direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×D19, C171⋊2C2, C19⋊5C18, C57.3C6, C3.(C3×D19), (C3×D19).2C3, SmallGroup(342,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C9×D19 |
Generators and relations for C9×D19
G = < a,b,c | a9=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C9×D19
►On 171 points
Generators in S171
(1 164 109 41 144 93 33 131 66)(2 165 110 42 145 94 34 132 67)(3 166 111 43 146 95 35 133 68)(4 167 112 44 147 77 36 115 69)(5 168 113 45 148 78 37 116 70)(6 169 114 46 149 79 38 117 71)(7 170 96 47 150 80 20 118 72)(8 171 97 48 151 81 21 119 73)(9 153 98 49 152 82 22 120 74)(10 154 99 50 134 83 23 121 75)(11 155 100 51 135 84 24 122 76)(12 156 101 52 136 85 25 123 58)(13 157 102 53 137 86 26 124 59)(14 158 103 54 138 87 27 125 60)(15 159 104 55 139 88 28 126 61)(16 160 105 56 140 89 29 127 62)(17 161 106 57 141 90 30 128 63)(18 162 107 39 142 91 31 129 64)(19 163 108 40 143 92 32 130 65)
(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 26)(21 25)(22 24)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(39 42)(40 41)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(58 73)(59 72)(60 71)(61 70)(62 69)(63 68)(64 67)(65 66)(74 76)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(90 95)(91 94)(92 93)(96 102)(97 101)(98 100)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(115 127)(116 126)(117 125)(118 124)(119 123)(120 122)(128 133)(129 132)(130 131)(135 152)(136 151)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(153 155)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 165)(163 164)
G:=sub<Sym(171)| (1,164,109,41,144,93,33,131,66)(2,165,110,42,145,94,34,132,67)(3,166,111,43,146,95,35,133,68)(4,167,112,44,147,77,36,115,69)(5,168,113,45,148,78,37,116,70)(6,169,114,46,149,79,38,117,71)(7,170,96,47,150,80,20,118,72)(8,171,97,48,151,81,21,119,73)(9,153,98,49,152,82,22,120,74)(10,154,99,50,134,83,23,121,75)(11,155,100,51,135,84,24,122,76)(12,156,101,52,136,85,25,123,58)(13,157,102,53,137,86,26,124,59)(14,158,103,54,138,87,27,125,60)(15,159,104,55,139,88,28,126,61)(16,160,105,56,140,89,29,127,62)(17,161,106,57,141,90,30,128,63)(18,162,107,39,142,91,31,129,64)(19,163,108,40,143,92,32,130,65), (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)(20,26)(21,25)(22,24)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(39,42)(40,41)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(74,76)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(90,95)(91,94)(92,93)(96,102)(97,101)(98,100)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(115,127)(116,126)(117,125)(118,124)(119,123)(120,122)(128,133)(129,132)(130,131)(135,152)(136,151)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(153,155)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)>;
G:=Group( (1,164,109,41,144,93,33,131,66)(2,165,110,42,145,94,34,132,67)(3,166,111,43,146,95,35,133,68)(4,167,112,44,147,77,36,115,69)(5,168,113,45,148,78,37,116,70)(6,169,114,46,149,79,38,117,71)(7,170,96,47,150,80,20,118,72)(8,171,97,48,151,81,21,119,73)(9,153,98,49,152,82,22,120,74)(10,154,99,50,134,83,23,121,75)(11,155,100,51,135,84,24,122,76)(12,156,101,52,136,85,25,123,58)(13,157,102,53,137,86,26,124,59)(14,158,103,54,138,87,27,125,60)(15,159,104,55,139,88,28,126,61)(16,160,105,56,140,89,29,127,62)(17,161,106,57,141,90,30,128,63)(18,162,107,39,142,91,31,129,64)(19,163,108,40,143,92,32,130,65), (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)(20,26)(21,25)(22,24)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(39,42)(40,41)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)(74,76)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(90,95)(91,94)(92,93)(96,102)(97,101)(98,100)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(115,127)(116,126)(117,125)(118,124)(119,123)(120,122)(128,133)(129,132)(130,131)(135,152)(136,151)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(153,155)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164) );
G=PermutationGroup([[(1,164,109,41,144,93,33,131,66),(2,165,110,42,145,94,34,132,67),(3,166,111,43,146,95,35,133,68),(4,167,112,44,147,77,36,115,69),(5,168,113,45,148,78,37,116,70),(6,169,114,46,149,79,38,117,71),(7,170,96,47,150,80,20,118,72),(8,171,97,48,151,81,21,119,73),(9,153,98,49,152,82,22,120,74),(10,154,99,50,134,83,23,121,75),(11,155,100,51,135,84,24,122,76),(12,156,101,52,136,85,25,123,58),(13,157,102,53,137,86,26,124,59),(14,158,103,54,138,87,27,125,60),(15,159,104,55,139,88,28,126,61),(16,160,105,56,140,89,29,127,62),(17,161,106,57,141,90,30,128,63),(18,162,107,39,142,91,31,129,64),(19,163,108,40,143,92,32,130,65)], [(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),(20,26),(21,25),(22,24),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(39,42),(40,41),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(58,73),(59,72),(60,71),(61,70),(62,69),(63,68),(64,67),(65,66),(74,76),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(90,95),(91,94),(92,93),(96,102),(97,101),(98,100),(103,114),(104,113),(105,112),(106,111),(107,110),(108,109),(115,127),(116,126),(117,125),(118,124),(119,123),(120,122),(128,133),(129,132),(130,131),(135,152),(136,151),(137,150),(138,149),(139,148),(140,147),(141,146),(142,145),(143,144),(153,155),(156,171),(157,170),(158,169),(159,168),(160,167),(161,166),(162,165),(163,164)]])
99 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | ··· | 9F | 18A | ··· | 18F | 19A | ··· | 19I | 57A | ··· | 57R | 171A | ··· | 171BB |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 19 | ··· | 19 | 57 | ··· | 57 | 171 | ··· | 171 |
| size | 1 | 19 | 1 | 1 | 19 | 19 | 1 | ··· | 1 | 19 | ··· | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | D19 | C3×D19 | C9×D19 |
| kernel | C9×D19 | C171 | C3×D19 | C57 | D19 | C19 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 9 | 18 | 54 |
Matrix representation of C9×D19 ►in GL2(𝔽37) generated by
| 34 | 0 |
| 0 | 34 |
| 27 | 1 |
| 4 | 18 |
| 18 | 16 |
| 33 | 19 |
G:=sub<GL(2,GF(37))| [34,0,0,34],[27,4,1,18],[18,33,16,19] >;
C9×D19 in GAP, Magma, Sage, TeX
C_9\times D_{19} % in TeX
G:=Group("C9xD19"); // GroupNames label
G:=SmallGroup(342,4);
// by ID
G=gap.SmallGroup(342,4);
# by ID
G:=PCGroup([4,-2,-3,-3,-19,29,5187]);
// Polycyclic
G:=Group<a,b,c|a^9=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export