direct product, abelian, monomial, 3-elementary
Aliases: C3×C126, SmallGroup(378,44)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C126 |
| C1 — C3×C126 |
| C1 — C3×C126 |
Generators and relations for C3×C126
G = < a,b | a3=b126=1, ab=ba >
Smallest permutation representation of C3×C126
►Regular action on 378 points
Generators in S378
(1 267 133)(2 268 134)(3 269 135)(4 270 136)(5 271 137)(6 272 138)(7 273 139)(8 274 140)(9 275 141)(10 276 142)(11 277 143)(12 278 144)(13 279 145)(14 280 146)(15 281 147)(16 282 148)(17 283 149)(18 284 150)(19 285 151)(20 286 152)(21 287 153)(22 288 154)(23 289 155)(24 290 156)(25 291 157)(26 292 158)(27 293 159)(28 294 160)(29 295 161)(30 296 162)(31 297 163)(32 298 164)(33 299 165)(34 300 166)(35 301 167)(36 302 168)(37 303 169)(38 304 170)(39 305 171)(40 306 172)(41 307 173)(42 308 174)(43 309 175)(44 310 176)(45 311 177)(46 312 178)(47 313 179)(48 314 180)(49 315 181)(50 316 182)(51 317 183)(52 318 184)(53 319 185)(54 320 186)(55 321 187)(56 322 188)(57 323 189)(58 324 190)(59 325 191)(60 326 192)(61 327 193)(62 328 194)(63 329 195)(64 330 196)(65 331 197)(66 332 198)(67 333 199)(68 334 200)(69 335 201)(70 336 202)(71 337 203)(72 338 204)(73 339 205)(74 340 206)(75 341 207)(76 342 208)(77 343 209)(78 344 210)(79 345 211)(80 346 212)(81 347 213)(82 348 214)(83 349 215)(84 350 216)(85 351 217)(86 352 218)(87 353 219)(88 354 220)(89 355 221)(90 356 222)(91 357 223)(92 358 224)(93 359 225)(94 360 226)(95 361 227)(96 362 228)(97 363 229)(98 364 230)(99 365 231)(100 366 232)(101 367 233)(102 368 234)(103 369 235)(104 370 236)(105 371 237)(106 372 238)(107 373 239)(108 374 240)(109 375 241)(110 376 242)(111 377 243)(112 378 244)(113 253 245)(114 254 246)(115 255 247)(116 256 248)(117 257 249)(118 258 250)(119 259 251)(120 260 252)(121 261 127)(122 262 128)(123 263 129)(124 264 130)(125 265 131)(126 266 132)
(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)
G:=sub<Sym(378)| (1,267,133)(2,268,134)(3,269,135)(4,270,136)(5,271,137)(6,272,138)(7,273,139)(8,274,140)(9,275,141)(10,276,142)(11,277,143)(12,278,144)(13,279,145)(14,280,146)(15,281,147)(16,282,148)(17,283,149)(18,284,150)(19,285,151)(20,286,152)(21,287,153)(22,288,154)(23,289,155)(24,290,156)(25,291,157)(26,292,158)(27,293,159)(28,294,160)(29,295,161)(30,296,162)(31,297,163)(32,298,164)(33,299,165)(34,300,166)(35,301,167)(36,302,168)(37,303,169)(38,304,170)(39,305,171)(40,306,172)(41,307,173)(42,308,174)(43,309,175)(44,310,176)(45,311,177)(46,312,178)(47,313,179)(48,314,180)(49,315,181)(50,316,182)(51,317,183)(52,318,184)(53,319,185)(54,320,186)(55,321,187)(56,322,188)(57,323,189)(58,324,190)(59,325,191)(60,326,192)(61,327,193)(62,328,194)(63,329,195)(64,330,196)(65,331,197)(66,332,198)(67,333,199)(68,334,200)(69,335,201)(70,336,202)(71,337,203)(72,338,204)(73,339,205)(74,340,206)(75,341,207)(76,342,208)(77,343,209)(78,344,210)(79,345,211)(80,346,212)(81,347,213)(82,348,214)(83,349,215)(84,350,216)(85,351,217)(86,352,218)(87,353,219)(88,354,220)(89,355,221)(90,356,222)(91,357,223)(92,358,224)(93,359,225)(94,360,226)(95,361,227)(96,362,228)(97,363,229)(98,364,230)(99,365,231)(100,366,232)(101,367,233)(102,368,234)(103,369,235)(104,370,236)(105,371,237)(106,372,238)(107,373,239)(108,374,240)(109,375,241)(110,376,242)(111,377,243)(112,378,244)(113,253,245)(114,254,246)(115,255,247)(116,256,248)(117,257,249)(118,258,250)(119,259,251)(120,260,252)(121,261,127)(122,262,128)(123,263,129)(124,264,130)(125,265,131)(126,266,132), (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)>;
G:=Group( (1,267,133)(2,268,134)(3,269,135)(4,270,136)(5,271,137)(6,272,138)(7,273,139)(8,274,140)(9,275,141)(10,276,142)(11,277,143)(12,278,144)(13,279,145)(14,280,146)(15,281,147)(16,282,148)(17,283,149)(18,284,150)(19,285,151)(20,286,152)(21,287,153)(22,288,154)(23,289,155)(24,290,156)(25,291,157)(26,292,158)(27,293,159)(28,294,160)(29,295,161)(30,296,162)(31,297,163)(32,298,164)(33,299,165)(34,300,166)(35,301,167)(36,302,168)(37,303,169)(38,304,170)(39,305,171)(40,306,172)(41,307,173)(42,308,174)(43,309,175)(44,310,176)(45,311,177)(46,312,178)(47,313,179)(48,314,180)(49,315,181)(50,316,182)(51,317,183)(52,318,184)(53,319,185)(54,320,186)(55,321,187)(56,322,188)(57,323,189)(58,324,190)(59,325,191)(60,326,192)(61,327,193)(62,328,194)(63,329,195)(64,330,196)(65,331,197)(66,332,198)(67,333,199)(68,334,200)(69,335,201)(70,336,202)(71,337,203)(72,338,204)(73,339,205)(74,340,206)(75,341,207)(76,342,208)(77,343,209)(78,344,210)(79,345,211)(80,346,212)(81,347,213)(82,348,214)(83,349,215)(84,350,216)(85,351,217)(86,352,218)(87,353,219)(88,354,220)(89,355,221)(90,356,222)(91,357,223)(92,358,224)(93,359,225)(94,360,226)(95,361,227)(96,362,228)(97,363,229)(98,364,230)(99,365,231)(100,366,232)(101,367,233)(102,368,234)(103,369,235)(104,370,236)(105,371,237)(106,372,238)(107,373,239)(108,374,240)(109,375,241)(110,376,242)(111,377,243)(112,378,244)(113,253,245)(114,254,246)(115,255,247)(116,256,248)(117,257,249)(118,258,250)(119,259,251)(120,260,252)(121,261,127)(122,262,128)(123,263,129)(124,264,130)(125,265,131)(126,266,132), (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) );
G=PermutationGroup([[(1,267,133),(2,268,134),(3,269,135),(4,270,136),(5,271,137),(6,272,138),(7,273,139),(8,274,140),(9,275,141),(10,276,142),(11,277,143),(12,278,144),(13,279,145),(14,280,146),(15,281,147),(16,282,148),(17,283,149),(18,284,150),(19,285,151),(20,286,152),(21,287,153),(22,288,154),(23,289,155),(24,290,156),(25,291,157),(26,292,158),(27,293,159),(28,294,160),(29,295,161),(30,296,162),(31,297,163),(32,298,164),(33,299,165),(34,300,166),(35,301,167),(36,302,168),(37,303,169),(38,304,170),(39,305,171),(40,306,172),(41,307,173),(42,308,174),(43,309,175),(44,310,176),(45,311,177),(46,312,178),(47,313,179),(48,314,180),(49,315,181),(50,316,182),(51,317,183),(52,318,184),(53,319,185),(54,320,186),(55,321,187),(56,322,188),(57,323,189),(58,324,190),(59,325,191),(60,326,192),(61,327,193),(62,328,194),(63,329,195),(64,330,196),(65,331,197),(66,332,198),(67,333,199),(68,334,200),(69,335,201),(70,336,202),(71,337,203),(72,338,204),(73,339,205),(74,340,206),(75,341,207),(76,342,208),(77,343,209),(78,344,210),(79,345,211),(80,346,212),(81,347,213),(82,348,214),(83,349,215),(84,350,216),(85,351,217),(86,352,218),(87,353,219),(88,354,220),(89,355,221),(90,356,222),(91,357,223),(92,358,224),(93,359,225),(94,360,226),(95,361,227),(96,362,228),(97,363,229),(98,364,230),(99,365,231),(100,366,232),(101,367,233),(102,368,234),(103,369,235),(104,370,236),(105,371,237),(106,372,238),(107,373,239),(108,374,240),(109,375,241),(110,376,242),(111,377,243),(112,378,244),(113,253,245),(114,254,246),(115,255,247),(116,256,248),(117,257,249),(118,258,250),(119,259,251),(120,260,252),(121,261,127),(122,262,128),(123,263,129),(124,264,130),(125,265,131),(126,266,132)], [(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)]])
378 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 7A | ··· | 7F | 9A | ··· | 9R | 14A | ··· | 14F | 18A | ··· | 18R | 21A | ··· | 21AV | 42A | ··· | 42AV | 63A | ··· | 63DD | 126A | ··· | 126DD |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 7 | ··· | 7 | 9 | ··· | 9 | 14 | ··· | 14 | 18 | ··· | 18 | 21 | ··· | 21 | 42 | ··· | 42 | 63 | ··· | 63 | 126 | ··· | 126 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
378 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C7 | C9 | C14 | C18 | C21 | C21 | C42 | C42 | C63 | C126 |
| kernel | C3×C126 | C3×C63 | C126 | C3×C42 | C63 | C3×C21 | C3×C18 | C42 | C3×C9 | C21 | C18 | C3×C6 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 6 | 18 | 6 | 18 | 36 | 12 | 36 | 12 | 108 | 108 |
Matrix representation of C3×C126 ►in GL2(𝔽127) generated by
| 19 | 0 |
| 0 | 1 |
| 118 | 0 |
| 0 | 29 |
G:=sub<GL(2,GF(127))| [19,0,0,1],[118,0,0,29] >;
C3×C126 in GAP, Magma, Sage, TeX
C_3\times C_{126} % in TeX
G:=Group("C3xC126"); // GroupNames label
G:=SmallGroup(378,44);
// by ID
G=gap.SmallGroup(378,44);
# by ID
G:=PCGroup([5,-2,-3,-3,-7,-3,636]);
// Polycyclic
G:=Group<a,b|a^3=b^126=1,a*b=b*a>;
// generators/relations
Export