direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C59, C3⋊C118, C177⋊3C2, SmallGroup(354,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C59 |
Generators and relations for S3×C59
G = < a,b,c | a59=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C59
►On 177 points
Generators in S177
(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)(119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177)
(1 124 61)(2 125 62)(3 126 63)(4 127 64)(5 128 65)(6 129 66)(7 130 67)(8 131 68)(9 132 69)(10 133 70)(11 134 71)(12 135 72)(13 136 73)(14 137 74)(15 138 75)(16 139 76)(17 140 77)(18 141 78)(19 142 79)(20 143 80)(21 144 81)(22 145 82)(23 146 83)(24 147 84)(25 148 85)(26 149 86)(27 150 87)(28 151 88)(29 152 89)(30 153 90)(31 154 91)(32 155 92)(33 156 93)(34 157 94)(35 158 95)(36 159 96)(37 160 97)(38 161 98)(39 162 99)(40 163 100)(41 164 101)(42 165 102)(43 166 103)(44 167 104)(45 168 105)(46 169 106)(47 170 107)(48 171 108)(49 172 109)(50 173 110)(51 174 111)(52 175 112)(53 176 113)(54 177 114)(55 119 115)(56 120 116)(57 121 117)(58 122 118)(59 123 60)
(60 123)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 144)(82 145)(83 146)(84 147)(85 148)(86 149)(87 150)(88 151)(89 152)(90 153)(91 154)(92 155)(93 156)(94 157)(95 158)(96 159)(97 160)(98 161)(99 162)(100 163)(101 164)(102 165)(103 166)(104 167)(105 168)(106 169)(107 170)(108 171)(109 172)(110 173)(111 174)(112 175)(113 176)(114 177)(115 119)(116 120)(117 121)(118 122)
G:=sub<Sym(177)| (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)(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177), (1,124,61)(2,125,62)(3,126,63)(4,127,64)(5,128,65)(6,129,66)(7,130,67)(8,131,68)(9,132,69)(10,133,70)(11,134,71)(12,135,72)(13,136,73)(14,137,74)(15,138,75)(16,139,76)(17,140,77)(18,141,78)(19,142,79)(20,143,80)(21,144,81)(22,145,82)(23,146,83)(24,147,84)(25,148,85)(26,149,86)(27,150,87)(28,151,88)(29,152,89)(30,153,90)(31,154,91)(32,155,92)(33,156,93)(34,157,94)(35,158,95)(36,159,96)(37,160,97)(38,161,98)(39,162,99)(40,163,100)(41,164,101)(42,165,102)(43,166,103)(44,167,104)(45,168,105)(46,169,106)(47,170,107)(48,171,108)(49,172,109)(50,173,110)(51,174,111)(52,175,112)(53,176,113)(54,177,114)(55,119,115)(56,120,116)(57,121,117)(58,122,118)(59,123,60), (60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(85,148)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,119)(116,120)(117,121)(118,122)>;
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)(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177), (1,124,61)(2,125,62)(3,126,63)(4,127,64)(5,128,65)(6,129,66)(7,130,67)(8,131,68)(9,132,69)(10,133,70)(11,134,71)(12,135,72)(13,136,73)(14,137,74)(15,138,75)(16,139,76)(17,140,77)(18,141,78)(19,142,79)(20,143,80)(21,144,81)(22,145,82)(23,146,83)(24,147,84)(25,148,85)(26,149,86)(27,150,87)(28,151,88)(29,152,89)(30,153,90)(31,154,91)(32,155,92)(33,156,93)(34,157,94)(35,158,95)(36,159,96)(37,160,97)(38,161,98)(39,162,99)(40,163,100)(41,164,101)(42,165,102)(43,166,103)(44,167,104)(45,168,105)(46,169,106)(47,170,107)(48,171,108)(49,172,109)(50,173,110)(51,174,111)(52,175,112)(53,176,113)(54,177,114)(55,119,115)(56,120,116)(57,121,117)(58,122,118)(59,123,60), (60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,145)(83,146)(84,147)(85,148)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,119)(116,120)(117,121)(118,122) );
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),(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177)], [(1,124,61),(2,125,62),(3,126,63),(4,127,64),(5,128,65),(6,129,66),(7,130,67),(8,131,68),(9,132,69),(10,133,70),(11,134,71),(12,135,72),(13,136,73),(14,137,74),(15,138,75),(16,139,76),(17,140,77),(18,141,78),(19,142,79),(20,143,80),(21,144,81),(22,145,82),(23,146,83),(24,147,84),(25,148,85),(26,149,86),(27,150,87),(28,151,88),(29,152,89),(30,153,90),(31,154,91),(32,155,92),(33,156,93),(34,157,94),(35,158,95),(36,159,96),(37,160,97),(38,161,98),(39,162,99),(40,163,100),(41,164,101),(42,165,102),(43,166,103),(44,167,104),(45,168,105),(46,169,106),(47,170,107),(48,171,108),(49,172,109),(50,173,110),(51,174,111),(52,175,112),(53,176,113),(54,177,114),(55,119,115),(56,120,116),(57,121,117),(58,122,118),(59,123,60)], [(60,123),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,133),(71,134),(72,135),(73,136),(74,137),(75,138),(76,139),(77,140),(78,141),(79,142),(80,143),(81,144),(82,145),(83,146),(84,147),(85,148),(86,149),(87,150),(88,151),(89,152),(90,153),(91,154),(92,155),(93,156),(94,157),(95,158),(96,159),(97,160),(98,161),(99,162),(100,163),(101,164),(102,165),(103,166),(104,167),(105,168),(106,169),(107,170),(108,171),(109,172),(110,173),(111,174),(112,175),(113,176),(114,177),(115,119),(116,120),(117,121),(118,122)]])
177 conjugacy classes
| class | 1 | 2 | 3 | 59A | ··· | 59BF | 118A | ··· | 118BF | 177A | ··· | 177BF |
| order | 1 | 2 | 3 | 59 | ··· | 59 | 118 | ··· | 118 | 177 | ··· | 177 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
177 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C59 | C118 | S3 | S3×C59 |
| kernel | S3×C59 | C177 | S3 | C3 | C59 | C1 |
| # reps | 1 | 1 | 58 | 58 | 1 | 58 |
Matrix representation of S3×C59 ►in GL2(𝔽709) generated by
| 664 | 0 |
| 0 | 664 |
| 0 | 708 |
| 1 | 708 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(709))| [664,0,0,664],[0,1,708,708],[0,1,1,0] >;
S3×C59 in GAP, Magma, Sage, TeX
S_3\times C_{59} % in TeX
G:=Group("S3xC59"); // GroupNames label
G:=SmallGroup(354,1);
// by ID
G=gap.SmallGroup(354,1);
# by ID
G:=PCGroup([3,-2,-59,-3,2126]);
// Polycyclic
G:=Group<a,b,c|a^59=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export