Copied to
clipboard

## G = S3×C32×C9order 486 = 2·35

### Direct product of C32×C9 and S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C32×C9
G = < a,b,c,d,e | a3=b3=c9=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 596 in 348 conjugacy classes, 150 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3×S3, C3×C6, C3×C9, C3×C9, C33, C33, C33, S3×C9, C3×C18, S3×C32, C32×C6, C32×C9, C32×C9, C32×C9, C34, S3×C3×C9, C32×C18, S3×C33, C33×C9, S3×C32×C9
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, C33, S3×C9, C3×C18, S3×C32, C32×C6, C32×C9, S3×C3×C9, C32×C18, S3×C33, S3×C32×C9

Smallest permutation representation of S3×C32×C9
On 162 points
Generators in S162
(1 59 35)(2 60 36)(3 61 28)(4 62 29)(5 63 30)(6 55 31)(7 56 32)(8 57 33)(9 58 34)(10 67 43)(11 68 44)(12 69 45)(13 70 37)(14 71 38)(15 72 39)(16 64 40)(17 65 41)(18 66 42)(19 76 52)(20 77 53)(21 78 54)(22 79 46)(23 80 47)(24 81 48)(25 73 49)(26 74 50)(27 75 51)(82 139 115)(83 140 116)(84 141 117)(85 142 109)(86 143 110)(87 144 111)(88 136 112)(89 137 113)(90 138 114)(91 148 124)(92 149 125)(93 150 126)(94 151 118)(95 152 119)(96 153 120)(97 145 121)(98 146 122)(99 147 123)(100 157 133)(101 158 134)(102 159 135)(103 160 127)(104 161 128)(105 162 129)(106 154 130)(107 155 131)(108 156 132)
(1 26 14)(2 27 15)(3 19 16)(4 20 17)(5 21 18)(6 22 10)(7 23 11)(8 24 12)(9 25 13)(28 52 40)(29 53 41)(30 54 42)(31 46 43)(32 47 44)(33 48 45)(34 49 37)(35 50 38)(36 51 39)(55 79 67)(56 80 68)(57 81 69)(58 73 70)(59 74 71)(60 75 72)(61 76 64)(62 77 65)(63 78 66)(82 106 94)(83 107 95)(84 108 96)(85 100 97)(86 101 98)(87 102 99)(88 103 91)(89 104 92)(90 105 93)(109 133 121)(110 134 122)(111 135 123)(112 127 124)(113 128 125)(114 129 126)(115 130 118)(116 131 119)(117 132 120)(136 160 148)(137 161 149)(138 162 150)(139 154 151)(140 155 152)(141 156 153)(142 157 145)(143 158 146)(144 159 147)
(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 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)(136 142 139)(137 143 140)(138 144 141)(145 151 148)(146 152 149)(147 153 150)(154 160 157)(155 161 158)(156 162 159)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 82)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)

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

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

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

243 conjugacy classes

 class 1 2 3A ··· 3Z 3AA ··· 3BA 6A ··· 6Z 9A ··· 9BB 9BC ··· 9DD 18A ··· 18BB 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 1 1 2 2 2 2 type + + + image C1 C2 C3 C3 C6 C6 C9 C18 S3 C3×S3 C3×S3 S3×C9 kernel S3×C32×C9 C33×C9 S3×C3×C9 S3×C33 C32×C9 C34 S3×C32 C33 C32×C9 C3×C9 C33 C32 # reps 1 1 24 2 24 2 54 54 1 24 2 54

Matrix representation of S3×C32×C9 in GL4(𝔽19) generated by

 1 0 0 0 0 11 0 0 0 0 7 0 0 0 0 7
,
 7 0 0 0 0 7 0 0 0 0 11 0 0 0 0 11
,
 6 0 0 0 0 6 0 0 0 0 4 0 0 0 0 4
,
 1 0 0 0 0 1 0 0 0 0 7 0 0 0 0 11
,
 18 0 0 0 0 18 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(19))| [1,0,0,0,0,11,0,0,0,0,7,0,0,0,0,7],[7,0,0,0,0,7,0,0,0,0,11,0,0,0,0,11],[6,0,0,0,0,6,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,7,0,0,0,0,11],[18,0,0,0,0,18,0,0,0,0,0,1,0,0,1,0] >;

S3×C32×C9 in GAP, Magma, Sage, TeX

S_3\times C_3^2\times C_9
% in TeX

G:=Group("S3xC3^2xC9");
// GroupNames label

G:=SmallGroup(486,221);
// by ID

G=gap.SmallGroup(486,221);
# by ID

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

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

׿
×
𝔽