direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic26, C39⋊3Q8, C52.5C6, C156.3C2, C6.13D26, C12.3D13, C78.13C22, Dic13.3C6, C4.(C3×D13), C13⋊3(C3×Q8), C26.9(C2×C6), C2.3(C6×D13), (C3×Dic13).2C2, SmallGroup(312,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic26
G = < a,b,c | a3=b52=1, c2=b26, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic26
►Regular action on 312 points
Generators in S312
(1 175 275)(2 176 276)(3 177 277)(4 178 278)(5 179 279)(6 180 280)(7 181 281)(8 182 282)(9 183 283)(10 184 284)(11 185 285)(12 186 286)(13 187 287)(14 188 288)(15 189 289)(16 190 290)(17 191 291)(18 192 292)(19 193 293)(20 194 294)(21 195 295)(22 196 296)(23 197 297)(24 198 298)(25 199 299)(26 200 300)(27 201 301)(28 202 302)(29 203 303)(30 204 304)(31 205 305)(32 206 306)(33 207 307)(34 208 308)(35 157 309)(36 158 310)(37 159 311)(38 160 312)(39 161 261)(40 162 262)(41 163 263)(42 164 264)(43 165 265)(44 166 266)(45 167 267)(46 168 268)(47 169 269)(48 170 270)(49 171 271)(50 172 272)(51 173 273)(52 174 274)(53 126 246)(54 127 247)(55 128 248)(56 129 249)(57 130 250)(58 131 251)(59 132 252)(60 133 253)(61 134 254)(62 135 255)(63 136 256)(64 137 257)(65 138 258)(66 139 259)(67 140 260)(68 141 209)(69 142 210)(70 143 211)(71 144 212)(72 145 213)(73 146 214)(74 147 215)(75 148 216)(76 149 217)(77 150 218)(78 151 219)(79 152 220)(80 153 221)(81 154 222)(82 155 223)(83 156 224)(84 105 225)(85 106 226)(86 107 227)(87 108 228)(88 109 229)(89 110 230)(90 111 231)(91 112 232)(92 113 233)(93 114 234)(94 115 235)(95 116 236)(96 117 237)(97 118 238)(98 119 239)(99 120 240)(100 121 241)(101 122 242)(102 123 243)(103 124 244)(104 125 245)
(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)
(1 62 27 88)(2 61 28 87)(3 60 29 86)(4 59 30 85)(5 58 31 84)(6 57 32 83)(7 56 33 82)(8 55 34 81)(9 54 35 80)(10 53 36 79)(11 104 37 78)(12 103 38 77)(13 102 39 76)(14 101 40 75)(15 100 41 74)(16 99 42 73)(17 98 43 72)(18 97 44 71)(19 96 45 70)(20 95 46 69)(21 94 47 68)(22 93 48 67)(23 92 49 66)(24 91 50 65)(25 90 51 64)(26 89 52 63)(105 179 131 205)(106 178 132 204)(107 177 133 203)(108 176 134 202)(109 175 135 201)(110 174 136 200)(111 173 137 199)(112 172 138 198)(113 171 139 197)(114 170 140 196)(115 169 141 195)(116 168 142 194)(117 167 143 193)(118 166 144 192)(119 165 145 191)(120 164 146 190)(121 163 147 189)(122 162 148 188)(123 161 149 187)(124 160 150 186)(125 159 151 185)(126 158 152 184)(127 157 153 183)(128 208 154 182)(129 207 155 181)(130 206 156 180)(209 295 235 269)(210 294 236 268)(211 293 237 267)(212 292 238 266)(213 291 239 265)(214 290 240 264)(215 289 241 263)(216 288 242 262)(217 287 243 261)(218 286 244 312)(219 285 245 311)(220 284 246 310)(221 283 247 309)(222 282 248 308)(223 281 249 307)(224 280 250 306)(225 279 251 305)(226 278 252 304)(227 277 253 303)(228 276 254 302)(229 275 255 301)(230 274 256 300)(231 273 257 299)(232 272 258 298)(233 271 259 297)(234 270 260 296)
G:=sub<Sym(312)| (1,175,275)(2,176,276)(3,177,277)(4,178,278)(5,179,279)(6,180,280)(7,181,281)(8,182,282)(9,183,283)(10,184,284)(11,185,285)(12,186,286)(13,187,287)(14,188,288)(15,189,289)(16,190,290)(17,191,291)(18,192,292)(19,193,293)(20,194,294)(21,195,295)(22,196,296)(23,197,297)(24,198,298)(25,199,299)(26,200,300)(27,201,301)(28,202,302)(29,203,303)(30,204,304)(31,205,305)(32,206,306)(33,207,307)(34,208,308)(35,157,309)(36,158,310)(37,159,311)(38,160,312)(39,161,261)(40,162,262)(41,163,263)(42,164,264)(43,165,265)(44,166,266)(45,167,267)(46,168,268)(47,169,269)(48,170,270)(49,171,271)(50,172,272)(51,173,273)(52,174,274)(53,126,246)(54,127,247)(55,128,248)(56,129,249)(57,130,250)(58,131,251)(59,132,252)(60,133,253)(61,134,254)(62,135,255)(63,136,256)(64,137,257)(65,138,258)(66,139,259)(67,140,260)(68,141,209)(69,142,210)(70,143,211)(71,144,212)(72,145,213)(73,146,214)(74,147,215)(75,148,216)(76,149,217)(77,150,218)(78,151,219)(79,152,220)(80,153,221)(81,154,222)(82,155,223)(83,156,224)(84,105,225)(85,106,226)(86,107,227)(87,108,228)(88,109,229)(89,110,230)(90,111,231)(91,112,232)(92,113,233)(93,114,234)(94,115,235)(95,116,236)(96,117,237)(97,118,238)(98,119,239)(99,120,240)(100,121,241)(101,122,242)(102,123,243)(103,124,244)(104,125,245), (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), (1,62,27,88)(2,61,28,87)(3,60,29,86)(4,59,30,85)(5,58,31,84)(6,57,32,83)(7,56,33,82)(8,55,34,81)(9,54,35,80)(10,53,36,79)(11,104,37,78)(12,103,38,77)(13,102,39,76)(14,101,40,75)(15,100,41,74)(16,99,42,73)(17,98,43,72)(18,97,44,71)(19,96,45,70)(20,95,46,69)(21,94,47,68)(22,93,48,67)(23,92,49,66)(24,91,50,65)(25,90,51,64)(26,89,52,63)(105,179,131,205)(106,178,132,204)(107,177,133,203)(108,176,134,202)(109,175,135,201)(110,174,136,200)(111,173,137,199)(112,172,138,198)(113,171,139,197)(114,170,140,196)(115,169,141,195)(116,168,142,194)(117,167,143,193)(118,166,144,192)(119,165,145,191)(120,164,146,190)(121,163,147,189)(122,162,148,188)(123,161,149,187)(124,160,150,186)(125,159,151,185)(126,158,152,184)(127,157,153,183)(128,208,154,182)(129,207,155,181)(130,206,156,180)(209,295,235,269)(210,294,236,268)(211,293,237,267)(212,292,238,266)(213,291,239,265)(214,290,240,264)(215,289,241,263)(216,288,242,262)(217,287,243,261)(218,286,244,312)(219,285,245,311)(220,284,246,310)(221,283,247,309)(222,282,248,308)(223,281,249,307)(224,280,250,306)(225,279,251,305)(226,278,252,304)(227,277,253,303)(228,276,254,302)(229,275,255,301)(230,274,256,300)(231,273,257,299)(232,272,258,298)(233,271,259,297)(234,270,260,296)>;
G:=Group( (1,175,275)(2,176,276)(3,177,277)(4,178,278)(5,179,279)(6,180,280)(7,181,281)(8,182,282)(9,183,283)(10,184,284)(11,185,285)(12,186,286)(13,187,287)(14,188,288)(15,189,289)(16,190,290)(17,191,291)(18,192,292)(19,193,293)(20,194,294)(21,195,295)(22,196,296)(23,197,297)(24,198,298)(25,199,299)(26,200,300)(27,201,301)(28,202,302)(29,203,303)(30,204,304)(31,205,305)(32,206,306)(33,207,307)(34,208,308)(35,157,309)(36,158,310)(37,159,311)(38,160,312)(39,161,261)(40,162,262)(41,163,263)(42,164,264)(43,165,265)(44,166,266)(45,167,267)(46,168,268)(47,169,269)(48,170,270)(49,171,271)(50,172,272)(51,173,273)(52,174,274)(53,126,246)(54,127,247)(55,128,248)(56,129,249)(57,130,250)(58,131,251)(59,132,252)(60,133,253)(61,134,254)(62,135,255)(63,136,256)(64,137,257)(65,138,258)(66,139,259)(67,140,260)(68,141,209)(69,142,210)(70,143,211)(71,144,212)(72,145,213)(73,146,214)(74,147,215)(75,148,216)(76,149,217)(77,150,218)(78,151,219)(79,152,220)(80,153,221)(81,154,222)(82,155,223)(83,156,224)(84,105,225)(85,106,226)(86,107,227)(87,108,228)(88,109,229)(89,110,230)(90,111,231)(91,112,232)(92,113,233)(93,114,234)(94,115,235)(95,116,236)(96,117,237)(97,118,238)(98,119,239)(99,120,240)(100,121,241)(101,122,242)(102,123,243)(103,124,244)(104,125,245), (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), (1,62,27,88)(2,61,28,87)(3,60,29,86)(4,59,30,85)(5,58,31,84)(6,57,32,83)(7,56,33,82)(8,55,34,81)(9,54,35,80)(10,53,36,79)(11,104,37,78)(12,103,38,77)(13,102,39,76)(14,101,40,75)(15,100,41,74)(16,99,42,73)(17,98,43,72)(18,97,44,71)(19,96,45,70)(20,95,46,69)(21,94,47,68)(22,93,48,67)(23,92,49,66)(24,91,50,65)(25,90,51,64)(26,89,52,63)(105,179,131,205)(106,178,132,204)(107,177,133,203)(108,176,134,202)(109,175,135,201)(110,174,136,200)(111,173,137,199)(112,172,138,198)(113,171,139,197)(114,170,140,196)(115,169,141,195)(116,168,142,194)(117,167,143,193)(118,166,144,192)(119,165,145,191)(120,164,146,190)(121,163,147,189)(122,162,148,188)(123,161,149,187)(124,160,150,186)(125,159,151,185)(126,158,152,184)(127,157,153,183)(128,208,154,182)(129,207,155,181)(130,206,156,180)(209,295,235,269)(210,294,236,268)(211,293,237,267)(212,292,238,266)(213,291,239,265)(214,290,240,264)(215,289,241,263)(216,288,242,262)(217,287,243,261)(218,286,244,312)(219,285,245,311)(220,284,246,310)(221,283,247,309)(222,282,248,308)(223,281,249,307)(224,280,250,306)(225,279,251,305)(226,278,252,304)(227,277,253,303)(228,276,254,302)(229,275,255,301)(230,274,256,300)(231,273,257,299)(232,272,258,298)(233,271,259,297)(234,270,260,296) );
G=PermutationGroup([[(1,175,275),(2,176,276),(3,177,277),(4,178,278),(5,179,279),(6,180,280),(7,181,281),(8,182,282),(9,183,283),(10,184,284),(11,185,285),(12,186,286),(13,187,287),(14,188,288),(15,189,289),(16,190,290),(17,191,291),(18,192,292),(19,193,293),(20,194,294),(21,195,295),(22,196,296),(23,197,297),(24,198,298),(25,199,299),(26,200,300),(27,201,301),(28,202,302),(29,203,303),(30,204,304),(31,205,305),(32,206,306),(33,207,307),(34,208,308),(35,157,309),(36,158,310),(37,159,311),(38,160,312),(39,161,261),(40,162,262),(41,163,263),(42,164,264),(43,165,265),(44,166,266),(45,167,267),(46,168,268),(47,169,269),(48,170,270),(49,171,271),(50,172,272),(51,173,273),(52,174,274),(53,126,246),(54,127,247),(55,128,248),(56,129,249),(57,130,250),(58,131,251),(59,132,252),(60,133,253),(61,134,254),(62,135,255),(63,136,256),(64,137,257),(65,138,258),(66,139,259),(67,140,260),(68,141,209),(69,142,210),(70,143,211),(71,144,212),(72,145,213),(73,146,214),(74,147,215),(75,148,216),(76,149,217),(77,150,218),(78,151,219),(79,152,220),(80,153,221),(81,154,222),(82,155,223),(83,156,224),(84,105,225),(85,106,226),(86,107,227),(87,108,228),(88,109,229),(89,110,230),(90,111,231),(91,112,232),(92,113,233),(93,114,234),(94,115,235),(95,116,236),(96,117,237),(97,118,238),(98,119,239),(99,120,240),(100,121,241),(101,122,242),(102,123,243),(103,124,244),(104,125,245)], [(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)], [(1,62,27,88),(2,61,28,87),(3,60,29,86),(4,59,30,85),(5,58,31,84),(6,57,32,83),(7,56,33,82),(8,55,34,81),(9,54,35,80),(10,53,36,79),(11,104,37,78),(12,103,38,77),(13,102,39,76),(14,101,40,75),(15,100,41,74),(16,99,42,73),(17,98,43,72),(18,97,44,71),(19,96,45,70),(20,95,46,69),(21,94,47,68),(22,93,48,67),(23,92,49,66),(24,91,50,65),(25,90,51,64),(26,89,52,63),(105,179,131,205),(106,178,132,204),(107,177,133,203),(108,176,134,202),(109,175,135,201),(110,174,136,200),(111,173,137,199),(112,172,138,198),(113,171,139,197),(114,170,140,196),(115,169,141,195),(116,168,142,194),(117,167,143,193),(118,166,144,192),(119,165,145,191),(120,164,146,190),(121,163,147,189),(122,162,148,188),(123,161,149,187),(124,160,150,186),(125,159,151,185),(126,158,152,184),(127,157,153,183),(128,208,154,182),(129,207,155,181),(130,206,156,180),(209,295,235,269),(210,294,236,268),(211,293,237,267),(212,292,238,266),(213,291,239,265),(214,290,240,264),(215,289,241,263),(216,288,242,262),(217,287,243,261),(218,286,244,312),(219,285,245,311),(220,284,246,310),(221,283,247,309),(222,282,248,308),(223,281,249,307),(224,280,250,306),(225,279,251,305),(226,278,252,304),(227,277,253,303),(228,276,254,302),(229,275,255,301),(230,274,256,300),(231,273,257,299),(232,272,258,298),(233,271,259,297),(234,270,260,296)]])
87 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 12A | 12B | 12C | 12D | 12E | 12F | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39L | 52A | ··· | 52L | 78A | ··· | 78L | 156A | ··· | 156X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 1 | 1 | 2 | 26 | 26 | 1 | 1 | 2 | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
87 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | Q8 | C3×Q8 | D13 | D26 | C3×D13 | Dic26 | C6×D13 | C3×Dic26 |
| kernel | C3×Dic26 | C3×Dic13 | C156 | Dic26 | Dic13 | C52 | C39 | C13 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 2 | 6 | 6 | 12 | 12 | 12 | 24 |
Matrix representation of C3×Dic26 ►in GL2(𝔽157) generated by
| 144 | 0 |
| 0 | 144 |
| 6 | 106 |
| 64 | 58 |
| 136 | 41 |
| 39 | 21 |
G:=sub<GL(2,GF(157))| [144,0,0,144],[6,64,106,58],[136,39,41,21] >;
C3×Dic26 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{26} % in TeX
G:=Group("C3xDic26"); // GroupNames label
G:=SmallGroup(312,27);
// by ID
G=gap.SmallGroup(312,27);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-13,60,141,66,7204]);
// Polycyclic
G:=Group<a,b,c|a^3=b^52=1,c^2=b^26,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export