direct product, metabelian, supersoluble, monomial, A-group
Aliases: C32×D27, C33.6D9, C27⋊3(C3×C6), (C3×C27)⋊14C6, (C32×C27)⋊3C2, C9.3(S3×C32), C3.2(C32×D9), (C32×C9).24S3, C32.15(C3×D9), (C3×C9).55(C3×S3), SmallGroup(486,111)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C32×D27 |
Generators and relations for C32×D27
G = < a,b,c,d | a3=b3=c27=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 312 in 72 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3×C6, C27, C27, C3×C9, C3×C9, C33, D27, C3×D9, S3×C32, C3×C27, C3×C27, C32×C9, C3×D27, C32×D9, C32×C27, C32×D27
Quotients: C1, C2, C3, S3, C6, C32, D9, C3×S3, C3×C6, D27, C3×D9, S3×C32, C3×D27, C32×D9, C32×D27
►On 162 points
Generators in S162
(1 39 56)(2 40 57)(3 41 58)(4 42 59)(5 43 60)(6 44 61)(7 45 62)(8 46 63)(9 47 64)(10 48 65)(11 49 66)(12 50 67)(13 51 68)(14 52 69)(15 53 70)(16 54 71)(17 28 72)(18 29 73)(19 30 74)(20 31 75)(21 32 76)(22 33 77)(23 34 78)(24 35 79)(25 36 80)(26 37 81)(27 38 55)(82 117 159)(83 118 160)(84 119 161)(85 120 162)(86 121 136)(87 122 137)(88 123 138)(89 124 139)(90 125 140)(91 126 141)(92 127 142)(93 128 143)(94 129 144)(95 130 145)(96 131 146)(97 132 147)(98 133 148)(99 134 149)(100 135 150)(101 109 151)(102 110 152)(103 111 153)(104 112 154)(105 113 155)(106 114 156)(107 115 157)(108 116 158)
(1 74 48)(2 75 49)(3 76 50)(4 77 51)(5 78 52)(6 79 53)(7 80 54)(8 81 28)(9 55 29)(10 56 30)(11 57 31)(12 58 32)(13 59 33)(14 60 34)(15 61 35)(16 62 36)(17 63 37)(18 64 38)(19 65 39)(20 66 40)(21 67 41)(22 68 42)(23 69 43)(24 70 44)(25 71 45)(26 72 46)(27 73 47)(82 141 135)(83 142 109)(84 143 110)(85 144 111)(86 145 112)(87 146 113)(88 147 114)(89 148 115)(90 149 116)(91 150 117)(92 151 118)(93 152 119)(94 153 120)(95 154 121)(96 155 122)(97 156 123)(98 157 124)(99 158 125)(100 159 126)(101 160 127)(102 161 128)(103 162 129)(104 136 130)(105 137 131)(106 138 132)(107 139 133)(108 140 134)
(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 89)(2 88)(3 87)(4 86)(5 85)(6 84)(7 83)(8 82)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 102)(16 101)(17 100)(18 99)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 90)(28 135)(29 134)(30 133)(31 132)(32 131)(33 130)(34 129)(35 128)(36 127)(37 126)(38 125)(39 124)(40 123)(41 122)(42 121)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 140)(56 139)(57 138)(58 137)(59 136)(60 162)(61 161)(62 160)(63 159)(64 158)(65 157)(66 156)(67 155)(68 154)(69 153)(70 152)(71 151)(72 150)(73 149)(74 148)(75 147)(76 146)(77 145)(78 144)(79 143)(80 142)(81 141)
G:=sub<Sym(162)| (1,39,56)(2,40,57)(3,41,58)(4,42,59)(5,43,60)(6,44,61)(7,45,62)(8,46,63)(9,47,64)(10,48,65)(11,49,66)(12,50,67)(13,51,68)(14,52,69)(15,53,70)(16,54,71)(17,28,72)(18,29,73)(19,30,74)(20,31,75)(21,32,76)(22,33,77)(23,34,78)(24,35,79)(25,36,80)(26,37,81)(27,38,55)(82,117,159)(83,118,160)(84,119,161)(85,120,162)(86,121,136)(87,122,137)(88,123,138)(89,124,139)(90,125,140)(91,126,141)(92,127,142)(93,128,143)(94,129,144)(95,130,145)(96,131,146)(97,132,147)(98,133,148)(99,134,149)(100,135,150)(101,109,151)(102,110,152)(103,111,153)(104,112,154)(105,113,155)(106,114,156)(107,115,157)(108,116,158), (1,74,48)(2,75,49)(3,76,50)(4,77,51)(5,78,52)(6,79,53)(7,80,54)(8,81,28)(9,55,29)(10,56,30)(11,57,31)(12,58,32)(13,59,33)(14,60,34)(15,61,35)(16,62,36)(17,63,37)(18,64,38)(19,65,39)(20,66,40)(21,67,41)(22,68,42)(23,69,43)(24,70,44)(25,71,45)(26,72,46)(27,73,47)(82,141,135)(83,142,109)(84,143,110)(85,144,111)(86,145,112)(87,146,113)(88,147,114)(89,148,115)(90,149,116)(91,150,117)(92,151,118)(93,152,119)(94,153,120)(95,154,121)(96,155,122)(97,156,123)(98,157,124)(99,158,125)(100,159,126)(101,160,127)(102,161,128)(103,162,129)(104,136,130)(105,137,131)(106,138,132)(107,139,133)(108,140,134), (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), (1,89)(2,88)(3,87)(4,86)(5,85)(6,84)(7,83)(8,82)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,140)(56,139)(57,138)(58,137)(59,136)(60,162)(61,161)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142)(81,141)>;
G:=Group( (1,39,56)(2,40,57)(3,41,58)(4,42,59)(5,43,60)(6,44,61)(7,45,62)(8,46,63)(9,47,64)(10,48,65)(11,49,66)(12,50,67)(13,51,68)(14,52,69)(15,53,70)(16,54,71)(17,28,72)(18,29,73)(19,30,74)(20,31,75)(21,32,76)(22,33,77)(23,34,78)(24,35,79)(25,36,80)(26,37,81)(27,38,55)(82,117,159)(83,118,160)(84,119,161)(85,120,162)(86,121,136)(87,122,137)(88,123,138)(89,124,139)(90,125,140)(91,126,141)(92,127,142)(93,128,143)(94,129,144)(95,130,145)(96,131,146)(97,132,147)(98,133,148)(99,134,149)(100,135,150)(101,109,151)(102,110,152)(103,111,153)(104,112,154)(105,113,155)(106,114,156)(107,115,157)(108,116,158), (1,74,48)(2,75,49)(3,76,50)(4,77,51)(5,78,52)(6,79,53)(7,80,54)(8,81,28)(9,55,29)(10,56,30)(11,57,31)(12,58,32)(13,59,33)(14,60,34)(15,61,35)(16,62,36)(17,63,37)(18,64,38)(19,65,39)(20,66,40)(21,67,41)(22,68,42)(23,69,43)(24,70,44)(25,71,45)(26,72,46)(27,73,47)(82,141,135)(83,142,109)(84,143,110)(85,144,111)(86,145,112)(87,146,113)(88,147,114)(89,148,115)(90,149,116)(91,150,117)(92,151,118)(93,152,119)(94,153,120)(95,154,121)(96,155,122)(97,156,123)(98,157,124)(99,158,125)(100,159,126)(101,160,127)(102,161,128)(103,162,129)(104,136,130)(105,137,131)(106,138,132)(107,139,133)(108,140,134), (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), (1,89)(2,88)(3,87)(4,86)(5,85)(6,84)(7,83)(8,82)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,140)(56,139)(57,138)(58,137)(59,136)(60,162)(61,161)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142)(81,141) );
G=PermutationGroup([[(1,39,56),(2,40,57),(3,41,58),(4,42,59),(5,43,60),(6,44,61),(7,45,62),(8,46,63),(9,47,64),(10,48,65),(11,49,66),(12,50,67),(13,51,68),(14,52,69),(15,53,70),(16,54,71),(17,28,72),(18,29,73),(19,30,74),(20,31,75),(21,32,76),(22,33,77),(23,34,78),(24,35,79),(25,36,80),(26,37,81),(27,38,55),(82,117,159),(83,118,160),(84,119,161),(85,120,162),(86,121,136),(87,122,137),(88,123,138),(89,124,139),(90,125,140),(91,126,141),(92,127,142),(93,128,143),(94,129,144),(95,130,145),(96,131,146),(97,132,147),(98,133,148),(99,134,149),(100,135,150),(101,109,151),(102,110,152),(103,111,153),(104,112,154),(105,113,155),(106,114,156),(107,115,157),(108,116,158)], [(1,74,48),(2,75,49),(3,76,50),(4,77,51),(5,78,52),(6,79,53),(7,80,54),(8,81,28),(9,55,29),(10,56,30),(11,57,31),(12,58,32),(13,59,33),(14,60,34),(15,61,35),(16,62,36),(17,63,37),(18,64,38),(19,65,39),(20,66,40),(21,67,41),(22,68,42),(23,69,43),(24,70,44),(25,71,45),(26,72,46),(27,73,47),(82,141,135),(83,142,109),(84,143,110),(85,144,111),(86,145,112),(87,146,113),(88,147,114),(89,148,115),(90,149,116),(91,150,117),(92,151,118),(93,152,119),(94,153,120),(95,154,121),(96,155,122),(97,156,123),(98,157,124),(99,158,125),(100,159,126),(101,160,127),(102,161,128),(103,162,129),(104,136,130),(105,137,131),(106,138,132),(107,139,133),(108,140,134)], [(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)], [(1,89),(2,88),(3,87),(4,86),(5,85),(6,84),(7,83),(8,82),(9,108),(10,107),(11,106),(12,105),(13,104),(14,103),(15,102),(16,101),(17,100),(18,99),(19,98),(20,97),(21,96),(22,95),(23,94),(24,93),(25,92),(26,91),(27,90),(28,135),(29,134),(30,133),(31,132),(32,131),(33,130),(34,129),(35,128),(36,127),(37,126),(38,125),(39,124),(40,123),(41,122),(42,121),(43,120),(44,119),(45,118),(46,117),(47,116),(48,115),(49,114),(50,113),(51,112),(52,111),(53,110),(54,109),(55,140),(56,139),(57,138),(58,137),(59,136),(60,162),(61,161),(62,160),(63,159),(64,158),(65,157),(66,156),(67,155),(68,154),(69,153),(70,152),(71,151),(72,150),(73,149),(74,148),(75,147),(76,146),(77,145),(78,144),(79,143),(80,142),(81,141)]])
135 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 6A | ··· | 6H | 9A | ··· | 9AA | 27A | ··· | 27CC |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 27 | ··· | 27 |
| size | 1 | 27 | 1 | ··· | 1 | 2 | ··· | 2 | 27 | ··· | 27 | 2 | ··· | 2 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | D9 | D27 | C3×D9 | C3×D27 |
| kernel | C32×D27 | C32×C27 | C3×D27 | C3×C27 | C32×C9 | C3×C9 | C33 | C32 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 1 | 8 | 3 | 9 | 24 | 72 |
Matrix representation of C32×D27 ►in GL3(𝔽109) generated by
| 63 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 63 | 0 | 0 |
| 0 | 45 | 0 |
| 0 | 0 | 45 |
| 1 | 0 | 0 |
| 0 | 81 | 0 |
| 0 | 106 | 35 |
| 1 | 0 | 0 |
| 0 | 23 | 62 |
| 0 | 104 | 86 |
G:=sub<GL(3,GF(109))| [63,0,0,0,1,0,0,0,1],[63,0,0,0,45,0,0,0,45],[1,0,0,0,81,106,0,0,35],[1,0,0,0,23,104,0,62,86] >;
C32×D27 in GAP, Magma, Sage, TeX
C_3^2\times D_{27} % in TeX
G:=Group("C3^2xD27"); // GroupNames label
G:=SmallGroup(486,111);
// by ID
G=gap.SmallGroup(486,111);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,2163,381,8104,208,11669]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^27=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