direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C6×D19, C57⋊3C23, C114⋊3C22, C38⋊3(C2×C6), (C2×C38)⋊11C6, (C2×C114)⋊5C2, C19⋊3(C22×C6), SmallGroup(456,51)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C2×C6×D19 |
Generators and relations for C2×C6×D19
G = < a,b,c,d | a2=b6=c19=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 460 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C6, C6, C23, C2×C6, C2×C6, C19, C22×C6, D19, C38, C57, D38, C2×C38, C3×D19, C114, C22×D19, C6×D19, C2×C114, C2×C6×D19
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C22×C6, D19, D38, C3×D19, C22×D19, C6×D19, C2×C6×D19
►On 228 points
Generators in S228
(1 185)(2 186)(3 187)(4 188)(5 189)(6 190)(7 172)(8 173)(9 174)(10 175)(11 176)(12 177)(13 178)(14 179)(15 180)(16 181)(17 182)(18 183)(19 184)(20 200)(21 201)(22 202)(23 203)(24 204)(25 205)(26 206)(27 207)(28 208)(29 209)(30 191)(31 192)(32 193)(33 194)(34 195)(35 196)(36 197)(37 198)(38 199)(39 210)(40 211)(41 212)(42 213)(43 214)(44 215)(45 216)(46 217)(47 218)(48 219)(49 220)(50 221)(51 222)(52 223)(53 224)(54 225)(55 226)(56 227)(57 228)(58 125)(59 126)(60 127)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 121)(74 122)(75 123)(76 124)(77 143)(78 144)(79 145)(80 146)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 141)(95 142)(96 168)(97 169)(98 170)(99 171)(100 153)(101 154)(102 155)(103 156)(104 157)(105 158)(106 159)(107 160)(108 161)(109 162)(110 163)(111 164)(112 165)(113 166)(114 167)
(1 85 41 62 37 102)(2 86 42 63 38 103)(3 87 43 64 20 104)(4 88 44 65 21 105)(5 89 45 66 22 106)(6 90 46 67 23 107)(7 91 47 68 24 108)(8 92 48 69 25 109)(9 93 49 70 26 110)(10 94 50 71 27 111)(11 95 51 72 28 112)(12 77 52 73 29 113)(13 78 53 74 30 114)(14 79 54 75 31 96)(15 80 55 76 32 97)(16 81 56 58 33 98)(17 82 57 59 34 99)(18 83 39 60 35 100)(19 84 40 61 36 101)(115 203 160 190 137 217)(116 204 161 172 138 218)(117 205 162 173 139 219)(118 206 163 174 140 220)(119 207 164 175 141 221)(120 208 165 176 142 222)(121 209 166 177 143 223)(122 191 167 178 144 224)(123 192 168 179 145 225)(124 193 169 180 146 226)(125 194 170 181 147 227)(126 195 171 182 148 228)(127 196 153 183 149 210)(128 197 154 184 150 211)(129 198 155 185 151 212)(130 199 156 186 152 213)(131 200 157 187 134 214)(132 201 158 188 135 215)(133 202 159 189 136 216)
(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)
(1 128)(2 127)(3 126)(4 125)(5 124)(6 123)(7 122)(8 121)(9 120)(10 119)(11 118)(12 117)(13 116)(14 115)(15 133)(16 132)(17 131)(18 130)(19 129)(20 148)(21 147)(22 146)(23 145)(24 144)(25 143)(26 142)(27 141)(28 140)(29 139)(30 138)(31 137)(32 136)(33 135)(34 134)(35 152)(36 151)(37 150)(38 149)(39 156)(40 155)(41 154)(42 153)(43 171)(44 170)(45 169)(46 168)(47 167)(48 166)(49 165)(50 164)(51 163)(52 162)(53 161)(54 160)(55 159)(56 158)(57 157)(58 188)(59 187)(60 186)(61 185)(62 184)(63 183)(64 182)(65 181)(66 180)(67 179)(68 178)(69 177)(70 176)(71 175)(72 174)(73 173)(74 172)(75 190)(76 189)(77 205)(78 204)(79 203)(80 202)(81 201)(82 200)(83 199)(84 198)(85 197)(86 196)(87 195)(88 194)(89 193)(90 192)(91 191)(92 209)(93 208)(94 207)(95 206)(96 217)(97 216)(98 215)(99 214)(100 213)(101 212)(102 211)(103 210)(104 228)(105 227)(106 226)(107 225)(108 224)(109 223)(110 222)(111 221)(112 220)(113 219)(114 218)
G:=sub<Sym(228)| (1,185)(2,186)(3,187)(4,188)(5,189)(6,190)(7,172)(8,173)(9,174)(10,175)(11,176)(12,177)(13,178)(14,179)(15,180)(16,181)(17,182)(18,183)(19,184)(20,200)(21,201)(22,202)(23,203)(24,204)(25,205)(26,206)(27,207)(28,208)(29,209)(30,191)(31,192)(32,193)(33,194)(34,195)(35,196)(36,197)(37,198)(38,199)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,217)(47,218)(48,219)(49,220)(50,221)(51,222)(52,223)(53,224)(54,225)(55,226)(56,227)(57,228)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,168)(97,169)(98,170)(99,171)(100,153)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,161)(109,162)(110,163)(111,164)(112,165)(113,166)(114,167), (1,85,41,62,37,102)(2,86,42,63,38,103)(3,87,43,64,20,104)(4,88,44,65,21,105)(5,89,45,66,22,106)(6,90,46,67,23,107)(7,91,47,68,24,108)(8,92,48,69,25,109)(9,93,49,70,26,110)(10,94,50,71,27,111)(11,95,51,72,28,112)(12,77,52,73,29,113)(13,78,53,74,30,114)(14,79,54,75,31,96)(15,80,55,76,32,97)(16,81,56,58,33,98)(17,82,57,59,34,99)(18,83,39,60,35,100)(19,84,40,61,36,101)(115,203,160,190,137,217)(116,204,161,172,138,218)(117,205,162,173,139,219)(118,206,163,174,140,220)(119,207,164,175,141,221)(120,208,165,176,142,222)(121,209,166,177,143,223)(122,191,167,178,144,224)(123,192,168,179,145,225)(124,193,169,180,146,226)(125,194,170,181,147,227)(126,195,171,182,148,228)(127,196,153,183,149,210)(128,197,154,184,150,211)(129,198,155,185,151,212)(130,199,156,186,152,213)(131,200,157,187,134,214)(132,201,158,188,135,215)(133,202,159,189,136,216), (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), (1,128)(2,127)(3,126)(4,125)(5,124)(6,123)(7,122)(8,121)(9,120)(10,119)(11,118)(12,117)(13,116)(14,115)(15,133)(16,132)(17,131)(18,130)(19,129)(20,148)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,140)(29,139)(30,138)(31,137)(32,136)(33,135)(34,134)(35,152)(36,151)(37,150)(38,149)(39,156)(40,155)(41,154)(42,153)(43,171)(44,170)(45,169)(46,168)(47,167)(48,166)(49,165)(50,164)(51,163)(52,162)(53,161)(54,160)(55,159)(56,158)(57,157)(58,188)(59,187)(60,186)(61,185)(62,184)(63,183)(64,182)(65,181)(66,180)(67,179)(68,178)(69,177)(70,176)(71,175)(72,174)(73,173)(74,172)(75,190)(76,189)(77,205)(78,204)(79,203)(80,202)(81,201)(82,200)(83,199)(84,198)(85,197)(86,196)(87,195)(88,194)(89,193)(90,192)(91,191)(92,209)(93,208)(94,207)(95,206)(96,217)(97,216)(98,215)(99,214)(100,213)(101,212)(102,211)(103,210)(104,228)(105,227)(106,226)(107,225)(108,224)(109,223)(110,222)(111,221)(112,220)(113,219)(114,218)>;
G:=Group( (1,185)(2,186)(3,187)(4,188)(5,189)(6,190)(7,172)(8,173)(9,174)(10,175)(11,176)(12,177)(13,178)(14,179)(15,180)(16,181)(17,182)(18,183)(19,184)(20,200)(21,201)(22,202)(23,203)(24,204)(25,205)(26,206)(27,207)(28,208)(29,209)(30,191)(31,192)(32,193)(33,194)(34,195)(35,196)(36,197)(37,198)(38,199)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,217)(47,218)(48,219)(49,220)(50,221)(51,222)(52,223)(53,224)(54,225)(55,226)(56,227)(57,228)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,168)(97,169)(98,170)(99,171)(100,153)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,161)(109,162)(110,163)(111,164)(112,165)(113,166)(114,167), (1,85,41,62,37,102)(2,86,42,63,38,103)(3,87,43,64,20,104)(4,88,44,65,21,105)(5,89,45,66,22,106)(6,90,46,67,23,107)(7,91,47,68,24,108)(8,92,48,69,25,109)(9,93,49,70,26,110)(10,94,50,71,27,111)(11,95,51,72,28,112)(12,77,52,73,29,113)(13,78,53,74,30,114)(14,79,54,75,31,96)(15,80,55,76,32,97)(16,81,56,58,33,98)(17,82,57,59,34,99)(18,83,39,60,35,100)(19,84,40,61,36,101)(115,203,160,190,137,217)(116,204,161,172,138,218)(117,205,162,173,139,219)(118,206,163,174,140,220)(119,207,164,175,141,221)(120,208,165,176,142,222)(121,209,166,177,143,223)(122,191,167,178,144,224)(123,192,168,179,145,225)(124,193,169,180,146,226)(125,194,170,181,147,227)(126,195,171,182,148,228)(127,196,153,183,149,210)(128,197,154,184,150,211)(129,198,155,185,151,212)(130,199,156,186,152,213)(131,200,157,187,134,214)(132,201,158,188,135,215)(133,202,159,189,136,216), (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), (1,128)(2,127)(3,126)(4,125)(5,124)(6,123)(7,122)(8,121)(9,120)(10,119)(11,118)(12,117)(13,116)(14,115)(15,133)(16,132)(17,131)(18,130)(19,129)(20,148)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,140)(29,139)(30,138)(31,137)(32,136)(33,135)(34,134)(35,152)(36,151)(37,150)(38,149)(39,156)(40,155)(41,154)(42,153)(43,171)(44,170)(45,169)(46,168)(47,167)(48,166)(49,165)(50,164)(51,163)(52,162)(53,161)(54,160)(55,159)(56,158)(57,157)(58,188)(59,187)(60,186)(61,185)(62,184)(63,183)(64,182)(65,181)(66,180)(67,179)(68,178)(69,177)(70,176)(71,175)(72,174)(73,173)(74,172)(75,190)(76,189)(77,205)(78,204)(79,203)(80,202)(81,201)(82,200)(83,199)(84,198)(85,197)(86,196)(87,195)(88,194)(89,193)(90,192)(91,191)(92,209)(93,208)(94,207)(95,206)(96,217)(97,216)(98,215)(99,214)(100,213)(101,212)(102,211)(103,210)(104,228)(105,227)(106,226)(107,225)(108,224)(109,223)(110,222)(111,221)(112,220)(113,219)(114,218) );
G=PermutationGroup([[(1,185),(2,186),(3,187),(4,188),(5,189),(6,190),(7,172),(8,173),(9,174),(10,175),(11,176),(12,177),(13,178),(14,179),(15,180),(16,181),(17,182),(18,183),(19,184),(20,200),(21,201),(22,202),(23,203),(24,204),(25,205),(26,206),(27,207),(28,208),(29,209),(30,191),(31,192),(32,193),(33,194),(34,195),(35,196),(36,197),(37,198),(38,199),(39,210),(40,211),(41,212),(42,213),(43,214),(44,215),(45,216),(46,217),(47,218),(48,219),(49,220),(50,221),(51,222),(52,223),(53,224),(54,225),(55,226),(56,227),(57,228),(58,125),(59,126),(60,127),(61,128),(62,129),(63,130),(64,131),(65,132),(66,133),(67,115),(68,116),(69,117),(70,118),(71,119),(72,120),(73,121),(74,122),(75,123),(76,124),(77,143),(78,144),(79,145),(80,146),(81,147),(82,148),(83,149),(84,150),(85,151),(86,152),(87,134),(88,135),(89,136),(90,137),(91,138),(92,139),(93,140),(94,141),(95,142),(96,168),(97,169),(98,170),(99,171),(100,153),(101,154),(102,155),(103,156),(104,157),(105,158),(106,159),(107,160),(108,161),(109,162),(110,163),(111,164),(112,165),(113,166),(114,167)], [(1,85,41,62,37,102),(2,86,42,63,38,103),(3,87,43,64,20,104),(4,88,44,65,21,105),(5,89,45,66,22,106),(6,90,46,67,23,107),(7,91,47,68,24,108),(8,92,48,69,25,109),(9,93,49,70,26,110),(10,94,50,71,27,111),(11,95,51,72,28,112),(12,77,52,73,29,113),(13,78,53,74,30,114),(14,79,54,75,31,96),(15,80,55,76,32,97),(16,81,56,58,33,98),(17,82,57,59,34,99),(18,83,39,60,35,100),(19,84,40,61,36,101),(115,203,160,190,137,217),(116,204,161,172,138,218),(117,205,162,173,139,219),(118,206,163,174,140,220),(119,207,164,175,141,221),(120,208,165,176,142,222),(121,209,166,177,143,223),(122,191,167,178,144,224),(123,192,168,179,145,225),(124,193,169,180,146,226),(125,194,170,181,147,227),(126,195,171,182,148,228),(127,196,153,183,149,210),(128,197,154,184,150,211),(129,198,155,185,151,212),(130,199,156,186,152,213),(131,200,157,187,134,214),(132,201,158,188,135,215),(133,202,159,189,136,216)], [(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)], [(1,128),(2,127),(3,126),(4,125),(5,124),(6,123),(7,122),(8,121),(9,120),(10,119),(11,118),(12,117),(13,116),(14,115),(15,133),(16,132),(17,131),(18,130),(19,129),(20,148),(21,147),(22,146),(23,145),(24,144),(25,143),(26,142),(27,141),(28,140),(29,139),(30,138),(31,137),(32,136),(33,135),(34,134),(35,152),(36,151),(37,150),(38,149),(39,156),(40,155),(41,154),(42,153),(43,171),(44,170),(45,169),(46,168),(47,167),(48,166),(49,165),(50,164),(51,163),(52,162),(53,161),(54,160),(55,159),(56,158),(57,157),(58,188),(59,187),(60,186),(61,185),(62,184),(63,183),(64,182),(65,181),(66,180),(67,179),(68,178),(69,177),(70,176),(71,175),(72,174),(73,173),(74,172),(75,190),(76,189),(77,205),(78,204),(79,203),(80,202),(81,201),(82,200),(83,199),(84,198),(85,197),(86,196),(87,195),(88,194),(89,193),(90,192),(91,191),(92,209),(93,208),(94,207),(95,206),(96,217),(97,216),(98,215),(99,214),(100,213),(101,212),(102,211),(103,210),(104,228),(105,227),(106,226),(107,225),(108,224),(109,223),(110,222),(111,221),(112,220),(113,219),(114,218)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 6A | ··· | 6F | 6G | ··· | 6N | 19A | ··· | 19I | 38A | ··· | 38AA | 57A | ··· | 57R | 114A | ··· | 114BB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 1 | 1 | 1 | ··· | 1 | 19 | ··· | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D19 | D38 | C3×D19 | C6×D19 |
| kernel | C2×C6×D19 | C6×D19 | C2×C114 | C22×D19 | D38 | C2×C38 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 2 | 12 | 2 | 9 | 27 | 18 | 54 |
Matrix representation of C2×C6×D19 ►in GL3(𝔽229) generated by
| 1 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 228 |
| 228 | 0 | 0 |
| 0 | 94 | 0 |
| 0 | 0 | 94 |
| 1 | 0 | 0 |
| 0 | 60 | 61 |
| 0 | 228 | 228 |
| 1 | 0 | 0 |
| 0 | 1 | 61 |
| 0 | 0 | 228 |
G:=sub<GL(3,GF(229))| [1,0,0,0,228,0,0,0,228],[228,0,0,0,94,0,0,0,94],[1,0,0,0,60,228,0,61,228],[1,0,0,0,1,0,0,61,228] >;
C2×C6×D19 in GAP, Magma, Sage, TeX
C_2\times C_6\times D_{19} % in TeX
G:=Group("C2xC6xD19"); // GroupNames label
G:=SmallGroup(456,51);
// by ID
G=gap.SmallGroup(456,51);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-19,10804]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^19=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations