),(34,314,211),(35,315,212),(36,316,213),(37,317,214),(38,318,215),(39,319,216),(40,320,217),(41,321,218),(42,322,219),(43,323,220),(44,324,221),(45,325,222),(46,326,223),(47,327,224),(48,328,225),(49,329,226),(50,330,227),(51,331,228),(52,332,229),(53,333,230),(54,334,231),(55,335,232),(56,336,233),(57,337,234),(58,338,235),(59,339,236),(60,340,237),(61,341,238),(62,342,239),(63,343,240),(64,344,241),(65,345,242),(66,346,243),(67,347,244),(68,348,245),(69,349,246),(70,350,247),(71,351,248),(72,352,249),(73,353,250),(74,354,251),(75,355,252),(76,356,253),(77,357,254),(78,358,255),(79,359,256),(80,360,257),(81,361,258),(82,362,130),(83,363,131),(84,364,132),(85,365,133),(86,366,134),(87,367,135),(88,368,136),(89,369,137),(90,370,138),(91,371,139),(92,372,140),(93,373,141),(94,374,142),(95,375,143),(96,376,144),(97,377,145),(98,378,146),(99,379,147),(100,380,148),(101,381,149),(102,382,150),(103,383,151),(104,384,152),(105,385,153),(106,386,154),(107,387,155),(108,259,156),(109,260,157),(110,261,158),(111,262,159),(112,263,160),(113,264,161),(114,265,162),(115,266,163),(116,267,164),(117,268,165),(118,269,166),(119,270,167),(120,271,168),(121,272,169),(122,273,170),(123,274,171),(124,275,172),(125,276,173),(126,277,174),(127,278,175),(128,279,176),(129,280,177)], [(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)]])
387 conjugacy classes
| class | 1 | 3A | ··· | 3H | 43A | ··· | 43AP | 129A | ··· | 129LX |
| order | 1 | 3 | ··· | 3 | 43 | ··· | 43 | 129 | ··· | 129 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
387 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C43 | C129 |
| kernel | C3×C129 | C129 | C32 | C3 |
| # reps | 1 | 8 | 42 | 336 |
Matrix representation of C3×C129 ►in GL2(𝔽1033) generated by
| 195 | 0 |
| 0 | 195 |
| 578 | 0 |
| 0 | 72 |
G:=sub<GL(2,GF(1033))| [195,0,0,195],[578,0,0,72] >;
C3×C129 in GAP, Magma, Sage, TeX
C_3\times C_{129} % in TeX
G:=Group("C3xC129"); // GroupNames label
G:=SmallGroup(387,4);
// by ID
G=gap.SmallGroup(387,4);
# by ID
G:=PCGroup([3,-3,-3,-43]);
// Polycyclic
G:=Group<a,b|a^3=b^129=1,a*b=b*a>;
// generators/relations
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