direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3×D17, C6.1D34, C34.1D6, D34.2S3, Dic51⋊2C2, C102.1C22, C51⋊4(C2×C4), C3⋊3(C4×D17), (C3×D17)⋊1C4, C2.1(S3×D17), C17⋊2(C2×Dic3), (C6×D17).1C2, (Dic3×C17)⋊1C2, SmallGroup(408,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — Dic3×D17 |
Generators and relations for Dic3×D17
G = < a,b,c,d | a6=c17=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of Dic3×D17
►On 204 points
Generators in S204
(1 95 21 53 48 84)(2 96 22 54 49 85)(3 97 23 55 50 69)(4 98 24 56 51 70)(5 99 25 57 35 71)(6 100 26 58 36 72)(7 101 27 59 37 73)(8 102 28 60 38 74)(9 86 29 61 39 75)(10 87 30 62 40 76)(11 88 31 63 41 77)(12 89 32 64 42 78)(13 90 33 65 43 79)(14 91 34 66 44 80)(15 92 18 67 45 81)(16 93 19 68 46 82)(17 94 20 52 47 83)(103 180 153 169 135 197)(104 181 137 170 136 198)(105 182 138 154 120 199)(106 183 139 155 121 200)(107 184 140 156 122 201)(108 185 141 157 123 202)(109 186 142 158 124 203)(110 187 143 159 125 204)(111 171 144 160 126 188)(112 172 145 161 127 189)(113 173 146 162 128 190)(114 174 147 163 129 191)(115 175 148 164 130 192)(116 176 149 165 131 193)(117 177 150 166 132 194)(118 178 151 167 133 195)(119 179 152 168 134 196)
(1 160 53 111)(2 161 54 112)(3 162 55 113)(4 163 56 114)(5 164 57 115)(6 165 58 116)(7 166 59 117)(8 167 60 118)(9 168 61 119)(10 169 62 103)(11 170 63 104)(12 154 64 105)(13 155 65 106)(14 156 66 107)(15 157 67 108)(16 158 68 109)(17 159 52 110)(18 185 81 123)(19 186 82 124)(20 187 83 125)(21 171 84 126)(22 172 85 127)(23 173 69 128)(24 174 70 129)(25 175 71 130)(26 176 72 131)(27 177 73 132)(28 178 74 133)(29 179 75 134)(30 180 76 135)(31 181 77 136)(32 182 78 120)(33 183 79 121)(34 184 80 122)(35 192 99 148)(36 193 100 149)(37 194 101 150)(38 195 102 151)(39 196 86 152)(40 197 87 153)(41 198 88 137)(42 199 89 138)(43 200 90 139)(44 201 91 140)(45 202 92 141)(46 203 93 142)(47 204 94 143)(48 188 95 144)(49 189 96 145)(50 190 97 146)(51 191 98 147)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 23)(19 22)(20 21)(24 34)(25 33)(26 32)(27 31)(28 30)(35 43)(36 42)(37 41)(38 40)(44 51)(45 50)(46 49)(47 48)(52 53)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(82 85)(83 84)(87 102)(88 101)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(120 131)(121 130)(122 129)(123 128)(124 127)(125 126)(132 136)(133 135)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(151 153)(154 165)(155 164)(156 163)(157 162)(158 161)(159 160)(166 170)(167 169)(171 187)(172 186)(173 185)(174 184)(175 183)(176 182)(177 181)(178 180)(188 204)(189 203)(190 202)(191 201)(192 200)(193 199)(194 198)(195 197)
G:=sub<Sym(204)| (1,95,21,53,48,84)(2,96,22,54,49,85)(3,97,23,55,50,69)(4,98,24,56,51,70)(5,99,25,57,35,71)(6,100,26,58,36,72)(7,101,27,59,37,73)(8,102,28,60,38,74)(9,86,29,61,39,75)(10,87,30,62,40,76)(11,88,31,63,41,77)(12,89,32,64,42,78)(13,90,33,65,43,79)(14,91,34,66,44,80)(15,92,18,67,45,81)(16,93,19,68,46,82)(17,94,20,52,47,83)(103,180,153,169,135,197)(104,181,137,170,136,198)(105,182,138,154,120,199)(106,183,139,155,121,200)(107,184,140,156,122,201)(108,185,141,157,123,202)(109,186,142,158,124,203)(110,187,143,159,125,204)(111,171,144,160,126,188)(112,172,145,161,127,189)(113,173,146,162,128,190)(114,174,147,163,129,191)(115,175,148,164,130,192)(116,176,149,165,131,193)(117,177,150,166,132,194)(118,178,151,167,133,195)(119,179,152,168,134,196), (1,160,53,111)(2,161,54,112)(3,162,55,113)(4,163,56,114)(5,164,57,115)(6,165,58,116)(7,166,59,117)(8,167,60,118)(9,168,61,119)(10,169,62,103)(11,170,63,104)(12,154,64,105)(13,155,65,106)(14,156,66,107)(15,157,67,108)(16,158,68,109)(17,159,52,110)(18,185,81,123)(19,186,82,124)(20,187,83,125)(21,171,84,126)(22,172,85,127)(23,173,69,128)(24,174,70,129)(25,175,71,130)(26,176,72,131)(27,177,73,132)(28,178,74,133)(29,179,75,134)(30,180,76,135)(31,181,77,136)(32,182,78,120)(33,183,79,121)(34,184,80,122)(35,192,99,148)(36,193,100,149)(37,194,101,150)(38,195,102,151)(39,196,86,152)(40,197,87,153)(41,198,88,137)(42,199,89,138)(43,200,90,139)(44,201,91,140)(45,202,92,141)(46,203,93,142)(47,204,94,143)(48,188,95,144)(49,189,96,145)(50,190,97,146)(51,191,98,147), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,23)(19,22)(20,21)(24,34)(25,33)(26,32)(27,31)(28,30)(35,43)(36,42)(37,41)(38,40)(44,51)(45,50)(46,49)(47,48)(52,53)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(82,85)(83,84)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)(120,131)(121,130)(122,129)(123,128)(124,127)(125,126)(132,136)(133,135)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(151,153)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(166,170)(167,169)(171,187)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,180)(188,204)(189,203)(190,202)(191,201)(192,200)(193,199)(194,198)(195,197)>;
G:=Group( (1,95,21,53,48,84)(2,96,22,54,49,85)(3,97,23,55,50,69)(4,98,24,56,51,70)(5,99,25,57,35,71)(6,100,26,58,36,72)(7,101,27,59,37,73)(8,102,28,60,38,74)(9,86,29,61,39,75)(10,87,30,62,40,76)(11,88,31,63,41,77)(12,89,32,64,42,78)(13,90,33,65,43,79)(14,91,34,66,44,80)(15,92,18,67,45,81)(16,93,19,68,46,82)(17,94,20,52,47,83)(103,180,153,169,135,197)(104,181,137,170,136,198)(105,182,138,154,120,199)(106,183,139,155,121,200)(107,184,140,156,122,201)(108,185,141,157,123,202)(109,186,142,158,124,203)(110,187,143,159,125,204)(111,171,144,160,126,188)(112,172,145,161,127,189)(113,173,146,162,128,190)(114,174,147,163,129,191)(115,175,148,164,130,192)(116,176,149,165,131,193)(117,177,150,166,132,194)(118,178,151,167,133,195)(119,179,152,168,134,196), (1,160,53,111)(2,161,54,112)(3,162,55,113)(4,163,56,114)(5,164,57,115)(6,165,58,116)(7,166,59,117)(8,167,60,118)(9,168,61,119)(10,169,62,103)(11,170,63,104)(12,154,64,105)(13,155,65,106)(14,156,66,107)(15,157,67,108)(16,158,68,109)(17,159,52,110)(18,185,81,123)(19,186,82,124)(20,187,83,125)(21,171,84,126)(22,172,85,127)(23,173,69,128)(24,174,70,129)(25,175,71,130)(26,176,72,131)(27,177,73,132)(28,178,74,133)(29,179,75,134)(30,180,76,135)(31,181,77,136)(32,182,78,120)(33,183,79,121)(34,184,80,122)(35,192,99,148)(36,193,100,149)(37,194,101,150)(38,195,102,151)(39,196,86,152)(40,197,87,153)(41,198,88,137)(42,199,89,138)(43,200,90,139)(44,201,91,140)(45,202,92,141)(46,203,93,142)(47,204,94,143)(48,188,95,144)(49,189,96,145)(50,190,97,146)(51,191,98,147), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,23)(19,22)(20,21)(24,34)(25,33)(26,32)(27,31)(28,30)(35,43)(36,42)(37,41)(38,40)(44,51)(45,50)(46,49)(47,48)(52,53)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(82,85)(83,84)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)(120,131)(121,130)(122,129)(123,128)(124,127)(125,126)(132,136)(133,135)(137,150)(138,149)(139,148)(140,147)(141,146)(142,145)(143,144)(151,153)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(166,170)(167,169)(171,187)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,180)(188,204)(189,203)(190,202)(191,201)(192,200)(193,199)(194,198)(195,197) );
G=PermutationGroup([[(1,95,21,53,48,84),(2,96,22,54,49,85),(3,97,23,55,50,69),(4,98,24,56,51,70),(5,99,25,57,35,71),(6,100,26,58,36,72),(7,101,27,59,37,73),(8,102,28,60,38,74),(9,86,29,61,39,75),(10,87,30,62,40,76),(11,88,31,63,41,77),(12,89,32,64,42,78),(13,90,33,65,43,79),(14,91,34,66,44,80),(15,92,18,67,45,81),(16,93,19,68,46,82),(17,94,20,52,47,83),(103,180,153,169,135,197),(104,181,137,170,136,198),(105,182,138,154,120,199),(106,183,139,155,121,200),(107,184,140,156,122,201),(108,185,141,157,123,202),(109,186,142,158,124,203),(110,187,143,159,125,204),(111,171,144,160,126,188),(112,172,145,161,127,189),(113,173,146,162,128,190),(114,174,147,163,129,191),(115,175,148,164,130,192),(116,176,149,165,131,193),(117,177,150,166,132,194),(118,178,151,167,133,195),(119,179,152,168,134,196)], [(1,160,53,111),(2,161,54,112),(3,162,55,113),(4,163,56,114),(5,164,57,115),(6,165,58,116),(7,166,59,117),(8,167,60,118),(9,168,61,119),(10,169,62,103),(11,170,63,104),(12,154,64,105),(13,155,65,106),(14,156,66,107),(15,157,67,108),(16,158,68,109),(17,159,52,110),(18,185,81,123),(19,186,82,124),(20,187,83,125),(21,171,84,126),(22,172,85,127),(23,173,69,128),(24,174,70,129),(25,175,71,130),(26,176,72,131),(27,177,73,132),(28,178,74,133),(29,179,75,134),(30,180,76,135),(31,181,77,136),(32,182,78,120),(33,183,79,121),(34,184,80,122),(35,192,99,148),(36,193,100,149),(37,194,101,150),(38,195,102,151),(39,196,86,152),(40,197,87,153),(41,198,88,137),(42,199,89,138),(43,200,90,139),(44,201,91,140),(45,202,92,141),(46,203,93,142),(47,204,94,143),(48,188,95,144),(49,189,96,145),(50,190,97,146),(51,191,98,147)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,23),(19,22),(20,21),(24,34),(25,33),(26,32),(27,31),(28,30),(35,43),(36,42),(37,41),(38,40),(44,51),(45,50),(46,49),(47,48),(52,53),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76),(82,85),(83,84),(87,102),(88,101),(89,100),(90,99),(91,98),(92,97),(93,96),(94,95),(103,118),(104,117),(105,116),(106,115),(107,114),(108,113),(109,112),(110,111),(120,131),(121,130),(122,129),(123,128),(124,127),(125,126),(132,136),(133,135),(137,150),(138,149),(139,148),(140,147),(141,146),(142,145),(143,144),(151,153),(154,165),(155,164),(156,163),(157,162),(158,161),(159,160),(166,170),(167,169),(171,187),(172,186),(173,185),(174,184),(175,183),(176,182),(177,181),(178,180),(188,204),(189,203),(190,202),(191,201),(192,200),(193,199),(194,198),(195,197)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51H | 68A | ··· | 68P | 102A | ··· | 102H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 |
| size | 1 | 1 | 17 | 17 | 2 | 3 | 3 | 51 | 51 | 2 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 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 | Dic3 | D6 | D17 | D34 | C4×D17 | S3×D17 | Dic3×D17 |
| kernel | Dic3×D17 | Dic3×C17 | Dic51 | C6×D17 | C3×D17 | D34 | D17 | C34 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 8 | 8 | 16 | 8 | 8 |
Matrix representation of Dic3×D17 ►in GL4(𝔽409) generated by
| 408 | 0 | 0 | 0 |
| 0 | 408 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 408 | 408 |
| 266 | 0 | 0 | 0 |
| 0 | 266 | 0 | 0 |
| 0 | 0 | 151 | 179 |
| 0 | 0 | 28 | 258 |
| 104 | 1 | 0 | 0 |
| 368 | 94 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 94 | 408 | 0 | 0 |
| 246 | 315 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(409))| [408,0,0,0,0,408,0,0,0,0,0,408,0,0,1,408],[266,0,0,0,0,266,0,0,0,0,151,28,0,0,179,258],[104,368,0,0,1,94,0,0,0,0,1,0,0,0,0,1],[94,246,0,0,408,315,0,0,0,0,1,0,0,0,0,1] >;
Dic3×D17 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times D_{17} % in TeX
G:=Group("Dic3xD17"); // GroupNames label
G:=SmallGroup(408,7);
// by ID
G=gap.SmallGroup(408,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,26,168,9604]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^17=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export