direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C53, C3⋊C106, C159⋊3C2, SmallGroup(318,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C53 |
Generators and relations for S3×C53
G = < a,b,c | a53=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C53
►On 159 points
Generators in S159
(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)
(1 83 141)(2 84 142)(3 85 143)(4 86 144)(5 87 145)(6 88 146)(7 89 147)(8 90 148)(9 91 149)(10 92 150)(11 93 151)(12 94 152)(13 95 153)(14 96 154)(15 97 155)(16 98 156)(17 99 157)(18 100 158)(19 101 159)(20 102 107)(21 103 108)(22 104 109)(23 105 110)(24 106 111)(25 54 112)(26 55 113)(27 56 114)(28 57 115)(29 58 116)(30 59 117)(31 60 118)(32 61 119)(33 62 120)(34 63 121)(35 64 122)(36 65 123)(37 66 124)(38 67 125)(39 68 126)(40 69 127)(41 70 128)(42 71 129)(43 72 130)(44 73 131)(45 74 132)(46 75 133)(47 76 134)(48 77 135)(49 78 136)(50 79 137)(51 80 138)(52 81 139)(53 82 140)
(54 112)(55 113)(56 114)(57 115)(58 116)(59 117)(60 118)(61 119)(62 120)(63 121)(64 122)(65 123)(66 124)(67 125)(68 126)(69 127)(70 128)(71 129)(72 130)(73 131)(74 132)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)(81 139)(82 140)(83 141)(84 142)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 154)(97 155)(98 156)(99 157)(100 158)(101 159)(102 107)(103 108)(104 109)(105 110)(106 111)
G:=sub<Sym(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), (1,83,141)(2,84,142)(3,85,143)(4,86,144)(5,87,145)(6,88,146)(7,89,147)(8,90,148)(9,91,149)(10,92,150)(11,93,151)(12,94,152)(13,95,153)(14,96,154)(15,97,155)(16,98,156)(17,99,157)(18,100,158)(19,101,159)(20,102,107)(21,103,108)(22,104,109)(23,105,110)(24,106,111)(25,54,112)(26,55,113)(27,56,114)(28,57,115)(29,58,116)(30,59,117)(31,60,118)(32,61,119)(33,62,120)(34,63,121)(35,64,122)(36,65,123)(37,66,124)(38,67,125)(39,68,126)(40,69,127)(41,70,128)(42,71,129)(43,72,130)(44,73,131)(45,74,132)(46,75,133)(47,76,134)(48,77,135)(49,78,136)(50,79,137)(51,80,138)(52,81,139)(53,82,140), (54,112)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130)(73,131)(74,132)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,155)(98,156)(99,157)(100,158)(101,159)(102,107)(103,108)(104,109)(105,110)(106,111)>;
G:=Group( (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), (1,83,141)(2,84,142)(3,85,143)(4,86,144)(5,87,145)(6,88,146)(7,89,147)(8,90,148)(9,91,149)(10,92,150)(11,93,151)(12,94,152)(13,95,153)(14,96,154)(15,97,155)(16,98,156)(17,99,157)(18,100,158)(19,101,159)(20,102,107)(21,103,108)(22,104,109)(23,105,110)(24,106,111)(25,54,112)(26,55,113)(27,56,114)(28,57,115)(29,58,116)(30,59,117)(31,60,118)(32,61,119)(33,62,120)(34,63,121)(35,64,122)(36,65,123)(37,66,124)(38,67,125)(39,68,126)(40,69,127)(41,70,128)(42,71,129)(43,72,130)(44,73,131)(45,74,132)(46,75,133)(47,76,134)(48,77,135)(49,78,136)(50,79,137)(51,80,138)(52,81,139)(53,82,140), (54,112)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130)(73,131)(74,132)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(81,139)(82,140)(83,141)(84,142)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,155)(98,156)(99,157)(100,158)(101,159)(102,107)(103,108)(104,109)(105,110)(106,111) );
G=PermutationGroup([[(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)], [(1,83,141),(2,84,142),(3,85,143),(4,86,144),(5,87,145),(6,88,146),(7,89,147),(8,90,148),(9,91,149),(10,92,150),(11,93,151),(12,94,152),(13,95,153),(14,96,154),(15,97,155),(16,98,156),(17,99,157),(18,100,158),(19,101,159),(20,102,107),(21,103,108),(22,104,109),(23,105,110),(24,106,111),(25,54,112),(26,55,113),(27,56,114),(28,57,115),(29,58,116),(30,59,117),(31,60,118),(32,61,119),(33,62,120),(34,63,121),(35,64,122),(36,65,123),(37,66,124),(38,67,125),(39,68,126),(40,69,127),(41,70,128),(42,71,129),(43,72,130),(44,73,131),(45,74,132),(46,75,133),(47,76,134),(48,77,135),(49,78,136),(50,79,137),(51,80,138),(52,81,139),(53,82,140)], [(54,112),(55,113),(56,114),(57,115),(58,116),(59,117),(60,118),(61,119),(62,120),(63,121),(64,122),(65,123),(66,124),(67,125),(68,126),(69,127),(70,128),(71,129),(72,130),(73,131),(74,132),(75,133),(76,134),(77,135),(78,136),(79,137),(80,138),(81,139),(82,140),(83,141),(84,142),(85,143),(86,144),(87,145),(88,146),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,153),(96,154),(97,155),(98,156),(99,157),(100,158),(101,159),(102,107),(103,108),(104,109),(105,110),(106,111)]])
159 conjugacy classes
| class | 1 | 2 | 3 | 53A | ··· | 53AZ | 106A | ··· | 106AZ | 159A | ··· | 159AZ |
| order | 1 | 2 | 3 | 53 | ··· | 53 | 106 | ··· | 106 | 159 | ··· | 159 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
159 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C53 | C106 | S3 | S3×C53 |
| kernel | S3×C53 | C159 | S3 | C3 | C53 | C1 |
| # reps | 1 | 1 | 52 | 52 | 1 | 52 |
Matrix representation of S3×C53 ►in GL2(𝔽3181) generated by
| 2027 | 0 |
| 0 | 2027 |
| 0 | 3180 |
| 1 | 3180 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(3181))| [2027,0,0,2027],[0,1,3180,3180],[0,1,1,0] >;
S3×C53 in GAP, Magma, Sage, TeX
S_3\times C_{53} % in TeX
G:=Group("S3xC53"); // GroupNames label
G:=SmallGroup(318,1);
// by ID
G=gap.SmallGroup(318,1);
# by ID
G:=PCGroup([3,-2,-53,-3,1910]);
// Polycyclic
G:=Group<a,b,c|a^53=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export