direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C65, C3⋊C130, C195⋊7C2, C39⋊3C10, C15⋊3C26, SmallGroup(390,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C65 |
Generators and relations for S3×C65
G = < a,b,c | a65=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C65
►On 195 points
Generators in S195
(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)(131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195)
(1 96 157)(2 97 158)(3 98 159)(4 99 160)(5 100 161)(6 101 162)(7 102 163)(8 103 164)(9 104 165)(10 105 166)(11 106 167)(12 107 168)(13 108 169)(14 109 170)(15 110 171)(16 111 172)(17 112 173)(18 113 174)(19 114 175)(20 115 176)(21 116 177)(22 117 178)(23 118 179)(24 119 180)(25 120 181)(26 121 182)(27 122 183)(28 123 184)(29 124 185)(30 125 186)(31 126 187)(32 127 188)(33 128 189)(34 129 190)(35 130 191)(36 66 192)(37 67 193)(38 68 194)(39 69 195)(40 70 131)(41 71 132)(42 72 133)(43 73 134)(44 74 135)(45 75 136)(46 76 137)(47 77 138)(48 78 139)(49 79 140)(50 80 141)(51 81 142)(52 82 143)(53 83 144)(54 84 145)(55 85 146)(56 86 147)(57 87 148)(58 88 149)(59 89 150)(60 90 151)(61 91 152)(62 92 153)(63 93 154)(64 94 155)(65 95 156)
(66 192)(67 193)(68 194)(69 195)(70 131)(71 132)(72 133)(73 134)(74 135)(75 136)(76 137)(77 138)(78 139)(79 140)(80 141)(81 142)(82 143)(83 144)(84 145)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 155)(95 156)(96 157)(97 158)(98 159)(99 160)(100 161)(101 162)(102 163)(103 164)(104 165)(105 166)(106 167)(107 168)(108 169)(109 170)(110 171)(111 172)(112 173)(113 174)(114 175)(115 176)(116 177)(117 178)(118 179)(119 180)(120 181)(121 182)(122 183)(123 184)(124 185)(125 186)(126 187)(127 188)(128 189)(129 190)(130 191)
G:=sub<Sym(195)| (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195), (1,96,157)(2,97,158)(3,98,159)(4,99,160)(5,100,161)(6,101,162)(7,102,163)(8,103,164)(9,104,165)(10,105,166)(11,106,167)(12,107,168)(13,108,169)(14,109,170)(15,110,171)(16,111,172)(17,112,173)(18,113,174)(19,114,175)(20,115,176)(21,116,177)(22,117,178)(23,118,179)(24,119,180)(25,120,181)(26,121,182)(27,122,183)(28,123,184)(29,124,185)(30,125,186)(31,126,187)(32,127,188)(33,128,189)(34,129,190)(35,130,191)(36,66,192)(37,67,193)(38,68,194)(39,69,195)(40,70,131)(41,71,132)(42,72,133)(43,73,134)(44,74,135)(45,75,136)(46,76,137)(47,77,138)(48,78,139)(49,79,140)(50,80,141)(51,81,142)(52,82,143)(53,83,144)(54,84,145)(55,85,146)(56,86,147)(57,87,148)(58,88,149)(59,89,150)(60,90,151)(61,91,152)(62,92,153)(63,93,154)(64,94,155)(65,95,156), (66,192)(67,193)(68,194)(69,195)(70,131)(71,132)(72,133)(73,134)(74,135)(75,136)(76,137)(77,138)(78,139)(79,140)(80,141)(81,142)(82,143)(83,144)(84,145)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,155)(95,156)(96,157)(97,158)(98,159)(99,160)(100,161)(101,162)(102,163)(103,164)(104,165)(105,166)(106,167)(107,168)(108,169)(109,170)(110,171)(111,172)(112,173)(113,174)(114,175)(115,176)(116,177)(117,178)(118,179)(119,180)(120,181)(121,182)(122,183)(123,184)(124,185)(125,186)(126,187)(127,188)(128,189)(129,190)(130,191)>;
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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195), (1,96,157)(2,97,158)(3,98,159)(4,99,160)(5,100,161)(6,101,162)(7,102,163)(8,103,164)(9,104,165)(10,105,166)(11,106,167)(12,107,168)(13,108,169)(14,109,170)(15,110,171)(16,111,172)(17,112,173)(18,113,174)(19,114,175)(20,115,176)(21,116,177)(22,117,178)(23,118,179)(24,119,180)(25,120,181)(26,121,182)(27,122,183)(28,123,184)(29,124,185)(30,125,186)(31,126,187)(32,127,188)(33,128,189)(34,129,190)(35,130,191)(36,66,192)(37,67,193)(38,68,194)(39,69,195)(40,70,131)(41,71,132)(42,72,133)(43,73,134)(44,74,135)(45,75,136)(46,76,137)(47,77,138)(48,78,139)(49,79,140)(50,80,141)(51,81,142)(52,82,143)(53,83,144)(54,84,145)(55,85,146)(56,86,147)(57,87,148)(58,88,149)(59,89,150)(60,90,151)(61,91,152)(62,92,153)(63,93,154)(64,94,155)(65,95,156), (66,192)(67,193)(68,194)(69,195)(70,131)(71,132)(72,133)(73,134)(74,135)(75,136)(76,137)(77,138)(78,139)(79,140)(80,141)(81,142)(82,143)(83,144)(84,145)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,155)(95,156)(96,157)(97,158)(98,159)(99,160)(100,161)(101,162)(102,163)(103,164)(104,165)(105,166)(106,167)(107,168)(108,169)(109,170)(110,171)(111,172)(112,173)(113,174)(114,175)(115,176)(116,177)(117,178)(118,179)(119,180)(120,181)(121,182)(122,183)(123,184)(124,185)(125,186)(126,187)(127,188)(128,189)(129,190)(130,191) );
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),(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195)], [(1,96,157),(2,97,158),(3,98,159),(4,99,160),(5,100,161),(6,101,162),(7,102,163),(8,103,164),(9,104,165),(10,105,166),(11,106,167),(12,107,168),(13,108,169),(14,109,170),(15,110,171),(16,111,172),(17,112,173),(18,113,174),(19,114,175),(20,115,176),(21,116,177),(22,117,178),(23,118,179),(24,119,180),(25,120,181),(26,121,182),(27,122,183),(28,123,184),(29,124,185),(30,125,186),(31,126,187),(32,127,188),(33,128,189),(34,129,190),(35,130,191),(36,66,192),(37,67,193),(38,68,194),(39,69,195),(40,70,131),(41,71,132),(42,72,133),(43,73,134),(44,74,135),(45,75,136),(46,76,137),(47,77,138),(48,78,139),(49,79,140),(50,80,141),(51,81,142),(52,82,143),(53,83,144),(54,84,145),(55,85,146),(56,86,147),(57,87,148),(58,88,149),(59,89,150),(60,90,151),(61,91,152),(62,92,153),(63,93,154),(64,94,155),(65,95,156)], [(66,192),(67,193),(68,194),(69,195),(70,131),(71,132),(72,133),(73,134),(74,135),(75,136),(76,137),(77,138),(78,139),(79,140),(80,141),(81,142),(82,143),(83,144),(84,145),(85,146),(86,147),(87,148),(88,149),(89,150),(90,151),(91,152),(92,153),(93,154),(94,155),(95,156),(96,157),(97,158),(98,159),(99,160),(100,161),(101,162),(102,163),(103,164),(104,165),(105,166),(106,167),(107,168),(108,169),(109,170),(110,171),(111,172),(112,173),(113,174),(114,175),(115,176),(116,177),(117,178),(118,179),(119,180),(120,181),(121,182),(122,183),(123,184),(124,185),(125,186),(126,187),(127,188),(128,189),(129,190),(130,191)]])
195 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 13A | ··· | 13L | 15A | 15B | 15C | 15D | 26A | ··· | 26L | 39A | ··· | 39L | 65A | ··· | 65AV | 130A | ··· | 130AV | 195A | ··· | 195AV |
| order | 1 | 2 | 3 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 13 | ··· | 13 | 15 | 15 | 15 | 15 | 26 | ··· | 26 | 39 | ··· | 39 | 65 | ··· | 65 | 130 | ··· | 130 | 195 | ··· | 195 |
| size | 1 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
195 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C5 | C10 | C13 | C26 | C65 | C130 | S3 | C5×S3 | S3×C13 | S3×C65 |
| kernel | S3×C65 | C195 | S3×C13 | C39 | C5×S3 | C15 | S3 | C3 | C65 | C13 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 12 | 12 | 48 | 48 | 1 | 4 | 12 | 48 |
Matrix representation of S3×C65 ►in GL2(𝔽1171) generated by
| 1009 | 0 |
| 0 | 1009 |
| 0 | 1170 |
| 1 | 1170 |
| 1 | 1170 |
| 0 | 1170 |
G:=sub<GL(2,GF(1171))| [1009,0,0,1009],[0,1,1170,1170],[1,0,1170,1170] >;
S3×C65 in GAP, Magma, Sage, TeX
S_3\times C_{65} % in TeX
G:=Group("S3xC65"); // GroupNames label
G:=SmallGroup(390,8);
// by ID
G=gap.SmallGroup(390,8);
# by ID
G:=PCGroup([4,-2,-5,-13,-3,4163]);
// Polycyclic
G:=Group<a,b,c|a^65=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export