direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D81, C81⋊3C6, C32.2D27, (C3×C81)⋊2C2, C9.2(C3×D9), (C3×C9).7D9, C27.1(C3×S3), (C3×C27).4S3, C3.2(C3×D27), SmallGroup(486,32)
Series: Derived ►Chief ►Lower central ►Upper central
| C81 — C3×D81 |
Generators and relations for C3×D81
G = < a,b,c | a3=b81=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D81
►On 162 points
Generators in S162
(1 28 55)(2 29 56)(3 30 57)(4 31 58)(5 32 59)(6 33 60)(7 34 61)(8 35 62)(9 36 63)(10 37 64)(11 38 65)(12 39 66)(13 40 67)(14 41 68)(15 42 69)(16 43 70)(17 44 71)(18 45 72)(19 46 73)(20 47 74)(21 48 75)(22 49 76)(23 50 77)(24 51 78)(25 52 79)(26 53 80)(27 54 81)(82 136 109)(83 137 110)(84 138 111)(85 139 112)(86 140 113)(87 141 114)(88 142 115)(89 143 116)(90 144 117)(91 145 118)(92 146 119)(93 147 120)(94 148 121)(95 149 122)(96 150 123)(97 151 124)(98 152 125)(99 153 126)(100 154 127)(101 155 128)(102 156 129)(103 157 130)(104 158 131)(105 159 132)(106 160 133)(107 161 134)(108 162 135)
(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)
(1 124)(2 123)(3 122)(4 121)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 112)(14 111)(15 110)(16 109)(17 108)(18 107)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 99)(27 98)(28 97)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 88)(38 87)(39 86)(40 85)(41 84)(42 83)(43 82)(44 162)(45 161)(46 160)(47 159)(48 158)(49 157)(50 156)(51 155)(52 154)(53 153)(54 152)(55 151)(56 150)(57 149)(58 148)(59 147)(60 146)(61 145)(62 144)(63 143)(64 142)(65 141)(66 140)(67 139)(68 138)(69 137)(70 136)(71 135)(72 134)(73 133)(74 132)(75 131)(76 130)(77 129)(78 128)(79 127)(80 126)(81 125)
G:=sub<Sym(162)| (1,28,55)(2,29,56)(3,30,57)(4,31,58)(5,32,59)(6,33,60)(7,34,61)(8,35,62)(9,36,63)(10,37,64)(11,38,65)(12,39,66)(13,40,67)(14,41,68)(15,42,69)(16,43,70)(17,44,71)(18,45,72)(19,46,73)(20,47,74)(21,48,75)(22,49,76)(23,50,77)(24,51,78)(25,52,79)(26,53,80)(27,54,81)(82,136,109)(83,137,110)(84,138,111)(85,139,112)(86,140,113)(87,141,114)(88,142,115)(89,143,116)(90,144,117)(91,145,118)(92,146,119)(93,147,120)(94,148,121)(95,149,122)(96,150,123)(97,151,124)(98,152,125)(99,153,126)(100,154,127)(101,155,128)(102,156,129)(103,157,130)(104,158,131)(105,159,132)(106,160,133)(107,161,134)(108,162,135), (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), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,162)(45,161)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,149)(58,148)(59,147)(60,146)(61,145)(62,144)(63,143)(64,142)(65,141)(66,140)(67,139)(68,138)(69,137)(70,136)(71,135)(72,134)(73,133)(74,132)(75,131)(76,130)(77,129)(78,128)(79,127)(80,126)(81,125)>;
G:=Group( (1,28,55)(2,29,56)(3,30,57)(4,31,58)(5,32,59)(6,33,60)(7,34,61)(8,35,62)(9,36,63)(10,37,64)(11,38,65)(12,39,66)(13,40,67)(14,41,68)(15,42,69)(16,43,70)(17,44,71)(18,45,72)(19,46,73)(20,47,74)(21,48,75)(22,49,76)(23,50,77)(24,51,78)(25,52,79)(26,53,80)(27,54,81)(82,136,109)(83,137,110)(84,138,111)(85,139,112)(86,140,113)(87,141,114)(88,142,115)(89,143,116)(90,144,117)(91,145,118)(92,146,119)(93,147,120)(94,148,121)(95,149,122)(96,150,123)(97,151,124)(98,152,125)(99,153,126)(100,154,127)(101,155,128)(102,156,129)(103,157,130)(104,158,131)(105,159,132)(106,160,133)(107,161,134)(108,162,135), (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), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,162)(45,161)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,149)(58,148)(59,147)(60,146)(61,145)(62,144)(63,143)(64,142)(65,141)(66,140)(67,139)(68,138)(69,137)(70,136)(71,135)(72,134)(73,133)(74,132)(75,131)(76,130)(77,129)(78,128)(79,127)(80,126)(81,125) );
G=PermutationGroup([[(1,28,55),(2,29,56),(3,30,57),(4,31,58),(5,32,59),(6,33,60),(7,34,61),(8,35,62),(9,36,63),(10,37,64),(11,38,65),(12,39,66),(13,40,67),(14,41,68),(15,42,69),(16,43,70),(17,44,71),(18,45,72),(19,46,73),(20,47,74),(21,48,75),(22,49,76),(23,50,77),(24,51,78),(25,52,79),(26,53,80),(27,54,81),(82,136,109),(83,137,110),(84,138,111),(85,139,112),(86,140,113),(87,141,114),(88,142,115),(89,143,116),(90,144,117),(91,145,118),(92,146,119),(93,147,120),(94,148,121),(95,149,122),(96,150,123),(97,151,124),(98,152,125),(99,153,126),(100,154,127),(101,155,128),(102,156,129),(103,157,130),(104,158,131),(105,159,132),(106,160,133),(107,161,134),(108,162,135)], [(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)], [(1,124),(2,123),(3,122),(4,121),(5,120),(6,119),(7,118),(8,117),(9,116),(10,115),(11,114),(12,113),(13,112),(14,111),(15,110),(16,109),(17,108),(18,107),(19,106),(20,105),(21,104),(22,103),(23,102),(24,101),(25,100),(26,99),(27,98),(28,97),(29,96),(30,95),(31,94),(32,93),(33,92),(34,91),(35,90),(36,89),(37,88),(38,87),(39,86),(40,85),(41,84),(42,83),(43,82),(44,162),(45,161),(46,160),(47,159),(48,158),(49,157),(50,156),(51,155),(52,154),(53,153),(54,152),(55,151),(56,150),(57,149),(58,148),(59,147),(60,146),(61,145),(62,144),(63,143),(64,142),(65,141),(66,140),(67,139),(68,138),(69,137),(70,136),(71,135),(72,134),(73,133),(74,132),(75,131),(76,130),(77,129),(78,128),(79,127),(80,126),(81,125)]])
126 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 9A | ··· | 9I | 27A | ··· | 27AA | 81A | ··· | 81CC |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 27 | ··· | 27 | 81 | ··· | 81 |
| size | 1 | 81 | 1 | 1 | 2 | 2 | 2 | 81 | 81 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | D9 | C3×D9 | D27 | D81 | C3×D27 | C3×D81 |
| kernel | C3×D81 | C3×C81 | D81 | C81 | C3×C27 | C27 | C3×C9 | C9 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 3 | 6 | 9 | 27 | 18 | 54 |
Matrix representation of C3×D81 ►in GL2(𝔽163) generated by
| 104 | 0 |
| 0 | 104 |
| 152 | 0 |
| 0 | 74 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(163))| [104,0,0,104],[152,0,0,74],[0,1,1,0] >;
C3×D81 in GAP, Magma, Sage, TeX
C_3\times D_{81} % in TeX
G:=Group("C3xD81"); // GroupNames label
G:=SmallGroup(486,32);
// by ID
G=gap.SmallGroup(486,32);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,542,284,2163,381,8104,208,11669]);
// Polycyclic
G:=Group<a,b,c|a^3=b^81=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export