direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D9×C19, C9⋊C38, C171⋊3C2, C57.2S3, C3.(S3×C19), SmallGroup(342,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C19 |
Generators and relations for D9×C19
G = < a,b,c | a19=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D9×C19
►On 171 points
Generators in S171
(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 47 134 35 163 113 69 130 82)(2 48 135 36 164 114 70 131 83)(3 49 136 37 165 96 71 132 84)(4 50 137 38 166 97 72 133 85)(5 51 138 20 167 98 73 115 86)(6 52 139 21 168 99 74 116 87)(7 53 140 22 169 100 75 117 88)(8 54 141 23 170 101 76 118 89)(9 55 142 24 171 102 58 119 90)(10 56 143 25 153 103 59 120 91)(11 57 144 26 154 104 60 121 92)(12 39 145 27 155 105 61 122 93)(13 40 146 28 156 106 62 123 94)(14 41 147 29 157 107 63 124 95)(15 42 148 30 158 108 64 125 77)(16 43 149 31 159 109 65 126 78)(17 44 150 32 160 110 66 127 79)(18 45 151 33 161 111 67 128 80)(19 46 152 34 162 112 68 129 81)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 77)(16 78)(17 79)(18 80)(19 81)(20 98)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 96)(38 97)(39 122)(40 123)(41 124)(42 125)(43 126)(44 127)(45 128)(46 129)(47 130)(48 131)(49 132)(50 133)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 121)(58 142)(59 143)(60 144)(61 145)(62 146)(63 147)(64 148)(65 149)(66 150)(67 151)(68 152)(69 134)(70 135)(71 136)(72 137)(73 138)(74 139)(75 140)(76 141)
G:=sub<Sym(171)| (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), (1,47,134,35,163,113,69,130,82)(2,48,135,36,164,114,70,131,83)(3,49,136,37,165,96,71,132,84)(4,50,137,38,166,97,72,133,85)(5,51,138,20,167,98,73,115,86)(6,52,139,21,168,99,74,116,87)(7,53,140,22,169,100,75,117,88)(8,54,141,23,170,101,76,118,89)(9,55,142,24,171,102,58,119,90)(10,56,143,25,153,103,59,120,91)(11,57,144,26,154,104,60,121,92)(12,39,145,27,155,105,61,122,93)(13,40,146,28,156,106,62,123,94)(14,41,147,29,157,107,63,124,95)(15,42,148,30,158,108,64,125,77)(16,43,149,31,159,109,65,126,78)(17,44,150,32,160,110,66,127,79)(18,45,151,33,161,111,67,128,80)(19,46,152,34,162,112,68,129,81), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,77)(16,78)(17,79)(18,80)(19,81)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,96)(38,97)(39,122)(40,123)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,131)(49,132)(50,133)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,121)(58,142)(59,143)(60,144)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,134)(70,135)(71,136)(72,137)(73,138)(74,139)(75,140)(76,141)>;
G:=Group( (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), (1,47,134,35,163,113,69,130,82)(2,48,135,36,164,114,70,131,83)(3,49,136,37,165,96,71,132,84)(4,50,137,38,166,97,72,133,85)(5,51,138,20,167,98,73,115,86)(6,52,139,21,168,99,74,116,87)(7,53,140,22,169,100,75,117,88)(8,54,141,23,170,101,76,118,89)(9,55,142,24,171,102,58,119,90)(10,56,143,25,153,103,59,120,91)(11,57,144,26,154,104,60,121,92)(12,39,145,27,155,105,61,122,93)(13,40,146,28,156,106,62,123,94)(14,41,147,29,157,107,63,124,95)(15,42,148,30,158,108,64,125,77)(16,43,149,31,159,109,65,126,78)(17,44,150,32,160,110,66,127,79)(18,45,151,33,161,111,67,128,80)(19,46,152,34,162,112,68,129,81), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,77)(16,78)(17,79)(18,80)(19,81)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,96)(38,97)(39,122)(40,123)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,131)(49,132)(50,133)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,121)(58,142)(59,143)(60,144)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,134)(70,135)(71,136)(72,137)(73,138)(74,139)(75,140)(76,141) );
G=PermutationGroup([[(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)], [(1,47,134,35,163,113,69,130,82),(2,48,135,36,164,114,70,131,83),(3,49,136,37,165,96,71,132,84),(4,50,137,38,166,97,72,133,85),(5,51,138,20,167,98,73,115,86),(6,52,139,21,168,99,74,116,87),(7,53,140,22,169,100,75,117,88),(8,54,141,23,170,101,76,118,89),(9,55,142,24,171,102,58,119,90),(10,56,143,25,153,103,59,120,91),(11,57,144,26,154,104,60,121,92),(12,39,145,27,155,105,61,122,93),(13,40,146,28,156,106,62,123,94),(14,41,147,29,157,107,63,124,95),(15,42,148,30,158,108,64,125,77),(16,43,149,31,159,109,65,126,78),(17,44,150,32,160,110,66,127,79),(18,45,151,33,161,111,67,128,80),(19,46,152,34,162,112,68,129,81)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,77),(16,78),(17,79),(18,80),(19,81),(20,98),(21,99),(22,100),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,107),(30,108),(31,109),(32,110),(33,111),(34,112),(35,113),(36,114),(37,96),(38,97),(39,122),(40,123),(41,124),(42,125),(43,126),(44,127),(45,128),(46,129),(47,130),(48,131),(49,132),(50,133),(51,115),(52,116),(53,117),(54,118),(55,119),(56,120),(57,121),(58,142),(59,143),(60,144),(61,145),(62,146),(63,147),(64,148),(65,149),(66,150),(67,151),(68,152),(69,134),(70,135),(71,136),(72,137),(73,138),(74,139),(75,140),(76,141)]])
114 conjugacy classes
| class | 1 | 2 | 3 | 9A | 9B | 9C | 19A | ··· | 19R | 38A | ··· | 38R | 57A | ··· | 57R | 171A | ··· | 171BB |
| order | 1 | 2 | 3 | 9 | 9 | 9 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 171 | ··· | 171 |
| size | 1 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C19 | C38 | S3 | D9 | S3×C19 | D9×C19 |
| kernel | D9×C19 | C171 | D9 | C9 | C57 | C19 | C3 | C1 |
| # reps | 1 | 1 | 18 | 18 | 1 | 3 | 18 | 54 |
Matrix representation of D9×C19 ►in GL2(𝔽2053) generated by
| 149 | 0 |
| 0 | 149 |
| 1647 | 974 |
| 1079 | 673 |
| 1079 | 406 |
| 1380 | 974 |
G:=sub<GL(2,GF(2053))| [149,0,0,149],[1647,1079,974,673],[1079,1380,406,974] >;
D9×C19 in GAP, Magma, Sage, TeX
D_9\times C_{19} % in TeX
G:=Group("D9xC19"); // GroupNames label
G:=SmallGroup(342,3);
// by ID
G=gap.SmallGroup(342,3);
# by ID
G:=PCGroup([4,-2,-19,-3,-3,2282,82,3651]);
// Polycyclic
G:=Group<a,b,c|a^19=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export