direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13×C3⋊C8, C3⋊C104, C39⋊5C8, C6.C52, C78.5C4, C52.4S3, C156.6C2, C12.2C26, C26.3Dic3, C4.2(S3×C13), C2.(Dic3×C13), SmallGroup(312,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C13×C3⋊C8 |
Generators and relations for C13×C3⋊C8
G = < a,b,c | a13=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C13×C3⋊C8
►Regular action on 312 points
Generators in S312
(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 45 139)(2 46 140)(3 47 141)(4 48 142)(5 49 143)(6 50 131)(7 51 132)(8 52 133)(9 40 134)(10 41 135)(11 42 136)(12 43 137)(13 44 138)(14 92 246)(15 93 247)(16 94 235)(17 95 236)(18 96 237)(19 97 238)(20 98 239)(21 99 240)(22 100 241)(23 101 242)(24 102 243)(25 103 244)(26 104 245)(27 88 68)(28 89 69)(29 90 70)(30 91 71)(31 79 72)(32 80 73)(33 81 74)(34 82 75)(35 83 76)(36 84 77)(37 85 78)(38 86 66)(39 87 67)(53 158 213)(54 159 214)(55 160 215)(56 161 216)(57 162 217)(58 163 218)(59 164 219)(60 165 220)(61 166 221)(62 167 209)(63 168 210)(64 169 211)(65 157 212)(105 296 198)(106 297 199)(107 298 200)(108 299 201)(109 287 202)(110 288 203)(111 289 204)(112 290 205)(113 291 206)(114 292 207)(115 293 208)(116 294 196)(117 295 197)(118 279 250)(119 280 251)(120 281 252)(121 282 253)(122 283 254)(123 284 255)(124 285 256)(125 286 257)(126 274 258)(127 275 259)(128 276 260)(129 277 248)(130 278 249)(144 302 230)(145 303 231)(146 304 232)(147 305 233)(148 306 234)(149 307 222)(150 308 223)(151 309 224)(152 310 225)(153 311 226)(154 312 227)(155 300 228)(156 301 229)(170 273 189)(171 261 190)(172 262 191)(173 263 192)(174 264 193)(175 265 194)(176 266 195)(177 267 183)(178 268 184)(179 269 185)(180 270 186)(181 271 187)(182 272 188)
(1 286 84 103 296 160 150 172)(2 274 85 104 297 161 151 173)(3 275 86 92 298 162 152 174)(4 276 87 93 299 163 153 175)(5 277 88 94 287 164 154 176)(6 278 89 95 288 165 155 177)(7 279 90 96 289 166 156 178)(8 280 91 97 290 167 144 179)(9 281 79 98 291 168 145 180)(10 282 80 99 292 169 146 181)(11 283 81 100 293 157 147 182)(12 284 82 101 294 158 148 170)(13 285 83 102 295 159 149 171)(14 200 57 310 193 47 127 66)(15 201 58 311 194 48 128 67)(16 202 59 312 195 49 129 68)(17 203 60 300 183 50 130 69)(18 204 61 301 184 51 118 70)(19 205 62 302 185 52 119 71)(20 206 63 303 186 40 120 72)(21 207 64 304 187 41 121 73)(22 208 65 305 188 42 122 74)(23 196 53 306 189 43 123 75)(24 197 54 307 190 44 124 76)(25 198 55 308 191 45 125 77)(26 199 56 309 192 46 126 78)(27 235 109 219 227 266 143 248)(28 236 110 220 228 267 131 249)(29 237 111 221 229 268 132 250)(30 238 112 209 230 269 133 251)(31 239 113 210 231 270 134 252)(32 240 114 211 232 271 135 253)(33 241 115 212 233 272 136 254)(34 242 116 213 234 273 137 255)(35 243 117 214 222 261 138 256)(36 244 105 215 223 262 139 257)(37 245 106 216 224 263 140 258)(38 246 107 217 225 264 141 259)(39 247 108 218 226 265 142 260)
G:=sub<Sym(312)| (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), (1,45,139)(2,46,140)(3,47,141)(4,48,142)(5,49,143)(6,50,131)(7,51,132)(8,52,133)(9,40,134)(10,41,135)(11,42,136)(12,43,137)(13,44,138)(14,92,246)(15,93,247)(16,94,235)(17,95,236)(18,96,237)(19,97,238)(20,98,239)(21,99,240)(22,100,241)(23,101,242)(24,102,243)(25,103,244)(26,104,245)(27,88,68)(28,89,69)(29,90,70)(30,91,71)(31,79,72)(32,80,73)(33,81,74)(34,82,75)(35,83,76)(36,84,77)(37,85,78)(38,86,66)(39,87,67)(53,158,213)(54,159,214)(55,160,215)(56,161,216)(57,162,217)(58,163,218)(59,164,219)(60,165,220)(61,166,221)(62,167,209)(63,168,210)(64,169,211)(65,157,212)(105,296,198)(106,297,199)(107,298,200)(108,299,201)(109,287,202)(110,288,203)(111,289,204)(112,290,205)(113,291,206)(114,292,207)(115,293,208)(116,294,196)(117,295,197)(118,279,250)(119,280,251)(120,281,252)(121,282,253)(122,283,254)(123,284,255)(124,285,256)(125,286,257)(126,274,258)(127,275,259)(128,276,260)(129,277,248)(130,278,249)(144,302,230)(145,303,231)(146,304,232)(147,305,233)(148,306,234)(149,307,222)(150,308,223)(151,309,224)(152,310,225)(153,311,226)(154,312,227)(155,300,228)(156,301,229)(170,273,189)(171,261,190)(172,262,191)(173,263,192)(174,264,193)(175,265,194)(176,266,195)(177,267,183)(178,268,184)(179,269,185)(180,270,186)(181,271,187)(182,272,188), (1,286,84,103,296,160,150,172)(2,274,85,104,297,161,151,173)(3,275,86,92,298,162,152,174)(4,276,87,93,299,163,153,175)(5,277,88,94,287,164,154,176)(6,278,89,95,288,165,155,177)(7,279,90,96,289,166,156,178)(8,280,91,97,290,167,144,179)(9,281,79,98,291,168,145,180)(10,282,80,99,292,169,146,181)(11,283,81,100,293,157,147,182)(12,284,82,101,294,158,148,170)(13,285,83,102,295,159,149,171)(14,200,57,310,193,47,127,66)(15,201,58,311,194,48,128,67)(16,202,59,312,195,49,129,68)(17,203,60,300,183,50,130,69)(18,204,61,301,184,51,118,70)(19,205,62,302,185,52,119,71)(20,206,63,303,186,40,120,72)(21,207,64,304,187,41,121,73)(22,208,65,305,188,42,122,74)(23,196,53,306,189,43,123,75)(24,197,54,307,190,44,124,76)(25,198,55,308,191,45,125,77)(26,199,56,309,192,46,126,78)(27,235,109,219,227,266,143,248)(28,236,110,220,228,267,131,249)(29,237,111,221,229,268,132,250)(30,238,112,209,230,269,133,251)(31,239,113,210,231,270,134,252)(32,240,114,211,232,271,135,253)(33,241,115,212,233,272,136,254)(34,242,116,213,234,273,137,255)(35,243,117,214,222,261,138,256)(36,244,105,215,223,262,139,257)(37,245,106,216,224,263,140,258)(38,246,107,217,225,264,141,259)(39,247,108,218,226,265,142,260)>;
G:=Group( (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), (1,45,139)(2,46,140)(3,47,141)(4,48,142)(5,49,143)(6,50,131)(7,51,132)(8,52,133)(9,40,134)(10,41,135)(11,42,136)(12,43,137)(13,44,138)(14,92,246)(15,93,247)(16,94,235)(17,95,236)(18,96,237)(19,97,238)(20,98,239)(21,99,240)(22,100,241)(23,101,242)(24,102,243)(25,103,244)(26,104,245)(27,88,68)(28,89,69)(29,90,70)(30,91,71)(31,79,72)(32,80,73)(33,81,74)(34,82,75)(35,83,76)(36,84,77)(37,85,78)(38,86,66)(39,87,67)(53,158,213)(54,159,214)(55,160,215)(56,161,216)(57,162,217)(58,163,218)(59,164,219)(60,165,220)(61,166,221)(62,167,209)(63,168,210)(64,169,211)(65,157,212)(105,296,198)(106,297,199)(107,298,200)(108,299,201)(109,287,202)(110,288,203)(111,289,204)(112,290,205)(113,291,206)(114,292,207)(115,293,208)(116,294,196)(117,295,197)(118,279,250)(119,280,251)(120,281,252)(121,282,253)(122,283,254)(123,284,255)(124,285,256)(125,286,257)(126,274,258)(127,275,259)(128,276,260)(129,277,248)(130,278,249)(144,302,230)(145,303,231)(146,304,232)(147,305,233)(148,306,234)(149,307,222)(150,308,223)(151,309,224)(152,310,225)(153,311,226)(154,312,227)(155,300,228)(156,301,229)(170,273,189)(171,261,190)(172,262,191)(173,263,192)(174,264,193)(175,265,194)(176,266,195)(177,267,183)(178,268,184)(179,269,185)(180,270,186)(181,271,187)(182,272,188), (1,286,84,103,296,160,150,172)(2,274,85,104,297,161,151,173)(3,275,86,92,298,162,152,174)(4,276,87,93,299,163,153,175)(5,277,88,94,287,164,154,176)(6,278,89,95,288,165,155,177)(7,279,90,96,289,166,156,178)(8,280,91,97,290,167,144,179)(9,281,79,98,291,168,145,180)(10,282,80,99,292,169,146,181)(11,283,81,100,293,157,147,182)(12,284,82,101,294,158,148,170)(13,285,83,102,295,159,149,171)(14,200,57,310,193,47,127,66)(15,201,58,311,194,48,128,67)(16,202,59,312,195,49,129,68)(17,203,60,300,183,50,130,69)(18,204,61,301,184,51,118,70)(19,205,62,302,185,52,119,71)(20,206,63,303,186,40,120,72)(21,207,64,304,187,41,121,73)(22,208,65,305,188,42,122,74)(23,196,53,306,189,43,123,75)(24,197,54,307,190,44,124,76)(25,198,55,308,191,45,125,77)(26,199,56,309,192,46,126,78)(27,235,109,219,227,266,143,248)(28,236,110,220,228,267,131,249)(29,237,111,221,229,268,132,250)(30,238,112,209,230,269,133,251)(31,239,113,210,231,270,134,252)(32,240,114,211,232,271,135,253)(33,241,115,212,233,272,136,254)(34,242,116,213,234,273,137,255)(35,243,117,214,222,261,138,256)(36,244,105,215,223,262,139,257)(37,245,106,216,224,263,140,258)(38,246,107,217,225,264,141,259)(39,247,108,218,226,265,142,260) );
G=PermutationGroup([[(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)], [(1,45,139),(2,46,140),(3,47,141),(4,48,142),(5,49,143),(6,50,131),(7,51,132),(8,52,133),(9,40,134),(10,41,135),(11,42,136),(12,43,137),(13,44,138),(14,92,246),(15,93,247),(16,94,235),(17,95,236),(18,96,237),(19,97,238),(20,98,239),(21,99,240),(22,100,241),(23,101,242),(24,102,243),(25,103,244),(26,104,245),(27,88,68),(28,89,69),(29,90,70),(30,91,71),(31,79,72),(32,80,73),(33,81,74),(34,82,75),(35,83,76),(36,84,77),(37,85,78),(38,86,66),(39,87,67),(53,158,213),(54,159,214),(55,160,215),(56,161,216),(57,162,217),(58,163,218),(59,164,219),(60,165,220),(61,166,221),(62,167,209),(63,168,210),(64,169,211),(65,157,212),(105,296,198),(106,297,199),(107,298,200),(108,299,201),(109,287,202),(110,288,203),(111,289,204),(112,290,205),(113,291,206),(114,292,207),(115,293,208),(116,294,196),(117,295,197),(118,279,250),(119,280,251),(120,281,252),(121,282,253),(122,283,254),(123,284,255),(124,285,256),(125,286,257),(126,274,258),(127,275,259),(128,276,260),(129,277,248),(130,278,249),(144,302,230),(145,303,231),(146,304,232),(147,305,233),(148,306,234),(149,307,222),(150,308,223),(151,309,224),(152,310,225),(153,311,226),(154,312,227),(155,300,228),(156,301,229),(170,273,189),(171,261,190),(172,262,191),(173,263,192),(174,264,193),(175,265,194),(176,266,195),(177,267,183),(178,268,184),(179,269,185),(180,270,186),(181,271,187),(182,272,188)], [(1,286,84,103,296,160,150,172),(2,274,85,104,297,161,151,173),(3,275,86,92,298,162,152,174),(4,276,87,93,299,163,153,175),(5,277,88,94,287,164,154,176),(6,278,89,95,288,165,155,177),(7,279,90,96,289,166,156,178),(8,280,91,97,290,167,144,179),(9,281,79,98,291,168,145,180),(10,282,80,99,292,169,146,181),(11,283,81,100,293,157,147,182),(12,284,82,101,294,158,148,170),(13,285,83,102,295,159,149,171),(14,200,57,310,193,47,127,66),(15,201,58,311,194,48,128,67),(16,202,59,312,195,49,129,68),(17,203,60,300,183,50,130,69),(18,204,61,301,184,51,118,70),(19,205,62,302,185,52,119,71),(20,206,63,303,186,40,120,72),(21,207,64,304,187,41,121,73),(22,208,65,305,188,42,122,74),(23,196,53,306,189,43,123,75),(24,197,54,307,190,44,124,76),(25,198,55,308,191,45,125,77),(26,199,56,309,192,46,126,78),(27,235,109,219,227,266,143,248),(28,236,110,220,228,267,131,249),(29,237,111,221,229,268,132,250),(30,238,112,209,230,269,133,251),(31,239,113,210,231,270,134,252),(32,240,114,211,232,271,135,253),(33,241,115,212,233,272,136,254),(34,242,116,213,234,273,137,255),(35,243,117,214,222,261,138,256),(36,244,105,215,223,262,139,257),(37,245,106,216,224,263,140,258),(38,246,107,217,225,264,141,259),(39,247,108,218,226,265,142,260)]])
156 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39L | 52A | ··· | 52X | 78A | ··· | 78L | 104A | ··· | 104AV | 156A | ··· | 156X |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 | 104 | ··· | 104 | 156 | ··· | 156 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C4 | C8 | C13 | C26 | C52 | C104 | S3 | Dic3 | C3⋊C8 | S3×C13 | Dic3×C13 | C13×C3⋊C8 |
| kernel | C13×C3⋊C8 | C156 | C78 | C39 | C3⋊C8 | C12 | C6 | C3 | C52 | C26 | C13 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 12 | 12 | 24 | 48 | 1 | 1 | 2 | 12 | 12 | 24 |
Matrix representation of C13×C3⋊C8 ►in GL3(𝔽313) generated by
| 1 | 0 | 0 |
| 0 | 44 | 0 |
| 0 | 0 | 44 |
| 1 | 0 | 0 |
| 0 | 0 | 312 |
| 0 | 1 | 312 |
| 5 | 0 | 0 |
| 0 | 175 | 271 |
| 0 | 133 | 138 |
G:=sub<GL(3,GF(313))| [1,0,0,0,44,0,0,0,44],[1,0,0,0,0,1,0,312,312],[5,0,0,0,175,133,0,271,138] >;
C13×C3⋊C8 in GAP, Magma, Sage, TeX
C_{13}\times C_3\rtimes C_8 % in TeX
G:=Group("C13xC3:C8"); // GroupNames label
G:=SmallGroup(312,3);
// by ID
G=gap.SmallGroup(312,3);
# by ID
G:=PCGroup([5,-2,-13,-2,-2,-3,130,42,5204]);
// Polycyclic
G:=Group<a,b,c|a^13=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export