direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C7⋊C27, C14⋊C27, C7⋊2C54, C42.C9, C126.C3, C63.2C6, C21.2C18, C6.(C7⋊C9), C18.(C7⋊C3), C9.(C2×C7⋊C3), C3.(C2×C7⋊C9), SmallGroup(378,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C7⋊C27 |
Generators and relations for C2×C7⋊C27
G = < a,b,c | a2=b7=c27=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C2×C7⋊C27
►Regular action on 378 points
Generators in S378
(1 252)(2 253)(3 254)(4 255)(5 256)(6 257)(7 258)(8 259)(9 260)(10 261)(11 262)(12 263)(13 264)(14 265)(15 266)(16 267)(17 268)(18 269)(19 270)(20 244)(21 245)(22 246)(23 247)(24 248)(25 249)(26 250)(27 251)(28 157)(29 158)(30 159)(31 160)(32 161)(33 162)(34 136)(35 137)(36 138)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 155)(54 156)(55 307)(56 308)(57 309)(58 310)(59 311)(60 312)(61 313)(62 314)(63 315)(64 316)(65 317)(66 318)(67 319)(68 320)(69 321)(70 322)(71 323)(72 324)(73 298)(74 299)(75 300)(76 301)(77 302)(78 303)(79 304)(80 305)(81 306)(82 229)(83 230)(84 231)(85 232)(86 233)(87 234)(88 235)(89 236)(90 237)(91 238)(92 239)(93 240)(94 241)(95 242)(96 243)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 225)(106 226)(107 227)(108 228)(109 284)(110 285)(111 286)(112 287)(113 288)(114 289)(115 290)(116 291)(117 292)(118 293)(119 294)(120 295)(121 296)(122 297)(123 271)(124 272)(125 273)(126 274)(127 275)(128 276)(129 277)(130 278)(131 279)(132 280)(133 281)(134 282)(135 283)(163 214)(164 215)(165 216)(166 190)(167 191)(168 192)(169 193)(170 194)(171 195)(172 196)(173 197)(174 198)(175 199)(176 200)(177 201)(178 202)(179 203)(180 204)(181 205)(182 206)(183 207)(184 208)(185 209)(186 210)(187 211)(188 212)(189 213)(325 378)(326 352)(327 353)(328 354)(329 355)(330 356)(331 357)(332 358)(333 359)(334 360)(335 361)(336 362)(337 363)(338 364)(339 365)(340 366)(341 367)(342 368)(343 369)(344 370)(345 371)(346 372)(347 373)(348 374)(349 375)(350 376)(351 377)
(1 72 160 297 195 327 103)(2 196 73 328 161 104 271)(3 162 197 105 74 272 329)(4 75 136 273 198 330 106)(5 199 76 331 137 107 274)(6 138 200 108 77 275 332)(7 78 139 276 201 333 82)(8 202 79 334 140 83 277)(9 141 203 84 80 278 335)(10 81 142 279 204 336 85)(11 205 55 337 143 86 280)(12 144 206 87 56 281 338)(13 57 145 282 207 339 88)(14 208 58 340 146 89 283)(15 147 209 90 59 284 341)(16 60 148 285 210 342 91)(17 211 61 343 149 92 286)(18 150 212 93 62 287 344)(19 63 151 288 213 345 94)(20 214 64 346 152 95 289)(21 153 215 96 65 290 347)(22 66 154 291 216 348 97)(23 190 67 349 155 98 292)(24 156 191 99 68 293 350)(25 69 157 294 192 351 100)(26 193 70 325 158 101 295)(27 159 194 102 71 296 326)(28 119 168 377 220 249 321)(29 221 120 250 169 322 378)(30 170 222 323 121 352 251)(31 122 171 353 223 252 324)(32 224 123 253 172 298 354)(33 173 225 299 124 355 254)(34 125 174 356 226 255 300)(35 227 126 256 175 301 357)(36 176 228 302 127 358 257)(37 128 177 359 229 258 303)(38 230 129 259 178 304 360)(39 179 231 305 130 361 260)(40 131 180 362 232 261 306)(41 233 132 262 181 307 363)(42 182 234 308 133 364 263)(43 134 183 365 235 264 309)(44 236 135 265 184 310 366)(45 185 237 311 109 367 266)(46 110 186 368 238 267 312)(47 239 111 268 187 313 369)(48 188 240 314 112 370 269)(49 113 189 371 241 270 315)(50 242 114 244 163 316 372)(51 164 243 317 115 373 245)(52 116 165 374 217 246 318)(53 218 117 247 166 319 375)(54 167 219 320 118 376 248)
(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)| (1,252)(2,253)(3,254)(4,255)(5,256)(6,257)(7,258)(8,259)(9,260)(10,261)(11,262)(12,263)(13,264)(14,265)(15,266)(16,267)(17,268)(18,269)(19,270)(20,244)(21,245)(22,246)(23,247)(24,248)(25,249)(26,250)(27,251)(28,157)(29,158)(30,159)(31,160)(32,161)(33,162)(34,136)(35,137)(36,138)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,307)(56,308)(57,309)(58,310)(59,311)(60,312)(61,313)(62,314)(63,315)(64,316)(65,317)(66,318)(67,319)(68,320)(69,321)(70,322)(71,323)(72,324)(73,298)(74,299)(75,300)(76,301)(77,302)(78,303)(79,304)(80,305)(81,306)(82,229)(83,230)(84,231)(85,232)(86,233)(87,234)(88,235)(89,236)(90,237)(91,238)(92,239)(93,240)(94,241)(95,242)(96,243)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,284)(110,285)(111,286)(112,287)(113,288)(114,289)(115,290)(116,291)(117,292)(118,293)(119,294)(120,295)(121,296)(122,297)(123,271)(124,272)(125,273)(126,274)(127,275)(128,276)(129,277)(130,278)(131,279)(132,280)(133,281)(134,282)(135,283)(163,214)(164,215)(165,216)(166,190)(167,191)(168,192)(169,193)(170,194)(171,195)(172,196)(173,197)(174,198)(175,199)(176,200)(177,201)(178,202)(179,203)(180,204)(181,205)(182,206)(183,207)(184,208)(185,209)(186,210)(187,211)(188,212)(189,213)(325,378)(326,352)(327,353)(328,354)(329,355)(330,356)(331,357)(332,358)(333,359)(334,360)(335,361)(336,362)(337,363)(338,364)(339,365)(340,366)(341,367)(342,368)(343,369)(344,370)(345,371)(346,372)(347,373)(348,374)(349,375)(350,376)(351,377), (1,72,160,297,195,327,103)(2,196,73,328,161,104,271)(3,162,197,105,74,272,329)(4,75,136,273,198,330,106)(5,199,76,331,137,107,274)(6,138,200,108,77,275,332)(7,78,139,276,201,333,82)(8,202,79,334,140,83,277)(9,141,203,84,80,278,335)(10,81,142,279,204,336,85)(11,205,55,337,143,86,280)(12,144,206,87,56,281,338)(13,57,145,282,207,339,88)(14,208,58,340,146,89,283)(15,147,209,90,59,284,341)(16,60,148,285,210,342,91)(17,211,61,343,149,92,286)(18,150,212,93,62,287,344)(19,63,151,288,213,345,94)(20,214,64,346,152,95,289)(21,153,215,96,65,290,347)(22,66,154,291,216,348,97)(23,190,67,349,155,98,292)(24,156,191,99,68,293,350)(25,69,157,294,192,351,100)(26,193,70,325,158,101,295)(27,159,194,102,71,296,326)(28,119,168,377,220,249,321)(29,221,120,250,169,322,378)(30,170,222,323,121,352,251)(31,122,171,353,223,252,324)(32,224,123,253,172,298,354)(33,173,225,299,124,355,254)(34,125,174,356,226,255,300)(35,227,126,256,175,301,357)(36,176,228,302,127,358,257)(37,128,177,359,229,258,303)(38,230,129,259,178,304,360)(39,179,231,305,130,361,260)(40,131,180,362,232,261,306)(41,233,132,262,181,307,363)(42,182,234,308,133,364,263)(43,134,183,365,235,264,309)(44,236,135,265,184,310,366)(45,185,237,311,109,367,266)(46,110,186,368,238,267,312)(47,239,111,268,187,313,369)(48,188,240,314,112,370,269)(49,113,189,371,241,270,315)(50,242,114,244,163,316,372)(51,164,243,317,115,373,245)(52,116,165,374,217,246,318)(53,218,117,247,166,319,375)(54,167,219,320,118,376,248), (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( (1,252)(2,253)(3,254)(4,255)(5,256)(6,257)(7,258)(8,259)(9,260)(10,261)(11,262)(12,263)(13,264)(14,265)(15,266)(16,267)(17,268)(18,269)(19,270)(20,244)(21,245)(22,246)(23,247)(24,248)(25,249)(26,250)(27,251)(28,157)(29,158)(30,159)(31,160)(32,161)(33,162)(34,136)(35,137)(36,138)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,307)(56,308)(57,309)(58,310)(59,311)(60,312)(61,313)(62,314)(63,315)(64,316)(65,317)(66,318)(67,319)(68,320)(69,321)(70,322)(71,323)(72,324)(73,298)(74,299)(75,300)(76,301)(77,302)(78,303)(79,304)(80,305)(81,306)(82,229)(83,230)(84,231)(85,232)(86,233)(87,234)(88,235)(89,236)(90,237)(91,238)(92,239)(93,240)(94,241)(95,242)(96,243)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,284)(110,285)(111,286)(112,287)(113,288)(114,289)(115,290)(116,291)(117,292)(118,293)(119,294)(120,295)(121,296)(122,297)(123,271)(124,272)(125,273)(126,274)(127,275)(128,276)(129,277)(130,278)(131,279)(132,280)(133,281)(134,282)(135,283)(163,214)(164,215)(165,216)(166,190)(167,191)(168,192)(169,193)(170,194)(171,195)(172,196)(173,197)(174,198)(175,199)(176,200)(177,201)(178,202)(179,203)(180,204)(181,205)(182,206)(183,207)(184,208)(185,209)(186,210)(187,211)(188,212)(189,213)(325,378)(326,352)(327,353)(328,354)(329,355)(330,356)(331,357)(332,358)(333,359)(334,360)(335,361)(336,362)(337,363)(338,364)(339,365)(340,366)(341,367)(342,368)(343,369)(344,370)(345,371)(346,372)(347,373)(348,374)(349,375)(350,376)(351,377), (1,72,160,297,195,327,103)(2,196,73,328,161,104,271)(3,162,197,105,74,272,329)(4,75,136,273,198,330,106)(5,199,76,331,137,107,274)(6,138,200,108,77,275,332)(7,78,139,276,201,333,82)(8,202,79,334,140,83,277)(9,141,203,84,80,278,335)(10,81,142,279,204,336,85)(11,205,55,337,143,86,280)(12,144,206,87,56,281,338)(13,57,145,282,207,339,88)(14,208,58,340,146,89,283)(15,147,209,90,59,284,341)(16,60,148,285,210,342,91)(17,211,61,343,149,92,286)(18,150,212,93,62,287,344)(19,63,151,288,213,345,94)(20,214,64,346,152,95,289)(21,153,215,96,65,290,347)(22,66,154,291,216,348,97)(23,190,67,349,155,98,292)(24,156,191,99,68,293,350)(25,69,157,294,192,351,100)(26,193,70,325,158,101,295)(27,159,194,102,71,296,326)(28,119,168,377,220,249,321)(29,221,120,250,169,322,378)(30,170,222,323,121,352,251)(31,122,171,353,223,252,324)(32,224,123,253,172,298,354)(33,173,225,299,124,355,254)(34,125,174,356,226,255,300)(35,227,126,256,175,301,357)(36,176,228,302,127,358,257)(37,128,177,359,229,258,303)(38,230,129,259,178,304,360)(39,179,231,305,130,361,260)(40,131,180,362,232,261,306)(41,233,132,262,181,307,363)(42,182,234,308,133,364,263)(43,134,183,365,235,264,309)(44,236,135,265,184,310,366)(45,185,237,311,109,367,266)(46,110,186,368,238,267,312)(47,239,111,268,187,313,369)(48,188,240,314,112,370,269)(49,113,189,371,241,270,315)(50,242,114,244,163,316,372)(51,164,243,317,115,373,245)(52,116,165,374,217,246,318)(53,218,117,247,166,319,375)(54,167,219,320,118,376,248), (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([[(1,252),(2,253),(3,254),(4,255),(5,256),(6,257),(7,258),(8,259),(9,260),(10,261),(11,262),(12,263),(13,264),(14,265),(15,266),(16,267),(17,268),(18,269),(19,270),(20,244),(21,245),(22,246),(23,247),(24,248),(25,249),(26,250),(27,251),(28,157),(29,158),(30,159),(31,160),(32,161),(33,162),(34,136),(35,137),(36,138),(37,139),(38,140),(39,141),(40,142),(41,143),(42,144),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(49,151),(50,152),(51,153),(52,154),(53,155),(54,156),(55,307),(56,308),(57,309),(58,310),(59,311),(60,312),(61,313),(62,314),(63,315),(64,316),(65,317),(66,318),(67,319),(68,320),(69,321),(70,322),(71,323),(72,324),(73,298),(74,299),(75,300),(76,301),(77,302),(78,303),(79,304),(80,305),(81,306),(82,229),(83,230),(84,231),(85,232),(86,233),(87,234),(88,235),(89,236),(90,237),(91,238),(92,239),(93,240),(94,241),(95,242),(96,243),(97,217),(98,218),(99,219),(100,220),(101,221),(102,222),(103,223),(104,224),(105,225),(106,226),(107,227),(108,228),(109,284),(110,285),(111,286),(112,287),(113,288),(114,289),(115,290),(116,291),(117,292),(118,293),(119,294),(120,295),(121,296),(122,297),(123,271),(124,272),(125,273),(126,274),(127,275),(128,276),(129,277),(130,278),(131,279),(132,280),(133,281),(134,282),(135,283),(163,214),(164,215),(165,216),(166,190),(167,191),(168,192),(169,193),(170,194),(171,195),(172,196),(173,197),(174,198),(175,199),(176,200),(177,201),(178,202),(179,203),(180,204),(181,205),(182,206),(183,207),(184,208),(185,209),(186,210),(187,211),(188,212),(189,213),(325,378),(326,352),(327,353),(328,354),(329,355),(330,356),(331,357),(332,358),(333,359),(334,360),(335,361),(336,362),(337,363),(338,364),(339,365),(340,366),(341,367),(342,368),(343,369),(344,370),(345,371),(346,372),(347,373),(348,374),(349,375),(350,376),(351,377)], [(1,72,160,297,195,327,103),(2,196,73,328,161,104,271),(3,162,197,105,74,272,329),(4,75,136,273,198,330,106),(5,199,76,331,137,107,274),(6,138,200,108,77,275,332),(7,78,139,276,201,333,82),(8,202,79,334,140,83,277),(9,141,203,84,80,278,335),(10,81,142,279,204,336,85),(11,205,55,337,143,86,280),(12,144,206,87,56,281,338),(13,57,145,282,207,339,88),(14,208,58,340,146,89,283),(15,147,209,90,59,284,341),(16,60,148,285,210,342,91),(17,211,61,343,149,92,286),(18,150,212,93,62,287,344),(19,63,151,288,213,345,94),(20,214,64,346,152,95,289),(21,153,215,96,65,290,347),(22,66,154,291,216,348,97),(23,190,67,349,155,98,292),(24,156,191,99,68,293,350),(25,69,157,294,192,351,100),(26,193,70,325,158,101,295),(27,159,194,102,71,296,326),(28,119,168,377,220,249,321),(29,221,120,250,169,322,378),(30,170,222,323,121,352,251),(31,122,171,353,223,252,324),(32,224,123,253,172,298,354),(33,173,225,299,124,355,254),(34,125,174,356,226,255,300),(35,227,126,256,175,301,357),(36,176,228,302,127,358,257),(37,128,177,359,229,258,303),(38,230,129,259,178,304,360),(39,179,231,305,130,361,260),(40,131,180,362,232,261,306),(41,233,132,262,181,307,363),(42,182,234,308,133,364,263),(43,134,183,365,235,264,309),(44,236,135,265,184,310,366),(45,185,237,311,109,367,266),(46,110,186,368,238,267,312),(47,239,111,268,187,313,369),(48,188,240,314,112,370,269),(49,113,189,371,241,270,315),(50,242,114,244,163,316,372),(51,164,243,317,115,373,245),(52,116,165,374,217,246,318),(53,218,117,247,166,319,375),(54,167,219,320,118,376,248)], [(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)]])
90 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 9A | ··· | 9F | 14A | 14B | 18A | ··· | 18F | 21A | 21B | 21C | 21D | 27A | ··· | 27R | 42A | 42B | 42C | 42D | 54A | ··· | 54R | 63A | ··· | 63L | 126A | ··· | 126L |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 7 | 9 | ··· | 9 | 14 | 14 | 18 | ··· | 18 | 21 | 21 | 21 | 21 | 27 | ··· | 27 | 42 | 42 | 42 | 42 | 54 | ··· | 54 | 63 | ··· | 63 | 126 | ··· | 126 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 7 | ··· | 7 | 3 | 3 | 3 | 3 | 7 | ··· | 7 | 3 | ··· | 3 | 3 | ··· | 3 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 | C7⋊C3 | C2×C7⋊C3 | C7⋊C9 | C2×C7⋊C9 | C7⋊C27 | C2×C7⋊C27 |
| kernel | C2×C7⋊C27 | C7⋊C27 | C126 | C63 | C42 | C21 | C14 | C7 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 2 | 2 | 4 | 4 | 12 | 12 |
Matrix representation of C2×C7⋊C27 ►in GL4(𝔽379) generated by
| 378 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 27 | 28 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 284 | 185 | 133 |
| 0 | 2 | 223 | 296 |
| 0 | 133 | 104 | 251 |
G:=sub<GL(4,GF(379))| [378,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,27,1,0,0,28,0,1,0,1,0,0],[1,0,0,0,0,284,2,133,0,185,223,104,0,133,296,251] >;
C2×C7⋊C27 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes C_{27} % in TeX
G:=Group("C2xC7:C27"); // GroupNames label
G:=SmallGroup(378,2);
// by ID
G=gap.SmallGroup(378,2);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,36,57,1359]);
// Polycyclic
G:=Group<a,b,c|a^2=b^7=c^27=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
Export