direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C67, C3⋊C134, C201⋊3C2, SmallGroup(402,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C67 |
Generators and relations for S3×C67
G = < a,b,c | a67=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C67
►On 201 points
Generators in S201
(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)(135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201)
(1 73 183)(2 74 184)(3 75 185)(4 76 186)(5 77 187)(6 78 188)(7 79 189)(8 80 190)(9 81 191)(10 82 192)(11 83 193)(12 84 194)(13 85 195)(14 86 196)(15 87 197)(16 88 198)(17 89 199)(18 90 200)(19 91 201)(20 92 135)(21 93 136)(22 94 137)(23 95 138)(24 96 139)(25 97 140)(26 98 141)(27 99 142)(28 100 143)(29 101 144)(30 102 145)(31 103 146)(32 104 147)(33 105 148)(34 106 149)(35 107 150)(36 108 151)(37 109 152)(38 110 153)(39 111 154)(40 112 155)(41 113 156)(42 114 157)(43 115 158)(44 116 159)(45 117 160)(46 118 161)(47 119 162)(48 120 163)(49 121 164)(50 122 165)(51 123 166)(52 124 167)(53 125 168)(54 126 169)(55 127 170)(56 128 171)(57 129 172)(58 130 173)(59 131 174)(60 132 175)(61 133 176)(62 134 177)(63 68 178)(64 69 179)(65 70 180)(66 71 181)(67 72 182)
(68 178)(69 179)(70 180)(71 181)(72 182)(73 183)(74 184)(75 185)(76 186)(77 187)(78 188)(79 189)(80 190)(81 191)(82 192)(83 193)(84 194)(85 195)(86 196)(87 197)(88 198)(89 199)(90 200)(91 201)(92 135)(93 136)(94 137)(95 138)(96 139)(97 140)(98 141)(99 142)(100 143)(101 144)(102 145)(103 146)(104 147)(105 148)(106 149)(107 150)(108 151)(109 152)(110 153)(111 154)(112 155)(113 156)(114 157)(115 158)(116 159)(117 160)(118 161)(119 162)(120 163)(121 164)(122 165)(123 166)(124 167)(125 168)(126 169)(127 170)(128 171)(129 172)(130 173)(131 174)(132 175)(133 176)(134 177)
G:=sub<Sym(201)| (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)(135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201), (1,73,183)(2,74,184)(3,75,185)(4,76,186)(5,77,187)(6,78,188)(7,79,189)(8,80,190)(9,81,191)(10,82,192)(11,83,193)(12,84,194)(13,85,195)(14,86,196)(15,87,197)(16,88,198)(17,89,199)(18,90,200)(19,91,201)(20,92,135)(21,93,136)(22,94,137)(23,95,138)(24,96,139)(25,97,140)(26,98,141)(27,99,142)(28,100,143)(29,101,144)(30,102,145)(31,103,146)(32,104,147)(33,105,148)(34,106,149)(35,107,150)(36,108,151)(37,109,152)(38,110,153)(39,111,154)(40,112,155)(41,113,156)(42,114,157)(43,115,158)(44,116,159)(45,117,160)(46,118,161)(47,119,162)(48,120,163)(49,121,164)(50,122,165)(51,123,166)(52,124,167)(53,125,168)(54,126,169)(55,127,170)(56,128,171)(57,129,172)(58,130,173)(59,131,174)(60,132,175)(61,133,176)(62,134,177)(63,68,178)(64,69,179)(65,70,180)(66,71,181)(67,72,182), (68,178)(69,179)(70,180)(71,181)(72,182)(73,183)(74,184)(75,185)(76,186)(77,187)(78,188)(79,189)(80,190)(81,191)(82,192)(83,193)(84,194)(85,195)(86,196)(87,197)(88,198)(89,199)(90,200)(91,201)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,141)(99,142)(100,143)(101,144)(102,145)(103,146)(104,147)(105,148)(106,149)(107,150)(108,151)(109,152)(110,153)(111,154)(112,155)(113,156)(114,157)(115,158)(116,159)(117,160)(118,161)(119,162)(120,163)(121,164)(122,165)(123,166)(124,167)(125,168)(126,169)(127,170)(128,171)(129,172)(130,173)(131,174)(132,175)(133,176)(134,177)>;
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)(135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201), (1,73,183)(2,74,184)(3,75,185)(4,76,186)(5,77,187)(6,78,188)(7,79,189)(8,80,190)(9,81,191)(10,82,192)(11,83,193)(12,84,194)(13,85,195)(14,86,196)(15,87,197)(16,88,198)(17,89,199)(18,90,200)(19,91,201)(20,92,135)(21,93,136)(22,94,137)(23,95,138)(24,96,139)(25,97,140)(26,98,141)(27,99,142)(28,100,143)(29,101,144)(30,102,145)(31,103,146)(32,104,147)(33,105,148)(34,106,149)(35,107,150)(36,108,151)(37,109,152)(38,110,153)(39,111,154)(40,112,155)(41,113,156)(42,114,157)(43,115,158)(44,116,159)(45,117,160)(46,118,161)(47,119,162)(48,120,163)(49,121,164)(50,122,165)(51,123,166)(52,124,167)(53,125,168)(54,126,169)(55,127,170)(56,128,171)(57,129,172)(58,130,173)(59,131,174)(60,132,175)(61,133,176)(62,134,177)(63,68,178)(64,69,179)(65,70,180)(66,71,181)(67,72,182), (68,178)(69,179)(70,180)(71,181)(72,182)(73,183)(74,184)(75,185)(76,186)(77,187)(78,188)(79,189)(80,190)(81,191)(82,192)(83,193)(84,194)(85,195)(86,196)(87,197)(88,198)(89,199)(90,200)(91,201)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,141)(99,142)(100,143)(101,144)(102,145)(103,146)(104,147)(105,148)(106,149)(107,150)(108,151)(109,152)(110,153)(111,154)(112,155)(113,156)(114,157)(115,158)(116,159)(117,160)(118,161)(119,162)(120,163)(121,164)(122,165)(123,166)(124,167)(125,168)(126,169)(127,170)(128,171)(129,172)(130,173)(131,174)(132,175)(133,176)(134,177) );
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),(135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201)], [(1,73,183),(2,74,184),(3,75,185),(4,76,186),(5,77,187),(6,78,188),(7,79,189),(8,80,190),(9,81,191),(10,82,192),(11,83,193),(12,84,194),(13,85,195),(14,86,196),(15,87,197),(16,88,198),(17,89,199),(18,90,200),(19,91,201),(20,92,135),(21,93,136),(22,94,137),(23,95,138),(24,96,139),(25,97,140),(26,98,141),(27,99,142),(28,100,143),(29,101,144),(30,102,145),(31,103,146),(32,104,147),(33,105,148),(34,106,149),(35,107,150),(36,108,151),(37,109,152),(38,110,153),(39,111,154),(40,112,155),(41,113,156),(42,114,157),(43,115,158),(44,116,159),(45,117,160),(46,118,161),(47,119,162),(48,120,163),(49,121,164),(50,122,165),(51,123,166),(52,124,167),(53,125,168),(54,126,169),(55,127,170),(56,128,171),(57,129,172),(58,130,173),(59,131,174),(60,132,175),(61,133,176),(62,134,177),(63,68,178),(64,69,179),(65,70,180),(66,71,181),(67,72,182)], [(68,178),(69,179),(70,180),(71,181),(72,182),(73,183),(74,184),(75,185),(76,186),(77,187),(78,188),(79,189),(80,190),(81,191),(82,192),(83,193),(84,194),(85,195),(86,196),(87,197),(88,198),(89,199),(90,200),(91,201),(92,135),(93,136),(94,137),(95,138),(96,139),(97,140),(98,141),(99,142),(100,143),(101,144),(102,145),(103,146),(104,147),(105,148),(106,149),(107,150),(108,151),(109,152),(110,153),(111,154),(112,155),(113,156),(114,157),(115,158),(116,159),(117,160),(118,161),(119,162),(120,163),(121,164),(122,165),(123,166),(124,167),(125,168),(126,169),(127,170),(128,171),(129,172),(130,173),(131,174),(132,175),(133,176),(134,177)]])
201 conjugacy classes
| class | 1 | 2 | 3 | 67A | ··· | 67BN | 134A | ··· | 134BN | 201A | ··· | 201BN |
| order | 1 | 2 | 3 | 67 | ··· | 67 | 134 | ··· | 134 | 201 | ··· | 201 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
201 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C67 | C134 | S3 | S3×C67 |
| kernel | S3×C67 | C201 | S3 | C3 | C67 | C1 |
| # reps | 1 | 1 | 66 | 66 | 1 | 66 |
Matrix representation of S3×C67 ►in GL2(𝔽1609) generated by
| 893 | 0 |
| 0 | 893 |
| 1608 | 1608 |
| 1 | 0 |
| 1 | 0 |
| 1608 | 1608 |
G:=sub<GL(2,GF(1609))| [893,0,0,893],[1608,1,1608,0],[1,1608,0,1608] >;
S3×C67 in GAP, Magma, Sage, TeX
S_3\times C_{67} % in TeX
G:=Group("S3xC67"); // GroupNames label
G:=SmallGroup(402,3);
// by ID
G=gap.SmallGroup(402,3);
# by ID
G:=PCGroup([3,-2,-67,-3,2414]);
// Polycyclic
G:=Group<a,b,c|a^67=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export