direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C71, C3⋊C142, C213⋊3C2, SmallGroup(426,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C71 |
Generators and relations for S3×C71
G = < a,b,c | a71=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C71
►On 213 points
Generators in S213
(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)(143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213)
(1 194 83)(2 195 84)(3 196 85)(4 197 86)(5 198 87)(6 199 88)(7 200 89)(8 201 90)(9 202 91)(10 203 92)(11 204 93)(12 205 94)(13 206 95)(14 207 96)(15 208 97)(16 209 98)(17 210 99)(18 211 100)(19 212 101)(20 213 102)(21 143 103)(22 144 104)(23 145 105)(24 146 106)(25 147 107)(26 148 108)(27 149 109)(28 150 110)(29 151 111)(30 152 112)(31 153 113)(32 154 114)(33 155 115)(34 156 116)(35 157 117)(36 158 118)(37 159 119)(38 160 120)(39 161 121)(40 162 122)(41 163 123)(42 164 124)(43 165 125)(44 166 126)(45 167 127)(46 168 128)(47 169 129)(48 170 130)(49 171 131)(50 172 132)(51 173 133)(52 174 134)(53 175 135)(54 176 136)(55 177 137)(56 178 138)(57 179 139)(58 180 140)(59 181 141)(60 182 142)(61 183 72)(62 184 73)(63 185 74)(64 186 75)(65 187 76)(66 188 77)(67 189 78)(68 190 79)(69 191 80)(70 192 81)(71 193 82)
(72 183)(73 184)(74 185)(75 186)(76 187)(77 188)(78 189)(79 190)(80 191)(81 192)(82 193)(83 194)(84 195)(85 196)(86 197)(87 198)(88 199)(89 200)(90 201)(91 202)(92 203)(93 204)(94 205)(95 206)(96 207)(97 208)(98 209)(99 210)(100 211)(101 212)(102 213)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)(121 161)(122 162)(123 163)(124 164)(125 165)(126 166)(127 167)(128 168)(129 169)(130 170)(131 171)(132 172)(133 173)(134 174)(135 175)(136 176)(137 177)(138 178)(139 179)(140 180)(141 181)(142 182)
G:=sub<Sym(213)| (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)(143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213), (1,194,83)(2,195,84)(3,196,85)(4,197,86)(5,198,87)(6,199,88)(7,200,89)(8,201,90)(9,202,91)(10,203,92)(11,204,93)(12,205,94)(13,206,95)(14,207,96)(15,208,97)(16,209,98)(17,210,99)(18,211,100)(19,212,101)(20,213,102)(21,143,103)(22,144,104)(23,145,105)(24,146,106)(25,147,107)(26,148,108)(27,149,109)(28,150,110)(29,151,111)(30,152,112)(31,153,113)(32,154,114)(33,155,115)(34,156,116)(35,157,117)(36,158,118)(37,159,119)(38,160,120)(39,161,121)(40,162,122)(41,163,123)(42,164,124)(43,165,125)(44,166,126)(45,167,127)(46,168,128)(47,169,129)(48,170,130)(49,171,131)(50,172,132)(51,173,133)(52,174,134)(53,175,135)(54,176,136)(55,177,137)(56,178,138)(57,179,139)(58,180,140)(59,181,141)(60,182,142)(61,183,72)(62,184,73)(63,185,74)(64,186,75)(65,187,76)(66,188,77)(67,189,78)(68,190,79)(69,191,80)(70,192,81)(71,193,82), (72,183)(73,184)(74,185)(75,186)(76,187)(77,188)(78,189)(79,190)(80,191)(81,192)(82,193)(83,194)(84,195)(85,196)(86,197)(87,198)(88,199)(89,200)(90,201)(91,202)(92,203)(93,204)(94,205)(95,206)(96,207)(97,208)(98,209)(99,210)(100,211)(101,212)(102,213)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160)(121,161)(122,162)(123,163)(124,164)(125,165)(126,166)(127,167)(128,168)(129,169)(130,170)(131,171)(132,172)(133,173)(134,174)(135,175)(136,176)(137,177)(138,178)(139,179)(140,180)(141,181)(142,182)>;
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)(143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213), (1,194,83)(2,195,84)(3,196,85)(4,197,86)(5,198,87)(6,199,88)(7,200,89)(8,201,90)(9,202,91)(10,203,92)(11,204,93)(12,205,94)(13,206,95)(14,207,96)(15,208,97)(16,209,98)(17,210,99)(18,211,100)(19,212,101)(20,213,102)(21,143,103)(22,144,104)(23,145,105)(24,146,106)(25,147,107)(26,148,108)(27,149,109)(28,150,110)(29,151,111)(30,152,112)(31,153,113)(32,154,114)(33,155,115)(34,156,116)(35,157,117)(36,158,118)(37,159,119)(38,160,120)(39,161,121)(40,162,122)(41,163,123)(42,164,124)(43,165,125)(44,166,126)(45,167,127)(46,168,128)(47,169,129)(48,170,130)(49,171,131)(50,172,132)(51,173,133)(52,174,134)(53,175,135)(54,176,136)(55,177,137)(56,178,138)(57,179,139)(58,180,140)(59,181,141)(60,182,142)(61,183,72)(62,184,73)(63,185,74)(64,186,75)(65,187,76)(66,188,77)(67,189,78)(68,190,79)(69,191,80)(70,192,81)(71,193,82), (72,183)(73,184)(74,185)(75,186)(76,187)(77,188)(78,189)(79,190)(80,191)(81,192)(82,193)(83,194)(84,195)(85,196)(86,197)(87,198)(88,199)(89,200)(90,201)(91,202)(92,203)(93,204)(94,205)(95,206)(96,207)(97,208)(98,209)(99,210)(100,211)(101,212)(102,213)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160)(121,161)(122,162)(123,163)(124,164)(125,165)(126,166)(127,167)(128,168)(129,169)(130,170)(131,171)(132,172)(133,173)(134,174)(135,175)(136,176)(137,177)(138,178)(139,179)(140,180)(141,181)(142,182) );
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),(143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213)], [(1,194,83),(2,195,84),(3,196,85),(4,197,86),(5,198,87),(6,199,88),(7,200,89),(8,201,90),(9,202,91),(10,203,92),(11,204,93),(12,205,94),(13,206,95),(14,207,96),(15,208,97),(16,209,98),(17,210,99),(18,211,100),(19,212,101),(20,213,102),(21,143,103),(22,144,104),(23,145,105),(24,146,106),(25,147,107),(26,148,108),(27,149,109),(28,150,110),(29,151,111),(30,152,112),(31,153,113),(32,154,114),(33,155,115),(34,156,116),(35,157,117),(36,158,118),(37,159,119),(38,160,120),(39,161,121),(40,162,122),(41,163,123),(42,164,124),(43,165,125),(44,166,126),(45,167,127),(46,168,128),(47,169,129),(48,170,130),(49,171,131),(50,172,132),(51,173,133),(52,174,134),(53,175,135),(54,176,136),(55,177,137),(56,178,138),(57,179,139),(58,180,140),(59,181,141),(60,182,142),(61,183,72),(62,184,73),(63,185,74),(64,186,75),(65,187,76),(66,188,77),(67,189,78),(68,190,79),(69,191,80),(70,192,81),(71,193,82)], [(72,183),(73,184),(74,185),(75,186),(76,187),(77,188),(78,189),(79,190),(80,191),(81,192),(82,193),(83,194),(84,195),(85,196),(86,197),(87,198),(88,199),(89,200),(90,201),(91,202),(92,203),(93,204),(94,205),(95,206),(96,207),(97,208),(98,209),(99,210),(100,211),(101,212),(102,213),(103,143),(104,144),(105,145),(106,146),(107,147),(108,148),(109,149),(110,150),(111,151),(112,152),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160),(121,161),(122,162),(123,163),(124,164),(125,165),(126,166),(127,167),(128,168),(129,169),(130,170),(131,171),(132,172),(133,173),(134,174),(135,175),(136,176),(137,177),(138,178),(139,179),(140,180),(141,181),(142,182)]])
213 conjugacy classes
| class | 1 | 2 | 3 | 71A | ··· | 71BR | 142A | ··· | 142BR | 213A | ··· | 213BR |
| order | 1 | 2 | 3 | 71 | ··· | 71 | 142 | ··· | 142 | 213 | ··· | 213 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
213 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C71 | C142 | S3 | S3×C71 |
| kernel | S3×C71 | C213 | S3 | C3 | C71 | C1 |
| # reps | 1 | 1 | 70 | 70 | 1 | 70 |
Matrix representation of S3×C71 ►in GL2(𝔽853) generated by
| 541 | 0 |
| 0 | 541 |
| 0 | 852 |
| 1 | 852 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(853))| [541,0,0,541],[0,1,852,852],[0,1,1,0] >;
S3×C71 in GAP, Magma, Sage, TeX
S_3\times C_{71} % in TeX
G:=Group("S3xC71"); // GroupNames label
G:=SmallGroup(426,1);
// by ID
G=gap.SmallGroup(426,1);
# by ID
G:=PCGroup([3,-2,-71,-3,2558]);
// Polycyclic
G:=Group<a,b,c|a^71=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export