direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C13⋊C8, C39⋊2C8, C13⋊3C24, C78.2C4, C26.3C12, Dic13.4C6, C6.2(C13⋊C4), (C3×Dic13).4C2, C2.(C3×C13⋊C4), SmallGroup(312,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C3×C13⋊C8 |
Generators and relations for C3×C13⋊C8
G = < a,b,c | a3=b13=c8=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C3×C13⋊C8
►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)(27 137 82)(28 138 83)(29 139 84)(30 140 85)(31 141 86)(32 142 87)(33 143 88)(34 131 89)(35 132 90)(36 133 91)(37 134 79)(38 135 80)(39 136 81)(40 154 104)(41 155 92)(42 156 93)(43 144 94)(44 145 95)(45 146 96)(46 147 97)(47 148 98)(48 149 99)(49 150 100)(50 151 101)(51 152 102)(52 153 103)(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)(170 275 230)(171 276 231)(172 277 232)(173 278 233)(174 279 234)(175 280 222)(176 281 223)(177 282 224)(178 283 225)(179 284 226)(180 285 227)(181 286 228)(182 274 229)(183 289 241)(184 290 242)(185 291 243)(186 292 244)(187 293 245)(188 294 246)(189 295 247)(190 296 235)(191 297 236)(192 298 237)(193 299 238)(194 287 239)(195 288 240)(196 304 253)(197 305 254)(198 306 255)(199 307 256)(200 308 257)(201 309 258)(202 310 259)(203 311 260)(204 312 248)(205 300 249)(206 301 250)(207 302 251)(208 303 252)
(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)
(1 198 41 177 14 187 34 167)(2 206 40 182 15 195 33 159)(3 201 52 174 16 190 32 164)(4 196 51 179 17 185 31 169)(5 204 50 171 18 193 30 161)(6 199 49 176 19 188 29 166)(7 207 48 181 20 183 28 158)(8 202 47 173 21 191 27 163)(9 197 46 178 22 186 39 168)(10 205 45 170 23 194 38 160)(11 200 44 175 24 189 37 165)(12 208 43 180 25 184 36 157)(13 203 42 172 26 192 35 162)(53 255 92 224 66 245 89 219)(54 250 104 229 67 240 88 211)(55 258 103 234 68 235 87 216)(56 253 102 226 69 243 86 221)(57 248 101 231 70 238 85 213)(58 256 100 223 71 246 84 218)(59 251 99 228 72 241 83 210)(60 259 98 233 73 236 82 215)(61 254 97 225 74 244 81 220)(62 249 96 230 75 239 80 212)(63 257 95 222 76 247 79 217)(64 252 94 227 77 242 91 209)(65 260 93 232 78 237 90 214)(105 306 155 282 118 293 131 263)(106 301 154 274 119 288 143 268)(107 309 153 279 120 296 142 273)(108 304 152 284 121 291 141 265)(109 312 151 276 122 299 140 270)(110 307 150 281 123 294 139 262)(111 302 149 286 124 289 138 267)(112 310 148 278 125 297 137 272)(113 305 147 283 126 292 136 264)(114 300 146 275 127 287 135 269)(115 308 145 280 128 295 134 261)(116 303 144 285 129 290 133 266)(117 311 156 277 130 298 132 271)
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)(27,137,82)(28,138,83)(29,139,84)(30,140,85)(31,141,86)(32,142,87)(33,143,88)(34,131,89)(35,132,90)(36,133,91)(37,134,79)(38,135,80)(39,136,81)(40,154,104)(41,155,92)(42,156,93)(43,144,94)(44,145,95)(45,146,96)(46,147,97)(47,148,98)(48,149,99)(49,150,100)(50,151,101)(51,152,102)(52,153,103)(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)(170,275,230)(171,276,231)(172,277,232)(173,278,233)(174,279,234)(175,280,222)(176,281,223)(177,282,224)(178,283,225)(179,284,226)(180,285,227)(181,286,228)(182,274,229)(183,289,241)(184,290,242)(185,291,243)(186,292,244)(187,293,245)(188,294,246)(189,295,247)(190,296,235)(191,297,236)(192,298,237)(193,299,238)(194,287,239)(195,288,240)(196,304,253)(197,305,254)(198,306,255)(199,307,256)(200,308,257)(201,309,258)(202,310,259)(203,311,260)(204,312,248)(205,300,249)(206,301,250)(207,302,251)(208,303,252), (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), (1,198,41,177,14,187,34,167)(2,206,40,182,15,195,33,159)(3,201,52,174,16,190,32,164)(4,196,51,179,17,185,31,169)(5,204,50,171,18,193,30,161)(6,199,49,176,19,188,29,166)(7,207,48,181,20,183,28,158)(8,202,47,173,21,191,27,163)(9,197,46,178,22,186,39,168)(10,205,45,170,23,194,38,160)(11,200,44,175,24,189,37,165)(12,208,43,180,25,184,36,157)(13,203,42,172,26,192,35,162)(53,255,92,224,66,245,89,219)(54,250,104,229,67,240,88,211)(55,258,103,234,68,235,87,216)(56,253,102,226,69,243,86,221)(57,248,101,231,70,238,85,213)(58,256,100,223,71,246,84,218)(59,251,99,228,72,241,83,210)(60,259,98,233,73,236,82,215)(61,254,97,225,74,244,81,220)(62,249,96,230,75,239,80,212)(63,257,95,222,76,247,79,217)(64,252,94,227,77,242,91,209)(65,260,93,232,78,237,90,214)(105,306,155,282,118,293,131,263)(106,301,154,274,119,288,143,268)(107,309,153,279,120,296,142,273)(108,304,152,284,121,291,141,265)(109,312,151,276,122,299,140,270)(110,307,150,281,123,294,139,262)(111,302,149,286,124,289,138,267)(112,310,148,278,125,297,137,272)(113,305,147,283,126,292,136,264)(114,300,146,275,127,287,135,269)(115,308,145,280,128,295,134,261)(116,303,144,285,129,290,133,266)(117,311,156,277,130,298,132,271)>;
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)(27,137,82)(28,138,83)(29,139,84)(30,140,85)(31,141,86)(32,142,87)(33,143,88)(34,131,89)(35,132,90)(36,133,91)(37,134,79)(38,135,80)(39,136,81)(40,154,104)(41,155,92)(42,156,93)(43,144,94)(44,145,95)(45,146,96)(46,147,97)(47,148,98)(48,149,99)(49,150,100)(50,151,101)(51,152,102)(52,153,103)(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)(170,275,230)(171,276,231)(172,277,232)(173,278,233)(174,279,234)(175,280,222)(176,281,223)(177,282,224)(178,283,225)(179,284,226)(180,285,227)(181,286,228)(182,274,229)(183,289,241)(184,290,242)(185,291,243)(186,292,244)(187,293,245)(188,294,246)(189,295,247)(190,296,235)(191,297,236)(192,298,237)(193,299,238)(194,287,239)(195,288,240)(196,304,253)(197,305,254)(198,306,255)(199,307,256)(200,308,257)(201,309,258)(202,310,259)(203,311,260)(204,312,248)(205,300,249)(206,301,250)(207,302,251)(208,303,252), (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), (1,198,41,177,14,187,34,167)(2,206,40,182,15,195,33,159)(3,201,52,174,16,190,32,164)(4,196,51,179,17,185,31,169)(5,204,50,171,18,193,30,161)(6,199,49,176,19,188,29,166)(7,207,48,181,20,183,28,158)(8,202,47,173,21,191,27,163)(9,197,46,178,22,186,39,168)(10,205,45,170,23,194,38,160)(11,200,44,175,24,189,37,165)(12,208,43,180,25,184,36,157)(13,203,42,172,26,192,35,162)(53,255,92,224,66,245,89,219)(54,250,104,229,67,240,88,211)(55,258,103,234,68,235,87,216)(56,253,102,226,69,243,86,221)(57,248,101,231,70,238,85,213)(58,256,100,223,71,246,84,218)(59,251,99,228,72,241,83,210)(60,259,98,233,73,236,82,215)(61,254,97,225,74,244,81,220)(62,249,96,230,75,239,80,212)(63,257,95,222,76,247,79,217)(64,252,94,227,77,242,91,209)(65,260,93,232,78,237,90,214)(105,306,155,282,118,293,131,263)(106,301,154,274,119,288,143,268)(107,309,153,279,120,296,142,273)(108,304,152,284,121,291,141,265)(109,312,151,276,122,299,140,270)(110,307,150,281,123,294,139,262)(111,302,149,286,124,289,138,267)(112,310,148,278,125,297,137,272)(113,305,147,283,126,292,136,264)(114,300,146,275,127,287,135,269)(115,308,145,280,128,295,134,261)(116,303,144,285,129,290,133,266)(117,311,156,277,130,298,132,271) );
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),(27,137,82),(28,138,83),(29,139,84),(30,140,85),(31,141,86),(32,142,87),(33,143,88),(34,131,89),(35,132,90),(36,133,91),(37,134,79),(38,135,80),(39,136,81),(40,154,104),(41,155,92),(42,156,93),(43,144,94),(44,145,95),(45,146,96),(46,147,97),(47,148,98),(48,149,99),(49,150,100),(50,151,101),(51,152,102),(52,153,103),(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),(170,275,230),(171,276,231),(172,277,232),(173,278,233),(174,279,234),(175,280,222),(176,281,223),(177,282,224),(178,283,225),(179,284,226),(180,285,227),(181,286,228),(182,274,229),(183,289,241),(184,290,242),(185,291,243),(186,292,244),(187,293,245),(188,294,246),(189,295,247),(190,296,235),(191,297,236),(192,298,237),(193,299,238),(194,287,239),(195,288,240),(196,304,253),(197,305,254),(198,306,255),(199,307,256),(200,308,257),(201,309,258),(202,310,259),(203,311,260),(204,312,248),(205,300,249),(206,301,250),(207,302,251),(208,303,252)], [(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)], [(1,198,41,177,14,187,34,167),(2,206,40,182,15,195,33,159),(3,201,52,174,16,190,32,164),(4,196,51,179,17,185,31,169),(5,204,50,171,18,193,30,161),(6,199,49,176,19,188,29,166),(7,207,48,181,20,183,28,158),(8,202,47,173,21,191,27,163),(9,197,46,178,22,186,39,168),(10,205,45,170,23,194,38,160),(11,200,44,175,24,189,37,165),(12,208,43,180,25,184,36,157),(13,203,42,172,26,192,35,162),(53,255,92,224,66,245,89,219),(54,250,104,229,67,240,88,211),(55,258,103,234,68,235,87,216),(56,253,102,226,69,243,86,221),(57,248,101,231,70,238,85,213),(58,256,100,223,71,246,84,218),(59,251,99,228,72,241,83,210),(60,259,98,233,73,236,82,215),(61,254,97,225,74,244,81,220),(62,249,96,230,75,239,80,212),(63,257,95,222,76,247,79,217),(64,252,94,227,77,242,91,209),(65,260,93,232,78,237,90,214),(105,306,155,282,118,293,131,263),(106,301,154,274,119,288,143,268),(107,309,153,279,120,296,142,273),(108,304,152,284,121,291,141,265),(109,312,151,276,122,299,140,270),(110,307,150,281,123,294,139,262),(111,302,149,286,124,289,138,267),(112,310,148,278,125,297,137,272),(113,305,147,283,126,292,136,264),(114,300,146,275,127,287,135,269),(115,308,145,280,128,295,134,261),(116,303,144,285,129,290,133,266),(117,311,156,277,130,298,132,271)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 13A | 13B | 13C | 24A | ··· | 24H | 26A | 26B | 26C | 39A | ··· | 39F | 78A | ··· | 78F |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 13 | 13 | 13 | 24 | ··· | 24 | 26 | 26 | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | 1 | 13 | 13 | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 4 | 4 | 4 | 13 | ··· | 13 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | C13⋊C4 | C13⋊C8 | C3×C13⋊C4 | C3×C13⋊C8 |
| kernel | C3×C13⋊C8 | C3×Dic13 | C13⋊C8 | C78 | Dic13 | C39 | C26 | C13 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 3 | 3 | 6 | 6 |
Matrix representation of C3×C13⋊C8 ►in GL5(𝔽313)
| 214 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 31 | 282 | 242 | 312 |
| 0 | 32 | 282 | 242 | 312 |
| 0 | 31 | 283 | 242 | 312 |
| 0 | 31 | 282 | 243 | 312 |
| 312 | 0 | 0 | 0 | 0 |
| 0 | 179 | 242 | 201 | 111 |
| 0 | 48 | 10 | 144 | 75 |
| 0 | 103 | 197 | 245 | 227 |
| 0 | 180 | 259 | 309 | 192 |
G:=sub<GL(5,GF(313))| [214,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,31,32,31,31,0,282,282,283,282,0,242,242,242,243,0,312,312,312,312],[312,0,0,0,0,0,179,48,103,180,0,242,10,197,259,0,201,144,245,309,0,111,75,227,192] >;
C3×C13⋊C8 in GAP, Magma, Sage, TeX
C_3\times C_{13}\rtimes C_8 % in TeX
G:=Group("C3xC13:C8"); // GroupNames label
G:=SmallGroup(312,13);
// by ID
G=gap.SmallGroup(312,13);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,-13,30,42,4804,1214]);
// 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^5>;
// generators/relations
Export