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## G = S3×C3×C27order 486 = 2·35

### Direct product of C3×C27 and S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C3×C27
G = < a,b,c,d | a3=b27=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 128 in 78 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C9, C32, C32, C32, C18, C3×S3, C3×S3, C3×C6, C27, C27, C3×C9, C3×C9, C3×C9, C33, C54, S3×C9, S3×C9, C3×C18, S3×C32, C3×C27, C3×C27, C3×C27, C32×C9, S3×C27, C3×C54, S3×C3×C9, C32×C27, S3×C3×C27
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C27, C3×C9, C54, S3×C9, C3×C18, S3×C32, C3×C27, S3×C27, C3×C54, S3×C3×C9, S3×C3×C27

Smallest permutation representation of S3×C3×C27
On 162 points
Generators in S162
(1 61 32)(2 62 33)(3 63 34)(4 64 35)(5 65 36)(6 66 37)(7 67 38)(8 68 39)(9 69 40)(10 70 41)(11 71 42)(12 72 43)(13 73 44)(14 74 45)(15 75 46)(16 76 47)(17 77 48)(18 78 49)(19 79 50)(20 80 51)(21 81 52)(22 55 53)(23 56 54)(24 57 28)(25 58 29)(26 59 30)(27 60 31)(82 146 134)(83 147 135)(84 148 109)(85 149 110)(86 150 111)(87 151 112)(88 152 113)(89 153 114)(90 154 115)(91 155 116)(92 156 117)(93 157 118)(94 158 119)(95 159 120)(96 160 121)(97 161 122)(98 162 123)(99 136 124)(100 137 125)(101 138 126)(102 139 127)(103 140 128)(104 141 129)(105 142 130)(106 143 131)(107 144 132)(108 145 133)
(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 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)(28 37 46)(29 38 47)(30 39 48)(31 40 49)(32 41 50)(33 42 51)(34 43 52)(35 44 53)(36 45 54)(55 64 73)(56 65 74)(57 66 75)(58 67 76)(59 68 77)(60 69 78)(61 70 79)(62 71 80)(63 72 81)(82 100 91)(83 101 92)(84 102 93)(85 103 94)(86 104 95)(87 105 96)(88 106 97)(89 107 98)(90 108 99)(109 127 118)(110 128 119)(111 129 120)(112 130 121)(113 131 122)(114 132 123)(115 133 124)(116 134 125)(117 135 126)(136 154 145)(137 155 146)(138 156 147)(139 157 148)(140 158 149)(141 159 150)(142 160 151)(143 161 152)(144 162 153)
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 97)(15 98)(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 82)(27 83)(28 132)(29 133)(30 134)(31 135)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 142)(56 143)(57 144)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(73 160)(74 161)(75 162)(76 136)(77 137)(78 138)(79 139)(80 140)(81 141)

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

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

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

243 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 6A ··· 6H 9A ··· 9R 9S ··· 9AJ 18A ··· 18R 27A ··· 27BB 27BC ··· 27DD 54A ··· 54BB order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 27 ··· 27 27 ··· 27 54 ··· 54 size 1 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 1 1 2 2 2 2 2 2 type + + + image C1 C2 C3 C3 C6 C6 C9 C9 C18 C18 C27 C54 S3 C3×S3 C3×S3 S3×C9 S3×C9 S3×C27 kernel S3×C3×C27 C32×C27 S3×C27 S3×C3×C9 C3×C27 C32×C9 S3×C9 S3×C32 C3×C9 C33 C3×S3 C32 C3×C27 C27 C3×C9 C9 C32 C3 # reps 1 1 6 2 6 2 12 6 12 6 54 54 1 6 2 12 6 54

Matrix representation of S3×C3×C27 in GL3(𝔽109) generated by

 1 0 0 0 63 0 0 0 63
,
 25 0 0 0 78 0 0 0 78
,
 1 0 0 0 45 0 0 0 63
,
 108 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(109))| [1,0,0,0,63,0,0,0,63],[25,0,0,0,78,0,0,0,78],[1,0,0,0,45,0,0,0,63],[108,0,0,0,0,1,0,1,0] >;

S3×C3×C27 in GAP, Magma, Sage, TeX

S_3\times C_3\times C_{27}
% in TeX

G:=Group("S3xC3xC27");
// GroupNames label

G:=SmallGroup(486,112);
// by ID

G=gap.SmallGroup(486,112);
# by ID

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

G:=Group<a,b,c,d|a^3=b^27=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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