direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C3×C27, C32⋊3C54, C33.6C18, C3⋊(C3×C54), C9.7(S3×C9), (C3×C27)⋊13C6, (S3×C9).3C9, (C32×C27)⋊1C2, (C3×C9).10C18, C9.6(S3×C32), (S3×C9).2C32, (S3×C32).2C9, C32.21(S3×C9), (C32×C9).27C6, C32.16(C3×C18), C3.7(S3×C3×C9), (S3×C3×C9).5C3, (C3×S3).4(C3×C9), (C3×C9).37(C3×C6), (C3×C9).63(C3×S3), SmallGroup(486,112)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C3×C27 |
Generators and relations for S3×C3×C27
G = < a,b,c,d | a3=b27=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 128 in 78 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C9, C32, C32, C32, C18, C3×S3, C3×S3, C3×C6, C27, C27, C3×C9, C3×C9, C3×C9, C33, C54, S3×C9, S3×C9, C3×C18, S3×C32, C3×C27, C3×C27, C3×C27, C32×C9, S3×C27, C3×C54, S3×C3×C9, C32×C27, S3×C3×C27
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C27, C3×C9, C54, S3×C9, C3×C18, S3×C32, C3×C27, S3×C27, C3×C54, S3×C3×C9, S3×C3×C27
►On 162 points
Generators in S162
(1 61 32)(2 62 33)(3 63 34)(4 64 35)(5 65 36)(6 66 37)(7 67 38)(8 68 39)(9 69 40)(10 70 41)(11 71 42)(12 72 43)(13 73 44)(14 74 45)(15 75 46)(16 76 47)(17 77 48)(18 78 49)(19 79 50)(20 80 51)(21 81 52)(22 55 53)(23 56 54)(24 57 28)(25 58 29)(26 59 30)(27 60 31)(82 146 134)(83 147 135)(84 148 109)(85 149 110)(86 150 111)(87 151 112)(88 152 113)(89 153 114)(90 154 115)(91 155 116)(92 156 117)(93 157 118)(94 158 119)(95 159 120)(96 160 121)(97 161 122)(98 162 123)(99 136 124)(100 137 125)(101 138 126)(102 139 127)(103 140 128)(104 141 129)(105 142 130)(106 143 131)(107 144 132)(108 145 133)
(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 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)(28 37 46)(29 38 47)(30 39 48)(31 40 49)(32 41 50)(33 42 51)(34 43 52)(35 44 53)(36 45 54)(55 64 73)(56 65 74)(57 66 75)(58 67 76)(59 68 77)(60 69 78)(61 70 79)(62 71 80)(63 72 81)(82 100 91)(83 101 92)(84 102 93)(85 103 94)(86 104 95)(87 105 96)(88 106 97)(89 107 98)(90 108 99)(109 127 118)(110 128 119)(111 129 120)(112 130 121)(113 131 122)(114 132 123)(115 133 124)(116 134 125)(117 135 126)(136 154 145)(137 155 146)(138 156 147)(139 157 148)(140 158 149)(141 159 150)(142 160 151)(143 161 152)(144 162 153)
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 97)(15 98)(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 82)(27 83)(28 132)(29 133)(30 134)(31 135)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 142)(56 143)(57 144)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(73 160)(74 161)(75 162)(76 136)(77 137)(78 138)(79 139)(80 140)(81 141)
G:=sub<Sym(162)| (1,61,32)(2,62,33)(3,63,34)(4,64,35)(5,65,36)(6,66,37)(7,67,38)(8,68,39)(9,69,40)(10,70,41)(11,71,42)(12,72,43)(13,73,44)(14,74,45)(15,75,46)(16,76,47)(17,77,48)(18,78,49)(19,79,50)(20,80,51)(21,81,52)(22,55,53)(23,56,54)(24,57,28)(25,58,29)(26,59,30)(27,60,31)(82,146,134)(83,147,135)(84,148,109)(85,149,110)(86,150,111)(87,151,112)(88,152,113)(89,153,114)(90,154,115)(91,155,116)(92,156,117)(93,157,118)(94,158,119)(95,159,120)(96,160,121)(97,161,122)(98,162,123)(99,136,124)(100,137,125)(101,138,126)(102,139,127)(103,140,128)(104,141,129)(105,142,130)(106,143,131)(107,144,132)(108,145,133), (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), (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,37,46)(29,38,47)(30,39,48)(31,40,49)(32,41,50)(33,42,51)(34,43,52)(35,44,53)(36,45,54)(55,64,73)(56,65,74)(57,66,75)(58,67,76)(59,68,77)(60,69,78)(61,70,79)(62,71,80)(63,72,81)(82,100,91)(83,101,92)(84,102,93)(85,103,94)(86,104,95)(87,105,96)(88,106,97)(89,107,98)(90,108,99)(109,127,118)(110,128,119)(111,129,120)(112,130,121)(113,131,122)(114,132,123)(115,133,124)(116,134,125)(117,135,126)(136,154,145)(137,155,146)(138,156,147)(139,157,148)(140,158,149)(141,159,150)(142,160,151)(143,161,152)(144,162,153), (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,82)(27,83)(28,132)(29,133)(30,134)(31,135)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,118)(42,119)(43,120)(44,121)(45,122)(46,123)(47,124)(48,125)(49,126)(50,127)(51,128)(52,129)(53,130)(54,131)(55,142)(56,143)(57,144)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,158)(72,159)(73,160)(74,161)(75,162)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)>;
G:=Group( (1,61,32)(2,62,33)(3,63,34)(4,64,35)(5,65,36)(6,66,37)(7,67,38)(8,68,39)(9,69,40)(10,70,41)(11,71,42)(12,72,43)(13,73,44)(14,74,45)(15,75,46)(16,76,47)(17,77,48)(18,78,49)(19,79,50)(20,80,51)(21,81,52)(22,55,53)(23,56,54)(24,57,28)(25,58,29)(26,59,30)(27,60,31)(82,146,134)(83,147,135)(84,148,109)(85,149,110)(86,150,111)(87,151,112)(88,152,113)(89,153,114)(90,154,115)(91,155,116)(92,156,117)(93,157,118)(94,158,119)(95,159,120)(96,160,121)(97,161,122)(98,162,123)(99,136,124)(100,137,125)(101,138,126)(102,139,127)(103,140,128)(104,141,129)(105,142,130)(106,143,131)(107,144,132)(108,145,133), (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), (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,37,46)(29,38,47)(30,39,48)(31,40,49)(32,41,50)(33,42,51)(34,43,52)(35,44,53)(36,45,54)(55,64,73)(56,65,74)(57,66,75)(58,67,76)(59,68,77)(60,69,78)(61,70,79)(62,71,80)(63,72,81)(82,100,91)(83,101,92)(84,102,93)(85,103,94)(86,104,95)(87,105,96)(88,106,97)(89,107,98)(90,108,99)(109,127,118)(110,128,119)(111,129,120)(112,130,121)(113,131,122)(114,132,123)(115,133,124)(116,134,125)(117,135,126)(136,154,145)(137,155,146)(138,156,147)(139,157,148)(140,158,149)(141,159,150)(142,160,151)(143,161,152)(144,162,153), (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,82)(27,83)(28,132)(29,133)(30,134)(31,135)(32,109)(33,110)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,118)(42,119)(43,120)(44,121)(45,122)(46,123)(47,124)(48,125)(49,126)(50,127)(51,128)(52,129)(53,130)(54,131)(55,142)(56,143)(57,144)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,158)(72,159)(73,160)(74,161)(75,162)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141) );
G=PermutationGroup([[(1,61,32),(2,62,33),(3,63,34),(4,64,35),(5,65,36),(6,66,37),(7,67,38),(8,68,39),(9,69,40),(10,70,41),(11,71,42),(12,72,43),(13,73,44),(14,74,45),(15,75,46),(16,76,47),(17,77,48),(18,78,49),(19,79,50),(20,80,51),(21,81,52),(22,55,53),(23,56,54),(24,57,28),(25,58,29),(26,59,30),(27,60,31),(82,146,134),(83,147,135),(84,148,109),(85,149,110),(86,150,111),(87,151,112),(88,152,113),(89,153,114),(90,154,115),(91,155,116),(92,156,117),(93,157,118),(94,158,119),(95,159,120),(96,160,121),(97,161,122),(98,162,123),(99,136,124),(100,137,125),(101,138,126),(102,139,127),(103,140,128),(104,141,129),(105,142,130),(106,143,131),(107,144,132),(108,145,133)], [(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)], [(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,25),(8,17,26),(9,18,27),(28,37,46),(29,38,47),(30,39,48),(31,40,49),(32,41,50),(33,42,51),(34,43,52),(35,44,53),(36,45,54),(55,64,73),(56,65,74),(57,66,75),(58,67,76),(59,68,77),(60,69,78),(61,70,79),(62,71,80),(63,72,81),(82,100,91),(83,101,92),(84,102,93),(85,103,94),(86,104,95),(87,105,96),(88,106,97),(89,107,98),(90,108,99),(109,127,118),(110,128,119),(111,129,120),(112,130,121),(113,131,122),(114,132,123),(115,133,124),(116,134,125),(117,135,126),(136,154,145),(137,155,146),(138,156,147),(139,157,148),(140,158,149),(141,159,150),(142,160,151),(143,161,152),(144,162,153)], [(1,84),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,91),(9,92),(10,93),(11,94),(12,95),(13,96),(14,97),(15,98),(16,99),(17,100),(18,101),(19,102),(20,103),(21,104),(22,105),(23,106),(24,107),(25,108),(26,82),(27,83),(28,132),(29,133),(30,134),(31,135),(32,109),(33,110),(34,111),(35,112),(36,113),(37,114),(38,115),(39,116),(40,117),(41,118),(42,119),(43,120),(44,121),(45,122),(46,123),(47,124),(48,125),(49,126),(50,127),(51,128),(52,129),(53,130),(54,131),(55,142),(56,143),(57,144),(58,145),(59,146),(60,147),(61,148),(62,149),(63,150),(64,151),(65,152),(66,153),(67,154),(68,155),(69,156),(70,157),(71,158),(72,159),(73,160),(74,161),(75,162),(76,136),(77,137),(78,138),(79,139),(80,140),(81,141)]])
243 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 6A | ··· | 6H | 9A | ··· | 9R | 9S | ··· | 9AJ | 18A | ··· | 18R | 27A | ··· | 27BB | 27BC | ··· | 27DD | 54A | ··· | 54BB |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C9 | C18 | C18 | C27 | C54 | S3 | C3×S3 | C3×S3 | S3×C9 | S3×C9 | S3×C27 |
| kernel | S3×C3×C27 | C32×C27 | S3×C27 | S3×C3×C9 | C3×C27 | C32×C9 | S3×C9 | S3×C32 | C3×C9 | C33 | C3×S3 | C32 | C3×C27 | C27 | C3×C9 | C9 | C32 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 12 | 6 | 12 | 6 | 54 | 54 | 1 | 6 | 2 | 12 | 6 | 54 |
Matrix representation of S3×C3×C27 ►in GL3(𝔽109) generated by
| 1 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 63 |
| 25 | 0 | 0 |
| 0 | 78 | 0 |
| 0 | 0 | 78 |
| 1 | 0 | 0 |
| 0 | 45 | 0 |
| 0 | 0 | 63 |
| 108 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(109))| [1,0,0,0,63,0,0,0,63],[25,0,0,0,78,0,0,0,78],[1,0,0,0,45,0,0,0,63],[108,0,0,0,0,1,0,1,0] >;
S3×C3×C27 in GAP, Magma, Sage, TeX
S_3\times C_3\times C_{27} % in TeX
G:=Group("S3xC3xC27"); // GroupNames label
G:=SmallGroup(486,112);
// by ID
G=gap.SmallGroup(486,112);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,68,93,11669]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations