direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C13×Dic6, C39⋊4Q8, C52.3S3, C156.5C2, C12.1C26, C26.13D6, Dic3.C26, C78.18C22, C3⋊(Q8×C13), C4.(S3×C13), C2.3(S3×C26), C6.1(C2×C26), (Dic3×C13).2C2, SmallGroup(312,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×Dic6
G = < a,b,c | a13=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C13×Dic6
►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)
(1 308 37 206 187 110 247 233 93 287 58 71)(2 309 38 207 188 111 235 234 94 288 59 72)(3 310 39 208 189 112 236 222 95 289 60 73)(4 311 27 196 190 113 237 223 96 290 61 74)(5 312 28 197 191 114 238 224 97 291 62 75)(6 300 29 198 192 115 239 225 98 292 63 76)(7 301 30 199 193 116 240 226 99 293 64 77)(8 302 31 200 194 117 241 227 100 294 65 78)(9 303 32 201 195 105 242 228 101 295 53 66)(10 304 33 202 183 106 243 229 102 296 54 67)(11 305 34 203 184 107 244 230 103 297 55 68)(12 306 35 204 185 108 245 231 104 298 56 69)(13 307 36 205 186 109 246 232 92 299 57 70)(14 215 82 174 266 285 121 158 249 135 156 52)(15 216 83 175 267 286 122 159 250 136 144 40)(16 217 84 176 268 274 123 160 251 137 145 41)(17 218 85 177 269 275 124 161 252 138 146 42)(18 219 86 178 270 276 125 162 253 139 147 43)(19 220 87 179 271 277 126 163 254 140 148 44)(20 221 88 180 272 278 127 164 255 141 149 45)(21 209 89 181 273 279 128 165 256 142 150 46)(22 210 90 182 261 280 129 166 257 143 151 47)(23 211 91 170 262 281 130 167 258 131 152 48)(24 212 79 171 263 282 118 168 259 132 153 49)(25 213 80 172 264 283 119 169 260 133 154 50)(26 214 81 173 265 284 120 157 248 134 155 51)
(1 17 247 124)(2 18 235 125)(3 19 236 126)(4 20 237 127)(5 21 238 128)(6 22 239 129)(7 23 240 130)(8 24 241 118)(9 25 242 119)(10 26 243 120)(11 14 244 121)(12 15 245 122)(13 16 246 123)(27 149 96 272)(28 150 97 273)(29 151 98 261)(30 152 99 262)(31 153 100 263)(32 154 101 264)(33 155 102 265)(34 156 103 266)(35 144 104 267)(36 145 92 268)(37 146 93 269)(38 147 94 270)(39 148 95 271)(40 231 286 306)(41 232 274 307)(42 233 275 308)(43 234 276 309)(44 222 277 310)(45 223 278 311)(46 224 279 312)(47 225 280 300)(48 226 281 301)(49 227 282 302)(50 228 283 303)(51 229 284 304)(52 230 285 305)(53 80 195 260)(54 81 183 248)(55 82 184 249)(56 83 185 250)(57 84 186 251)(58 85 187 252)(59 86 188 253)(60 87 189 254)(61 88 190 255)(62 89 191 256)(63 90 192 257)(64 91 193 258)(65 79 194 259)(66 213 105 169)(67 214 106 157)(68 215 107 158)(69 216 108 159)(70 217 109 160)(71 218 110 161)(72 219 111 162)(73 220 112 163)(74 221 113 164)(75 209 114 165)(76 210 115 166)(77 211 116 167)(78 212 117 168)(131 293 170 199)(132 294 171 200)(133 295 172 201)(134 296 173 202)(135 297 174 203)(136 298 175 204)(137 299 176 205)(138 287 177 206)(139 288 178 207)(140 289 179 208)(141 290 180 196)(142 291 181 197)(143 292 182 198)
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), (1,308,37,206,187,110,247,233,93,287,58,71)(2,309,38,207,188,111,235,234,94,288,59,72)(3,310,39,208,189,112,236,222,95,289,60,73)(4,311,27,196,190,113,237,223,96,290,61,74)(5,312,28,197,191,114,238,224,97,291,62,75)(6,300,29,198,192,115,239,225,98,292,63,76)(7,301,30,199,193,116,240,226,99,293,64,77)(8,302,31,200,194,117,241,227,100,294,65,78)(9,303,32,201,195,105,242,228,101,295,53,66)(10,304,33,202,183,106,243,229,102,296,54,67)(11,305,34,203,184,107,244,230,103,297,55,68)(12,306,35,204,185,108,245,231,104,298,56,69)(13,307,36,205,186,109,246,232,92,299,57,70)(14,215,82,174,266,285,121,158,249,135,156,52)(15,216,83,175,267,286,122,159,250,136,144,40)(16,217,84,176,268,274,123,160,251,137,145,41)(17,218,85,177,269,275,124,161,252,138,146,42)(18,219,86,178,270,276,125,162,253,139,147,43)(19,220,87,179,271,277,126,163,254,140,148,44)(20,221,88,180,272,278,127,164,255,141,149,45)(21,209,89,181,273,279,128,165,256,142,150,46)(22,210,90,182,261,280,129,166,257,143,151,47)(23,211,91,170,262,281,130,167,258,131,152,48)(24,212,79,171,263,282,118,168,259,132,153,49)(25,213,80,172,264,283,119,169,260,133,154,50)(26,214,81,173,265,284,120,157,248,134,155,51), (1,17,247,124)(2,18,235,125)(3,19,236,126)(4,20,237,127)(5,21,238,128)(6,22,239,129)(7,23,240,130)(8,24,241,118)(9,25,242,119)(10,26,243,120)(11,14,244,121)(12,15,245,122)(13,16,246,123)(27,149,96,272)(28,150,97,273)(29,151,98,261)(30,152,99,262)(31,153,100,263)(32,154,101,264)(33,155,102,265)(34,156,103,266)(35,144,104,267)(36,145,92,268)(37,146,93,269)(38,147,94,270)(39,148,95,271)(40,231,286,306)(41,232,274,307)(42,233,275,308)(43,234,276,309)(44,222,277,310)(45,223,278,311)(46,224,279,312)(47,225,280,300)(48,226,281,301)(49,227,282,302)(50,228,283,303)(51,229,284,304)(52,230,285,305)(53,80,195,260)(54,81,183,248)(55,82,184,249)(56,83,185,250)(57,84,186,251)(58,85,187,252)(59,86,188,253)(60,87,189,254)(61,88,190,255)(62,89,191,256)(63,90,192,257)(64,91,193,258)(65,79,194,259)(66,213,105,169)(67,214,106,157)(68,215,107,158)(69,216,108,159)(70,217,109,160)(71,218,110,161)(72,219,111,162)(73,220,112,163)(74,221,113,164)(75,209,114,165)(76,210,115,166)(77,211,116,167)(78,212,117,168)(131,293,170,199)(132,294,171,200)(133,295,172,201)(134,296,173,202)(135,297,174,203)(136,298,175,204)(137,299,176,205)(138,287,177,206)(139,288,178,207)(140,289,179,208)(141,290,180,196)(142,291,181,197)(143,292,182,198)>;
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), (1,308,37,206,187,110,247,233,93,287,58,71)(2,309,38,207,188,111,235,234,94,288,59,72)(3,310,39,208,189,112,236,222,95,289,60,73)(4,311,27,196,190,113,237,223,96,290,61,74)(5,312,28,197,191,114,238,224,97,291,62,75)(6,300,29,198,192,115,239,225,98,292,63,76)(7,301,30,199,193,116,240,226,99,293,64,77)(8,302,31,200,194,117,241,227,100,294,65,78)(9,303,32,201,195,105,242,228,101,295,53,66)(10,304,33,202,183,106,243,229,102,296,54,67)(11,305,34,203,184,107,244,230,103,297,55,68)(12,306,35,204,185,108,245,231,104,298,56,69)(13,307,36,205,186,109,246,232,92,299,57,70)(14,215,82,174,266,285,121,158,249,135,156,52)(15,216,83,175,267,286,122,159,250,136,144,40)(16,217,84,176,268,274,123,160,251,137,145,41)(17,218,85,177,269,275,124,161,252,138,146,42)(18,219,86,178,270,276,125,162,253,139,147,43)(19,220,87,179,271,277,126,163,254,140,148,44)(20,221,88,180,272,278,127,164,255,141,149,45)(21,209,89,181,273,279,128,165,256,142,150,46)(22,210,90,182,261,280,129,166,257,143,151,47)(23,211,91,170,262,281,130,167,258,131,152,48)(24,212,79,171,263,282,118,168,259,132,153,49)(25,213,80,172,264,283,119,169,260,133,154,50)(26,214,81,173,265,284,120,157,248,134,155,51), (1,17,247,124)(2,18,235,125)(3,19,236,126)(4,20,237,127)(5,21,238,128)(6,22,239,129)(7,23,240,130)(8,24,241,118)(9,25,242,119)(10,26,243,120)(11,14,244,121)(12,15,245,122)(13,16,246,123)(27,149,96,272)(28,150,97,273)(29,151,98,261)(30,152,99,262)(31,153,100,263)(32,154,101,264)(33,155,102,265)(34,156,103,266)(35,144,104,267)(36,145,92,268)(37,146,93,269)(38,147,94,270)(39,148,95,271)(40,231,286,306)(41,232,274,307)(42,233,275,308)(43,234,276,309)(44,222,277,310)(45,223,278,311)(46,224,279,312)(47,225,280,300)(48,226,281,301)(49,227,282,302)(50,228,283,303)(51,229,284,304)(52,230,285,305)(53,80,195,260)(54,81,183,248)(55,82,184,249)(56,83,185,250)(57,84,186,251)(58,85,187,252)(59,86,188,253)(60,87,189,254)(61,88,190,255)(62,89,191,256)(63,90,192,257)(64,91,193,258)(65,79,194,259)(66,213,105,169)(67,214,106,157)(68,215,107,158)(69,216,108,159)(70,217,109,160)(71,218,110,161)(72,219,111,162)(73,220,112,163)(74,221,113,164)(75,209,114,165)(76,210,115,166)(77,211,116,167)(78,212,117,168)(131,293,170,199)(132,294,171,200)(133,295,172,201)(134,296,173,202)(135,297,174,203)(136,298,175,204)(137,299,176,205)(138,287,177,206)(139,288,178,207)(140,289,179,208)(141,290,180,196)(142,291,181,197)(143,292,182,198) );
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)], [(1,308,37,206,187,110,247,233,93,287,58,71),(2,309,38,207,188,111,235,234,94,288,59,72),(3,310,39,208,189,112,236,222,95,289,60,73),(4,311,27,196,190,113,237,223,96,290,61,74),(5,312,28,197,191,114,238,224,97,291,62,75),(6,300,29,198,192,115,239,225,98,292,63,76),(7,301,30,199,193,116,240,226,99,293,64,77),(8,302,31,200,194,117,241,227,100,294,65,78),(9,303,32,201,195,105,242,228,101,295,53,66),(10,304,33,202,183,106,243,229,102,296,54,67),(11,305,34,203,184,107,244,230,103,297,55,68),(12,306,35,204,185,108,245,231,104,298,56,69),(13,307,36,205,186,109,246,232,92,299,57,70),(14,215,82,174,266,285,121,158,249,135,156,52),(15,216,83,175,267,286,122,159,250,136,144,40),(16,217,84,176,268,274,123,160,251,137,145,41),(17,218,85,177,269,275,124,161,252,138,146,42),(18,219,86,178,270,276,125,162,253,139,147,43),(19,220,87,179,271,277,126,163,254,140,148,44),(20,221,88,180,272,278,127,164,255,141,149,45),(21,209,89,181,273,279,128,165,256,142,150,46),(22,210,90,182,261,280,129,166,257,143,151,47),(23,211,91,170,262,281,130,167,258,131,152,48),(24,212,79,171,263,282,118,168,259,132,153,49),(25,213,80,172,264,283,119,169,260,133,154,50),(26,214,81,173,265,284,120,157,248,134,155,51)], [(1,17,247,124),(2,18,235,125),(3,19,236,126),(4,20,237,127),(5,21,238,128),(6,22,239,129),(7,23,240,130),(8,24,241,118),(9,25,242,119),(10,26,243,120),(11,14,244,121),(12,15,245,122),(13,16,246,123),(27,149,96,272),(28,150,97,273),(29,151,98,261),(30,152,99,262),(31,153,100,263),(32,154,101,264),(33,155,102,265),(34,156,103,266),(35,144,104,267),(36,145,92,268),(37,146,93,269),(38,147,94,270),(39,148,95,271),(40,231,286,306),(41,232,274,307),(42,233,275,308),(43,234,276,309),(44,222,277,310),(45,223,278,311),(46,224,279,312),(47,225,280,300),(48,226,281,301),(49,227,282,302),(50,228,283,303),(51,229,284,304),(52,230,285,305),(53,80,195,260),(54,81,183,248),(55,82,184,249),(56,83,185,250),(57,84,186,251),(58,85,187,252),(59,86,188,253),(60,87,189,254),(61,88,190,255),(62,89,191,256),(63,90,192,257),(64,91,193,258),(65,79,194,259),(66,213,105,169),(67,214,106,157),(68,215,107,158),(69,216,108,159),(70,217,109,160),(71,218,110,161),(72,219,111,162),(73,220,112,163),(74,221,113,164),(75,209,114,165),(76,210,115,166),(77,211,116,167),(78,212,117,168),(131,293,170,199),(132,294,171,200),(133,295,172,201),(134,296,173,202),(135,297,174,203),(136,298,175,204),(137,299,176,205),(138,287,177,206),(139,288,178,207),(140,289,179,208),(141,290,180,196),(142,291,181,197),(143,292,182,198)]])
117 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 12A | 12B | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39L | 52A | ··· | 52L | 52M | ··· | 52AJ | 78A | ··· | 78L | 156A | ··· | 156X |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 12 | 12 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 52 | ··· | 52 | 78 | ··· | 78 | 156 | ··· | 156 |
| size | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | S3 | Q8 | D6 | Dic6 | S3×C13 | Q8×C13 | S3×C26 | C13×Dic6 |
| kernel | C13×Dic6 | Dic3×C13 | C156 | Dic6 | Dic3 | C12 | C52 | C39 | C26 | C13 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 12 | 24 | 12 | 1 | 1 | 1 | 2 | 12 | 12 | 12 | 24 |
Matrix representation of C13×Dic6 ►in GL2(𝔽157) generated by
| 39 | 0 |
| 0 | 39 |
| 48 | 24 |
| 133 | 24 |
| 127 | 85 |
| 115 | 30 |
G:=sub<GL(2,GF(157))| [39,0,0,39],[48,133,24,24],[127,115,85,30] >;
C13×Dic6 in GAP, Magma, Sage, TeX
C_{13}\times {\rm Dic}_6 % in TeX
G:=Group("C13xDic6"); // GroupNames label
G:=SmallGroup(312,32);
// by ID
G=gap.SmallGroup(312,32);
# by ID
G:=PCGroup([5,-2,-2,-13,-2,-3,260,541,266,5204]);
// Polycyclic
G:=Group<a,b,c|a^13=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export