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## G = S3×C81order 486 = 2·35

### Direct product of C81 and S3

Aliases: S3×C81, C3⋊C162, C32.2C54, (C3×C81)⋊1C2, (S3×C27).C3, (C3×S3).C27, C9.6(S3×C9), (S3×C9).2C9, C3.4(S3×C27), C27.4(C3×S3), (C3×C9).9C18, (C3×C27).5C6, SmallGroup(486,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C81
 Chief series C1 — C3 — C32 — C3×C9 — C3×C27 — C3×C81 — S3×C81
 Lower central C3 — S3×C81
 Upper central C1 — C81

Generators and relations for S3×C81
G = < a,b,c | a81=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of S3×C81
On 162 points
Generators in S162
(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 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)
(1 136)(2 137)(3 138)(4 139)(5 140)(6 141)(7 142)(8 143)(9 144)(10 145)(11 146)(12 147)(13 148)(14 149)(15 150)(16 151)(17 152)(18 153)(19 154)(20 155)(21 156)(22 157)(23 158)(24 159)(25 160)(26 161)(27 162)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 121)(68 122)(69 123)(70 124)(71 125)(72 126)(73 127)(74 128)(75 129)(76 130)(77 131)(78 132)(79 133)(80 134)(81 135)

G:=sub<Sym(162)| (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), (1,28,55)(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), (1,136)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,144)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,151)(17,152)(18,153)(19,154)(20,155)(21,156)(22,157)(23,158)(24,159)(25,160)(26,161)(27,162)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134)(81,135)>;

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), (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), (1,136)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,144)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,151)(17,152)(18,153)(19,154)(20,155)(21,156)(22,157)(23,158)(24,159)(25,160)(26,161)(27,162)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134)(81,135) );

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)], [(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)], [(1,136),(2,137),(3,138),(4,139),(5,140),(6,141),(7,142),(8,143),(9,144),(10,145),(11,146),(12,147),(13,148),(14,149),(15,150),(16,151),(17,152),(18,153),(19,154),(20,155),(21,156),(22,157),(23,158),(24,159),(25,160),(26,161),(27,162),(28,82),(29,83),(30,84),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114),(61,115),(62,116),(63,117),(64,118),(65,119),(66,120),(67,121),(68,122),(69,123),(70,124),(71,125),(72,126),(73,127),(74,128),(75,129),(76,130),(77,131),(78,132),(79,133),(80,134),(81,135)])

243 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 6A 6B 9A ··· 9F 9G ··· 9L 18A ··· 18F 27A ··· 27R 27S ··· 27AJ 54A ··· 54R 81A ··· 81BB 81BC ··· 81DD 162A ··· 162BB order 1 2 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 27 ··· 27 27 ··· 27 54 ··· 54 81 ··· 81 81 ··· 81 162 ··· 162 size 1 3 1 1 2 2 2 3 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

243 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + image C1 C2 C3 C6 C9 C18 C27 C54 C81 C162 S3 C3×S3 S3×C9 S3×C27 S3×C81 kernel S3×C81 C3×C81 S3×C27 C3×C27 S3×C9 C3×C9 C3×S3 C32 S3 C3 C81 C27 C9 C3 C1 # reps 1 1 2 2 6 6 18 18 54 54 1 2 6 18 54

Matrix representation of S3×C81 in GL2(𝔽163) generated by

 131 0 0 131
,
 104 0 0 58
,
 0 1 1 0
G:=sub<GL(2,GF(163))| [131,0,0,131],[104,0,0,58],[0,1,1,0] >;

S3×C81 in GAP, Magma, Sage, TeX

S_3\times C_{81}
% in TeX

G:=Group("S3xC81");
// GroupNames label

G:=SmallGroup(486,33);
// by ID

G=gap.SmallGroup(486,33);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,68,93,11669]);
// Polycyclic

G:=Group<a,b,c|a^81=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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