direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C2×C81⋊C3, C162⋊C3, C9.C54, C54.C9, C81⋊2C6, C27.C18, C18.C27, C32.C54, C54.2C32, (C3×C6).C27, (C3×C18).5C9, (C3×C54).3C3, C27.1(C3×C6), C6.3(C3×C27), C3.3(C3×C54), (C3×C27).4C6, C18.6(C3×C9), C9.6(C3×C18), (C3×C9).7C18, SmallGroup(486,84)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C81⋊C3
G = < a,b,c | a2=b81=c3=1, ab=ba, ac=ca, cbc-1=b28 >
Smallest permutation representation of C2×C81⋊C3
►On 162 points
Generators in S162
(1 162)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)(81 161)
(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)
(2 56 29)(3 30 57)(5 59 32)(6 33 60)(8 62 35)(9 36 63)(11 65 38)(12 39 66)(14 68 41)(15 42 69)(17 71 44)(18 45 72)(20 74 47)(21 48 75)(23 77 50)(24 51 78)(26 80 53)(27 54 81)(82 136 109)(83 110 137)(85 139 112)(86 113 140)(88 142 115)(89 116 143)(91 145 118)(92 119 146)(94 148 121)(95 122 149)(97 151 124)(98 125 152)(100 154 127)(101 128 155)(103 157 130)(104 131 158)(106 160 133)(107 134 161)
G:=sub<Sym(162)| (1,162)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)(81,161), (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), (2,56,29)(3,30,57)(5,59,32)(6,33,60)(8,62,35)(9,36,63)(11,65,38)(12,39,66)(14,68,41)(15,42,69)(17,71,44)(18,45,72)(20,74,47)(21,48,75)(23,77,50)(24,51,78)(26,80,53)(27,54,81)(82,136,109)(83,110,137)(85,139,112)(86,113,140)(88,142,115)(89,116,143)(91,145,118)(92,119,146)(94,148,121)(95,122,149)(97,151,124)(98,125,152)(100,154,127)(101,128,155)(103,157,130)(104,131,158)(106,160,133)(107,134,161)>;
G:=Group( (1,162)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)(81,161), (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), (2,56,29)(3,30,57)(5,59,32)(6,33,60)(8,62,35)(9,36,63)(11,65,38)(12,39,66)(14,68,41)(15,42,69)(17,71,44)(18,45,72)(20,74,47)(21,48,75)(23,77,50)(24,51,78)(26,80,53)(27,54,81)(82,136,109)(83,110,137)(85,139,112)(86,113,140)(88,142,115)(89,116,143)(91,145,118)(92,119,146)(94,148,121)(95,122,149)(97,151,124)(98,125,152)(100,154,127)(101,128,155)(103,157,130)(104,131,158)(106,160,133)(107,134,161) );
G=PermutationGroup([[(1,162),(2,82),(3,83),(4,84),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(17,97),(18,98),(19,99),(20,100),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160),(81,161)], [(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)], [(2,56,29),(3,30,57),(5,59,32),(6,33,60),(8,62,35),(9,36,63),(11,65,38),(12,39,66),(14,68,41),(15,42,69),(17,71,44),(18,45,72),(20,74,47),(21,48,75),(23,77,50),(24,51,78),(26,80,53),(27,54,81),(82,136,109),(83,110,137),(85,139,112),(86,113,140),(88,142,115),(89,116,143),(91,145,118),(92,119,146),(94,148,121),(95,122,149),(97,151,124),(98,125,152),(100,154,127),(101,128,155),(103,157,130),(104,131,158),(106,160,133),(107,134,161)]])
198 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 18A | ··· | 18F | 18G | 18H | 18I | 18J | 27A | ··· | 27R | 27S | ··· | 27AD | 54A | ··· | 54R | 54S | ··· | 54AD | 81A | ··· | 81BB | 162A | ··· | 162BB |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 18 | 18 | 18 | 18 | 27 | ··· | 27 | 27 | ··· | 27 | 54 | ··· | 54 | 54 | ··· | 54 | 81 | ··· | 81 | 162 | ··· | 162 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
198 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C9 | C18 | C18 | C27 | C27 | C54 | C54 | C81⋊C3 | C2×C81⋊C3 |
| kernel | C2×C81⋊C3 | C81⋊C3 | C162 | C3×C54 | C81 | C3×C27 | C54 | C3×C18 | C27 | C3×C9 | C18 | C3×C6 | C9 | C32 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 12 | 6 | 12 | 6 | 36 | 18 | 36 | 18 | 18 | 18 |
Matrix representation of C2×C81⋊C3 ►in GL4(𝔽163) generated by
| 162 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 58 | 0 | 0 | 0 |
| 0 | 15 | 1 | 25 |
| 0 | 146 | 0 | 6 |
| 0 | 96 | 0 | 148 |
| 58 | 0 | 0 | 0 |
| 0 | 1 | 0 | 14 |
| 0 | 0 | 104 | 127 |
| 0 | 0 | 0 | 58 |
G:=sub<GL(4,GF(163))| [162,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[58,0,0,0,0,15,146,96,0,1,0,0,0,25,6,148],[58,0,0,0,0,1,0,0,0,0,104,0,0,14,127,58] >;
C2×C81⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{81}\rtimes C_3 % in TeX
G:=Group("C2xC81:C3"); // GroupNames label
G:=SmallGroup(486,84);
// by ID
G=gap.SmallGroup(486,84);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,1520,93,118]);
// Polycyclic
G:=Group<a,b,c|a^2=b^81=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^28>;
// generators/relations
Export