direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C61, C3⋊C122, C183⋊3C2, SmallGroup(366,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C61 |
Generators and relations for S3×C61
G = < a,b,c | a61=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C61
►On 183 points
Generators in S183
(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)
(1 129 70)(2 130 71)(3 131 72)(4 132 73)(5 133 74)(6 134 75)(7 135 76)(8 136 77)(9 137 78)(10 138 79)(11 139 80)(12 140 81)(13 141 82)(14 142 83)(15 143 84)(16 144 85)(17 145 86)(18 146 87)(19 147 88)(20 148 89)(21 149 90)(22 150 91)(23 151 92)(24 152 93)(25 153 94)(26 154 95)(27 155 96)(28 156 97)(29 157 98)(30 158 99)(31 159 100)(32 160 101)(33 161 102)(34 162 103)(35 163 104)(36 164 105)(37 165 106)(38 166 107)(39 167 108)(40 168 109)(41 169 110)(42 170 111)(43 171 112)(44 172 113)(45 173 114)(46 174 115)(47 175 116)(48 176 117)(49 177 118)(50 178 119)(51 179 120)(52 180 121)(53 181 122)(54 182 62)(55 183 63)(56 123 64)(57 124 65)(58 125 66)(59 126 67)(60 127 68)(61 128 69)
(62 182)(63 183)(64 123)(65 124)(66 125)(67 126)(68 127)(69 128)(70 129)(71 130)(72 131)(73 132)(74 133)(75 134)(76 135)(77 136)(78 137)(79 138)(80 139)(81 140)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 163)(105 164)(106 165)(107 166)(108 167)(109 168)(110 169)(111 170)(112 171)(113 172)(114 173)(115 174)(116 175)(117 176)(118 177)(119 178)(120 179)(121 180)(122 181)
G:=sub<Sym(183)| (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), (1,129,70)(2,130,71)(3,131,72)(4,132,73)(5,133,74)(6,134,75)(7,135,76)(8,136,77)(9,137,78)(10,138,79)(11,139,80)(12,140,81)(13,141,82)(14,142,83)(15,143,84)(16,144,85)(17,145,86)(18,146,87)(19,147,88)(20,148,89)(21,149,90)(22,150,91)(23,151,92)(24,152,93)(25,153,94)(26,154,95)(27,155,96)(28,156,97)(29,157,98)(30,158,99)(31,159,100)(32,160,101)(33,161,102)(34,162,103)(35,163,104)(36,164,105)(37,165,106)(38,166,107)(39,167,108)(40,168,109)(41,169,110)(42,170,111)(43,171,112)(44,172,113)(45,173,114)(46,174,115)(47,175,116)(48,176,117)(49,177,118)(50,178,119)(51,179,120)(52,180,121)(53,181,122)(54,182,62)(55,183,63)(56,123,64)(57,124,65)(58,125,66)(59,126,67)(60,127,68)(61,128,69), (62,182)(63,183)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,169)(111,170)(112,171)(113,172)(114,173)(115,174)(116,175)(117,176)(118,177)(119,178)(120,179)(121,180)(122,181)>;
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), (1,129,70)(2,130,71)(3,131,72)(4,132,73)(5,133,74)(6,134,75)(7,135,76)(8,136,77)(9,137,78)(10,138,79)(11,139,80)(12,140,81)(13,141,82)(14,142,83)(15,143,84)(16,144,85)(17,145,86)(18,146,87)(19,147,88)(20,148,89)(21,149,90)(22,150,91)(23,151,92)(24,152,93)(25,153,94)(26,154,95)(27,155,96)(28,156,97)(29,157,98)(30,158,99)(31,159,100)(32,160,101)(33,161,102)(34,162,103)(35,163,104)(36,164,105)(37,165,106)(38,166,107)(39,167,108)(40,168,109)(41,169,110)(42,170,111)(43,171,112)(44,172,113)(45,173,114)(46,174,115)(47,175,116)(48,176,117)(49,177,118)(50,178,119)(51,179,120)(52,180,121)(53,181,122)(54,182,62)(55,183,63)(56,123,64)(57,124,65)(58,125,66)(59,126,67)(60,127,68)(61,128,69), (62,182)(63,183)(64,123)(65,124)(66,125)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,133)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,169)(111,170)(112,171)(113,172)(114,173)(115,174)(116,175)(117,176)(118,177)(119,178)(120,179)(121,180)(122,181) );
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)], [(1,129,70),(2,130,71),(3,131,72),(4,132,73),(5,133,74),(6,134,75),(7,135,76),(8,136,77),(9,137,78),(10,138,79),(11,139,80),(12,140,81),(13,141,82),(14,142,83),(15,143,84),(16,144,85),(17,145,86),(18,146,87),(19,147,88),(20,148,89),(21,149,90),(22,150,91),(23,151,92),(24,152,93),(25,153,94),(26,154,95),(27,155,96),(28,156,97),(29,157,98),(30,158,99),(31,159,100),(32,160,101),(33,161,102),(34,162,103),(35,163,104),(36,164,105),(37,165,106),(38,166,107),(39,167,108),(40,168,109),(41,169,110),(42,170,111),(43,171,112),(44,172,113),(45,173,114),(46,174,115),(47,175,116),(48,176,117),(49,177,118),(50,178,119),(51,179,120),(52,180,121),(53,181,122),(54,182,62),(55,183,63),(56,123,64),(57,124,65),(58,125,66),(59,126,67),(60,127,68),(61,128,69)], [(62,182),(63,183),(64,123),(65,124),(66,125),(67,126),(68,127),(69,128),(70,129),(71,130),(72,131),(73,132),(74,133),(75,134),(76,135),(77,136),(78,137),(79,138),(80,139),(81,140),(82,141),(83,142),(84,143),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159),(101,160),(102,161),(103,162),(104,163),(105,164),(106,165),(107,166),(108,167),(109,168),(110,169),(111,170),(112,171),(113,172),(114,173),(115,174),(116,175),(117,176),(118,177),(119,178),(120,179),(121,180),(122,181)]])
183 conjugacy classes
| class | 1 | 2 | 3 | 61A | ··· | 61BH | 122A | ··· | 122BH | 183A | ··· | 183BH |
| order | 1 | 2 | 3 | 61 | ··· | 61 | 122 | ··· | 122 | 183 | ··· | 183 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
183 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C61 | C122 | S3 | S3×C61 |
| kernel | S3×C61 | C183 | S3 | C3 | C61 | C1 |
| # reps | 1 | 1 | 60 | 60 | 1 | 60 |
Matrix representation of S3×C61 ►in GL2(𝔽367) generated by
| 204 | 0 |
| 0 | 204 |
| 366 | 366 |
| 1 | 0 |
| 1 | 0 |
| 366 | 366 |
G:=sub<GL(2,GF(367))| [204,0,0,204],[366,1,366,0],[1,366,0,366] >;
S3×C61 in GAP, Magma, Sage, TeX
S_3\times C_{61} % in TeX
G:=Group("S3xC61"); // GroupNames label
G:=SmallGroup(366,3);
// by ID
G=gap.SmallGroup(366,3);
# by ID
G:=PCGroup([3,-2,-61,-3,2198]);
// Polycyclic
G:=Group<a,b,c|a^61=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export