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## G = C2×C6×D19order 456 = 23·3·19

### Direct product of C2×C6 and D19

Aliases: C2×C6×D19, C573C23, C1143C22, C383(C2×C6), (C2×C38)⋊11C6, (C2×C114)⋊5C2, C193(C22×C6), SmallGroup(456,51)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — C2×C6×D19
 Chief series C1 — C19 — C57 — C3×D19 — C6×D19 — C2×C6×D19
 Lower central C19 — C2×C6×D19
 Upper central C1 — C2×C6

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

Smallest permutation representation of 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

׿
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𝔽