direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C32×D19, C57⋊6C6, (C3×C57)⋊3C2, C19⋊3(C3×C6), SmallGroup(342,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C32×D19 |
Generators and relations for C32×D19
G = < a,b,c,d | a3=b3=c19=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of C32×D19
►On 171 points
Generators in S171
(1 166 81)(2 167 82)(3 168 83)(4 169 84)(5 170 85)(6 171 86)(7 153 87)(8 154 88)(9 155 89)(10 156 90)(11 157 91)(12 158 92)(13 159 93)(14 160 94)(15 161 95)(16 162 77)(17 163 78)(18 164 79)(19 165 80)(20 117 101)(21 118 102)(22 119 103)(23 120 104)(24 121 105)(25 122 106)(26 123 107)(27 124 108)(28 125 109)(29 126 110)(30 127 111)(31 128 112)(32 129 113)(33 130 114)(34 131 96)(35 132 97)(36 133 98)(37 115 99)(38 116 100)(39 143 71)(40 144 72)(41 145 73)(42 146 74)(43 147 75)(44 148 76)(45 149 58)(46 150 59)(47 151 60)(48 152 61)(49 134 62)(50 135 63)(51 136 64)(52 137 65)(53 138 66)(54 139 67)(55 140 68)(56 141 69)(57 142 70)
(1 57 34)(2 39 35)(3 40 36)(4 41 37)(5 42 38)(6 43 20)(7 44 21)(8 45 22)(9 46 23)(10 47 24)(11 48 25)(12 49 26)(13 50 27)(14 51 28)(15 52 29)(16 53 30)(17 54 31)(18 55 32)(19 56 33)(58 103 88)(59 104 89)(60 105 90)(61 106 91)(62 107 92)(63 108 93)(64 109 94)(65 110 95)(66 111 77)(67 112 78)(68 113 79)(69 114 80)(70 96 81)(71 97 82)(72 98 83)(73 99 84)(74 100 85)(75 101 86)(76 102 87)(115 169 145)(116 170 146)(117 171 147)(118 153 148)(119 154 149)(120 155 150)(121 156 151)(122 157 152)(123 158 134)(124 159 135)(125 160 136)(126 161 137)(127 162 138)(128 163 139)(129 164 140)(130 165 141)(131 166 142)(132 167 143)(133 168 144)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 28)(21 27)(22 26)(23 25)(29 38)(30 37)(31 36)(32 35)(33 34)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(56 57)(58 62)(59 61)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(77 84)(78 83)(79 82)(80 81)(85 95)(86 94)(87 93)(88 92)(89 91)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(115 127)(116 126)(117 125)(118 124)(119 123)(120 122)(128 133)(129 132)(130 131)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)(141 142)(150 152)(153 159)(154 158)(155 157)(160 171)(161 170)(162 169)(163 168)(164 167)(165 166)
G:=sub<Sym(171)| (1,166,81)(2,167,82)(3,168,83)(4,169,84)(5,170,85)(6,171,86)(7,153,87)(8,154,88)(9,155,89)(10,156,90)(11,157,91)(12,158,92)(13,159,93)(14,160,94)(15,161,95)(16,162,77)(17,163,78)(18,164,79)(19,165,80)(20,117,101)(21,118,102)(22,119,103)(23,120,104)(24,121,105)(25,122,106)(26,123,107)(27,124,108)(28,125,109)(29,126,110)(30,127,111)(31,128,112)(32,129,113)(33,130,114)(34,131,96)(35,132,97)(36,133,98)(37,115,99)(38,116,100)(39,143,71)(40,144,72)(41,145,73)(42,146,74)(43,147,75)(44,148,76)(45,149,58)(46,150,59)(47,151,60)(48,152,61)(49,134,62)(50,135,63)(51,136,64)(52,137,65)(53,138,66)(54,139,67)(55,140,68)(56,141,69)(57,142,70), (1,57,34)(2,39,35)(3,40,36)(4,41,37)(5,42,38)(6,43,20)(7,44,21)(8,45,22)(9,46,23)(10,47,24)(11,48,25)(12,49,26)(13,50,27)(14,51,28)(15,52,29)(16,53,30)(17,54,31)(18,55,32)(19,56,33)(58,103,88)(59,104,89)(60,105,90)(61,106,91)(62,107,92)(63,108,93)(64,109,94)(65,110,95)(66,111,77)(67,112,78)(68,113,79)(69,114,80)(70,96,81)(71,97,82)(72,98,83)(73,99,84)(74,100,85)(75,101,86)(76,102,87)(115,169,145)(116,170,146)(117,171,147)(118,153,148)(119,154,149)(120,155,150)(121,156,151)(122,157,152)(123,158,134)(124,159,135)(125,160,136)(126,161,137)(127,162,138)(128,163,139)(129,164,140)(130,165,141)(131,166,142)(132,167,143)(133,168,144), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,28)(21,27)(22,26)(23,25)(29,38)(30,37)(31,36)(32,35)(33,34)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(56,57)(58,62)(59,61)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(77,84)(78,83)(79,82)(80,81)(85,95)(86,94)(87,93)(88,92)(89,91)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(115,127)(116,126)(117,125)(118,124)(119,123)(120,122)(128,133)(129,132)(130,131)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)(150,152)(153,159)(154,158)(155,157)(160,171)(161,170)(162,169)(163,168)(164,167)(165,166)>;
G:=Group( (1,166,81)(2,167,82)(3,168,83)(4,169,84)(5,170,85)(6,171,86)(7,153,87)(8,154,88)(9,155,89)(10,156,90)(11,157,91)(12,158,92)(13,159,93)(14,160,94)(15,161,95)(16,162,77)(17,163,78)(18,164,79)(19,165,80)(20,117,101)(21,118,102)(22,119,103)(23,120,104)(24,121,105)(25,122,106)(26,123,107)(27,124,108)(28,125,109)(29,126,110)(30,127,111)(31,128,112)(32,129,113)(33,130,114)(34,131,96)(35,132,97)(36,133,98)(37,115,99)(38,116,100)(39,143,71)(40,144,72)(41,145,73)(42,146,74)(43,147,75)(44,148,76)(45,149,58)(46,150,59)(47,151,60)(48,152,61)(49,134,62)(50,135,63)(51,136,64)(52,137,65)(53,138,66)(54,139,67)(55,140,68)(56,141,69)(57,142,70), (1,57,34)(2,39,35)(3,40,36)(4,41,37)(5,42,38)(6,43,20)(7,44,21)(8,45,22)(9,46,23)(10,47,24)(11,48,25)(12,49,26)(13,50,27)(14,51,28)(15,52,29)(16,53,30)(17,54,31)(18,55,32)(19,56,33)(58,103,88)(59,104,89)(60,105,90)(61,106,91)(62,107,92)(63,108,93)(64,109,94)(65,110,95)(66,111,77)(67,112,78)(68,113,79)(69,114,80)(70,96,81)(71,97,82)(72,98,83)(73,99,84)(74,100,85)(75,101,86)(76,102,87)(115,169,145)(116,170,146)(117,171,147)(118,153,148)(119,154,149)(120,155,150)(121,156,151)(122,157,152)(123,158,134)(124,159,135)(125,160,136)(126,161,137)(127,162,138)(128,163,139)(129,164,140)(130,165,141)(131,166,142)(132,167,143)(133,168,144), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,28)(21,27)(22,26)(23,25)(29,38)(30,37)(31,36)(32,35)(33,34)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(56,57)(58,62)(59,61)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(77,84)(78,83)(79,82)(80,81)(85,95)(86,94)(87,93)(88,92)(89,91)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(115,127)(116,126)(117,125)(118,124)(119,123)(120,122)(128,133)(129,132)(130,131)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143)(141,142)(150,152)(153,159)(154,158)(155,157)(160,171)(161,170)(162,169)(163,168)(164,167)(165,166) );
G=PermutationGroup([[(1,166,81),(2,167,82),(3,168,83),(4,169,84),(5,170,85),(6,171,86),(7,153,87),(8,154,88),(9,155,89),(10,156,90),(11,157,91),(12,158,92),(13,159,93),(14,160,94),(15,161,95),(16,162,77),(17,163,78),(18,164,79),(19,165,80),(20,117,101),(21,118,102),(22,119,103),(23,120,104),(24,121,105),(25,122,106),(26,123,107),(27,124,108),(28,125,109),(29,126,110),(30,127,111),(31,128,112),(32,129,113),(33,130,114),(34,131,96),(35,132,97),(36,133,98),(37,115,99),(38,116,100),(39,143,71),(40,144,72),(41,145,73),(42,146,74),(43,147,75),(44,148,76),(45,149,58),(46,150,59),(47,151,60),(48,152,61),(49,134,62),(50,135,63),(51,136,64),(52,137,65),(53,138,66),(54,139,67),(55,140,68),(56,141,69),(57,142,70)], [(1,57,34),(2,39,35),(3,40,36),(4,41,37),(5,42,38),(6,43,20),(7,44,21),(8,45,22),(9,46,23),(10,47,24),(11,48,25),(12,49,26),(13,50,27),(14,51,28),(15,52,29),(16,53,30),(17,54,31),(18,55,32),(19,56,33),(58,103,88),(59,104,89),(60,105,90),(61,106,91),(62,107,92),(63,108,93),(64,109,94),(65,110,95),(66,111,77),(67,112,78),(68,113,79),(69,114,80),(70,96,81),(71,97,82),(72,98,83),(73,99,84),(74,100,85),(75,101,86),(76,102,87),(115,169,145),(116,170,146),(117,171,147),(118,153,148),(119,154,149),(120,155,150),(121,156,151),(122,157,152),(123,158,134),(124,159,135),(125,160,136),(126,161,137),(127,162,138),(128,163,139),(129,164,140),(130,165,141),(131,166,142),(132,167,143),(133,168,144)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,28),(21,27),(22,26),(23,25),(29,38),(30,37),(31,36),(32,35),(33,34),(39,55),(40,54),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(56,57),(58,62),(59,61),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(77,84),(78,83),(79,82),(80,81),(85,95),(86,94),(87,93),(88,92),(89,91),(96,114),(97,113),(98,112),(99,111),(100,110),(101,109),(102,108),(103,107),(104,106),(115,127),(116,126),(117,125),(118,124),(119,123),(120,122),(128,133),(129,132),(130,131),(134,149),(135,148),(136,147),(137,146),(138,145),(139,144),(140,143),(141,142),(150,152),(153,159),(154,158),(155,157),(160,171),(161,170),(162,169),(163,168),(164,167),(165,166)]])
99 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 19A | ··· | 19I | 57A | ··· | 57BT |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 19 | ··· | 19 | 57 | ··· | 57 |
| size | 1 | 19 | 1 | ··· | 1 | 19 | ··· | 19 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D19 | C3×D19 |
| kernel | C32×D19 | C3×C57 | C3×D19 | C57 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 9 | 72 |
Matrix representation of C32×D19 ►in GL4(𝔽229) generated by
| 1 | 0 | 0 | 0 |
| 0 | 94 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 134 | 0 | 0 | 0 |
| 0 | 94 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 228 | 18 |
| 228 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(229))| [1,0,0,0,0,94,0,0,0,0,1,0,0,0,0,1],[134,0,0,0,0,94,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,228,0,0,1,18],[228,0,0,0,0,228,0,0,0,0,0,1,0,0,1,0] >;
C32×D19 in GAP, Magma, Sage, TeX
C_3^2\times D_{19} % in TeX
G:=Group("C3^2xD19"); // GroupNames label
G:=SmallGroup(342,13);
// by ID
G=gap.SmallGroup(342,13);
# by ID
G:=PCGroup([4,-2,-3,-3,-19,5187]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=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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