direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C58, C6⋊C58, C174⋊3C2, C87⋊4C22, C3⋊(C2×C58), SmallGroup(348,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C58 |
Generators and relations for S3×C58
G = < a,b,c | a58=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C58
►On 174 points
Generators in S174
(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)
(1 170 75)(2 171 76)(3 172 77)(4 173 78)(5 174 79)(6 117 80)(7 118 81)(8 119 82)(9 120 83)(10 121 84)(11 122 85)(12 123 86)(13 124 87)(14 125 88)(15 126 89)(16 127 90)(17 128 91)(18 129 92)(19 130 93)(20 131 94)(21 132 95)(22 133 96)(23 134 97)(24 135 98)(25 136 99)(26 137 100)(27 138 101)(28 139 102)(29 140 103)(30 141 104)(31 142 105)(32 143 106)(33 144 107)(34 145 108)(35 146 109)(36 147 110)(37 148 111)(38 149 112)(39 150 113)(40 151 114)(41 152 115)(42 153 116)(43 154 59)(44 155 60)(45 156 61)(46 157 62)(47 158 63)(48 159 64)(49 160 65)(50 161 66)(51 162 67)(52 163 68)(53 164 69)(54 165 70)(55 166 71)(56 167 72)(57 168 73)(58 169 74)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(59 125)(60 126)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(71 137)(72 138)(73 139)(74 140)(75 141)(76 142)(77 143)(78 144)(79 145)(80 146)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)(91 157)(92 158)(93 159)(94 160)(95 161)(96 162)(97 163)(98 164)(99 165)(100 166)(101 167)(102 168)(103 169)(104 170)(105 171)(106 172)(107 173)(108 174)(109 117)(110 118)(111 119)(112 120)(113 121)(114 122)(115 123)(116 124)
G:=sub<Sym(174)| (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), (1,170,75)(2,171,76)(3,172,77)(4,173,78)(5,174,79)(6,117,80)(7,118,81)(8,119,82)(9,120,83)(10,121,84)(11,122,85)(12,123,86)(13,124,87)(14,125,88)(15,126,89)(16,127,90)(17,128,91)(18,129,92)(19,130,93)(20,131,94)(21,132,95)(22,133,96)(23,134,97)(24,135,98)(25,136,99)(26,137,100)(27,138,101)(28,139,102)(29,140,103)(30,141,104)(31,142,105)(32,143,106)(33,144,107)(34,145,108)(35,146,109)(36,147,110)(37,148,111)(38,149,112)(39,150,113)(40,151,114)(41,152,115)(42,153,116)(43,154,59)(44,155,60)(45,156,61)(46,157,62)(47,158,63)(48,159,64)(49,160,65)(50,161,66)(51,162,67)(52,163,68)(53,164,69)(54,165,70)(55,166,71)(56,167,72)(57,168,73)(58,169,74), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(59,125)(60,126)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,169)(104,170)(105,171)(106,172)(107,173)(108,174)(109,117)(110,118)(111,119)(112,120)(113,121)(114,122)(115,123)(116,124)>;
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), (1,170,75)(2,171,76)(3,172,77)(4,173,78)(5,174,79)(6,117,80)(7,118,81)(8,119,82)(9,120,83)(10,121,84)(11,122,85)(12,123,86)(13,124,87)(14,125,88)(15,126,89)(16,127,90)(17,128,91)(18,129,92)(19,130,93)(20,131,94)(21,132,95)(22,133,96)(23,134,97)(24,135,98)(25,136,99)(26,137,100)(27,138,101)(28,139,102)(29,140,103)(30,141,104)(31,142,105)(32,143,106)(33,144,107)(34,145,108)(35,146,109)(36,147,110)(37,148,111)(38,149,112)(39,150,113)(40,151,114)(41,152,115)(42,153,116)(43,154,59)(44,155,60)(45,156,61)(46,157,62)(47,158,63)(48,159,64)(49,160,65)(50,161,66)(51,162,67)(52,163,68)(53,164,69)(54,165,70)(55,166,71)(56,167,72)(57,168,73)(58,169,74), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,58)(59,125)(60,126)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,169)(104,170)(105,171)(106,172)(107,173)(108,174)(109,117)(110,118)(111,119)(112,120)(113,121)(114,122)(115,123)(116,124) );
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)], [(1,170,75),(2,171,76),(3,172,77),(4,173,78),(5,174,79),(6,117,80),(7,118,81),(8,119,82),(9,120,83),(10,121,84),(11,122,85),(12,123,86),(13,124,87),(14,125,88),(15,126,89),(16,127,90),(17,128,91),(18,129,92),(19,130,93),(20,131,94),(21,132,95),(22,133,96),(23,134,97),(24,135,98),(25,136,99),(26,137,100),(27,138,101),(28,139,102),(29,140,103),(30,141,104),(31,142,105),(32,143,106),(33,144,107),(34,145,108),(35,146,109),(36,147,110),(37,148,111),(38,149,112),(39,150,113),(40,151,114),(41,152,115),(42,153,116),(43,154,59),(44,155,60),(45,156,61),(46,157,62),(47,158,63),(48,159,64),(49,160,65),(50,161,66),(51,162,67),(52,163,68),(53,164,69),(54,165,70),(55,166,71),(56,167,72),(57,168,73),(58,169,74)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,50),(22,51),(23,52),(24,53),(25,54),(26,55),(27,56),(28,57),(29,58),(59,125),(60,126),(61,127),(62,128),(63,129),(64,130),(65,131),(66,132),(67,133),(68,134),(69,135),(70,136),(71,137),(72,138),(73,139),(74,140),(75,141),(76,142),(77,143),(78,144),(79,145),(80,146),(81,147),(82,148),(83,149),(84,150),(85,151),(86,152),(87,153),(88,154),(89,155),(90,156),(91,157),(92,158),(93,159),(94,160),(95,161),(96,162),(97,163),(98,164),(99,165),(100,166),(101,167),(102,168),(103,169),(104,170),(105,171),(106,172),(107,173),(108,174),(109,117),(110,118),(111,119),(112,120),(113,121),(114,122),(115,123),(116,124)]])
174 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 29A | ··· | 29AB | 58A | ··· | 58AB | 58AC | ··· | 58CF | 87A | ··· | 87AB | 174A | ··· | 174AB |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 29 | ··· | 29 | 58 | ··· | 58 | 58 | ··· | 58 | 87 | ··· | 87 | 174 | ··· | 174 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
174 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C29 | C58 | C58 | S3 | D6 | S3×C29 | S3×C58 |
| kernel | S3×C58 | S3×C29 | C174 | D6 | S3 | C6 | C58 | C29 | C2 | C1 |
| # reps | 1 | 2 | 1 | 28 | 56 | 28 | 1 | 1 | 28 | 28 |
Matrix representation of S3×C58 ►in GL2(𝔽349) generated by
| 181 | 0 |
| 0 | 181 |
| 0 | 348 |
| 1 | 348 |
| 0 | 348 |
| 348 | 0 |
G:=sub<GL(2,GF(349))| [181,0,0,181],[0,1,348,348],[0,348,348,0] >;
S3×C58 in GAP, Magma, Sage, TeX
S_3\times C_{58} % in TeX
G:=Group("S3xC58"); // GroupNames label
G:=SmallGroup(348,10);
// by ID
G=gap.SmallGroup(348,10);
# by ID
G:=PCGroup([4,-2,-2,-29,-3,3715]);
// Polycyclic
G:=Group<a,b,c|a^58=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export