direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C62, C6⋊C62, C186⋊3C2, C93⋊4C22, C3⋊(C2×C62), SmallGroup(372,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C62 |
Generators and relations for S3×C62
G = < a,b,c | a62=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C62
►On 186 points
Generators in S186
(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)(125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186)
(1 145 104)(2 146 105)(3 147 106)(4 148 107)(5 149 108)(6 150 109)(7 151 110)(8 152 111)(9 153 112)(10 154 113)(11 155 114)(12 156 115)(13 157 116)(14 158 117)(15 159 118)(16 160 119)(17 161 120)(18 162 121)(19 163 122)(20 164 123)(21 165 124)(22 166 63)(23 167 64)(24 168 65)(25 169 66)(26 170 67)(27 171 68)(28 172 69)(29 173 70)(30 174 71)(31 175 72)(32 176 73)(33 177 74)(34 178 75)(35 179 76)(36 180 77)(37 181 78)(38 182 79)(39 183 80)(40 184 81)(41 185 82)(42 186 83)(43 125 84)(44 126 85)(45 127 86)(46 128 87)(47 129 88)(48 130 89)(49 131 90)(50 132 91)(51 133 92)(52 134 93)(53 135 94)(54 136 95)(55 137 96)(56 138 97)(57 139 98)(58 140 99)(59 141 100)(60 142 101)(61 143 102)(62 144 103)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)(73 145)(74 146)(75 147)(76 148)(77 149)(78 150)(79 151)(80 152)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 161)(90 162)(91 163)(92 164)(93 165)(94 166)(95 167)(96 168)(97 169)(98 170)(99 171)(100 172)(101 173)(102 174)(103 175)(104 176)(105 177)(106 178)(107 179)(108 180)(109 181)(110 182)(111 183)(112 184)(113 185)(114 186)(115 125)(116 126)(117 127)(118 128)(119 129)(120 130)(121 131)(122 132)(123 133)(124 134)
G:=sub<Sym(186)| (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)(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186), (1,145,104)(2,146,105)(3,147,106)(4,148,107)(5,149,108)(6,150,109)(7,151,110)(8,152,111)(9,153,112)(10,154,113)(11,155,114)(12,156,115)(13,157,116)(14,158,117)(15,159,118)(16,160,119)(17,161,120)(18,162,121)(19,163,122)(20,164,123)(21,165,124)(22,166,63)(23,167,64)(24,168,65)(25,169,66)(26,170,67)(27,171,68)(28,172,69)(29,173,70)(30,174,71)(31,175,72)(32,176,73)(33,177,74)(34,178,75)(35,179,76)(36,180,77)(37,181,78)(38,182,79)(39,183,80)(40,184,81)(41,185,82)(42,186,83)(43,125,84)(44,126,85)(45,127,86)(46,128,87)(47,129,88)(48,130,89)(49,131,90)(50,132,91)(51,133,92)(52,134,93)(53,135,94)(54,136,95)(55,137,96)(56,138,97)(57,139,98)(58,140,99)(59,141,100)(60,142,101)(61,143,102)(62,144,103), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,169)(98,170)(99,171)(100,172)(101,173)(102,174)(103,175)(104,176)(105,177)(106,178)(107,179)(108,180)(109,181)(110,182)(111,183)(112,184)(113,185)(114,186)(115,125)(116,126)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134)>;
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)(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186), (1,145,104)(2,146,105)(3,147,106)(4,148,107)(5,149,108)(6,150,109)(7,151,110)(8,152,111)(9,153,112)(10,154,113)(11,155,114)(12,156,115)(13,157,116)(14,158,117)(15,159,118)(16,160,119)(17,161,120)(18,162,121)(19,163,122)(20,164,123)(21,165,124)(22,166,63)(23,167,64)(24,168,65)(25,169,66)(26,170,67)(27,171,68)(28,172,69)(29,173,70)(30,174,71)(31,175,72)(32,176,73)(33,177,74)(34,178,75)(35,179,76)(36,180,77)(37,181,78)(38,182,79)(39,183,80)(40,184,81)(41,185,82)(42,186,83)(43,125,84)(44,126,85)(45,127,86)(46,128,87)(47,129,88)(48,130,89)(49,131,90)(50,132,91)(51,133,92)(52,134,93)(53,135,94)(54,136,95)(55,137,96)(56,138,97)(57,139,98)(58,140,99)(59,141,100)(60,142,101)(61,143,102)(62,144,103), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,169)(98,170)(99,171)(100,172)(101,173)(102,174)(103,175)(104,176)(105,177)(106,178)(107,179)(108,180)(109,181)(110,182)(111,183)(112,184)(113,185)(114,186)(115,125)(116,126)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134) );
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),(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186)], [(1,145,104),(2,146,105),(3,147,106),(4,148,107),(5,149,108),(6,150,109),(7,151,110),(8,152,111),(9,153,112),(10,154,113),(11,155,114),(12,156,115),(13,157,116),(14,158,117),(15,159,118),(16,160,119),(17,161,120),(18,162,121),(19,163,122),(20,164,123),(21,165,124),(22,166,63),(23,167,64),(24,168,65),(25,169,66),(26,170,67),(27,171,68),(28,172,69),(29,173,70),(30,174,71),(31,175,72),(32,176,73),(33,177,74),(34,178,75),(35,179,76),(36,180,77),(37,181,78),(38,182,79),(39,183,80),(40,184,81),(41,185,82),(42,186,83),(43,125,84),(44,126,85),(45,127,86),(46,128,87),(47,129,88),(48,130,89),(49,131,90),(50,132,91),(51,133,92),(52,134,93),(53,135,94),(54,136,95),(55,137,96),(56,138,97),(57,139,98),(58,140,99),(59,141,100),(60,142,101),(61,143,102),(62,144,103)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144),(73,145),(74,146),(75,147),(76,148),(77,149),(78,150),(79,151),(80,152),(81,153),(82,154),(83,155),(84,156),(85,157),(86,158),(87,159),(88,160),(89,161),(90,162),(91,163),(92,164),(93,165),(94,166),(95,167),(96,168),(97,169),(98,170),(99,171),(100,172),(101,173),(102,174),(103,175),(104,176),(105,177),(106,178),(107,179),(108,180),(109,181),(110,182),(111,183),(112,184),(113,185),(114,186),(115,125),(116,126),(117,127),(118,128),(119,129),(120,130),(121,131),(122,132),(123,133),(124,134)]])
186 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 31A | ··· | 31AD | 62A | ··· | 62AD | 62AE | ··· | 62CL | 93A | ··· | 93AD | 186A | ··· | 186AD |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 31 | ··· | 31 | 62 | ··· | 62 | 62 | ··· | 62 | 93 | ··· | 93 | 186 | ··· | 186 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
186 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C31 | C62 | C62 | S3 | D6 | S3×C31 | S3×C62 |
| kernel | S3×C62 | S3×C31 | C186 | D6 | S3 | C6 | C62 | C31 | C2 | C1 |
| # reps | 1 | 2 | 1 | 30 | 60 | 30 | 1 | 1 | 30 | 30 |
Matrix representation of S3×C62 ►in GL2(𝔽373) generated by
| 287 | 0 |
| 0 | 287 |
| 0 | 372 |
| 1 | 372 |
| 0 | 372 |
| 372 | 0 |
G:=sub<GL(2,GF(373))| [287,0,0,287],[0,1,372,372],[0,372,372,0] >;
S3×C62 in GAP, Magma, Sage, TeX
S_3\times C_{62} % in TeX
G:=Group("S3xC62"); // GroupNames label
G:=SmallGroup(372,13);
// by ID
G=gap.SmallGroup(372,13);
# by ID
G:=PCGroup([4,-2,-2,-31,-3,3971]);
// Polycyclic
G:=Group<a,b,c|a^62=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export