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G = C3×C126order 378 = 2·33·7

Abelian group of type [3,126]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C126, SmallGroup(378,44)

Series: Derived Chief Lower central Upper central

C1 — C3×C126
C1C3C21C3×C21C3×C63 — 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
(1 173 300)(2 174 301)(3 175 302)(4 176 303)(5 177 304)(6 178 305)(7 179 306)(8 180 307)(9 181 308)(10 182 309)(11 183 310)(12 184 311)(13 185 312)(14 186 313)(15 187 314)(16 188 315)(17 189 316)(18 190 317)(19 191 318)(20 192 319)(21 193 320)(22 194 321)(23 195 322)(24 196 323)(25 197 324)(26 198 325)(27 199 326)(28 200 327)(29 201 328)(30 202 329)(31 203 330)(32 204 331)(33 205 332)(34 206 333)(35 207 334)(36 208 335)(37 209 336)(38 210 337)(39 211 338)(40 212 339)(41 213 340)(42 214 341)(43 215 342)(44 216 343)(45 217 344)(46 218 345)(47 219 346)(48 220 347)(49 221 348)(50 222 349)(51 223 350)(52 224 351)(53 225 352)(54 226 353)(55 227 354)(56 228 355)(57 229 356)(58 230 357)(59 231 358)(60 232 359)(61 233 360)(62 234 361)(63 235 362)(64 236 363)(65 237 364)(66 238 365)(67 239 366)(68 240 367)(69 241 368)(70 242 369)(71 243 370)(72 244 371)(73 245 372)(74 246 373)(75 247 374)(76 248 375)(77 249 376)(78 250 377)(79 251 378)(80 252 253)(81 127 254)(82 128 255)(83 129 256)(84 130 257)(85 131 258)(86 132 259)(87 133 260)(88 134 261)(89 135 262)(90 136 263)(91 137 264)(92 138 265)(93 139 266)(94 140 267)(95 141 268)(96 142 269)(97 143 270)(98 144 271)(99 145 272)(100 146 273)(101 147 274)(102 148 275)(103 149 276)(104 150 277)(105 151 278)(106 152 279)(107 153 280)(108 154 281)(109 155 282)(110 156 283)(111 157 284)(112 158 285)(113 159 286)(114 160 287)(115 161 288)(116 162 289)(117 163 290)(118 164 291)(119 165 292)(120 166 293)(121 167 294)(122 168 295)(123 169 296)(124 170 297)(125 171 298)(126 172 299)
(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)| (1,173,300)(2,174,301)(3,175,302)(4,176,303)(5,177,304)(6,178,305)(7,179,306)(8,180,307)(9,181,308)(10,182,309)(11,183,310)(12,184,311)(13,185,312)(14,186,313)(15,187,314)(16,188,315)(17,189,316)(18,190,317)(19,191,318)(20,192,319)(21,193,320)(22,194,321)(23,195,322)(24,196,323)(25,197,324)(26,198,325)(27,199,326)(28,200,327)(29,201,328)(30,202,329)(31,203,330)(32,204,331)(33,205,332)(34,206,333)(35,207,334)(36,208,335)(37,209,336)(38,210,337)(39,211,338)(40,212,339)(41,213,340)(42,214,341)(43,215,342)(44,216,343)(45,217,344)(46,218,345)(47,219,346)(48,220,347)(49,221,348)(50,222,349)(51,223,350)(52,224,351)(53,225,352)(54,226,353)(55,227,354)(56,228,355)(57,229,356)(58,230,357)(59,231,358)(60,232,359)(61,233,360)(62,234,361)(63,235,362)(64,236,363)(65,237,364)(66,238,365)(67,239,366)(68,240,367)(69,241,368)(70,242,369)(71,243,370)(72,244,371)(73,245,372)(74,246,373)(75,247,374)(76,248,375)(77,249,376)(78,250,377)(79,251,378)(80,252,253)(81,127,254)(82,128,255)(83,129,256)(84,130,257)(85,131,258)(86,132,259)(87,133,260)(88,134,261)(89,135,262)(90,136,263)(91,137,264)(92,138,265)(93,139,266)(94,140,267)(95,141,268)(96,142,269)(97,143,270)(98,144,271)(99,145,272)(100,146,273)(101,147,274)(102,148,275)(103,149,276)(104,150,277)(105,151,278)(106,152,279)(107,153,280)(108,154,281)(109,155,282)(110,156,283)(111,157,284)(112,158,285)(113,159,286)(114,160,287)(115,161,288)(116,162,289)(117,163,290)(118,164,291)(119,165,292)(120,166,293)(121,167,294)(122,168,295)(123,169,296)(124,170,297)(125,171,298)(126,172,299), (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( (1,173,300)(2,174,301)(3,175,302)(4,176,303)(5,177,304)(6,178,305)(7,179,306)(8,180,307)(9,181,308)(10,182,309)(11,183,310)(12,184,311)(13,185,312)(14,186,313)(15,187,314)(16,188,315)(17,189,316)(18,190,317)(19,191,318)(20,192,319)(21,193,320)(22,194,321)(23,195,322)(24,196,323)(25,197,324)(26,198,325)(27,199,326)(28,200,327)(29,201,328)(30,202,329)(31,203,330)(32,204,331)(33,205,332)(34,206,333)(35,207,334)(36,208,335)(37,209,336)(38,210,337)(39,211,338)(40,212,339)(41,213,340)(42,214,341)(43,215,342)(44,216,343)(45,217,344)(46,218,345)(47,219,346)(48,220,347)(49,221,348)(50,222,349)(51,223,350)(52,224,351)(53,225,352)(54,226,353)(55,227,354)(56,228,355)(57,229,356)(58,230,357)(59,231,358)(60,232,359)(61,233,360)(62,234,361)(63,235,362)(64,236,363)(65,237,364)(66,238,365)(67,239,366)(68,240,367)(69,241,368)(70,242,369)(71,243,370)(72,244,371)(73,245,372)(74,246,373)(75,247,374)(76,248,375)(77,249,376)(78,250,377)(79,251,378)(80,252,253)(81,127,254)(82,128,255)(83,129,256)(84,130,257)(85,131,258)(86,132,259)(87,133,260)(88,134,261)(89,135,262)(90,136,263)(91,137,264)(92,138,265)(93,139,266)(94,140,267)(95,141,268)(96,142,269)(97,143,270)(98,144,271)(99,145,272)(100,146,273)(101,147,274)(102,148,275)(103,149,276)(104,150,277)(105,151,278)(106,152,279)(107,153,280)(108,154,281)(109,155,282)(110,156,283)(111,157,284)(112,158,285)(113,159,286)(114,160,287)(115,161,288)(116,162,289)(117,163,290)(118,164,291)(119,165,292)(120,166,293)(121,167,294)(122,168,295)(123,169,296)(124,170,297)(125,171,298)(126,172,299), (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([(1,173,300),(2,174,301),(3,175,302),(4,176,303),(5,177,304),(6,178,305),(7,179,306),(8,180,307),(9,181,308),(10,182,309),(11,183,310),(12,184,311),(13,185,312),(14,186,313),(15,187,314),(16,188,315),(17,189,316),(18,190,317),(19,191,318),(20,192,319),(21,193,320),(22,194,321),(23,195,322),(24,196,323),(25,197,324),(26,198,325),(27,199,326),(28,200,327),(29,201,328),(30,202,329),(31,203,330),(32,204,331),(33,205,332),(34,206,333),(35,207,334),(36,208,335),(37,209,336),(38,210,337),(39,211,338),(40,212,339),(41,213,340),(42,214,341),(43,215,342),(44,216,343),(45,217,344),(46,218,345),(47,219,346),(48,220,347),(49,221,348),(50,222,349),(51,223,350),(52,224,351),(53,225,352),(54,226,353),(55,227,354),(56,228,355),(57,229,356),(58,230,357),(59,231,358),(60,232,359),(61,233,360),(62,234,361),(63,235,362),(64,236,363),(65,237,364),(66,238,365),(67,239,366),(68,240,367),(69,241,368),(70,242,369),(71,243,370),(72,244,371),(73,245,372),(74,246,373),(75,247,374),(76,248,375),(77,249,376),(78,250,377),(79,251,378),(80,252,253),(81,127,254),(82,128,255),(83,129,256),(84,130,257),(85,131,258),(86,132,259),(87,133,260),(88,134,261),(89,135,262),(90,136,263),(91,137,264),(92,138,265),(93,139,266),(94,140,267),(95,141,268),(96,142,269),(97,143,270),(98,144,271),(99,145,272),(100,146,273),(101,147,274),(102,148,275),(103,149,276),(104,150,277),(105,151,278),(106,152,279),(107,153,280),(108,154,281),(109,155,282),(110,156,283),(111,157,284),(112,158,285),(113,159,286),(114,160,287),(115,161,288),(116,162,289),(117,163,290),(118,164,291),(119,165,292),(120,166,293),(121,167,294),(122,168,295),(123,169,296),(124,170,297),(125,171,298),(126,172,299)], [(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···3H6A···6H7A···7F9A···9R14A···14F18A···18R21A···21AV42A···42AV63A···63DD126A···126DD
order123···36···67···79···914···1418···1821···2142···4263···63126···126
size111···11···11···11···11···11···11···11···11···11···1

378 irreducible representations

dim1111111111111111
type++
imageC1C2C3C3C6C6C7C9C14C18C21C21C42C42C63C126
kernelC3×C126C3×C63C126C3×C42C63C3×C21C3×C18C42C3×C9C21C18C3×C6C9C32C6C3
# reps11626261861836123612108108

Matrix representation of C3×C126 in GL2(𝔽127) generated by

190
01
,
1180
029
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

Subgroup lattice of C3×C126 in TeX

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