direct product, metabelian, supersoluble, monomial, A-group
Aliases: D9×C33, C34.12S3, (C33×C9)⋊4C2, C9⋊3(C32×C6), (C32×C9)⋊39C6, C3.1(S3×C33), C33.81(C3×S3), C32.44(S3×C32), (C3×C9)⋊22(C3×C6), SmallGroup(486,220)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C33 |
Generators and relations for D9×C33
G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 760 in 324 conjugacy classes, 112 normal (8 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, D9, C3×S3, C3×C6, C3×C9, C3×C9, C33, C33, C33, C3×D9, S3×C32, C32×C6, C32×C9, C32×C9, C34, C32×D9, S3×C33, C33×C9, D9×C33
Quotients: C1, C2, C3, S3, C6, C32, D9, C3×S3, C3×C6, C33, C3×D9, S3×C32, C32×C6, C32×D9, S3×C33, D9×C33
►On 162 points
Generators in S162
(1 62 29)(2 63 30)(3 55 31)(4 56 32)(5 57 33)(6 58 34)(7 59 35)(8 60 36)(9 61 28)(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)(82 142 112)(83 143 113)(84 144 114)(85 136 115)(86 137 116)(87 138 117)(88 139 109)(89 140 110)(90 141 111)(91 151 121)(92 152 122)(93 153 123)(94 145 124)(95 146 125)(96 147 126)(97 148 118)(98 149 119)(99 150 120)(100 160 130)(101 161 131)(102 162 132)(103 154 133)(104 155 134)(105 156 135)(106 157 127)(107 158 128)(108 159 129)
(1 26 11)(2 27 12)(3 19 13)(4 20 14)(5 21 15)(6 22 16)(7 23 17)(8 24 18)(9 25 10)(28 49 43)(29 50 44)(30 51 45)(31 52 37)(32 53 38)(33 54 39)(34 46 40)(35 47 41)(36 48 42)(55 76 70)(56 77 71)(57 78 72)(58 79 64)(59 80 65)(60 81 66)(61 73 67)(62 74 68)(63 75 69)(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 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 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 84)(2 83)(3 82)(4 90)(5 89)(6 88)(7 87)(8 86)(9 85)(10 97)(11 96)(12 95)(13 94)(14 93)(15 92)(16 91)(17 99)(18 98)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 108)(27 107)(28 115)(29 114)(30 113)(31 112)(32 111)(33 110)(34 109)(35 117)(36 116)(37 124)(38 123)(39 122)(40 121)(41 120)(42 119)(43 118)(44 126)(45 125)(46 133)(47 132)(48 131)(49 130)(50 129)(51 128)(52 127)(53 135)(54 134)(55 142)(56 141)(57 140)(58 139)(59 138)(60 137)(61 136)(62 144)(63 143)(64 151)(65 150)(66 149)(67 148)(68 147)(69 146)(70 145)(71 153)(72 152)(73 160)(74 159)(75 158)(76 157)(77 156)(78 155)(79 154)(80 162)(81 161)
G:=sub<Sym(162)| (1,62,29)(2,63,30)(3,55,31)(4,56,32)(5,57,33)(6,58,34)(7,59,35)(8,60,36)(9,61,28)(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)(82,142,112)(83,143,113)(84,144,114)(85,136,115)(86,137,116)(87,138,117)(88,139,109)(89,140,110)(90,141,111)(91,151,121)(92,152,122)(93,153,123)(94,145,124)(95,146,125)(96,147,126)(97,148,118)(98,149,119)(99,150,120)(100,160,130)(101,161,131)(102,162,132)(103,154,133)(104,155,134)(105,156,135)(106,157,127)(107,158,128)(108,159,129), (1,26,11)(2,27,12)(3,19,13)(4,20,14)(5,21,15)(6,22,16)(7,23,17)(8,24,18)(9,25,10)(28,49,43)(29,50,44)(30,51,45)(31,52,37)(32,53,38)(33,54,39)(34,46,40)(35,47,41)(36,48,42)(55,76,70)(56,77,71)(57,78,72)(58,79,64)(59,80,65)(60,81,66)(61,73,67)(62,74,68)(63,75,69)(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), (1,4,7)(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), (1,2,3,4,5,6,7,8,9)(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), (1,84)(2,83)(3,82)(4,90)(5,89)(6,88)(7,87)(8,86)(9,85)(10,97)(11,96)(12,95)(13,94)(14,93)(15,92)(16,91)(17,99)(18,98)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,108)(27,107)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,117)(36,116)(37,124)(38,123)(39,122)(40,121)(41,120)(42,119)(43,118)(44,126)(45,125)(46,133)(47,132)(48,131)(49,130)(50,129)(51,128)(52,127)(53,135)(54,134)(55,142)(56,141)(57,140)(58,139)(59,138)(60,137)(61,136)(62,144)(63,143)(64,151)(65,150)(66,149)(67,148)(68,147)(69,146)(70,145)(71,153)(72,152)(73,160)(74,159)(75,158)(76,157)(77,156)(78,155)(79,154)(80,162)(81,161)>;
G:=Group( (1,62,29)(2,63,30)(3,55,31)(4,56,32)(5,57,33)(6,58,34)(7,59,35)(8,60,36)(9,61,28)(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)(82,142,112)(83,143,113)(84,144,114)(85,136,115)(86,137,116)(87,138,117)(88,139,109)(89,140,110)(90,141,111)(91,151,121)(92,152,122)(93,153,123)(94,145,124)(95,146,125)(96,147,126)(97,148,118)(98,149,119)(99,150,120)(100,160,130)(101,161,131)(102,162,132)(103,154,133)(104,155,134)(105,156,135)(106,157,127)(107,158,128)(108,159,129), (1,26,11)(2,27,12)(3,19,13)(4,20,14)(5,21,15)(6,22,16)(7,23,17)(8,24,18)(9,25,10)(28,49,43)(29,50,44)(30,51,45)(31,52,37)(32,53,38)(33,54,39)(34,46,40)(35,47,41)(36,48,42)(55,76,70)(56,77,71)(57,78,72)(58,79,64)(59,80,65)(60,81,66)(61,73,67)(62,74,68)(63,75,69)(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), (1,4,7)(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), (1,2,3,4,5,6,7,8,9)(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), (1,84)(2,83)(3,82)(4,90)(5,89)(6,88)(7,87)(8,86)(9,85)(10,97)(11,96)(12,95)(13,94)(14,93)(15,92)(16,91)(17,99)(18,98)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,108)(27,107)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,117)(36,116)(37,124)(38,123)(39,122)(40,121)(41,120)(42,119)(43,118)(44,126)(45,125)(46,133)(47,132)(48,131)(49,130)(50,129)(51,128)(52,127)(53,135)(54,134)(55,142)(56,141)(57,140)(58,139)(59,138)(60,137)(61,136)(62,144)(63,143)(64,151)(65,150)(66,149)(67,148)(68,147)(69,146)(70,145)(71,153)(72,152)(73,160)(74,159)(75,158)(76,157)(77,156)(78,155)(79,154)(80,162)(81,161) );
G=PermutationGroup([[(1,62,29),(2,63,30),(3,55,31),(4,56,32),(5,57,33),(6,58,34),(7,59,35),(8,60,36),(9,61,28),(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),(82,142,112),(83,143,113),(84,144,114),(85,136,115),(86,137,116),(87,138,117),(88,139,109),(89,140,110),(90,141,111),(91,151,121),(92,152,122),(93,153,123),(94,145,124),(95,146,125),(96,147,126),(97,148,118),(98,149,119),(99,150,120),(100,160,130),(101,161,131),(102,162,132),(103,154,133),(104,155,134),(105,156,135),(106,157,127),(107,158,128),(108,159,129)], [(1,26,11),(2,27,12),(3,19,13),(4,20,14),(5,21,15),(6,22,16),(7,23,17),(8,24,18),(9,25,10),(28,49,43),(29,50,44),(30,51,45),(31,52,37),(32,53,38),(33,54,39),(34,46,40),(35,47,41),(36,48,42),(55,76,70),(56,77,71),(57,78,72),(58,79,64),(59,80,65),(60,81,66),(61,73,67),(62,74,68),(63,75,69),(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)], [(1,4,7),(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)], [(1,2,3,4,5,6,7,8,9),(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)], [(1,84),(2,83),(3,82),(4,90),(5,89),(6,88),(7,87),(8,86),(9,85),(10,97),(11,96),(12,95),(13,94),(14,93),(15,92),(16,91),(17,99),(18,98),(19,106),(20,105),(21,104),(22,103),(23,102),(24,101),(25,100),(26,108),(27,107),(28,115),(29,114),(30,113),(31,112),(32,111),(33,110),(34,109),(35,117),(36,116),(37,124),(38,123),(39,122),(40,121),(41,120),(42,119),(43,118),(44,126),(45,125),(46,133),(47,132),(48,131),(49,130),(50,129),(51,128),(52,127),(53,135),(54,134),(55,142),(56,141),(57,140),(58,139),(59,138),(60,137),(61,136),(62,144),(63,143),(64,151),(65,150),(66,149),(67,148),(68,147),(69,146),(70,145),(71,153),(72,152),(73,160),(74,159),(75,158),(76,157),(77,156),(78,155),(79,154),(80,162),(81,161)]])
162 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3Z | 3AA | ··· | 3BA | 6A | ··· | 6Z | 9A | ··· | 9CC |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
| size | 1 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C3 | C6 | S3 | D9 | C3×S3 | C3×D9 |
| kernel | D9×C33 | C33×C9 | C32×D9 | C32×C9 | C34 | C33 | C33 | C32 |
| # reps | 1 | 1 | 26 | 26 | 1 | 3 | 26 | 78 |
Matrix representation of D9×C33 ►in GL4(𝔽19) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 11 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 17 |
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(19))| [7,0,0,0,0,7,0,0,0,0,11,0,0,0,0,11],[11,0,0,0,0,1,0,0,0,0,11,0,0,0,0,11],[7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,17],[18,0,0,0,0,18,0,0,0,0,0,9,0,0,17,0] >;
D9×C33 in GAP, Magma, Sage, TeX
D_9\times C_3^3
% in TeX
G:=Group("D9xC3^3"); // GroupNames label
G:=SmallGroup(486,220);
// by ID
G=gap.SmallGroup(486,220);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,8104,208,11669]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^9=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