direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C19⋊D4, C57⋊8D4, D38⋊5C6, C6.17D38, Dic19⋊4C6, C114.17C22, C19⋊5(C3×D4), (C2×C6)⋊1D19, (C2×C38)⋊10C6, (C2×C114)⋊4C2, (C6×D19)⋊5C2, C2.5(C6×D19), C38.13(C2×C6), C22⋊2(C3×D19), (C3×Dic19)⋊4C2, SmallGroup(456,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C19⋊D4
G = < a,b,c,d | a3=b19=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C3×C19⋊D4
►On 228 points
Generators in S228
(1 77 39)(2 78 40)(3 79 41)(4 80 42)(5 81 43)(6 82 44)(7 83 45)(8 84 46)(9 85 47)(10 86 48)(11 87 49)(12 88 50)(13 89 51)(14 90 52)(15 91 53)(16 92 54)(17 93 55)(18 94 56)(19 95 57)(20 96 58)(21 97 59)(22 98 60)(23 99 61)(24 100 62)(25 101 63)(26 102 64)(27 103 65)(28 104 66)(29 105 67)(30 106 68)(31 107 69)(32 108 70)(33 109 71)(34 110 72)(35 111 73)(36 112 74)(37 113 75)(38 114 76)(115 191 153)(116 192 154)(117 193 155)(118 194 156)(119 195 157)(120 196 158)(121 197 159)(122 198 160)(123 199 161)(124 200 162)(125 201 163)(126 202 164)(127 203 165)(128 204 166)(129 205 167)(130 206 168)(131 207 169)(132 208 170)(133 209 171)(134 210 172)(135 211 173)(136 212 174)(137 213 175)(138 214 176)(139 215 177)(140 216 178)(141 217 179)(142 218 180)(143 219 181)(144 220 182)(145 221 183)(146 222 184)(147 223 185)(148 224 186)(149 225 187)(150 226 188)(151 227 189)(152 228 190)
(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)
(1 115 20 134)(2 133 21 152)(3 132 22 151)(4 131 23 150)(5 130 24 149)(6 129 25 148)(7 128 26 147)(8 127 27 146)(9 126 28 145)(10 125 29 144)(11 124 30 143)(12 123 31 142)(13 122 32 141)(14 121 33 140)(15 120 34 139)(16 119 35 138)(17 118 36 137)(18 117 37 136)(19 116 38 135)(39 153 58 172)(40 171 59 190)(41 170 60 189)(42 169 61 188)(43 168 62 187)(44 167 63 186)(45 166 64 185)(46 165 65 184)(47 164 66 183)(48 163 67 182)(49 162 68 181)(50 161 69 180)(51 160 70 179)(52 159 71 178)(53 158 72 177)(54 157 73 176)(55 156 74 175)(56 155 75 174)(57 154 76 173)(77 191 96 210)(78 209 97 228)(79 208 98 227)(80 207 99 226)(81 206 100 225)(82 205 101 224)(83 204 102 223)(84 203 103 222)(85 202 104 221)(86 201 105 220)(87 200 106 219)(88 199 107 218)(89 198 108 217)(90 197 109 216)(91 196 110 215)(92 195 111 214)(93 194 112 213)(94 193 113 212)(95 192 114 211)
(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(40 57)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(115 134)(116 152)(117 151)(118 150)(119 149)(120 148)(121 147)(122 146)(123 145)(124 144)(125 143)(126 142)(127 141)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(153 172)(154 190)(155 189)(156 188)(157 187)(158 186)(159 185)(160 184)(161 183)(162 182)(163 181)(164 180)(165 179)(166 178)(167 177)(168 176)(169 175)(170 174)(171 173)(191 210)(192 228)(193 227)(194 226)(195 225)(196 224)(197 223)(198 222)(199 221)(200 220)(201 219)(202 218)(203 217)(204 216)(205 215)(206 214)(207 213)(208 212)(209 211)
G:=sub<Sym(228)| (1,77,39)(2,78,40)(3,79,41)(4,80,42)(5,81,43)(6,82,44)(7,83,45)(8,84,46)(9,85,47)(10,86,48)(11,87,49)(12,88,50)(13,89,51)(14,90,52)(15,91,53)(16,92,54)(17,93,55)(18,94,56)(19,95,57)(20,96,58)(21,97,59)(22,98,60)(23,99,61)(24,100,62)(25,101,63)(26,102,64)(27,103,65)(28,104,66)(29,105,67)(30,106,68)(31,107,69)(32,108,70)(33,109,71)(34,110,72)(35,111,73)(36,112,74)(37,113,75)(38,114,76)(115,191,153)(116,192,154)(117,193,155)(118,194,156)(119,195,157)(120,196,158)(121,197,159)(122,198,160)(123,199,161)(124,200,162)(125,201,163)(126,202,164)(127,203,165)(128,204,166)(129,205,167)(130,206,168)(131,207,169)(132,208,170)(133,209,171)(134,210,172)(135,211,173)(136,212,174)(137,213,175)(138,214,176)(139,215,177)(140,216,178)(141,217,179)(142,218,180)(143,219,181)(144,220,182)(145,221,183)(146,222,184)(147,223,185)(148,224,186)(149,225,187)(150,226,188)(151,227,189)(152,228,190), (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), (1,115,20,134)(2,133,21,152)(3,132,22,151)(4,131,23,150)(5,130,24,149)(6,129,25,148)(7,128,26,147)(8,127,27,146)(9,126,28,145)(10,125,29,144)(11,124,30,143)(12,123,31,142)(13,122,32,141)(14,121,33,140)(15,120,34,139)(16,119,35,138)(17,118,36,137)(18,117,37,136)(19,116,38,135)(39,153,58,172)(40,171,59,190)(41,170,60,189)(42,169,61,188)(43,168,62,187)(44,167,63,186)(45,166,64,185)(46,165,65,184)(47,164,66,183)(48,163,67,182)(49,162,68,181)(50,161,69,180)(51,160,70,179)(52,159,71,178)(53,158,72,177)(54,157,73,176)(55,156,74,175)(56,155,75,174)(57,154,76,173)(77,191,96,210)(78,209,97,228)(79,208,98,227)(80,207,99,226)(81,206,100,225)(82,205,101,224)(83,204,102,223)(84,203,103,222)(85,202,104,221)(86,201,105,220)(87,200,106,219)(88,199,107,218)(89,198,108,217)(90,197,109,216)(91,196,110,215)(92,195,111,214)(93,194,112,213)(94,193,113,212)(95,192,114,211), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(115,134)(116,152)(117,151)(118,150)(119,149)(120,148)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(153,172)(154,190)(155,189)(156,188)(157,187)(158,186)(159,185)(160,184)(161,183)(162,182)(163,181)(164,180)(165,179)(166,178)(167,177)(168,176)(169,175)(170,174)(171,173)(191,210)(192,228)(193,227)(194,226)(195,225)(196,224)(197,223)(198,222)(199,221)(200,220)(201,219)(202,218)(203,217)(204,216)(205,215)(206,214)(207,213)(208,212)(209,211)>;
G:=Group( (1,77,39)(2,78,40)(3,79,41)(4,80,42)(5,81,43)(6,82,44)(7,83,45)(8,84,46)(9,85,47)(10,86,48)(11,87,49)(12,88,50)(13,89,51)(14,90,52)(15,91,53)(16,92,54)(17,93,55)(18,94,56)(19,95,57)(20,96,58)(21,97,59)(22,98,60)(23,99,61)(24,100,62)(25,101,63)(26,102,64)(27,103,65)(28,104,66)(29,105,67)(30,106,68)(31,107,69)(32,108,70)(33,109,71)(34,110,72)(35,111,73)(36,112,74)(37,113,75)(38,114,76)(115,191,153)(116,192,154)(117,193,155)(118,194,156)(119,195,157)(120,196,158)(121,197,159)(122,198,160)(123,199,161)(124,200,162)(125,201,163)(126,202,164)(127,203,165)(128,204,166)(129,205,167)(130,206,168)(131,207,169)(132,208,170)(133,209,171)(134,210,172)(135,211,173)(136,212,174)(137,213,175)(138,214,176)(139,215,177)(140,216,178)(141,217,179)(142,218,180)(143,219,181)(144,220,182)(145,221,183)(146,222,184)(147,223,185)(148,224,186)(149,225,187)(150,226,188)(151,227,189)(152,228,190), (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), (1,115,20,134)(2,133,21,152)(3,132,22,151)(4,131,23,150)(5,130,24,149)(6,129,25,148)(7,128,26,147)(8,127,27,146)(9,126,28,145)(10,125,29,144)(11,124,30,143)(12,123,31,142)(13,122,32,141)(14,121,33,140)(15,120,34,139)(16,119,35,138)(17,118,36,137)(18,117,37,136)(19,116,38,135)(39,153,58,172)(40,171,59,190)(41,170,60,189)(42,169,61,188)(43,168,62,187)(44,167,63,186)(45,166,64,185)(46,165,65,184)(47,164,66,183)(48,163,67,182)(49,162,68,181)(50,161,69,180)(51,160,70,179)(52,159,71,178)(53,158,72,177)(54,157,73,176)(55,156,74,175)(56,155,75,174)(57,154,76,173)(77,191,96,210)(78,209,97,228)(79,208,98,227)(80,207,99,226)(81,206,100,225)(82,205,101,224)(83,204,102,223)(84,203,103,222)(85,202,104,221)(86,201,105,220)(87,200,106,219)(88,199,107,218)(89,198,108,217)(90,197,109,216)(91,196,110,215)(92,195,111,214)(93,194,112,213)(94,193,113,212)(95,192,114,211), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(115,134)(116,152)(117,151)(118,150)(119,149)(120,148)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(153,172)(154,190)(155,189)(156,188)(157,187)(158,186)(159,185)(160,184)(161,183)(162,182)(163,181)(164,180)(165,179)(166,178)(167,177)(168,176)(169,175)(170,174)(171,173)(191,210)(192,228)(193,227)(194,226)(195,225)(196,224)(197,223)(198,222)(199,221)(200,220)(201,219)(202,218)(203,217)(204,216)(205,215)(206,214)(207,213)(208,212)(209,211) );
G=PermutationGroup([[(1,77,39),(2,78,40),(3,79,41),(4,80,42),(5,81,43),(6,82,44),(7,83,45),(8,84,46),(9,85,47),(10,86,48),(11,87,49),(12,88,50),(13,89,51),(14,90,52),(15,91,53),(16,92,54),(17,93,55),(18,94,56),(19,95,57),(20,96,58),(21,97,59),(22,98,60),(23,99,61),(24,100,62),(25,101,63),(26,102,64),(27,103,65),(28,104,66),(29,105,67),(30,106,68),(31,107,69),(32,108,70),(33,109,71),(34,110,72),(35,111,73),(36,112,74),(37,113,75),(38,114,76),(115,191,153),(116,192,154),(117,193,155),(118,194,156),(119,195,157),(120,196,158),(121,197,159),(122,198,160),(123,199,161),(124,200,162),(125,201,163),(126,202,164),(127,203,165),(128,204,166),(129,205,167),(130,206,168),(131,207,169),(132,208,170),(133,209,171),(134,210,172),(135,211,173),(136,212,174),(137,213,175),(138,214,176),(139,215,177),(140,216,178),(141,217,179),(142,218,180),(143,219,181),(144,220,182),(145,221,183),(146,222,184),(147,223,185),(148,224,186),(149,225,187),(150,226,188),(151,227,189),(152,228,190)], [(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)], [(1,115,20,134),(2,133,21,152),(3,132,22,151),(4,131,23,150),(5,130,24,149),(6,129,25,148),(7,128,26,147),(8,127,27,146),(9,126,28,145),(10,125,29,144),(11,124,30,143),(12,123,31,142),(13,122,32,141),(14,121,33,140),(15,120,34,139),(16,119,35,138),(17,118,36,137),(18,117,37,136),(19,116,38,135),(39,153,58,172),(40,171,59,190),(41,170,60,189),(42,169,61,188),(43,168,62,187),(44,167,63,186),(45,166,64,185),(46,165,65,184),(47,164,66,183),(48,163,67,182),(49,162,68,181),(50,161,69,180),(51,160,70,179),(52,159,71,178),(53,158,72,177),(54,157,73,176),(55,156,74,175),(56,155,75,174),(57,154,76,173),(77,191,96,210),(78,209,97,228),(79,208,98,227),(80,207,99,226),(81,206,100,225),(82,205,101,224),(83,204,102,223),(84,203,103,222),(85,202,104,221),(86,201,105,220),(87,200,106,219),(88,199,107,218),(89,198,108,217),(90,197,109,216),(91,196,110,215),(92,195,111,214),(93,194,112,213),(94,193,113,212),(95,192,114,211)], [(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(40,57),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(59,76),(60,75),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(97,114),(98,113),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106),(115,134),(116,152),(117,151),(118,150),(119,149),(120,148),(121,147),(122,146),(123,145),(124,144),(125,143),(126,142),(127,141),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(153,172),(154,190),(155,189),(156,188),(157,187),(158,186),(159,185),(160,184),(161,183),(162,182),(163,181),(164,180),(165,179),(166,178),(167,177),(168,176),(169,175),(170,174),(171,173),(191,210),(192,228),(193,227),(194,226),(195,225),(196,224),(197,223),(198,222),(199,221),(200,220),(201,219),(202,218),(203,217),(204,216),(205,215),(206,214),(207,213),(208,212),(209,211)]])
123 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 19A | ··· | 19I | 38A | ··· | 38AA | 57A | ··· | 57R | 114A | ··· | 114BB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 2 | 38 | 1 | 1 | 38 | 1 | 1 | 2 | 2 | 38 | 38 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
123 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | C3×D4 | D19 | D38 | C3×D19 | C19⋊D4 | C6×D19 | C3×C19⋊D4 |
| kernel | C3×C19⋊D4 | C3×Dic19 | C6×D19 | C2×C114 | C19⋊D4 | Dic19 | D38 | C2×C38 | C57 | C19 | C2×C6 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 9 | 9 | 18 | 18 | 18 | 36 |
Matrix representation of C3×C19⋊D4 ►in GL2(𝔽229) generated by
| 94 | 0 |
| 0 | 94 |
| 53 | 228 |
| 1 | 0 |
| 121 | 30 |
| 31 | 108 |
| 1 | 0 |
| 53 | 228 |
G:=sub<GL(2,GF(229))| [94,0,0,94],[53,1,228,0],[121,31,30,108],[1,53,0,228] >;
C3×C19⋊D4 in GAP, Magma, Sage, TeX
C_3\times C_{19}\rtimes D_4 % in TeX
G:=Group("C3xC19:D4"); // GroupNames label
G:=SmallGroup(456,28);
// by ID
G=gap.SmallGroup(456,28);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-19,141,10804]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^19=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export