direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C32×C9, C33⋊7C18, C34.12C6, C3⋊(C32×C18), (C33×C9)⋊1C2, C32⋊5(C3×C18), (C32×C9)⋊37C6, C3.4(S3×C33), (C3×S3).1C33, (S3×C33).3C3, C33.88(C3×S3), C33.52(C3×C6), C32.13(C32×C6), C32.53(S3×C32), (S3×C32).10C32, (C3×C9)⋊21(C3×C6), SmallGroup(486,221)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×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
►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