direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C13⋊2C8, C39⋊4C8, C13⋊5C24, C78.4C4, C52.6C6, C156.4C2, C26.5C12, C12.4D13, C6.2Dic13, C4.2(C3×D13), C2.(C3×Dic13), SmallGroup(312,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C3×C13⋊2C8 |
Generators and relations for C3×C13⋊2C8
G = < a,b,c | a3=b13=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C13⋊2C8
►Regular action on 312 points
Generators in S312
(1 105 53)(2 106 54)(3 107 55)(4 108 56)(5 109 57)(6 110 58)(7 111 59)(8 112 60)(9 113 61)(10 114 62)(11 115 63)(12 116 64)(13 117 65)(14 118 66)(15 119 67)(16 120 68)(17 121 69)(18 122 70)(19 123 71)(20 124 72)(21 125 73)(22 126 74)(23 127 75)(24 128 76)(25 129 77)(26 130 78)(27 131 79)(28 132 80)(29 133 81)(30 134 82)(31 135 83)(32 136 84)(33 137 85)(34 138 86)(35 139 87)(36 140 88)(37 141 89)(38 142 90)(39 143 91)(40 144 92)(41 145 93)(42 146 94)(43 147 95)(44 148 96)(45 149 97)(46 150 98)(47 151 99)(48 152 100)(49 153 101)(50 154 102)(51 155 103)(52 156 104)(157 266 209)(158 267 210)(159 268 211)(160 269 212)(161 270 213)(162 271 214)(163 272 215)(164 273 216)(165 261 217)(166 262 218)(167 263 219)(168 264 220)(169 265 221)(170 284 234)(171 285 222)(172 286 223)(173 274 224)(174 275 225)(175 276 226)(176 277 227)(177 278 228)(178 279 229)(179 280 230)(180 281 231)(181 282 232)(182 283 233)(183 288 241)(184 289 242)(185 290 243)(186 291 244)(187 292 245)(188 293 246)(189 294 247)(190 295 235)(191 296 236)(192 297 237)(193 298 238)(194 299 239)(195 287 240)(196 312 254)(197 300 255)(198 301 256)(199 302 257)(200 303 258)(201 304 259)(202 305 260)(203 306 248)(204 307 249)(205 308 250)(206 309 251)(207 310 252)(208 311 253)
(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)
(1 197 40 172 14 185 27 157)(2 196 41 171 15 184 28 169)(3 208 42 170 16 183 29 168)(4 207 43 182 17 195 30 167)(5 206 44 181 18 194 31 166)(6 205 45 180 19 193 32 165)(7 204 46 179 20 192 33 164)(8 203 47 178 21 191 34 163)(9 202 48 177 22 190 35 162)(10 201 49 176 23 189 36 161)(11 200 50 175 24 188 37 160)(12 199 51 174 25 187 38 159)(13 198 52 173 26 186 39 158)(53 255 92 223 66 243 79 209)(54 254 93 222 67 242 80 221)(55 253 94 234 68 241 81 220)(56 252 95 233 69 240 82 219)(57 251 96 232 70 239 83 218)(58 250 97 231 71 238 84 217)(59 249 98 230 72 237 85 216)(60 248 99 229 73 236 86 215)(61 260 100 228 74 235 87 214)(62 259 101 227 75 247 88 213)(63 258 102 226 76 246 89 212)(64 257 103 225 77 245 90 211)(65 256 104 224 78 244 91 210)(105 300 144 286 118 290 131 266)(106 312 145 285 119 289 132 265)(107 311 146 284 120 288 133 264)(108 310 147 283 121 287 134 263)(109 309 148 282 122 299 135 262)(110 308 149 281 123 298 136 261)(111 307 150 280 124 297 137 273)(112 306 151 279 125 296 138 272)(113 305 152 278 126 295 139 271)(114 304 153 277 127 294 140 270)(115 303 154 276 128 293 141 269)(116 302 155 275 129 292 142 268)(117 301 156 274 130 291 143 267)
G:=sub<Sym(312)| (1,105,53)(2,106,54)(3,107,55)(4,108,56)(5,109,57)(6,110,58)(7,111,59)(8,112,60)(9,113,61)(10,114,62)(11,115,63)(12,116,64)(13,117,65)(14,118,66)(15,119,67)(16,120,68)(17,121,69)(18,122,70)(19,123,71)(20,124,72)(21,125,73)(22,126,74)(23,127,75)(24,128,76)(25,129,77)(26,130,78)(27,131,79)(28,132,80)(29,133,81)(30,134,82)(31,135,83)(32,136,84)(33,137,85)(34,138,86)(35,139,87)(36,140,88)(37,141,89)(38,142,90)(39,143,91)(40,144,92)(41,145,93)(42,146,94)(43,147,95)(44,148,96)(45,149,97)(46,150,98)(47,151,99)(48,152,100)(49,153,101)(50,154,102)(51,155,103)(52,156,104)(157,266,209)(158,267,210)(159,268,211)(160,269,212)(161,270,213)(162,271,214)(163,272,215)(164,273,216)(165,261,217)(166,262,218)(167,263,219)(168,264,220)(169,265,221)(170,284,234)(171,285,222)(172,286,223)(173,274,224)(174,275,225)(175,276,226)(176,277,227)(177,278,228)(178,279,229)(179,280,230)(180,281,231)(181,282,232)(182,283,233)(183,288,241)(184,289,242)(185,290,243)(186,291,244)(187,292,245)(188,293,246)(189,294,247)(190,295,235)(191,296,236)(192,297,237)(193,298,238)(194,299,239)(195,287,240)(196,312,254)(197,300,255)(198,301,256)(199,302,257)(200,303,258)(201,304,259)(202,305,260)(203,306,248)(204,307,249)(205,308,250)(206,309,251)(207,310,252)(208,311,253), (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), (1,197,40,172,14,185,27,157)(2,196,41,171,15,184,28,169)(3,208,42,170,16,183,29,168)(4,207,43,182,17,195,30,167)(5,206,44,181,18,194,31,166)(6,205,45,180,19,193,32,165)(7,204,46,179,20,192,33,164)(8,203,47,178,21,191,34,163)(9,202,48,177,22,190,35,162)(10,201,49,176,23,189,36,161)(11,200,50,175,24,188,37,160)(12,199,51,174,25,187,38,159)(13,198,52,173,26,186,39,158)(53,255,92,223,66,243,79,209)(54,254,93,222,67,242,80,221)(55,253,94,234,68,241,81,220)(56,252,95,233,69,240,82,219)(57,251,96,232,70,239,83,218)(58,250,97,231,71,238,84,217)(59,249,98,230,72,237,85,216)(60,248,99,229,73,236,86,215)(61,260,100,228,74,235,87,214)(62,259,101,227,75,247,88,213)(63,258,102,226,76,246,89,212)(64,257,103,225,77,245,90,211)(65,256,104,224,78,244,91,210)(105,300,144,286,118,290,131,266)(106,312,145,285,119,289,132,265)(107,311,146,284,120,288,133,264)(108,310,147,283,121,287,134,263)(109,309,148,282,122,299,135,262)(110,308,149,281,123,298,136,261)(111,307,150,280,124,297,137,273)(112,306,151,279,125,296,138,272)(113,305,152,278,126,295,139,271)(114,304,153,277,127,294,140,270)(115,303,154,276,128,293,141,269)(116,302,155,275,129,292,142,268)(117,301,156,274,130,291,143,267)>;
G:=Group( (1,105,53)(2,106,54)(3,107,55)(4,108,56)(5,109,57)(6,110,58)(7,111,59)(8,112,60)(9,113,61)(10,114,62)(11,115,63)(12,116,64)(13,117,65)(14,118,66)(15,119,67)(16,120,68)(17,121,69)(18,122,70)(19,123,71)(20,124,72)(21,125,73)(22,126,74)(23,127,75)(24,128,76)(25,129,77)(26,130,78)(27,131,79)(28,132,80)(29,133,81)(30,134,82)(31,135,83)(32,136,84)(33,137,85)(34,138,86)(35,139,87)(36,140,88)(37,141,89)(38,142,90)(39,143,91)(40,144,92)(41,145,93)(42,146,94)(43,147,95)(44,148,96)(45,149,97)(46,150,98)(47,151,99)(48,152,100)(49,153,101)(50,154,102)(51,155,103)(52,156,104)(157,266,209)(158,267,210)(159,268,211)(160,269,212)(161,270,213)(162,271,214)(163,272,215)(164,273,216)(165,261,217)(166,262,218)(167,263,219)(168,264,220)(169,265,221)(170,284,234)(171,285,222)(172,286,223)(173,274,224)(174,275,225)(175,276,226)(176,277,227)(177,278,228)(178,279,229)(179,280,230)(180,281,231)(181,282,232)(182,283,233)(183,288,241)(184,289,242)(185,290,243)(186,291,244)(187,292,245)(188,293,246)(189,294,247)(190,295,235)(191,296,236)(192,297,237)(193,298,238)(194,299,239)(195,287,240)(196,312,254)(197,300,255)(198,301,256)(199,302,257)(200,303,258)(201,304,259)(202,305,260)(203,306,248)(204,307,249)(205,308,250)(206,309,251)(207,310,252)(208,311,253), (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), (1,197,40,172,14,185,27,157)(2,196,41,171,15,184,28,169)(3,208,42,170,16,183,29,168)(4,207,43,182,17,195,30,167)(5,206,44,181,18,194,31,166)(6,205,45,180,19,193,32,165)(7,204,46,179,20,192,33,164)(8,203,47,178,21,191,34,163)(9,202,48,177,22,190,35,162)(10,201,49,176,23,189,36,161)(11,200,50,175,24,188,37,160)(12,199,51,174,25,187,38,159)(13,198,52,173,26,186,39,158)(53,255,92,223,66,243,79,209)(54,254,93,222,67,242,80,221)(55,253,94,234,68,241,81,220)(56,252,95,233,69,240,82,219)(57,251,96,232,70,239,83,218)(58,250,97,231,71,238,84,217)(59,249,98,230,72,237,85,216)(60,248,99,229,73,236,86,215)(61,260,100,228,74,235,87,214)(62,259,101,227,75,247,88,213)(63,258,102,226,76,246,89,212)(64,257,103,225,77,245,90,211)(65,256,104,224,78,244,91,210)(105,300,144,286,118,290,131,266)(106,312,145,285,119,289,132,265)(107,311,146,284,120,288,133,264)(108,310,147,283,121,287,134,263)(109,309,148,282,122,299,135,262)(110,308,149,281,123,298,136,261)(111,307,150,280,124,297,137,273)(112,306,151,279,125,296,138,272)(113,305,152,278,126,295,139,271)(114,304,153,277,127,294,140,270)(115,303,154,276,128,293,141,269)(116,302,155,275,129,292,142,268)(117,301,156,274,130,291,143,267) );
G=PermutationGroup([[(1,105,53),(2,106,54),(3,107,55),(4,108,56),(5,109,57),(6,110,58),(7,111,59),(8,112,60),(9,113,61),(10,114,62),(11,115,63),(12,116,64),(13,117,65),(14,118,66),(15,119,67),(16,120,68),(17,121,69),(18,122,70),(19,123,71),(20,124,72),(21,125,73),(22,126,74),(23,127,75),(24,128,76),(25,129,77),(26,130,78),(27,131,79),(28,132,80),(29,133,81),(30,134,82),(31,135,83),(32,136,84),(33,137,85),(34,138,86),(35,139,87),(36,140,88),(37,141,89),(38,142,90),(39,143,91),(40,144,92),(41,145,93),(42,146,94),(43,147,95),(44,148,96),(45,149,97),(46,150,98),(47,151,99),(48,152,100),(49,153,101),(50,154,102),(51,155,103),(52,156,104),(157,266,209),(158,267,210),(159,268,211),(160,269,212),(161,270,213),(162,271,214),(163,272,215),(164,273,216),(165,261,217),(166,262,218),(167,263,219),(168,264,220),(169,265,221),(170,284,234),(171,285,222),(172,286,223),(173,274,224),(174,275,225),(175,276,226),(176,277,227),(177,278,228),(178,279,229),(179,280,230),(180,281,231),(181,282,232),(182,283,233),(183,288,241),(184,289,242),(185,290,243),(186,291,244),(187,292,245),(188,293,246),(189,294,247),(190,295,235),(191,296,236),(192,297,237),(193,298,238),(194,299,239),(195,287,240),(196,312,254),(197,300,255),(198,301,256),(199,302,257),(200,303,258),(201,304,259),(202,305,260),(203,306,248),(204,307,249),(205,308,250),(206,309,251),(207,310,252),(208,311,253)], [(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)], [(1,197,40,172,14,185,27,157),(2,196,41,171,15,184,28,169),(3,208,42,170,16,183,29,168),(4,207,43,182,17,195,30,167),(5,206,44,181,18,194,31,166),(6,205,45,180,19,193,32,165),(7,204,46,179,20,192,33,164),(8,203,47,178,21,191,34,163),(9,202,48,177,22,190,35,162),(10,201,49,176,23,189,36,161),(11,200,50,175,24,188,37,160),(12,199,51,174,25,187,38,159),(13,198,52,173,26,186,39,158),(53,255,92,223,66,243,79,209),(54,254,93,222,67,242,80,221),(55,253,94,234,68,241,81,220),(56,252,95,233,69,240,82,219),(57,251,96,232,70,239,83,218),(58,250,97,231,71,238,84,217),(59,249,98,230,72,237,85,216),(60,248,99,229,73,236,86,215),(61,260,100,228,74,235,87,214),(62,259,101,227,75,247,88,213),(63,258,102,226,76,246,89,212),(64,257,103,225,77,245,90,211),(65,256,104,224,78,244,91,210),(105,300,144,286,118,290,131,266),(106,312,145,285,119,289,132,265),(107,311,146,284,120,288,133,264),(108,310,147,283,121,287,134,263),(109,309,148,282,122,299,135,262),(110,308,149,281,123,298,136,261),(111,307,150,280,124,297,137,273),(112,306,151,279,125,296,138,272),(113,305,152,278,126,295,139,271),(114,304,153,277,127,294,140,270),(115,303,154,276,128,293,141,269),(116,302,155,275,129,292,142,268),(117,301,156,274,130,291,143,267)]])
96 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 13A | ··· | 13F | 24A | ··· | 24H | 26A | ··· | 26F | 39A | ··· | 39L | 52A | ··· | 52L | 78A | ··· | 78L | 156A | ··· | 156X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 13 | ··· | 13 | 24 | ··· | 24 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | D13 | Dic13 | C3×D13 | C13⋊2C8 | C3×Dic13 | C3×C13⋊2C8 |
| kernel | C3×C13⋊2C8 | C156 | C13⋊2C8 | C78 | C52 | C39 | C26 | C13 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 6 | 6 | 12 | 12 | 12 | 24 |
Matrix representation of C3×C13⋊2C8 ►in GL4(𝔽5) generated by
| 2 | 4 | 2 | 4 |
| 1 | 4 | 0 | 3 |
| 0 | 1 | 0 | 4 |
| 1 | 0 | 1 | 2 |
| 1 | 3 | 2 | 3 |
| 0 | 1 | 4 | 4 |
| 2 | 0 | 1 | 3 |
| 3 | 2 | 2 | 1 |
| 3 | 1 | 4 | 3 |
| 0 | 2 | 2 | 3 |
| 4 | 1 | 0 | 2 |
| 4 | 2 | 2 | 0 |
G:=sub<GL(4,GF(5))| [2,1,0,1,4,4,1,0,2,0,0,1,4,3,4,2],[1,0,2,3,3,1,0,2,2,4,1,2,3,4,3,1],[3,0,4,4,1,2,1,2,4,2,0,2,3,3,2,0] >;
C3×C13⋊2C8 in GAP, Magma, Sage, TeX
C_3\times C_{13}\rtimes_2C_8 % in TeX
G:=Group("C3xC13:2C8"); // GroupNames label
G:=SmallGroup(312,4);
// by ID
G=gap.SmallGroup(312,4);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,-13,30,42,7204]);
// Polycyclic
G:=Group<a,b,c|a^3=b^13=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export