direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×Dic17, D6.D17, C34.2D6, C6.2D34, Dic51⋊3C2, C102.2C22, C17⋊4(C4×S3), C51⋊5(C2×C4), (S3×C34).C2, (S3×C17)⋊2C4, C2.2(S3×D17), C3⋊1(C2×Dic17), (C3×Dic17)⋊1C2, SmallGroup(408,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — S3×Dic17 |
Generators and relations for S3×Dic17
G = < a,b,c,d | a3=b2=c34=1, d2=c17, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of S3×Dic17
►On 204 points
Generators in S204
(1 50 70)(2 51 71)(3 52 72)(4 53 73)(5 54 74)(6 55 75)(7 56 76)(8 57 77)(9 58 78)(10 59 79)(11 60 80)(12 61 81)(13 62 82)(14 63 83)(15 64 84)(16 65 85)(17 66 86)(18 67 87)(19 68 88)(20 35 89)(21 36 90)(22 37 91)(23 38 92)(24 39 93)(25 40 94)(26 41 95)(27 42 96)(28 43 97)(29 44 98)(30 45 99)(31 46 100)(32 47 101)(33 48 102)(34 49 69)(103 149 187)(104 150 188)(105 151 189)(106 152 190)(107 153 191)(108 154 192)(109 155 193)(110 156 194)(111 157 195)(112 158 196)(113 159 197)(114 160 198)(115 161 199)(116 162 200)(117 163 201)(118 164 202)(119 165 203)(120 166 204)(121 167 171)(122 168 172)(123 169 173)(124 170 174)(125 137 175)(126 138 176)(127 139 177)(128 140 178)(129 141 179)(130 142 180)(131 143 181)(132 144 182)(133 145 183)(134 146 184)(135 147 185)(136 148 186)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 78)(42 79)(43 80)(44 81)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 97)(61 98)(62 99)(63 100)(64 101)(65 102)(66 69)(67 70)(68 71)(103 166)(104 167)(105 168)(106 169)(107 170)(108 137)(109 138)(110 139)(111 140)(112 141)(113 142)(114 143)(115 144)(116 145)(117 146)(118 147)(119 148)(120 149)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 161)(133 162)(134 163)(135 164)(136 165)(171 188)(172 189)(173 190)(174 191)(175 192)(176 193)(177 194)(178 195)(179 196)(180 197)(181 198)(182 199)(183 200)(184 201)(185 202)(186 203)(187 204)
(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)
(1 175 18 192)(2 174 19 191)(3 173 20 190)(4 172 21 189)(5 171 22 188)(6 204 23 187)(7 203 24 186)(8 202 25 185)(9 201 26 184)(10 200 27 183)(11 199 28 182)(12 198 29 181)(13 197 30 180)(14 196 31 179)(15 195 32 178)(16 194 33 177)(17 193 34 176)(35 106 52 123)(36 105 53 122)(37 104 54 121)(38 103 55 120)(39 136 56 119)(40 135 57 118)(41 134 58 117)(42 133 59 116)(43 132 60 115)(44 131 61 114)(45 130 62 113)(46 129 63 112)(47 128 64 111)(48 127 65 110)(49 126 66 109)(50 125 67 108)(51 124 68 107)(69 138 86 155)(70 137 87 154)(71 170 88 153)(72 169 89 152)(73 168 90 151)(74 167 91 150)(75 166 92 149)(76 165 93 148)(77 164 94 147)(78 163 95 146)(79 162 96 145)(80 161 97 144)(81 160 98 143)(82 159 99 142)(83 158 100 141)(84 157 101 140)(85 156 102 139)
G:=sub<Sym(204)| (1,50,70)(2,51,71)(3,52,72)(4,53,73)(5,54,74)(6,55,75)(7,56,76)(8,57,77)(9,58,78)(10,59,79)(11,60,80)(12,61,81)(13,62,82)(14,63,83)(15,64,84)(16,65,85)(17,66,86)(18,67,87)(19,68,88)(20,35,89)(21,36,90)(22,37,91)(23,38,92)(24,39,93)(25,40,94)(26,41,95)(27,42,96)(28,43,97)(29,44,98)(30,45,99)(31,46,100)(32,47,101)(33,48,102)(34,49,69)(103,149,187)(104,150,188)(105,151,189)(106,152,190)(107,153,191)(108,154,192)(109,155,193)(110,156,194)(111,157,195)(112,158,196)(113,159,197)(114,160,198)(115,161,199)(116,162,200)(117,163,201)(118,164,202)(119,165,203)(120,166,204)(121,167,171)(122,168,172)(123,169,173)(124,170,174)(125,137,175)(126,138,176)(127,139,177)(128,140,178)(129,141,179)(130,142,180)(131,143,181)(132,144,182)(133,145,183)(134,146,184)(135,147,185)(136,148,186), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,97)(61,98)(62,99)(63,100)(64,101)(65,102)(66,69)(67,70)(68,71)(103,166)(104,167)(105,168)(106,169)(107,170)(108,137)(109,138)(110,139)(111,140)(112,141)(113,142)(114,143)(115,144)(116,145)(117,146)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(171,188)(172,189)(173,190)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,197)(181,198)(182,199)(183,200)(184,201)(185,202)(186,203)(187,204), (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), (1,175,18,192)(2,174,19,191)(3,173,20,190)(4,172,21,189)(5,171,22,188)(6,204,23,187)(7,203,24,186)(8,202,25,185)(9,201,26,184)(10,200,27,183)(11,199,28,182)(12,198,29,181)(13,197,30,180)(14,196,31,179)(15,195,32,178)(16,194,33,177)(17,193,34,176)(35,106,52,123)(36,105,53,122)(37,104,54,121)(38,103,55,120)(39,136,56,119)(40,135,57,118)(41,134,58,117)(42,133,59,116)(43,132,60,115)(44,131,61,114)(45,130,62,113)(46,129,63,112)(47,128,64,111)(48,127,65,110)(49,126,66,109)(50,125,67,108)(51,124,68,107)(69,138,86,155)(70,137,87,154)(71,170,88,153)(72,169,89,152)(73,168,90,151)(74,167,91,150)(75,166,92,149)(76,165,93,148)(77,164,94,147)(78,163,95,146)(79,162,96,145)(80,161,97,144)(81,160,98,143)(82,159,99,142)(83,158,100,141)(84,157,101,140)(85,156,102,139)>;
G:=Group( (1,50,70)(2,51,71)(3,52,72)(4,53,73)(5,54,74)(6,55,75)(7,56,76)(8,57,77)(9,58,78)(10,59,79)(11,60,80)(12,61,81)(13,62,82)(14,63,83)(15,64,84)(16,65,85)(17,66,86)(18,67,87)(19,68,88)(20,35,89)(21,36,90)(22,37,91)(23,38,92)(24,39,93)(25,40,94)(26,41,95)(27,42,96)(28,43,97)(29,44,98)(30,45,99)(31,46,100)(32,47,101)(33,48,102)(34,49,69)(103,149,187)(104,150,188)(105,151,189)(106,152,190)(107,153,191)(108,154,192)(109,155,193)(110,156,194)(111,157,195)(112,158,196)(113,159,197)(114,160,198)(115,161,199)(116,162,200)(117,163,201)(118,164,202)(119,165,203)(120,166,204)(121,167,171)(122,168,172)(123,169,173)(124,170,174)(125,137,175)(126,138,176)(127,139,177)(128,140,178)(129,141,179)(130,142,180)(131,143,181)(132,144,182)(133,145,183)(134,146,184)(135,147,185)(136,148,186), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)(46,83)(47,84)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,97)(61,98)(62,99)(63,100)(64,101)(65,102)(66,69)(67,70)(68,71)(103,166)(104,167)(105,168)(106,169)(107,170)(108,137)(109,138)(110,139)(111,140)(112,141)(113,142)(114,143)(115,144)(116,145)(117,146)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(171,188)(172,189)(173,190)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,197)(181,198)(182,199)(183,200)(184,201)(185,202)(186,203)(187,204), (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), (1,175,18,192)(2,174,19,191)(3,173,20,190)(4,172,21,189)(5,171,22,188)(6,204,23,187)(7,203,24,186)(8,202,25,185)(9,201,26,184)(10,200,27,183)(11,199,28,182)(12,198,29,181)(13,197,30,180)(14,196,31,179)(15,195,32,178)(16,194,33,177)(17,193,34,176)(35,106,52,123)(36,105,53,122)(37,104,54,121)(38,103,55,120)(39,136,56,119)(40,135,57,118)(41,134,58,117)(42,133,59,116)(43,132,60,115)(44,131,61,114)(45,130,62,113)(46,129,63,112)(47,128,64,111)(48,127,65,110)(49,126,66,109)(50,125,67,108)(51,124,68,107)(69,138,86,155)(70,137,87,154)(71,170,88,153)(72,169,89,152)(73,168,90,151)(74,167,91,150)(75,166,92,149)(76,165,93,148)(77,164,94,147)(78,163,95,146)(79,162,96,145)(80,161,97,144)(81,160,98,143)(82,159,99,142)(83,158,100,141)(84,157,101,140)(85,156,102,139) );
G=PermutationGroup([[(1,50,70),(2,51,71),(3,52,72),(4,53,73),(5,54,74),(6,55,75),(7,56,76),(8,57,77),(9,58,78),(10,59,79),(11,60,80),(12,61,81),(13,62,82),(14,63,83),(15,64,84),(16,65,85),(17,66,86),(18,67,87),(19,68,88),(20,35,89),(21,36,90),(22,37,91),(23,38,92),(24,39,93),(25,40,94),(26,41,95),(27,42,96),(28,43,97),(29,44,98),(30,45,99),(31,46,100),(32,47,101),(33,48,102),(34,49,69),(103,149,187),(104,150,188),(105,151,189),(106,152,190),(107,153,191),(108,154,192),(109,155,193),(110,156,194),(111,157,195),(112,158,196),(113,159,197),(114,160,198),(115,161,199),(116,162,200),(117,163,201),(118,164,202),(119,165,203),(120,166,204),(121,167,171),(122,168,172),(123,169,173),(124,170,174),(125,137,175),(126,138,176),(127,139,177),(128,140,178),(129,141,179),(130,142,180),(131,143,181),(132,144,182),(133,145,183),(134,146,184),(135,147,185),(136,148,186)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(35,72),(36,73),(37,74),(38,75),(39,76),(40,77),(41,78),(42,79),(43,80),(44,81),(45,82),(46,83),(47,84),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93),(57,94),(58,95),(59,96),(60,97),(61,98),(62,99),(63,100),(64,101),(65,102),(66,69),(67,70),(68,71),(103,166),(104,167),(105,168),(106,169),(107,170),(108,137),(109,138),(110,139),(111,140),(112,141),(113,142),(114,143),(115,144),(116,145),(117,146),(118,147),(119,148),(120,149),(121,150),(122,151),(123,152),(124,153),(125,154),(126,155),(127,156),(128,157),(129,158),(130,159),(131,160),(132,161),(133,162),(134,163),(135,164),(136,165),(171,188),(172,189),(173,190),(174,191),(175,192),(176,193),(177,194),(178,195),(179,196),(180,197),(181,198),(182,199),(183,200),(184,201),(185,202),(186,203),(187,204)], [(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)], [(1,175,18,192),(2,174,19,191),(3,173,20,190),(4,172,21,189),(5,171,22,188),(6,204,23,187),(7,203,24,186),(8,202,25,185),(9,201,26,184),(10,200,27,183),(11,199,28,182),(12,198,29,181),(13,197,30,180),(14,196,31,179),(15,195,32,178),(16,194,33,177),(17,193,34,176),(35,106,52,123),(36,105,53,122),(37,104,54,121),(38,103,55,120),(39,136,56,119),(40,135,57,118),(41,134,58,117),(42,133,59,116),(43,132,60,115),(44,131,61,114),(45,130,62,113),(46,129,63,112),(47,128,64,111),(48,127,65,110),(49,126,66,109),(50,125,67,108),(51,124,68,107),(69,138,86,155),(70,137,87,154),(71,170,88,153),(72,169,89,152),(73,168,90,151),(74,167,91,150),(75,166,92,149),(76,165,93,148),(77,164,94,147),(78,163,95,146),(79,162,96,145),(80,161,97,144),(81,160,98,143),(82,159,99,142),(83,158,100,141),(84,157,101,140),(85,156,102,139)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 12A | 12B | 17A | ··· | 17H | 34A | ··· | 34H | 34I | ··· | 34X | 51A | ··· | 51H | 102A | ··· | 102H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 3 | 3 | 2 | 17 | 17 | 51 | 51 | 2 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4×S3 | D17 | Dic17 | D34 | S3×D17 | S3×Dic17 |
| kernel | S3×Dic17 | C3×Dic17 | Dic51 | S3×C34 | S3×C17 | Dic17 | C34 | C17 | D6 | S3 | C6 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 8 | 16 | 8 | 8 | 8 |
Matrix representation of S3×Dic17 ►in GL4(𝔽409) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 159 |
| 0 | 0 | 54 | 407 |
| 408 | 0 | 0 | 0 |
| 0 | 408 | 0 | 0 |
| 0 | 0 | 408 | 250 |
| 0 | 0 | 0 | 1 |
| 314 | 408 | 0 | 0 |
| 106 | 324 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 349 | 9 | 0 | 0 |
| 327 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(409))| [1,0,0,0,0,1,0,0,0,0,1,54,0,0,159,407],[408,0,0,0,0,408,0,0,0,0,408,0,0,0,250,1],[314,106,0,0,408,324,0,0,0,0,1,0,0,0,0,1],[349,327,0,0,9,60,0,0,0,0,1,0,0,0,0,1] >;
S3×Dic17 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_{17} % in TeX
G:=Group("S3xDic17"); // GroupNames label
G:=SmallGroup(408,8);
// by ID
G=gap.SmallGroup(408,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,20,168,9604]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^34=1,d^2=c^17,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export