direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3×D19, D38.S3, C6.1D38, C38.1D6, Dic57⋊2C2, C114.1C22, C57⋊1(C2×C4), (C3×D19)⋊C4, C3⋊3(C4×D19), (C6×D19).C2, C2.1(S3×D19), C19⋊1(C2×Dic3), (Dic3×C19)⋊1C2, SmallGroup(456,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C57 — Dic3×D19 |
Generators and relations for Dic3×D19
G = < a,b,c,d | a6=c19=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of Dic3×D19
►On 228 points
Generators in S228
(1 114 38 62 44 77)(2 96 20 63 45 78)(3 97 21 64 46 79)(4 98 22 65 47 80)(5 99 23 66 48 81)(6 100 24 67 49 82)(7 101 25 68 50 83)(8 102 26 69 51 84)(9 103 27 70 52 85)(10 104 28 71 53 86)(11 105 29 72 54 87)(12 106 30 73 55 88)(13 107 31 74 56 89)(14 108 32 75 57 90)(15 109 33 76 39 91)(16 110 34 58 40 92)(17 111 35 59 41 93)(18 112 36 60 42 94)(19 113 37 61 43 95)(115 204 157 180 139 214)(116 205 158 181 140 215)(117 206 159 182 141 216)(118 207 160 183 142 217)(119 208 161 184 143 218)(120 209 162 185 144 219)(121 191 163 186 145 220)(122 192 164 187 146 221)(123 193 165 188 147 222)(124 194 166 189 148 223)(125 195 167 190 149 224)(126 196 168 172 150 225)(127 197 169 173 151 226)(128 198 170 174 152 227)(129 199 171 175 134 228)(130 200 153 176 135 210)(131 201 154 177 136 211)(132 202 155 178 137 212)(133 203 156 179 138 213)
(1 181 62 116)(2 182 63 117)(3 183 64 118)(4 184 65 119)(5 185 66 120)(6 186 67 121)(7 187 68 122)(8 188 69 123)(9 189 70 124)(10 190 71 125)(11 172 72 126)(12 173 73 127)(13 174 74 128)(14 175 75 129)(15 176 76 130)(16 177 58 131)(17 178 59 132)(18 179 60 133)(19 180 61 115)(20 206 78 141)(21 207 79 142)(22 208 80 143)(23 209 81 144)(24 191 82 145)(25 192 83 146)(26 193 84 147)(27 194 85 148)(28 195 86 149)(29 196 87 150)(30 197 88 151)(31 198 89 152)(32 199 90 134)(33 200 91 135)(34 201 92 136)(35 202 93 137)(36 203 94 138)(37 204 95 139)(38 205 77 140)(39 210 109 153)(40 211 110 154)(41 212 111 155)(42 213 112 156)(43 214 113 157)(44 215 114 158)(45 216 96 159)(46 217 97 160)(47 218 98 161)(48 219 99 162)(49 220 100 163)(50 221 101 164)(51 222 102 165)(52 223 103 166)(53 224 104 167)(54 225 105 168)(55 226 106 169)(56 227 107 170)(57 228 108 171)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(37 38)(39 48)(40 47)(41 46)(42 45)(43 44)(49 57)(50 56)(51 55)(52 54)(58 65)(59 64)(60 63)(61 62)(66 76)(67 75)(68 74)(69 73)(70 72)(77 95)(78 94)(79 93)(80 92)(81 91)(82 90)(83 89)(84 88)(85 87)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)(113 114)(115 116)(117 133)(118 132)(119 131)(120 130)(121 129)(122 128)(123 127)(124 126)(134 145)(135 144)(136 143)(137 142)(138 141)(139 140)(146 152)(147 151)(148 150)(153 162)(154 161)(155 160)(156 159)(157 158)(163 171)(164 170)(165 169)(166 168)(172 189)(173 188)(174 187)(175 186)(176 185)(177 184)(178 183)(179 182)(180 181)(191 199)(192 198)(193 197)(194 196)(200 209)(201 208)(202 207)(203 206)(204 205)(210 219)(211 218)(212 217)(213 216)(214 215)(220 228)(221 227)(222 226)(223 225)
G:=sub<Sym(228)| (1,114,38,62,44,77)(2,96,20,63,45,78)(3,97,21,64,46,79)(4,98,22,65,47,80)(5,99,23,66,48,81)(6,100,24,67,49,82)(7,101,25,68,50,83)(8,102,26,69,51,84)(9,103,27,70,52,85)(10,104,28,71,53,86)(11,105,29,72,54,87)(12,106,30,73,55,88)(13,107,31,74,56,89)(14,108,32,75,57,90)(15,109,33,76,39,91)(16,110,34,58,40,92)(17,111,35,59,41,93)(18,112,36,60,42,94)(19,113,37,61,43,95)(115,204,157,180,139,214)(116,205,158,181,140,215)(117,206,159,182,141,216)(118,207,160,183,142,217)(119,208,161,184,143,218)(120,209,162,185,144,219)(121,191,163,186,145,220)(122,192,164,187,146,221)(123,193,165,188,147,222)(124,194,166,189,148,223)(125,195,167,190,149,224)(126,196,168,172,150,225)(127,197,169,173,151,226)(128,198,170,174,152,227)(129,199,171,175,134,228)(130,200,153,176,135,210)(131,201,154,177,136,211)(132,202,155,178,137,212)(133,203,156,179,138,213), (1,181,62,116)(2,182,63,117)(3,183,64,118)(4,184,65,119)(5,185,66,120)(6,186,67,121)(7,187,68,122)(8,188,69,123)(9,189,70,124)(10,190,71,125)(11,172,72,126)(12,173,73,127)(13,174,74,128)(14,175,75,129)(15,176,76,130)(16,177,58,131)(17,178,59,132)(18,179,60,133)(19,180,61,115)(20,206,78,141)(21,207,79,142)(22,208,80,143)(23,209,81,144)(24,191,82,145)(25,192,83,146)(26,193,84,147)(27,194,85,148)(28,195,86,149)(29,196,87,150)(30,197,88,151)(31,198,89,152)(32,199,90,134)(33,200,91,135)(34,201,92,136)(35,202,93,137)(36,203,94,138)(37,204,95,139)(38,205,77,140)(39,210,109,153)(40,211,110,154)(41,212,111,155)(42,213,112,156)(43,214,113,157)(44,215,114,158)(45,216,96,159)(46,217,97,160)(47,218,98,161)(48,219,99,162)(49,220,100,163)(50,221,101,164)(51,222,102,165)(52,223,103,166)(53,224,104,167)(54,225,105,168)(55,226,106,169)(56,227,107,170)(57,228,108,171), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44)(49,57)(50,56)(51,55)(52,54)(58,65)(59,64)(60,63)(61,62)(66,76)(67,75)(68,74)(69,73)(70,72)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(113,114)(115,116)(117,133)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126)(134,145)(135,144)(136,143)(137,142)(138,141)(139,140)(146,152)(147,151)(148,150)(153,162)(154,161)(155,160)(156,159)(157,158)(163,171)(164,170)(165,169)(166,168)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)(191,199)(192,198)(193,197)(194,196)(200,209)(201,208)(202,207)(203,206)(204,205)(210,219)(211,218)(212,217)(213,216)(214,215)(220,228)(221,227)(222,226)(223,225)>;
G:=Group( (1,114,38,62,44,77)(2,96,20,63,45,78)(3,97,21,64,46,79)(4,98,22,65,47,80)(5,99,23,66,48,81)(6,100,24,67,49,82)(7,101,25,68,50,83)(8,102,26,69,51,84)(9,103,27,70,52,85)(10,104,28,71,53,86)(11,105,29,72,54,87)(12,106,30,73,55,88)(13,107,31,74,56,89)(14,108,32,75,57,90)(15,109,33,76,39,91)(16,110,34,58,40,92)(17,111,35,59,41,93)(18,112,36,60,42,94)(19,113,37,61,43,95)(115,204,157,180,139,214)(116,205,158,181,140,215)(117,206,159,182,141,216)(118,207,160,183,142,217)(119,208,161,184,143,218)(120,209,162,185,144,219)(121,191,163,186,145,220)(122,192,164,187,146,221)(123,193,165,188,147,222)(124,194,166,189,148,223)(125,195,167,190,149,224)(126,196,168,172,150,225)(127,197,169,173,151,226)(128,198,170,174,152,227)(129,199,171,175,134,228)(130,200,153,176,135,210)(131,201,154,177,136,211)(132,202,155,178,137,212)(133,203,156,179,138,213), (1,181,62,116)(2,182,63,117)(3,183,64,118)(4,184,65,119)(5,185,66,120)(6,186,67,121)(7,187,68,122)(8,188,69,123)(9,189,70,124)(10,190,71,125)(11,172,72,126)(12,173,73,127)(13,174,74,128)(14,175,75,129)(15,176,76,130)(16,177,58,131)(17,178,59,132)(18,179,60,133)(19,180,61,115)(20,206,78,141)(21,207,79,142)(22,208,80,143)(23,209,81,144)(24,191,82,145)(25,192,83,146)(26,193,84,147)(27,194,85,148)(28,195,86,149)(29,196,87,150)(30,197,88,151)(31,198,89,152)(32,199,90,134)(33,200,91,135)(34,201,92,136)(35,202,93,137)(36,203,94,138)(37,204,95,139)(38,205,77,140)(39,210,109,153)(40,211,110,154)(41,212,111,155)(42,213,112,156)(43,214,113,157)(44,215,114,158)(45,216,96,159)(46,217,97,160)(47,218,98,161)(48,219,99,162)(49,220,100,163)(50,221,101,164)(51,222,102,165)(52,223,103,166)(53,224,104,167)(54,225,105,168)(55,226,106,169)(56,227,107,170)(57,228,108,171), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(37,38)(39,48)(40,47)(41,46)(42,45)(43,44)(49,57)(50,56)(51,55)(52,54)(58,65)(59,64)(60,63)(61,62)(66,76)(67,75)(68,74)(69,73)(70,72)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(113,114)(115,116)(117,133)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126)(134,145)(135,144)(136,143)(137,142)(138,141)(139,140)(146,152)(147,151)(148,150)(153,162)(154,161)(155,160)(156,159)(157,158)(163,171)(164,170)(165,169)(166,168)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)(191,199)(192,198)(193,197)(194,196)(200,209)(201,208)(202,207)(203,206)(204,205)(210,219)(211,218)(212,217)(213,216)(214,215)(220,228)(221,227)(222,226)(223,225) );
G=PermutationGroup([[(1,114,38,62,44,77),(2,96,20,63,45,78),(3,97,21,64,46,79),(4,98,22,65,47,80),(5,99,23,66,48,81),(6,100,24,67,49,82),(7,101,25,68,50,83),(8,102,26,69,51,84),(9,103,27,70,52,85),(10,104,28,71,53,86),(11,105,29,72,54,87),(12,106,30,73,55,88),(13,107,31,74,56,89),(14,108,32,75,57,90),(15,109,33,76,39,91),(16,110,34,58,40,92),(17,111,35,59,41,93),(18,112,36,60,42,94),(19,113,37,61,43,95),(115,204,157,180,139,214),(116,205,158,181,140,215),(117,206,159,182,141,216),(118,207,160,183,142,217),(119,208,161,184,143,218),(120,209,162,185,144,219),(121,191,163,186,145,220),(122,192,164,187,146,221),(123,193,165,188,147,222),(124,194,166,189,148,223),(125,195,167,190,149,224),(126,196,168,172,150,225),(127,197,169,173,151,226),(128,198,170,174,152,227),(129,199,171,175,134,228),(130,200,153,176,135,210),(131,201,154,177,136,211),(132,202,155,178,137,212),(133,203,156,179,138,213)], [(1,181,62,116),(2,182,63,117),(3,183,64,118),(4,184,65,119),(5,185,66,120),(6,186,67,121),(7,187,68,122),(8,188,69,123),(9,189,70,124),(10,190,71,125),(11,172,72,126),(12,173,73,127),(13,174,74,128),(14,175,75,129),(15,176,76,130),(16,177,58,131),(17,178,59,132),(18,179,60,133),(19,180,61,115),(20,206,78,141),(21,207,79,142),(22,208,80,143),(23,209,81,144),(24,191,82,145),(25,192,83,146),(26,193,84,147),(27,194,85,148),(28,195,86,149),(29,196,87,150),(30,197,88,151),(31,198,89,152),(32,199,90,134),(33,200,91,135),(34,201,92,136),(35,202,93,137),(36,203,94,138),(37,204,95,139),(38,205,77,140),(39,210,109,153),(40,211,110,154),(41,212,111,155),(42,213,112,156),(43,214,113,157),(44,215,114,158),(45,216,96,159),(46,217,97,160),(47,218,98,161),(48,219,99,162),(49,220,100,163),(50,221,101,164),(51,222,102,165),(52,223,103,166),(53,224,104,167),(54,225,105,168),(55,226,106,169),(56,227,107,170),(57,228,108,171)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(37,38),(39,48),(40,47),(41,46),(42,45),(43,44),(49,57),(50,56),(51,55),(52,54),(58,65),(59,64),(60,63),(61,62),(66,76),(67,75),(68,74),(69,73),(70,72),(77,95),(78,94),(79,93),(80,92),(81,91),(82,90),(83,89),(84,88),(85,87),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,105),(113,114),(115,116),(117,133),(118,132),(119,131),(120,130),(121,129),(122,128),(123,127),(124,126),(134,145),(135,144),(136,143),(137,142),(138,141),(139,140),(146,152),(147,151),(148,150),(153,162),(154,161),(155,160),(156,159),(157,158),(163,171),(164,170),(165,169),(166,168),(172,189),(173,188),(174,187),(175,186),(176,185),(177,184),(178,183),(179,182),(180,181),(191,199),(192,198),(193,197),(194,196),(200,209),(201,208),(202,207),(203,206),(204,205),(210,219),(211,218),(212,217),(213,216),(214,215),(220,228),(221,227),(222,226),(223,225)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 19A | ··· | 19I | 38A | ··· | 38I | 57A | ··· | 57I | 76A | ··· | 76R | 114A | ··· | 114I |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 |
| size | 1 | 1 | 19 | 19 | 2 | 3 | 3 | 57 | 57 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Dic3 | D6 | D19 | D38 | C4×D19 | S3×D19 | Dic3×D19 |
| kernel | Dic3×D19 | Dic3×C19 | Dic57 | C6×D19 | C3×D19 | D38 | D19 | C38 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 9 | 9 | 18 | 9 | 9 |
Matrix representation of Dic3×D19 ►in GL4(𝔽229) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 44 |
| 0 | 0 | 78 | 228 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 159 | 76 |
| 0 | 0 | 47 | 70 |
| 12 | 1 | 0 | 0 |
| 55 | 81 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 116 | 8 | 0 | 0 |
| 7 | 113 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(229))| [1,0,0,0,0,1,0,0,0,0,2,78,0,0,44,228],[1,0,0,0,0,1,0,0,0,0,159,47,0,0,76,70],[12,55,0,0,1,81,0,0,0,0,1,0,0,0,0,1],[116,7,0,0,8,113,0,0,0,0,1,0,0,0,0,1] >;
Dic3×D19 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times D_{19} % in TeX
G:=Group("Dic3xD19"); // GroupNames label
G:=SmallGroup(456,12);
// by ID
G=gap.SmallGroup(456,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,26,168,10804]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^19=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export