direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic3×C17, C3⋊C68, C51⋊5C4, C6.C34, C34.2S3, C102.3C2, C2.(S3×C17), SmallGroup(204,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C17 |
Generators and relations for Dic3×C17
G = < a,b,c | a17=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C17
►Regular action on 204 points
Generators in S204
(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 99 35 191 113 81)(2 100 36 192 114 82)(3 101 37 193 115 83)(4 102 38 194 116 84)(5 86 39 195 117 85)(6 87 40 196 118 69)(7 88 41 197 119 70)(8 89 42 198 103 71)(9 90 43 199 104 72)(10 91 44 200 105 73)(11 92 45 201 106 74)(12 93 46 202 107 75)(13 94 47 203 108 76)(14 95 48 204 109 77)(15 96 49 188 110 78)(16 97 50 189 111 79)(17 98 51 190 112 80)(18 154 149 182 64 129)(19 155 150 183 65 130)(20 156 151 184 66 131)(21 157 152 185 67 132)(22 158 153 186 68 133)(23 159 137 187 52 134)(24 160 138 171 53 135)(25 161 139 172 54 136)(26 162 140 173 55 120)(27 163 141 174 56 121)(28 164 142 175 57 122)(29 165 143 176 58 123)(30 166 144 177 59 124)(31 167 145 178 60 125)(32 168 146 179 61 126)(33 169 147 180 62 127)(34 170 148 181 63 128)
(1 148 191 128)(2 149 192 129)(3 150 193 130)(4 151 194 131)(5 152 195 132)(6 153 196 133)(7 137 197 134)(8 138 198 135)(9 139 199 136)(10 140 200 120)(11 141 201 121)(12 142 202 122)(13 143 203 123)(14 144 204 124)(15 145 188 125)(16 146 189 126)(17 147 190 127)(18 82 182 36)(19 83 183 37)(20 84 184 38)(21 85 185 39)(22 69 186 40)(23 70 187 41)(24 71 171 42)(25 72 172 43)(26 73 173 44)(27 74 174 45)(28 75 175 46)(29 76 176 47)(30 77 177 48)(31 78 178 49)(32 79 179 50)(33 80 180 51)(34 81 181 35)(52 88 159 119)(53 89 160 103)(54 90 161 104)(55 91 162 105)(56 92 163 106)(57 93 164 107)(58 94 165 108)(59 95 166 109)(60 96 167 110)(61 97 168 111)(62 98 169 112)(63 99 170 113)(64 100 154 114)(65 101 155 115)(66 102 156 116)(67 86 157 117)(68 87 158 118)
G:=sub<Sym(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), (1,99,35,191,113,81)(2,100,36,192,114,82)(3,101,37,193,115,83)(4,102,38,194,116,84)(5,86,39,195,117,85)(6,87,40,196,118,69)(7,88,41,197,119,70)(8,89,42,198,103,71)(9,90,43,199,104,72)(10,91,44,200,105,73)(11,92,45,201,106,74)(12,93,46,202,107,75)(13,94,47,203,108,76)(14,95,48,204,109,77)(15,96,49,188,110,78)(16,97,50,189,111,79)(17,98,51,190,112,80)(18,154,149,182,64,129)(19,155,150,183,65,130)(20,156,151,184,66,131)(21,157,152,185,67,132)(22,158,153,186,68,133)(23,159,137,187,52,134)(24,160,138,171,53,135)(25,161,139,172,54,136)(26,162,140,173,55,120)(27,163,141,174,56,121)(28,164,142,175,57,122)(29,165,143,176,58,123)(30,166,144,177,59,124)(31,167,145,178,60,125)(32,168,146,179,61,126)(33,169,147,180,62,127)(34,170,148,181,63,128), (1,148,191,128)(2,149,192,129)(3,150,193,130)(4,151,194,131)(5,152,195,132)(6,153,196,133)(7,137,197,134)(8,138,198,135)(9,139,199,136)(10,140,200,120)(11,141,201,121)(12,142,202,122)(13,143,203,123)(14,144,204,124)(15,145,188,125)(16,146,189,126)(17,147,190,127)(18,82,182,36)(19,83,183,37)(20,84,184,38)(21,85,185,39)(22,69,186,40)(23,70,187,41)(24,71,171,42)(25,72,172,43)(26,73,173,44)(27,74,174,45)(28,75,175,46)(29,76,176,47)(30,77,177,48)(31,78,178,49)(32,79,179,50)(33,80,180,51)(34,81,181,35)(52,88,159,119)(53,89,160,103)(54,90,161,104)(55,91,162,105)(56,92,163,106)(57,93,164,107)(58,94,165,108)(59,95,166,109)(60,96,167,110)(61,97,168,111)(62,98,169,112)(63,99,170,113)(64,100,154,114)(65,101,155,115)(66,102,156,116)(67,86,157,117)(68,87,158,118)>;
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), (1,99,35,191,113,81)(2,100,36,192,114,82)(3,101,37,193,115,83)(4,102,38,194,116,84)(5,86,39,195,117,85)(6,87,40,196,118,69)(7,88,41,197,119,70)(8,89,42,198,103,71)(9,90,43,199,104,72)(10,91,44,200,105,73)(11,92,45,201,106,74)(12,93,46,202,107,75)(13,94,47,203,108,76)(14,95,48,204,109,77)(15,96,49,188,110,78)(16,97,50,189,111,79)(17,98,51,190,112,80)(18,154,149,182,64,129)(19,155,150,183,65,130)(20,156,151,184,66,131)(21,157,152,185,67,132)(22,158,153,186,68,133)(23,159,137,187,52,134)(24,160,138,171,53,135)(25,161,139,172,54,136)(26,162,140,173,55,120)(27,163,141,174,56,121)(28,164,142,175,57,122)(29,165,143,176,58,123)(30,166,144,177,59,124)(31,167,145,178,60,125)(32,168,146,179,61,126)(33,169,147,180,62,127)(34,170,148,181,63,128), (1,148,191,128)(2,149,192,129)(3,150,193,130)(4,151,194,131)(5,152,195,132)(6,153,196,133)(7,137,197,134)(8,138,198,135)(9,139,199,136)(10,140,200,120)(11,141,201,121)(12,142,202,122)(13,143,203,123)(14,144,204,124)(15,145,188,125)(16,146,189,126)(17,147,190,127)(18,82,182,36)(19,83,183,37)(20,84,184,38)(21,85,185,39)(22,69,186,40)(23,70,187,41)(24,71,171,42)(25,72,172,43)(26,73,173,44)(27,74,174,45)(28,75,175,46)(29,76,176,47)(30,77,177,48)(31,78,178,49)(32,79,179,50)(33,80,180,51)(34,81,181,35)(52,88,159,119)(53,89,160,103)(54,90,161,104)(55,91,162,105)(56,92,163,106)(57,93,164,107)(58,94,165,108)(59,95,166,109)(60,96,167,110)(61,97,168,111)(62,98,169,112)(63,99,170,113)(64,100,154,114)(65,101,155,115)(66,102,156,116)(67,86,157,117)(68,87,158,118) );
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)], [(1,99,35,191,113,81),(2,100,36,192,114,82),(3,101,37,193,115,83),(4,102,38,194,116,84),(5,86,39,195,117,85),(6,87,40,196,118,69),(7,88,41,197,119,70),(8,89,42,198,103,71),(9,90,43,199,104,72),(10,91,44,200,105,73),(11,92,45,201,106,74),(12,93,46,202,107,75),(13,94,47,203,108,76),(14,95,48,204,109,77),(15,96,49,188,110,78),(16,97,50,189,111,79),(17,98,51,190,112,80),(18,154,149,182,64,129),(19,155,150,183,65,130),(20,156,151,184,66,131),(21,157,152,185,67,132),(22,158,153,186,68,133),(23,159,137,187,52,134),(24,160,138,171,53,135),(25,161,139,172,54,136),(26,162,140,173,55,120),(27,163,141,174,56,121),(28,164,142,175,57,122),(29,165,143,176,58,123),(30,166,144,177,59,124),(31,167,145,178,60,125),(32,168,146,179,61,126),(33,169,147,180,62,127),(34,170,148,181,63,128)], [(1,148,191,128),(2,149,192,129),(3,150,193,130),(4,151,194,131),(5,152,195,132),(6,153,196,133),(7,137,197,134),(8,138,198,135),(9,139,199,136),(10,140,200,120),(11,141,201,121),(12,142,202,122),(13,143,203,123),(14,144,204,124),(15,145,188,125),(16,146,189,126),(17,147,190,127),(18,82,182,36),(19,83,183,37),(20,84,184,38),(21,85,185,39),(22,69,186,40),(23,70,187,41),(24,71,171,42),(25,72,172,43),(26,73,173,44),(27,74,174,45),(28,75,175,46),(29,76,176,47),(30,77,177,48),(31,78,178,49),(32,79,179,50),(33,80,180,51),(34,81,181,35),(52,88,159,119),(53,89,160,103),(54,90,161,104),(55,91,162,105),(56,92,163,106),(57,93,164,107),(58,94,165,108),(59,95,166,109),(60,96,167,110),(61,97,168,111),(62,98,169,112),(63,99,170,113),(64,100,154,114),(65,101,155,115),(66,102,156,116),(67,86,157,117),(68,87,158,118)]])
Dic3×C17 is a maximal subgroup of
D51⋊2C4 C3⋊D68 C51⋊Q8 S3×C68
102 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 17A | ··· | 17P | 34A | ··· | 34P | 51A | ··· | 51P | 68A | ··· | 68AF | 102A | ··· | 102P |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 |
| size | 1 | 1 | 2 | 3 | 3 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C17 | C34 | C68 | S3 | Dic3 | S3×C17 | Dic3×C17 |
| kernel | Dic3×C17 | C102 | C51 | Dic3 | C6 | C3 | C34 | C17 | C2 | C1 |
| # reps | 1 | 1 | 2 | 16 | 16 | 32 | 1 | 1 | 16 | 16 |
Matrix representation of Dic3×C17 ►in GL2(𝔽409) generated by
| 30 | 0 |
| 0 | 30 |
| 1 | 408 |
| 1 | 0 |
| 50 | 371 |
| 12 | 359 |
G:=sub<GL(2,GF(409))| [30,0,0,30],[1,1,408,0],[50,12,371,359] >;
Dic3×C17 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{17} % in TeX
G:=Group("Dic3xC17"); // GroupNames label
G:=SmallGroup(204,1);
// by ID
G=gap.SmallGroup(204,1);
# by ID
G:=PCGroup([4,-2,-17,-2,-3,136,2179]);
// Polycyclic
G:=Group<a,b,c|a^17=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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