direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C27⋊S3, C32⋊3D27, C33.7D9, C3⋊(C3×D27), C27⋊3(C3×S3), C9.5(C3×D9), (C3×C27)⋊15C6, (C3×C27)⋊10S3, (C32×C27)⋊4C2, (C3×C9).10D9, (C32×C9).26S3, C32.18(C3×D9), C32.17(C9⋊S3), C3.6(C3×C9⋊S3), C9.3(C3×C3⋊S3), (C3×C9).60(C3×S3), (C3×C9).19(C3⋊S3), SmallGroup(486,160)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C27 — C3×C27⋊S3 |
Generators and relations for C3×C27⋊S3
G = < a,b,c,d | a3=b27=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 548 in 78 conjugacy classes, 30 normal (14 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C27, C27, C3×C9, C3×C9, C3×C9, C33, D27, C3×D9, C9⋊S3, C3×C3⋊S3, C3×C27, C3×C27, C3×C27, C32×C9, C3×D27, C27⋊S3, C3×C9⋊S3, C32×C27, C3×C27⋊S3
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, D27, C3×D9, C9⋊S3, C3×C3⋊S3, C3×D27, C27⋊S3, C3×C9⋊S3, C3×C27⋊S3
►On 162 points
Generators in S162
(1 96 80)(2 97 81)(3 98 55)(4 99 56)(5 100 57)(6 101 58)(7 102 59)(8 103 60)(9 104 61)(10 105 62)(11 106 63)(12 107 64)(13 108 65)(14 82 66)(15 83 67)(16 84 68)(17 85 69)(18 86 70)(19 87 71)(20 88 72)(21 89 73)(22 90 74)(23 91 75)(24 92 76)(25 93 77)(26 94 78)(27 95 79)(28 130 161)(29 131 162)(30 132 136)(31 133 137)(32 134 138)(33 135 139)(34 109 140)(35 110 141)(36 111 142)(37 112 143)(38 113 144)(39 114 145)(40 115 146)(41 116 147)(42 117 148)(43 118 149)(44 119 150)(45 120 151)(46 121 152)(47 122 153)(48 123 154)(49 124 155)(50 125 156)(51 126 157)(52 127 158)(53 128 159)(54 129 160)
(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 87 62)(2 88 63)(3 89 64)(4 90 65)(5 91 66)(6 92 67)(7 93 68)(8 94 69)(9 95 70)(10 96 71)(11 97 72)(12 98 73)(13 99 74)(14 100 75)(15 101 76)(16 102 77)(17 103 78)(18 104 79)(19 105 80)(20 106 81)(21 107 55)(22 108 56)(23 82 57)(24 83 58)(25 84 59)(26 85 60)(27 86 61)(28 152 112)(29 153 113)(30 154 114)(31 155 115)(32 156 116)(33 157 117)(34 158 118)(35 159 119)(36 160 120)(37 161 121)(38 162 122)(39 136 123)(40 137 124)(41 138 125)(42 139 126)(43 140 127)(44 141 128)(45 142 129)(46 143 130)(47 144 131)(48 145 132)(49 146 133)(50 147 134)(51 148 135)(52 149 109)(53 150 110)(54 151 111)
(1 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(55 146)(56 145)(57 144)(58 143)(59 142)(60 141)(61 140)(62 139)(63 138)(64 137)(65 136)(66 162)(67 161)(68 160)(69 159)(70 158)(71 157)(72 156)(73 155)(74 154)(75 153)(76 152)(77 151)(78 150)(79 149)(80 148)(81 147)(82 131)(83 130)(84 129)(85 128)(86 127)(87 126)(88 125)(89 124)(90 123)(91 122)(92 121)(93 120)(94 119)(95 118)(96 117)(97 116)(98 115)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 135)(106 134)(107 133)(108 132)
G:=sub<Sym(162)| (1,96,80)(2,97,81)(3,98,55)(4,99,56)(5,100,57)(6,101,58)(7,102,59)(8,103,60)(9,104,61)(10,105,62)(11,106,63)(12,107,64)(13,108,65)(14,82,66)(15,83,67)(16,84,68)(17,85,69)(18,86,70)(19,87,71)(20,88,72)(21,89,73)(22,90,74)(23,91,75)(24,92,76)(25,93,77)(26,94,78)(27,95,79)(28,130,161)(29,131,162)(30,132,136)(31,133,137)(32,134,138)(33,135,139)(34,109,140)(35,110,141)(36,111,142)(37,112,143)(38,113,144)(39,114,145)(40,115,146)(41,116,147)(42,117,148)(43,118,149)(44,119,150)(45,120,151)(46,121,152)(47,122,153)(48,123,154)(49,124,155)(50,125,156)(51,126,157)(52,127,158)(53,128,159)(54,129,160), (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), (1,87,62)(2,88,63)(3,89,64)(4,90,65)(5,91,66)(6,92,67)(7,93,68)(8,94,69)(9,95,70)(10,96,71)(11,97,72)(12,98,73)(13,99,74)(14,100,75)(15,101,76)(16,102,77)(17,103,78)(18,104,79)(19,105,80)(20,106,81)(21,107,55)(22,108,56)(23,82,57)(24,83,58)(25,84,59)(26,85,60)(27,86,61)(28,152,112)(29,153,113)(30,154,114)(31,155,115)(32,156,116)(33,157,117)(34,158,118)(35,159,119)(36,160,120)(37,161,121)(38,162,122)(39,136,123)(40,137,124)(41,138,125)(42,139,126)(43,140,127)(44,141,128)(45,142,129)(46,143,130)(47,144,131)(48,145,132)(49,146,133)(50,147,134)(51,148,135)(52,149,109)(53,150,110)(54,151,111), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(55,146)(56,145)(57,144)(58,143)(59,142)(60,141)(61,140)(62,139)(63,138)(64,137)(65,136)(66,162)(67,161)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,154)(75,153)(76,152)(77,151)(78,150)(79,149)(80,148)(81,147)(82,131)(83,130)(84,129)(85,128)(86,127)(87,126)(88,125)(89,124)(90,123)(91,122)(92,121)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,135)(106,134)(107,133)(108,132)>;
G:=Group( (1,96,80)(2,97,81)(3,98,55)(4,99,56)(5,100,57)(6,101,58)(7,102,59)(8,103,60)(9,104,61)(10,105,62)(11,106,63)(12,107,64)(13,108,65)(14,82,66)(15,83,67)(16,84,68)(17,85,69)(18,86,70)(19,87,71)(20,88,72)(21,89,73)(22,90,74)(23,91,75)(24,92,76)(25,93,77)(26,94,78)(27,95,79)(28,130,161)(29,131,162)(30,132,136)(31,133,137)(32,134,138)(33,135,139)(34,109,140)(35,110,141)(36,111,142)(37,112,143)(38,113,144)(39,114,145)(40,115,146)(41,116,147)(42,117,148)(43,118,149)(44,119,150)(45,120,151)(46,121,152)(47,122,153)(48,123,154)(49,124,155)(50,125,156)(51,126,157)(52,127,158)(53,128,159)(54,129,160), (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), (1,87,62)(2,88,63)(3,89,64)(4,90,65)(5,91,66)(6,92,67)(7,93,68)(8,94,69)(9,95,70)(10,96,71)(11,97,72)(12,98,73)(13,99,74)(14,100,75)(15,101,76)(16,102,77)(17,103,78)(18,104,79)(19,105,80)(20,106,81)(21,107,55)(22,108,56)(23,82,57)(24,83,58)(25,84,59)(26,85,60)(27,86,61)(28,152,112)(29,153,113)(30,154,114)(31,155,115)(32,156,116)(33,157,117)(34,158,118)(35,159,119)(36,160,120)(37,161,121)(38,162,122)(39,136,123)(40,137,124)(41,138,125)(42,139,126)(43,140,127)(44,141,128)(45,142,129)(46,143,130)(47,144,131)(48,145,132)(49,146,133)(50,147,134)(51,148,135)(52,149,109)(53,150,110)(54,151,111), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(55,146)(56,145)(57,144)(58,143)(59,142)(60,141)(61,140)(62,139)(63,138)(64,137)(65,136)(66,162)(67,161)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,154)(75,153)(76,152)(77,151)(78,150)(79,149)(80,148)(81,147)(82,131)(83,130)(84,129)(85,128)(86,127)(87,126)(88,125)(89,124)(90,123)(91,122)(92,121)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,135)(106,134)(107,133)(108,132) );
G=PermutationGroup([[(1,96,80),(2,97,81),(3,98,55),(4,99,56),(5,100,57),(6,101,58),(7,102,59),(8,103,60),(9,104,61),(10,105,62),(11,106,63),(12,107,64),(13,108,65),(14,82,66),(15,83,67),(16,84,68),(17,85,69),(18,86,70),(19,87,71),(20,88,72),(21,89,73),(22,90,74),(23,91,75),(24,92,76),(25,93,77),(26,94,78),(27,95,79),(28,130,161),(29,131,162),(30,132,136),(31,133,137),(32,134,138),(33,135,139),(34,109,140),(35,110,141),(36,111,142),(37,112,143),(38,113,144),(39,114,145),(40,115,146),(41,116,147),(42,117,148),(43,118,149),(44,119,150),(45,120,151),(46,121,152),(47,122,153),(48,123,154),(49,124,155),(50,125,156),(51,126,157),(52,127,158),(53,128,159),(54,129,160)], [(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)], [(1,87,62),(2,88,63),(3,89,64),(4,90,65),(5,91,66),(6,92,67),(7,93,68),(8,94,69),(9,95,70),(10,96,71),(11,97,72),(12,98,73),(13,99,74),(14,100,75),(15,101,76),(16,102,77),(17,103,78),(18,104,79),(19,105,80),(20,106,81),(21,107,55),(22,108,56),(23,82,57),(24,83,58),(25,84,59),(26,85,60),(27,86,61),(28,152,112),(29,153,113),(30,154,114),(31,155,115),(32,156,116),(33,157,117),(34,158,118),(35,159,119),(36,160,120),(37,161,121),(38,162,122),(39,136,123),(40,137,124),(41,138,125),(42,139,126),(43,140,127),(44,141,128),(45,142,129),(46,143,130),(47,144,131),(48,145,132),(49,146,133),(50,147,134),(51,148,135),(52,149,109),(53,150,110),(54,151,111)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,30),(14,29),(15,28),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(55,146),(56,145),(57,144),(58,143),(59,142),(60,141),(61,140),(62,139),(63,138),(64,137),(65,136),(66,162),(67,161),(68,160),(69,159),(70,158),(71,157),(72,156),(73,155),(74,154),(75,153),(76,152),(77,151),(78,150),(79,149),(80,148),(81,147),(82,131),(83,130),(84,129),(85,128),(86,127),(87,126),(88,125),(89,124),(90,123),(91,122),(92,121),(93,120),(94,119),(95,118),(96,117),(97,116),(98,115),(99,114),(100,113),(101,112),(102,111),(103,110),(104,109),(105,135),(106,134),(107,133),(108,132)]])
126 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 6A | 6B | 9A | ··· | 9AA | 27A | ··· | 27CC |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 9 | ··· | 9 | 27 | ··· | 27 |
| size | 1 | 81 | 1 | 1 | 2 | ··· | 2 | 81 | 81 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C3 | C6 | S3 | S3 | C3×S3 | D9 | C3×S3 | D9 | C3×D9 | D27 | C3×D9 | C3×D27 |
| kernel | C3×C27⋊S3 | C32×C27 | C27⋊S3 | C3×C27 | C3×C27 | C32×C9 | C27 | C3×C9 | C3×C9 | C33 | C9 | C32 | C32 | C3 |
| # reps | 1 | 1 | 2 | 2 | 3 | 1 | 6 | 6 | 2 | 3 | 12 | 27 | 6 | 54 |
Matrix representation of C3×C27⋊S3 ►in GL4(𝔽109) generated by
| 45 | 0 | 0 | 0 |
| 0 | 45 | 0 | 0 |
| 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 45 |
| 38 | 0 | 0 | 0 |
| 0 | 66 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 78 |
| 45 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 63 | 0 |
| 0 | 0 | 0 | 45 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(109))| [45,0,0,0,0,45,0,0,0,0,45,0,0,0,0,45],[38,0,0,0,0,66,0,0,0,0,7,0,0,0,0,78],[45,0,0,0,0,63,0,0,0,0,63,0,0,0,0,45],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3×C27⋊S3 in GAP, Magma, Sage, TeX
C_3\times C_{27}\rtimes S_3 % in TeX
G:=Group("C3xC27:S3"); // GroupNames label
G:=SmallGroup(486,160);
// by ID
G=gap.SmallGroup(486,160);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,824,867,8104,208,11669]);
// 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,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations