direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×Dic19, D6.D19, C6.2D38, C38.2D6, Dic57⋊3C2, C114.2C22, (S3×C19)⋊C4, C19⋊3(C4×S3), C57⋊2(C2×C4), (S3×C38).C2, C2.2(S3×D19), C3⋊1(C2×Dic19), (C3×Dic19)⋊1C2, SmallGroup(456,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C57 — S3×Dic19 |
Generators and relations for S3×Dic19
G = < a,b,c,d | a3=b2=c38=1, d2=c19, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of S3×Dic19
►On 228 points
Generators in S228
(1 170 76)(2 171 39)(3 172 40)(4 173 41)(5 174 42)(6 175 43)(7 176 44)(8 177 45)(9 178 46)(10 179 47)(11 180 48)(12 181 49)(13 182 50)(14 183 51)(15 184 52)(16 185 53)(17 186 54)(18 187 55)(19 188 56)(20 189 57)(21 190 58)(22 153 59)(23 154 60)(24 155 61)(25 156 62)(26 157 63)(27 158 64)(28 159 65)(29 160 66)(30 161 67)(31 162 68)(32 163 69)(33 164 70)(34 165 71)(35 166 72)(36 167 73)(37 168 74)(38 169 75)(77 212 145)(78 213 146)(79 214 147)(80 215 148)(81 216 149)(82 217 150)(83 218 151)(84 219 152)(85 220 115)(86 221 116)(87 222 117)(88 223 118)(89 224 119)(90 225 120)(91 226 121)(92 227 122)(93 228 123)(94 191 124)(95 192 125)(96 193 126)(97 194 127)(98 195 128)(99 196 129)(100 197 130)(101 198 131)(102 199 132)(103 200 133)(104 201 134)(105 202 135)(106 203 136)(107 204 137)(108 205 138)(109 206 139)(110 207 140)(111 208 141)(112 209 142)(113 210 143)(114 211 144)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(39 190)(40 153)(41 154)(42 155)(43 156)(44 157)(45 158)(46 159)(47 160)(48 161)(49 162)(50 163)(51 164)(52 165)(53 166)(54 167)(55 168)(56 169)(57 170)(58 171)(59 172)(60 173)(61 174)(62 175)(63 176)(64 177)(65 178)(66 179)(67 180)(68 181)(69 182)(70 183)(71 184)(72 185)(73 186)(74 187)(75 188)(76 189)(77 193)(78 194)(79 195)(80 196)(81 197)(82 198)(83 199)(84 200)(85 201)(86 202)(87 203)(88 204)(89 205)(90 206)(91 207)(92 208)(93 209)(94 210)(95 211)(96 212)(97 213)(98 214)(99 215)(100 216)(101 217)(102 218)(103 219)(104 220)(105 221)(106 222)(107 223)(108 224)(109 225)(110 226)(111 227)(112 228)(113 191)(114 192)(115 134)(116 135)(117 136)(118 137)(119 138)(120 139)(121 140)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)(133 152)
(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)
(1 133 20 152)(2 132 21 151)(3 131 22 150)(4 130 23 149)(5 129 24 148)(6 128 25 147)(7 127 26 146)(8 126 27 145)(9 125 28 144)(10 124 29 143)(11 123 30 142)(12 122 31 141)(13 121 32 140)(14 120 33 139)(15 119 34 138)(16 118 35 137)(17 117 36 136)(18 116 37 135)(19 115 38 134)(39 199 58 218)(40 198 59 217)(41 197 60 216)(42 196 61 215)(43 195 62 214)(44 194 63 213)(45 193 64 212)(46 192 65 211)(47 191 66 210)(48 228 67 209)(49 227 68 208)(50 226 69 207)(51 225 70 206)(52 224 71 205)(53 223 72 204)(54 222 73 203)(55 221 74 202)(56 220 75 201)(57 219 76 200)(77 177 96 158)(78 176 97 157)(79 175 98 156)(80 174 99 155)(81 173 100 154)(82 172 101 153)(83 171 102 190)(84 170 103 189)(85 169 104 188)(86 168 105 187)(87 167 106 186)(88 166 107 185)(89 165 108 184)(90 164 109 183)(91 163 110 182)(92 162 111 181)(93 161 112 180)(94 160 113 179)(95 159 114 178)
G:=sub<Sym(228)| (1,170,76)(2,171,39)(3,172,40)(4,173,41)(5,174,42)(6,175,43)(7,176,44)(8,177,45)(9,178,46)(10,179,47)(11,180,48)(12,181,49)(13,182,50)(14,183,51)(15,184,52)(16,185,53)(17,186,54)(18,187,55)(19,188,56)(20,189,57)(21,190,58)(22,153,59)(23,154,60)(24,155,61)(25,156,62)(26,157,63)(27,158,64)(28,159,65)(29,160,66)(30,161,67)(31,162,68)(32,163,69)(33,164,70)(34,165,71)(35,166,72)(36,167,73)(37,168,74)(38,169,75)(77,212,145)(78,213,146)(79,214,147)(80,215,148)(81,216,149)(82,217,150)(83,218,151)(84,219,152)(85,220,115)(86,221,116)(87,222,117)(88,223,118)(89,224,119)(90,225,120)(91,226,121)(92,227,122)(93,228,123)(94,191,124)(95,192,125)(96,193,126)(97,194,127)(98,195,128)(99,196,129)(100,197,130)(101,198,131)(102,199,132)(103,200,133)(104,201,134)(105,202,135)(106,203,136)(107,204,137)(108,205,138)(109,206,139)(110,207,140)(111,208,141)(112,209,142)(113,210,143)(114,211,144), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,190)(40,153)(41,154)(42,155)(43,156)(44,157)(45,158)(46,159)(47,160)(48,161)(49,162)(50,163)(51,164)(52,165)(53,166)(54,167)(55,168)(56,169)(57,170)(58,171)(59,172)(60,173)(61,174)(62,175)(63,176)(64,177)(65,178)(66,179)(67,180)(68,181)(69,182)(70,183)(71,184)(72,185)(73,186)(74,187)(75,188)(76,189)(77,193)(78,194)(79,195)(80,196)(81,197)(82,198)(83,199)(84,200)(85,201)(86,202)(87,203)(88,204)(89,205)(90,206)(91,207)(92,208)(93,209)(94,210)(95,211)(96,212)(97,213)(98,214)(99,215)(100,216)(101,217)(102,218)(103,219)(104,220)(105,221)(106,222)(107,223)(108,224)(109,225)(110,226)(111,227)(112,228)(113,191)(114,192)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152), (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), (1,133,20,152)(2,132,21,151)(3,131,22,150)(4,130,23,149)(5,129,24,148)(6,128,25,147)(7,127,26,146)(8,126,27,145)(9,125,28,144)(10,124,29,143)(11,123,30,142)(12,122,31,141)(13,121,32,140)(14,120,33,139)(15,119,34,138)(16,118,35,137)(17,117,36,136)(18,116,37,135)(19,115,38,134)(39,199,58,218)(40,198,59,217)(41,197,60,216)(42,196,61,215)(43,195,62,214)(44,194,63,213)(45,193,64,212)(46,192,65,211)(47,191,66,210)(48,228,67,209)(49,227,68,208)(50,226,69,207)(51,225,70,206)(52,224,71,205)(53,223,72,204)(54,222,73,203)(55,221,74,202)(56,220,75,201)(57,219,76,200)(77,177,96,158)(78,176,97,157)(79,175,98,156)(80,174,99,155)(81,173,100,154)(82,172,101,153)(83,171,102,190)(84,170,103,189)(85,169,104,188)(86,168,105,187)(87,167,106,186)(88,166,107,185)(89,165,108,184)(90,164,109,183)(91,163,110,182)(92,162,111,181)(93,161,112,180)(94,160,113,179)(95,159,114,178)>;
G:=Group( (1,170,76)(2,171,39)(3,172,40)(4,173,41)(5,174,42)(6,175,43)(7,176,44)(8,177,45)(9,178,46)(10,179,47)(11,180,48)(12,181,49)(13,182,50)(14,183,51)(15,184,52)(16,185,53)(17,186,54)(18,187,55)(19,188,56)(20,189,57)(21,190,58)(22,153,59)(23,154,60)(24,155,61)(25,156,62)(26,157,63)(27,158,64)(28,159,65)(29,160,66)(30,161,67)(31,162,68)(32,163,69)(33,164,70)(34,165,71)(35,166,72)(36,167,73)(37,168,74)(38,169,75)(77,212,145)(78,213,146)(79,214,147)(80,215,148)(81,216,149)(82,217,150)(83,218,151)(84,219,152)(85,220,115)(86,221,116)(87,222,117)(88,223,118)(89,224,119)(90,225,120)(91,226,121)(92,227,122)(93,228,123)(94,191,124)(95,192,125)(96,193,126)(97,194,127)(98,195,128)(99,196,129)(100,197,130)(101,198,131)(102,199,132)(103,200,133)(104,201,134)(105,202,135)(106,203,136)(107,204,137)(108,205,138)(109,206,139)(110,207,140)(111,208,141)(112,209,142)(113,210,143)(114,211,144), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(39,190)(40,153)(41,154)(42,155)(43,156)(44,157)(45,158)(46,159)(47,160)(48,161)(49,162)(50,163)(51,164)(52,165)(53,166)(54,167)(55,168)(56,169)(57,170)(58,171)(59,172)(60,173)(61,174)(62,175)(63,176)(64,177)(65,178)(66,179)(67,180)(68,181)(69,182)(70,183)(71,184)(72,185)(73,186)(74,187)(75,188)(76,189)(77,193)(78,194)(79,195)(80,196)(81,197)(82,198)(83,199)(84,200)(85,201)(86,202)(87,203)(88,204)(89,205)(90,206)(91,207)(92,208)(93,209)(94,210)(95,211)(96,212)(97,213)(98,214)(99,215)(100,216)(101,217)(102,218)(103,219)(104,220)(105,221)(106,222)(107,223)(108,224)(109,225)(110,226)(111,227)(112,228)(113,191)(114,192)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152), (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), (1,133,20,152)(2,132,21,151)(3,131,22,150)(4,130,23,149)(5,129,24,148)(6,128,25,147)(7,127,26,146)(8,126,27,145)(9,125,28,144)(10,124,29,143)(11,123,30,142)(12,122,31,141)(13,121,32,140)(14,120,33,139)(15,119,34,138)(16,118,35,137)(17,117,36,136)(18,116,37,135)(19,115,38,134)(39,199,58,218)(40,198,59,217)(41,197,60,216)(42,196,61,215)(43,195,62,214)(44,194,63,213)(45,193,64,212)(46,192,65,211)(47,191,66,210)(48,228,67,209)(49,227,68,208)(50,226,69,207)(51,225,70,206)(52,224,71,205)(53,223,72,204)(54,222,73,203)(55,221,74,202)(56,220,75,201)(57,219,76,200)(77,177,96,158)(78,176,97,157)(79,175,98,156)(80,174,99,155)(81,173,100,154)(82,172,101,153)(83,171,102,190)(84,170,103,189)(85,169,104,188)(86,168,105,187)(87,167,106,186)(88,166,107,185)(89,165,108,184)(90,164,109,183)(91,163,110,182)(92,162,111,181)(93,161,112,180)(94,160,113,179)(95,159,114,178) );
G=PermutationGroup([[(1,170,76),(2,171,39),(3,172,40),(4,173,41),(5,174,42),(6,175,43),(7,176,44),(8,177,45),(9,178,46),(10,179,47),(11,180,48),(12,181,49),(13,182,50),(14,183,51),(15,184,52),(16,185,53),(17,186,54),(18,187,55),(19,188,56),(20,189,57),(21,190,58),(22,153,59),(23,154,60),(24,155,61),(25,156,62),(26,157,63),(27,158,64),(28,159,65),(29,160,66),(30,161,67),(31,162,68),(32,163,69),(33,164,70),(34,165,71),(35,166,72),(36,167,73),(37,168,74),(38,169,75),(77,212,145),(78,213,146),(79,214,147),(80,215,148),(81,216,149),(82,217,150),(83,218,151),(84,219,152),(85,220,115),(86,221,116),(87,222,117),(88,223,118),(89,224,119),(90,225,120),(91,226,121),(92,227,122),(93,228,123),(94,191,124),(95,192,125),(96,193,126),(97,194,127),(98,195,128),(99,196,129),(100,197,130),(101,198,131),(102,199,132),(103,200,133),(104,201,134),(105,202,135),(106,203,136),(107,204,137),(108,205,138),(109,206,139),(110,207,140),(111,208,141),(112,209,142),(113,210,143),(114,211,144)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(39,190),(40,153),(41,154),(42,155),(43,156),(44,157),(45,158),(46,159),(47,160),(48,161),(49,162),(50,163),(51,164),(52,165),(53,166),(54,167),(55,168),(56,169),(57,170),(58,171),(59,172),(60,173),(61,174),(62,175),(63,176),(64,177),(65,178),(66,179),(67,180),(68,181),(69,182),(70,183),(71,184),(72,185),(73,186),(74,187),(75,188),(76,189),(77,193),(78,194),(79,195),(80,196),(81,197),(82,198),(83,199),(84,200),(85,201),(86,202),(87,203),(88,204),(89,205),(90,206),(91,207),(92,208),(93,209),(94,210),(95,211),(96,212),(97,213),(98,214),(99,215),(100,216),(101,217),(102,218),(103,219),(104,220),(105,221),(106,222),(107,223),(108,224),(109,225),(110,226),(111,227),(112,228),(113,191),(114,192),(115,134),(116,135),(117,136),(118,137),(119,138),(120,139),(121,140),(122,141),(123,142),(124,143),(125,144),(126,145),(127,146),(128,147),(129,148),(130,149),(131,150),(132,151),(133,152)], [(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)], [(1,133,20,152),(2,132,21,151),(3,131,22,150),(4,130,23,149),(5,129,24,148),(6,128,25,147),(7,127,26,146),(8,126,27,145),(9,125,28,144),(10,124,29,143),(11,123,30,142),(12,122,31,141),(13,121,32,140),(14,120,33,139),(15,119,34,138),(16,118,35,137),(17,117,36,136),(18,116,37,135),(19,115,38,134),(39,199,58,218),(40,198,59,217),(41,197,60,216),(42,196,61,215),(43,195,62,214),(44,194,63,213),(45,193,64,212),(46,192,65,211),(47,191,66,210),(48,228,67,209),(49,227,68,208),(50,226,69,207),(51,225,70,206),(52,224,71,205),(53,223,72,204),(54,222,73,203),(55,221,74,202),(56,220,75,201),(57,219,76,200),(77,177,96,158),(78,176,97,157),(79,175,98,156),(80,174,99,155),(81,173,100,154),(82,172,101,153),(83,171,102,190),(84,170,103,189),(85,169,104,188),(86,168,105,187),(87,167,106,186),(88,166,107,185),(89,165,108,184),(90,164,109,183),(91,163,110,182),(92,162,111,181),(93,161,112,180),(94,160,113,179),(95,159,114,178)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 19A | ··· | 19I | 38A | ··· | 38I | 38J | ··· | 38AA | 57A | ··· | 57I | 114A | ··· | 114I |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 3 | 3 | 2 | 19 | 19 | 57 | 57 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 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 | D6 | C4×S3 | D19 | Dic19 | D38 | S3×D19 | S3×Dic19 |
| kernel | S3×Dic19 | C3×Dic19 | Dic57 | S3×C38 | S3×C19 | Dic19 | C38 | C19 | D6 | S3 | C6 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 9 | 18 | 9 | 9 | 9 |
Matrix representation of S3×Dic19 ►in GL4(𝔽229) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 227 | 70 |
| 0 | 0 | 121 | 1 |
| 228 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 108 | 228 |
| 0 | 228 | 0 | 0 |
| 1 | 127 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 58 | 139 | 0 | 0 |
| 101 | 171 | 0 | 0 |
| 0 | 0 | 228 | 0 |
| 0 | 0 | 0 | 228 |
G:=sub<GL(4,GF(229))| [1,0,0,0,0,1,0,0,0,0,227,121,0,0,70,1],[228,0,0,0,0,228,0,0,0,0,1,108,0,0,0,228],[0,1,0,0,228,127,0,0,0,0,1,0,0,0,0,1],[58,101,0,0,139,171,0,0,0,0,228,0,0,0,0,228] >;
S3×Dic19 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_{19} % in TeX
G:=Group("S3xDic19"); // GroupNames label
G:=SmallGroup(456,13);
// by ID
G=gap.SmallGroup(456,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,20,168,10804]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^38=1,d^2=c^19,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export