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G = Dic3×D19order 456 = 23·3·19

Direct product of Dic3 and D19

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: Dic3×D19, D38.S3, C6.1D38, C38.1D6, Dic572C2, C114.1C22, C571(C2×C4), (C3×D19)⋊C4, C33(C4×D19), (C6×D19).C2, C2.1(S3×D19), C191(C2×Dic3), (Dic3×C19)⋊1C2, SmallGroup(456,12)

Series: Derived Chief Lower central Upper central

C1C57 — Dic3×D19
C1C19C57C114C6×D19 — Dic3×D19
C57 — Dic3×D19
C1C2

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 >

19C2
19C2
3C4
19C22
57C4
19C6
19C6
57C2×C4
19Dic3
19C2×C6
3C76
3Dic19
19C2×Dic3
3C4×D19

Smallest permutation representation of Dic3×D19
On 228 points
Generators in S228
(1 107 25 58 43 90)(2 108 26 59 44 91)(3 109 27 60 45 92)(4 110 28 61 46 93)(5 111 29 62 47 94)(6 112 30 63 48 95)(7 113 31 64 49 77)(8 114 32 65 50 78)(9 96 33 66 51 79)(10 97 34 67 52 80)(11 98 35 68 53 81)(12 99 36 69 54 82)(13 100 37 70 55 83)(14 101 38 71 56 84)(15 102 20 72 57 85)(16 103 21 73 39 86)(17 104 22 74 40 87)(18 105 23 75 41 88)(19 106 24 76 42 89)(115 191 160 181 139 225)(116 192 161 182 140 226)(117 193 162 183 141 227)(118 194 163 184 142 228)(119 195 164 185 143 210)(120 196 165 186 144 211)(121 197 166 187 145 212)(122 198 167 188 146 213)(123 199 168 189 147 214)(124 200 169 190 148 215)(125 201 170 172 149 216)(126 202 171 173 150 217)(127 203 153 174 151 218)(128 204 154 175 152 219)(129 205 155 176 134 220)(130 206 156 177 135 221)(131 207 157 178 136 222)(132 208 158 179 137 223)(133 209 159 180 138 224)
(1 175 58 128)(2 176 59 129)(3 177 60 130)(4 178 61 131)(5 179 62 132)(6 180 63 133)(7 181 64 115)(8 182 65 116)(9 183 66 117)(10 184 67 118)(11 185 68 119)(12 186 69 120)(13 187 70 121)(14 188 71 122)(15 189 72 123)(16 190 73 124)(17 172 74 125)(18 173 75 126)(19 174 76 127)(20 199 85 147)(21 200 86 148)(22 201 87 149)(23 202 88 150)(24 203 89 151)(25 204 90 152)(26 205 91 134)(27 206 92 135)(28 207 93 136)(29 208 94 137)(30 209 95 138)(31 191 77 139)(32 192 78 140)(33 193 79 141)(34 194 80 142)(35 195 81 143)(36 196 82 144)(37 197 83 145)(38 198 84 146)(39 215 103 169)(40 216 104 170)(41 217 105 171)(42 218 106 153)(43 219 107 154)(44 220 108 155)(45 221 109 156)(46 222 110 157)(47 223 111 158)(48 224 112 159)(49 225 113 160)(50 226 114 161)(51 227 96 162)(52 228 97 163)(53 210 98 164)(54 211 99 165)(55 212 100 166)(56 213 101 167)(57 214 102 168)
(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 29)(21 28)(22 27)(23 26)(24 25)(30 38)(31 37)(32 36)(33 35)(39 46)(40 45)(41 44)(42 43)(47 57)(48 56)(49 55)(50 54)(51 53)(58 76)(59 75)(60 74)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(77 83)(78 82)(79 81)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(96 98)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(115 121)(116 120)(117 119)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(134 150)(135 149)(136 148)(137 147)(138 146)(139 145)(140 144)(141 143)(151 152)(153 154)(155 171)(156 170)(157 169)(158 168)(159 167)(160 166)(161 165)(162 164)(172 177)(173 176)(174 175)(178 190)(179 189)(180 188)(181 187)(182 186)(183 185)(191 197)(192 196)(193 195)(198 209)(199 208)(200 207)(201 206)(202 205)(203 204)(210 227)(211 226)(212 225)(213 224)(214 223)(215 222)(216 221)(217 220)(218 219)

G:=sub<Sym(228)| (1,107,25,58,43,90)(2,108,26,59,44,91)(3,109,27,60,45,92)(4,110,28,61,46,93)(5,111,29,62,47,94)(6,112,30,63,48,95)(7,113,31,64,49,77)(8,114,32,65,50,78)(9,96,33,66,51,79)(10,97,34,67,52,80)(11,98,35,68,53,81)(12,99,36,69,54,82)(13,100,37,70,55,83)(14,101,38,71,56,84)(15,102,20,72,57,85)(16,103,21,73,39,86)(17,104,22,74,40,87)(18,105,23,75,41,88)(19,106,24,76,42,89)(115,191,160,181,139,225)(116,192,161,182,140,226)(117,193,162,183,141,227)(118,194,163,184,142,228)(119,195,164,185,143,210)(120,196,165,186,144,211)(121,197,166,187,145,212)(122,198,167,188,146,213)(123,199,168,189,147,214)(124,200,169,190,148,215)(125,201,170,172,149,216)(126,202,171,173,150,217)(127,203,153,174,151,218)(128,204,154,175,152,219)(129,205,155,176,134,220)(130,206,156,177,135,221)(131,207,157,178,136,222)(132,208,158,179,137,223)(133,209,159,180,138,224), (1,175,58,128)(2,176,59,129)(3,177,60,130)(4,178,61,131)(5,179,62,132)(6,180,63,133)(7,181,64,115)(8,182,65,116)(9,183,66,117)(10,184,67,118)(11,185,68,119)(12,186,69,120)(13,187,70,121)(14,188,71,122)(15,189,72,123)(16,190,73,124)(17,172,74,125)(18,173,75,126)(19,174,76,127)(20,199,85,147)(21,200,86,148)(22,201,87,149)(23,202,88,150)(24,203,89,151)(25,204,90,152)(26,205,91,134)(27,206,92,135)(28,207,93,136)(29,208,94,137)(30,209,95,138)(31,191,77,139)(32,192,78,140)(33,193,79,141)(34,194,80,142)(35,195,81,143)(36,196,82,144)(37,197,83,145)(38,198,84,146)(39,215,103,169)(40,216,104,170)(41,217,105,171)(42,218,106,153)(43,219,107,154)(44,220,108,155)(45,221,109,156)(46,222,110,157)(47,223,111,158)(48,224,112,159)(49,225,113,160)(50,226,114,161)(51,227,96,162)(52,228,97,163)(53,210,98,164)(54,211,99,165)(55,212,100,166)(56,213,101,167)(57,214,102,168), (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)(20,29)(21,28)(22,27)(23,26)(24,25)(30,38)(31,37)(32,36)(33,35)(39,46)(40,45)(41,44)(42,43)(47,57)(48,56)(49,55)(50,54)(51,53)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(77,83)(78,82)(79,81)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(96,98)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(115,121)(116,120)(117,119)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144)(141,143)(151,152)(153,154)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(172,177)(173,176)(174,175)(178,190)(179,189)(180,188)(181,187)(182,186)(183,185)(191,197)(192,196)(193,195)(198,209)(199,208)(200,207)(201,206)(202,205)(203,204)(210,227)(211,226)(212,225)(213,224)(214,223)(215,222)(216,221)(217,220)(218,219)>;

G:=Group( (1,107,25,58,43,90)(2,108,26,59,44,91)(3,109,27,60,45,92)(4,110,28,61,46,93)(5,111,29,62,47,94)(6,112,30,63,48,95)(7,113,31,64,49,77)(8,114,32,65,50,78)(9,96,33,66,51,79)(10,97,34,67,52,80)(11,98,35,68,53,81)(12,99,36,69,54,82)(13,100,37,70,55,83)(14,101,38,71,56,84)(15,102,20,72,57,85)(16,103,21,73,39,86)(17,104,22,74,40,87)(18,105,23,75,41,88)(19,106,24,76,42,89)(115,191,160,181,139,225)(116,192,161,182,140,226)(117,193,162,183,141,227)(118,194,163,184,142,228)(119,195,164,185,143,210)(120,196,165,186,144,211)(121,197,166,187,145,212)(122,198,167,188,146,213)(123,199,168,189,147,214)(124,200,169,190,148,215)(125,201,170,172,149,216)(126,202,171,173,150,217)(127,203,153,174,151,218)(128,204,154,175,152,219)(129,205,155,176,134,220)(130,206,156,177,135,221)(131,207,157,178,136,222)(132,208,158,179,137,223)(133,209,159,180,138,224), (1,175,58,128)(2,176,59,129)(3,177,60,130)(4,178,61,131)(5,179,62,132)(6,180,63,133)(7,181,64,115)(8,182,65,116)(9,183,66,117)(10,184,67,118)(11,185,68,119)(12,186,69,120)(13,187,70,121)(14,188,71,122)(15,189,72,123)(16,190,73,124)(17,172,74,125)(18,173,75,126)(19,174,76,127)(20,199,85,147)(21,200,86,148)(22,201,87,149)(23,202,88,150)(24,203,89,151)(25,204,90,152)(26,205,91,134)(27,206,92,135)(28,207,93,136)(29,208,94,137)(30,209,95,138)(31,191,77,139)(32,192,78,140)(33,193,79,141)(34,194,80,142)(35,195,81,143)(36,196,82,144)(37,197,83,145)(38,198,84,146)(39,215,103,169)(40,216,104,170)(41,217,105,171)(42,218,106,153)(43,219,107,154)(44,220,108,155)(45,221,109,156)(46,222,110,157)(47,223,111,158)(48,224,112,159)(49,225,113,160)(50,226,114,161)(51,227,96,162)(52,228,97,163)(53,210,98,164)(54,211,99,165)(55,212,100,166)(56,213,101,167)(57,214,102,168), (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)(20,29)(21,28)(22,27)(23,26)(24,25)(30,38)(31,37)(32,36)(33,35)(39,46)(40,45)(41,44)(42,43)(47,57)(48,56)(49,55)(50,54)(51,53)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(77,83)(78,82)(79,81)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(96,98)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(115,121)(116,120)(117,119)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144)(141,143)(151,152)(153,154)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(172,177)(173,176)(174,175)(178,190)(179,189)(180,188)(181,187)(182,186)(183,185)(191,197)(192,196)(193,195)(198,209)(199,208)(200,207)(201,206)(202,205)(203,204)(210,227)(211,226)(212,225)(213,224)(214,223)(215,222)(216,221)(217,220)(218,219) );

G=PermutationGroup([(1,107,25,58,43,90),(2,108,26,59,44,91),(3,109,27,60,45,92),(4,110,28,61,46,93),(5,111,29,62,47,94),(6,112,30,63,48,95),(7,113,31,64,49,77),(8,114,32,65,50,78),(9,96,33,66,51,79),(10,97,34,67,52,80),(11,98,35,68,53,81),(12,99,36,69,54,82),(13,100,37,70,55,83),(14,101,38,71,56,84),(15,102,20,72,57,85),(16,103,21,73,39,86),(17,104,22,74,40,87),(18,105,23,75,41,88),(19,106,24,76,42,89),(115,191,160,181,139,225),(116,192,161,182,140,226),(117,193,162,183,141,227),(118,194,163,184,142,228),(119,195,164,185,143,210),(120,196,165,186,144,211),(121,197,166,187,145,212),(122,198,167,188,146,213),(123,199,168,189,147,214),(124,200,169,190,148,215),(125,201,170,172,149,216),(126,202,171,173,150,217),(127,203,153,174,151,218),(128,204,154,175,152,219),(129,205,155,176,134,220),(130,206,156,177,135,221),(131,207,157,178,136,222),(132,208,158,179,137,223),(133,209,159,180,138,224)], [(1,175,58,128),(2,176,59,129),(3,177,60,130),(4,178,61,131),(5,179,62,132),(6,180,63,133),(7,181,64,115),(8,182,65,116),(9,183,66,117),(10,184,67,118),(11,185,68,119),(12,186,69,120),(13,187,70,121),(14,188,71,122),(15,189,72,123),(16,190,73,124),(17,172,74,125),(18,173,75,126),(19,174,76,127),(20,199,85,147),(21,200,86,148),(22,201,87,149),(23,202,88,150),(24,203,89,151),(25,204,90,152),(26,205,91,134),(27,206,92,135),(28,207,93,136),(29,208,94,137),(30,209,95,138),(31,191,77,139),(32,192,78,140),(33,193,79,141),(34,194,80,142),(35,195,81,143),(36,196,82,144),(37,197,83,145),(38,198,84,146),(39,215,103,169),(40,216,104,170),(41,217,105,171),(42,218,106,153),(43,219,107,154),(44,220,108,155),(45,221,109,156),(46,222,110,157),(47,223,111,158),(48,224,112,159),(49,225,113,160),(50,226,114,161),(51,227,96,162),(52,228,97,163),(53,210,98,164),(54,211,99,165),(55,212,100,166),(56,213,101,167),(57,214,102,168)], [(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),(20,29),(21,28),(22,27),(23,26),(24,25),(30,38),(31,37),(32,36),(33,35),(39,46),(40,45),(41,44),(42,43),(47,57),(48,56),(49,55),(50,54),(51,53),(58,76),(59,75),(60,74),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(77,83),(78,82),(79,81),(84,95),(85,94),(86,93),(87,92),(88,91),(89,90),(96,98),(99,114),(100,113),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(115,121),(116,120),(117,119),(122,133),(123,132),(124,131),(125,130),(126,129),(127,128),(134,150),(135,149),(136,148),(137,147),(138,146),(139,145),(140,144),(141,143),(151,152),(153,154),(155,171),(156,170),(157,169),(158,168),(159,167),(160,166),(161,165),(162,164),(172,177),(173,176),(174,175),(178,190),(179,189),(180,188),(181,187),(182,186),(183,185),(191,197),(192,196),(193,195),(198,209),(199,208),(200,207),(201,206),(202,205),(203,204),(210,227),(211,226),(212,225),(213,224),(214,223),(215,222),(216,221),(217,220),(218,219)])

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D6A6B6C19A···19I38A···38I57A···57I76A···76R114A···114I
order12223444466619···1938···3857···5776···76114···114
size1119192335757238382···22···24···46···64···4

66 irreducible representations

dim1111122222244
type+++++-++++-
imageC1C2C2C2C4S3Dic3D6D19D38C4×D19S3×D19Dic3×D19
kernelDic3×D19Dic3×C19Dic57C6×D19C3×D19D38D19C38Dic3C6C3C2C1
# reps11114121991899

Matrix representation of Dic3×D19 in GL4(𝔽229) generated by

1000
0100
00244
0078228
,
1000
0100
0015976
004770
,
12100
558100
0010
0001
,
116800
711300
0010
0001
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

Subgroup lattice of Dic3×D19 in TeX

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