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## G = S3×C92order 486 = 2·35

### Direct product of C92 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C92
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C3×C92 — S3×C92
 Lower central C3 — S3×C92
 Upper central C1 — C92

Generators and relations for S3×C92
G = < a,b,c,d | a9=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 218 in 132 conjugacy classes, 69 normal (9 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3×C6, C3×C9, C3×C9, C33, S3×C9, C3×C18, S3×C32, C92, C92, C32×C9, C9×C18, S3×C3×C9, C3×C92, S3×C92
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, S3×C9, C3×C18, S3×C32, C92, C9×C18, S3×C3×C9, S3×C92

Smallest permutation representation of S3×C92
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 55 71 133 25 30 54 97 15)(2 56 72 134 26 31 46 98 16)(3 57 64 135 27 32 47 99 17)(4 58 65 127 19 33 48 91 18)(5 59 66 128 20 34 49 92 10)(6 60 67 129 21 35 50 93 11)(7 61 68 130 22 36 51 94 12)(8 62 69 131 23 28 52 95 13)(9 63 70 132 24 29 53 96 14)(37 106 85 144 162 73 146 121 112)(38 107 86 136 154 74 147 122 113)(39 108 87 137 155 75 148 123 114)(40 100 88 138 156 76 149 124 115)(41 101 89 139 157 77 150 125 116)(42 102 90 140 158 78 151 126 117)(43 103 82 141 159 79 152 118 109)(44 104 83 142 160 80 153 119 110)(45 105 84 143 161 81 145 120 111)
(1 48 130)(2 49 131)(3 50 132)(4 51 133)(5 52 134)(6 53 135)(7 54 127)(8 46 128)(9 47 129)(10 28 72)(11 29 64)(12 30 65)(13 31 66)(14 32 67)(15 33 68)(16 34 69)(17 35 70)(18 36 71)(19 61 97)(20 62 98)(21 63 99)(22 55 91)(23 56 92)(24 57 93)(25 58 94)(26 59 95)(27 60 96)(37 141 149)(38 142 150)(39 143 151)(40 144 152)(41 136 153)(42 137 145)(43 138 146)(44 139 147)(45 140 148)(73 109 88)(74 110 89)(75 111 90)(76 112 82)(77 113 83)(78 114 84)(79 115 85)(80 116 86)(81 117 87)(100 162 118)(101 154 119)(102 155 120)(103 156 121)(104 157 122)(105 158 123)(106 159 124)(107 160 125)(108 161 126)
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 82)(9 83)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 148)(20 149)(21 150)(22 151)(23 152)(24 153)(25 145)(26 146)(27 147)(28 118)(29 119)(30 120)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 98)(38 99)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 112)(47 113)(48 114)(49 115)(50 116)(51 117)(52 109)(53 110)(54 111)(55 143)(56 144)(57 136)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 154)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 161)(72 162)(73 134)(74 135)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)(81 133)

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

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

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

243 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 6A ··· 6H 9A ··· 9BT 9BU ··· 9EN 18A ··· 18BT order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

243 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + + image C1 C2 C3 C6 C9 C18 S3 C3×S3 S3×C9 kernel S3×C92 C3×C92 S3×C3×C9 C32×C9 S3×C9 C3×C9 C92 C3×C9 C9 # reps 1 1 8 8 72 72 1 8 72

Matrix representation of S3×C92 in GL3(𝔽19) generated by

 7 0 0 0 9 0 0 0 9
,
 6 0 0 0 17 0 0 0 17
,
 1 0 0 0 11 0 0 7 7
,
 1 0 0 0 18 6 0 0 1
G:=sub<GL(3,GF(19))| [7,0,0,0,9,0,0,0,9],[6,0,0,0,17,0,0,0,17],[1,0,0,0,11,7,0,0,7],[1,0,0,0,18,0,0,6,1] >;

S3×C92 in GAP, Magma, Sage, TeX

S_3\times C_9^2
% in TeX

G:=Group("S3xC9^2");
// GroupNames label

G:=SmallGroup(486,92);
// by ID

G=gap.SmallGroup(486,92);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,68,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^9=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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