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## G = Dic3×C26order 312 = 23·3·13

### Direct product of C26 and Dic3

Aliases: Dic3×C26, C6⋊C52, C785C4, C26.16D6, C78.21C22, C32(C2×C52), (C2×C6).C26, C3912(C2×C4), C2.2(S3×C26), (C2×C26).2S3, (C2×C78).3C2, C6.4(C2×C26), C22.(S3×C13), SmallGroup(312,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C26
 Chief series C1 — C3 — C6 — C78 — Dic3×C13 — Dic3×C26
 Lower central C3 — Dic3×C26
 Upper central C1 — C2×C26

Generators and relations for Dic3×C26
G = < a,b,c | a26=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Dic3×C26
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)
(1 215 273 70 82 139)(2 216 274 71 83 140)(3 217 275 72 84 141)(4 218 276 73 85 142)(5 219 277 74 86 143)(6 220 278 75 87 144)(7 221 279 76 88 145)(8 222 280 77 89 146)(9 223 281 78 90 147)(10 224 282 53 91 148)(11 225 283 54 92 149)(12 226 284 55 93 150)(13 227 285 56 94 151)(14 228 286 57 95 152)(15 229 261 58 96 153)(16 230 262 59 97 154)(17 231 263 60 98 155)(18 232 264 61 99 156)(19 233 265 62 100 131)(20 234 266 63 101 132)(21 209 267 64 102 133)(22 210 268 65 103 134)(23 211 269 66 104 135)(24 212 270 67 79 136)(25 213 271 68 80 137)(26 214 272 69 81 138)(27 183 166 239 113 312)(28 184 167 240 114 287)(29 185 168 241 115 288)(30 186 169 242 116 289)(31 187 170 243 117 290)(32 188 171 244 118 291)(33 189 172 245 119 292)(34 190 173 246 120 293)(35 191 174 247 121 294)(36 192 175 248 122 295)(37 193 176 249 123 296)(38 194 177 250 124 297)(39 195 178 251 125 298)(40 196 179 252 126 299)(41 197 180 253 127 300)(42 198 181 254 128 301)(43 199 182 255 129 302)(44 200 157 256 130 303)(45 201 158 257 105 304)(46 202 159 258 106 305)(47 203 160 259 107 306)(48 204 161 260 108 307)(49 205 162 235 109 308)(50 206 163 236 110 309)(51 207 164 237 111 310)(52 208 165 238 112 311)
(1 192 70 122)(2 193 71 123)(3 194 72 124)(4 195 73 125)(5 196 74 126)(6 197 75 127)(7 198 76 128)(8 199 77 129)(9 200 78 130)(10 201 53 105)(11 202 54 106)(12 203 55 107)(13 204 56 108)(14 205 57 109)(15 206 58 110)(16 207 59 111)(17 208 60 112)(18 183 61 113)(19 184 62 114)(20 185 63 115)(21 186 64 116)(22 187 65 117)(23 188 66 118)(24 189 67 119)(25 190 68 120)(26 191 69 121)(27 99 239 232)(28 100 240 233)(29 101 241 234)(30 102 242 209)(31 103 243 210)(32 104 244 211)(33 79 245 212)(34 80 246 213)(35 81 247 214)(36 82 248 215)(37 83 249 216)(38 84 250 217)(39 85 251 218)(40 86 252 219)(41 87 253 220)(42 88 254 221)(43 89 255 222)(44 90 256 223)(45 91 257 224)(46 92 258 225)(47 93 259 226)(48 94 260 227)(49 95 235 228)(50 96 236 229)(51 97 237 230)(52 98 238 231)(131 167 265 287)(132 168 266 288)(133 169 267 289)(134 170 268 290)(135 171 269 291)(136 172 270 292)(137 173 271 293)(138 174 272 294)(139 175 273 295)(140 176 274 296)(141 177 275 297)(142 178 276 298)(143 179 277 299)(144 180 278 300)(145 181 279 301)(146 182 280 302)(147 157 281 303)(148 158 282 304)(149 159 283 305)(150 160 284 306)(151 161 285 307)(152 162 286 308)(153 163 261 309)(154 164 262 310)(155 165 263 311)(156 166 264 312)

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), (1,215,273,70,82,139)(2,216,274,71,83,140)(3,217,275,72,84,141)(4,218,276,73,85,142)(5,219,277,74,86,143)(6,220,278,75,87,144)(7,221,279,76,88,145)(8,222,280,77,89,146)(9,223,281,78,90,147)(10,224,282,53,91,148)(11,225,283,54,92,149)(12,226,284,55,93,150)(13,227,285,56,94,151)(14,228,286,57,95,152)(15,229,261,58,96,153)(16,230,262,59,97,154)(17,231,263,60,98,155)(18,232,264,61,99,156)(19,233,265,62,100,131)(20,234,266,63,101,132)(21,209,267,64,102,133)(22,210,268,65,103,134)(23,211,269,66,104,135)(24,212,270,67,79,136)(25,213,271,68,80,137)(26,214,272,69,81,138)(27,183,166,239,113,312)(28,184,167,240,114,287)(29,185,168,241,115,288)(30,186,169,242,116,289)(31,187,170,243,117,290)(32,188,171,244,118,291)(33,189,172,245,119,292)(34,190,173,246,120,293)(35,191,174,247,121,294)(36,192,175,248,122,295)(37,193,176,249,123,296)(38,194,177,250,124,297)(39,195,178,251,125,298)(40,196,179,252,126,299)(41,197,180,253,127,300)(42,198,181,254,128,301)(43,199,182,255,129,302)(44,200,157,256,130,303)(45,201,158,257,105,304)(46,202,159,258,106,305)(47,203,160,259,107,306)(48,204,161,260,108,307)(49,205,162,235,109,308)(50,206,163,236,110,309)(51,207,164,237,111,310)(52,208,165,238,112,311), (1,192,70,122)(2,193,71,123)(3,194,72,124)(4,195,73,125)(5,196,74,126)(6,197,75,127)(7,198,76,128)(8,199,77,129)(9,200,78,130)(10,201,53,105)(11,202,54,106)(12,203,55,107)(13,204,56,108)(14,205,57,109)(15,206,58,110)(16,207,59,111)(17,208,60,112)(18,183,61,113)(19,184,62,114)(20,185,63,115)(21,186,64,116)(22,187,65,117)(23,188,66,118)(24,189,67,119)(25,190,68,120)(26,191,69,121)(27,99,239,232)(28,100,240,233)(29,101,241,234)(30,102,242,209)(31,103,243,210)(32,104,244,211)(33,79,245,212)(34,80,246,213)(35,81,247,214)(36,82,248,215)(37,83,249,216)(38,84,250,217)(39,85,251,218)(40,86,252,219)(41,87,253,220)(42,88,254,221)(43,89,255,222)(44,90,256,223)(45,91,257,224)(46,92,258,225)(47,93,259,226)(48,94,260,227)(49,95,235,228)(50,96,236,229)(51,97,237,230)(52,98,238,231)(131,167,265,287)(132,168,266,288)(133,169,267,289)(134,170,268,290)(135,171,269,291)(136,172,270,292)(137,173,271,293)(138,174,272,294)(139,175,273,295)(140,176,274,296)(141,177,275,297)(142,178,276,298)(143,179,277,299)(144,180,278,300)(145,181,279,301)(146,182,280,302)(147,157,281,303)(148,158,282,304)(149,159,283,305)(150,160,284,306)(151,161,285,307)(152,162,286,308)(153,163,261,309)(154,164,262,310)(155,165,263,311)(156,166,264,312)>;

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), (1,215,273,70,82,139)(2,216,274,71,83,140)(3,217,275,72,84,141)(4,218,276,73,85,142)(5,219,277,74,86,143)(6,220,278,75,87,144)(7,221,279,76,88,145)(8,222,280,77,89,146)(9,223,281,78,90,147)(10,224,282,53,91,148)(11,225,283,54,92,149)(12,226,284,55,93,150)(13,227,285,56,94,151)(14,228,286,57,95,152)(15,229,261,58,96,153)(16,230,262,59,97,154)(17,231,263,60,98,155)(18,232,264,61,99,156)(19,233,265,62,100,131)(20,234,266,63,101,132)(21,209,267,64,102,133)(22,210,268,65,103,134)(23,211,269,66,104,135)(24,212,270,67,79,136)(25,213,271,68,80,137)(26,214,272,69,81,138)(27,183,166,239,113,312)(28,184,167,240,114,287)(29,185,168,241,115,288)(30,186,169,242,116,289)(31,187,170,243,117,290)(32,188,171,244,118,291)(33,189,172,245,119,292)(34,190,173,246,120,293)(35,191,174,247,121,294)(36,192,175,248,122,295)(37,193,176,249,123,296)(38,194,177,250,124,297)(39,195,178,251,125,298)(40,196,179,252,126,299)(41,197,180,253,127,300)(42,198,181,254,128,301)(43,199,182,255,129,302)(44,200,157,256,130,303)(45,201,158,257,105,304)(46,202,159,258,106,305)(47,203,160,259,107,306)(48,204,161,260,108,307)(49,205,162,235,109,308)(50,206,163,236,110,309)(51,207,164,237,111,310)(52,208,165,238,112,311), (1,192,70,122)(2,193,71,123)(3,194,72,124)(4,195,73,125)(5,196,74,126)(6,197,75,127)(7,198,76,128)(8,199,77,129)(9,200,78,130)(10,201,53,105)(11,202,54,106)(12,203,55,107)(13,204,56,108)(14,205,57,109)(15,206,58,110)(16,207,59,111)(17,208,60,112)(18,183,61,113)(19,184,62,114)(20,185,63,115)(21,186,64,116)(22,187,65,117)(23,188,66,118)(24,189,67,119)(25,190,68,120)(26,191,69,121)(27,99,239,232)(28,100,240,233)(29,101,241,234)(30,102,242,209)(31,103,243,210)(32,104,244,211)(33,79,245,212)(34,80,246,213)(35,81,247,214)(36,82,248,215)(37,83,249,216)(38,84,250,217)(39,85,251,218)(40,86,252,219)(41,87,253,220)(42,88,254,221)(43,89,255,222)(44,90,256,223)(45,91,257,224)(46,92,258,225)(47,93,259,226)(48,94,260,227)(49,95,235,228)(50,96,236,229)(51,97,237,230)(52,98,238,231)(131,167,265,287)(132,168,266,288)(133,169,267,289)(134,170,268,290)(135,171,269,291)(136,172,270,292)(137,173,271,293)(138,174,272,294)(139,175,273,295)(140,176,274,296)(141,177,275,297)(142,178,276,298)(143,179,277,299)(144,180,278,300)(145,181,279,301)(146,182,280,302)(147,157,281,303)(148,158,282,304)(149,159,283,305)(150,160,284,306)(151,161,285,307)(152,162,286,308)(153,163,261,309)(154,164,262,310)(155,165,263,311)(156,166,264,312) );

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)], [(1,215,273,70,82,139),(2,216,274,71,83,140),(3,217,275,72,84,141),(4,218,276,73,85,142),(5,219,277,74,86,143),(6,220,278,75,87,144),(7,221,279,76,88,145),(8,222,280,77,89,146),(9,223,281,78,90,147),(10,224,282,53,91,148),(11,225,283,54,92,149),(12,226,284,55,93,150),(13,227,285,56,94,151),(14,228,286,57,95,152),(15,229,261,58,96,153),(16,230,262,59,97,154),(17,231,263,60,98,155),(18,232,264,61,99,156),(19,233,265,62,100,131),(20,234,266,63,101,132),(21,209,267,64,102,133),(22,210,268,65,103,134),(23,211,269,66,104,135),(24,212,270,67,79,136),(25,213,271,68,80,137),(26,214,272,69,81,138),(27,183,166,239,113,312),(28,184,167,240,114,287),(29,185,168,241,115,288),(30,186,169,242,116,289),(31,187,170,243,117,290),(32,188,171,244,118,291),(33,189,172,245,119,292),(34,190,173,246,120,293),(35,191,174,247,121,294),(36,192,175,248,122,295),(37,193,176,249,123,296),(38,194,177,250,124,297),(39,195,178,251,125,298),(40,196,179,252,126,299),(41,197,180,253,127,300),(42,198,181,254,128,301),(43,199,182,255,129,302),(44,200,157,256,130,303),(45,201,158,257,105,304),(46,202,159,258,106,305),(47,203,160,259,107,306),(48,204,161,260,108,307),(49,205,162,235,109,308),(50,206,163,236,110,309),(51,207,164,237,111,310),(52,208,165,238,112,311)], [(1,192,70,122),(2,193,71,123),(3,194,72,124),(4,195,73,125),(5,196,74,126),(6,197,75,127),(7,198,76,128),(8,199,77,129),(9,200,78,130),(10,201,53,105),(11,202,54,106),(12,203,55,107),(13,204,56,108),(14,205,57,109),(15,206,58,110),(16,207,59,111),(17,208,60,112),(18,183,61,113),(19,184,62,114),(20,185,63,115),(21,186,64,116),(22,187,65,117),(23,188,66,118),(24,189,67,119),(25,190,68,120),(26,191,69,121),(27,99,239,232),(28,100,240,233),(29,101,241,234),(30,102,242,209),(31,103,243,210),(32,104,244,211),(33,79,245,212),(34,80,246,213),(35,81,247,214),(36,82,248,215),(37,83,249,216),(38,84,250,217),(39,85,251,218),(40,86,252,219),(41,87,253,220),(42,88,254,221),(43,89,255,222),(44,90,256,223),(45,91,257,224),(46,92,258,225),(47,93,259,226),(48,94,260,227),(49,95,235,228),(50,96,236,229),(51,97,237,230),(52,98,238,231),(131,167,265,287),(132,168,266,288),(133,169,267,289),(134,170,268,290),(135,171,269,291),(136,172,270,292),(137,173,271,293),(138,174,272,294),(139,175,273,295),(140,176,274,296),(141,177,275,297),(142,178,276,298),(143,179,277,299),(144,180,278,300),(145,181,279,301),(146,182,280,302),(147,157,281,303),(148,158,282,304),(149,159,283,305),(150,160,284,306),(151,161,285,307),(152,162,286,308),(153,163,261,309),(154,164,262,310),(155,165,263,311),(156,166,264,312)]])

156 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 13A ··· 13L 26A ··· 26AJ 39A ··· 39L 52A ··· 52AV 78A ··· 78AJ order 1 2 2 2 3 4 4 4 4 6 6 6 13 ··· 13 26 ··· 26 39 ··· 39 52 ··· 52 78 ··· 78 size 1 1 1 1 2 3 3 3 3 2 2 2 1 ··· 1 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 C2 C4 C13 C26 C26 C52 S3 Dic3 D6 S3×C13 Dic3×C13 S3×C26 kernel Dic3×C26 Dic3×C13 C2×C78 C78 C2×Dic3 Dic3 C2×C6 C6 C2×C26 C26 C26 C22 C2 C2 # reps 1 2 1 4 12 24 12 48 1 2 1 12 24 12

Matrix representation of Dic3×C26 in GL3(𝔽157) generated by

 156 0 0 0 130 0 0 0 130
,
 1 0 0 0 1 156 0 1 0
,
 156 0 0 0 13 39 0 52 144
G:=sub<GL(3,GF(157))| [156,0,0,0,130,0,0,0,130],[1,0,0,0,1,1,0,156,0],[156,0,0,0,13,52,0,39,144] >;

Dic3×C26 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{26}
% in TeX

G:=Group("Dic3xC26");
// GroupNames label

G:=SmallGroup(312,35);
// by ID

G=gap.SmallGroup(312,35);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-3,260,5204]);
// Polycyclic

G:=Group<a,b,c|a^26=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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