direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×Dic17, C51⋊4C4, C34.C6, C17⋊2C12, C6.2D17, C102.2C2, C2.(C3×D17), SmallGroup(204,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C3×Dic17 |
Generators and relations for C3×Dic17
G = < a,b,c | a3=b34=1, c2=b17, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic17
►Regular action on 204 points
Generators in S204
(1 77 43)(2 78 44)(3 79 45)(4 80 46)(5 81 47)(6 82 48)(7 83 49)(8 84 50)(9 85 51)(10 86 52)(11 87 53)(12 88 54)(13 89 55)(14 90 56)(15 91 57)(16 92 58)(17 93 59)(18 94 60)(19 95 61)(20 96 62)(21 97 63)(22 98 64)(23 99 65)(24 100 66)(25 101 67)(26 102 68)(27 69 35)(28 70 36)(29 71 37)(30 72 38)(31 73 39)(32 74 40)(33 75 41)(34 76 42)(103 188 154)(104 189 155)(105 190 156)(106 191 157)(107 192 158)(108 193 159)(109 194 160)(110 195 161)(111 196 162)(112 197 163)(113 198 164)(114 199 165)(115 200 166)(116 201 167)(117 202 168)(118 203 169)(119 204 170)(120 171 137)(121 172 138)(122 173 139)(123 174 140)(124 175 141)(125 176 142)(126 177 143)(127 178 144)(128 179 145)(129 180 146)(130 181 147)(131 182 148)(132 183 149)(133 184 150)(134 185 151)(135 186 152)(136 187 153)
(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)
(1 120 18 103)(2 119 19 136)(3 118 20 135)(4 117 21 134)(5 116 22 133)(6 115 23 132)(7 114 24 131)(8 113 25 130)(9 112 26 129)(10 111 27 128)(11 110 28 127)(12 109 29 126)(13 108 30 125)(14 107 31 124)(15 106 32 123)(16 105 33 122)(17 104 34 121)(35 145 52 162)(36 144 53 161)(37 143 54 160)(38 142 55 159)(39 141 56 158)(40 140 57 157)(41 139 58 156)(42 138 59 155)(43 137 60 154)(44 170 61 153)(45 169 62 152)(46 168 63 151)(47 167 64 150)(48 166 65 149)(49 165 66 148)(50 164 67 147)(51 163 68 146)(69 179 86 196)(70 178 87 195)(71 177 88 194)(72 176 89 193)(73 175 90 192)(74 174 91 191)(75 173 92 190)(76 172 93 189)(77 171 94 188)(78 204 95 187)(79 203 96 186)(80 202 97 185)(81 201 98 184)(82 200 99 183)(83 199 100 182)(84 198 101 181)(85 197 102 180)
G:=sub<Sym(204)| (1,77,43)(2,78,44)(3,79,45)(4,80,46)(5,81,47)(6,82,48)(7,83,49)(8,84,50)(9,85,51)(10,86,52)(11,87,53)(12,88,54)(13,89,55)(14,90,56)(15,91,57)(16,92,58)(17,93,59)(18,94,60)(19,95,61)(20,96,62)(21,97,63)(22,98,64)(23,99,65)(24,100,66)(25,101,67)(26,102,68)(27,69,35)(28,70,36)(29,71,37)(30,72,38)(31,73,39)(32,74,40)(33,75,41)(34,76,42)(103,188,154)(104,189,155)(105,190,156)(106,191,157)(107,192,158)(108,193,159)(109,194,160)(110,195,161)(111,196,162)(112,197,163)(113,198,164)(114,199,165)(115,200,166)(116,201,167)(117,202,168)(118,203,169)(119,204,170)(120,171,137)(121,172,138)(122,173,139)(123,174,140)(124,175,141)(125,176,142)(126,177,143)(127,178,144)(128,179,145)(129,180,146)(130,181,147)(131,182,148)(132,183,149)(133,184,150)(134,185,151)(135,186,152)(136,187,153), (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), (1,120,18,103)(2,119,19,136)(3,118,20,135)(4,117,21,134)(5,116,22,133)(6,115,23,132)(7,114,24,131)(8,113,25,130)(9,112,26,129)(10,111,27,128)(11,110,28,127)(12,109,29,126)(13,108,30,125)(14,107,31,124)(15,106,32,123)(16,105,33,122)(17,104,34,121)(35,145,52,162)(36,144,53,161)(37,143,54,160)(38,142,55,159)(39,141,56,158)(40,140,57,157)(41,139,58,156)(42,138,59,155)(43,137,60,154)(44,170,61,153)(45,169,62,152)(46,168,63,151)(47,167,64,150)(48,166,65,149)(49,165,66,148)(50,164,67,147)(51,163,68,146)(69,179,86,196)(70,178,87,195)(71,177,88,194)(72,176,89,193)(73,175,90,192)(74,174,91,191)(75,173,92,190)(76,172,93,189)(77,171,94,188)(78,204,95,187)(79,203,96,186)(80,202,97,185)(81,201,98,184)(82,200,99,183)(83,199,100,182)(84,198,101,181)(85,197,102,180)>;
G:=Group( (1,77,43)(2,78,44)(3,79,45)(4,80,46)(5,81,47)(6,82,48)(7,83,49)(8,84,50)(9,85,51)(10,86,52)(11,87,53)(12,88,54)(13,89,55)(14,90,56)(15,91,57)(16,92,58)(17,93,59)(18,94,60)(19,95,61)(20,96,62)(21,97,63)(22,98,64)(23,99,65)(24,100,66)(25,101,67)(26,102,68)(27,69,35)(28,70,36)(29,71,37)(30,72,38)(31,73,39)(32,74,40)(33,75,41)(34,76,42)(103,188,154)(104,189,155)(105,190,156)(106,191,157)(107,192,158)(108,193,159)(109,194,160)(110,195,161)(111,196,162)(112,197,163)(113,198,164)(114,199,165)(115,200,166)(116,201,167)(117,202,168)(118,203,169)(119,204,170)(120,171,137)(121,172,138)(122,173,139)(123,174,140)(124,175,141)(125,176,142)(126,177,143)(127,178,144)(128,179,145)(129,180,146)(130,181,147)(131,182,148)(132,183,149)(133,184,150)(134,185,151)(135,186,152)(136,187,153), (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), (1,120,18,103)(2,119,19,136)(3,118,20,135)(4,117,21,134)(5,116,22,133)(6,115,23,132)(7,114,24,131)(8,113,25,130)(9,112,26,129)(10,111,27,128)(11,110,28,127)(12,109,29,126)(13,108,30,125)(14,107,31,124)(15,106,32,123)(16,105,33,122)(17,104,34,121)(35,145,52,162)(36,144,53,161)(37,143,54,160)(38,142,55,159)(39,141,56,158)(40,140,57,157)(41,139,58,156)(42,138,59,155)(43,137,60,154)(44,170,61,153)(45,169,62,152)(46,168,63,151)(47,167,64,150)(48,166,65,149)(49,165,66,148)(50,164,67,147)(51,163,68,146)(69,179,86,196)(70,178,87,195)(71,177,88,194)(72,176,89,193)(73,175,90,192)(74,174,91,191)(75,173,92,190)(76,172,93,189)(77,171,94,188)(78,204,95,187)(79,203,96,186)(80,202,97,185)(81,201,98,184)(82,200,99,183)(83,199,100,182)(84,198,101,181)(85,197,102,180) );
G=PermutationGroup([[(1,77,43),(2,78,44),(3,79,45),(4,80,46),(5,81,47),(6,82,48),(7,83,49),(8,84,50),(9,85,51),(10,86,52),(11,87,53),(12,88,54),(13,89,55),(14,90,56),(15,91,57),(16,92,58),(17,93,59),(18,94,60),(19,95,61),(20,96,62),(21,97,63),(22,98,64),(23,99,65),(24,100,66),(25,101,67),(26,102,68),(27,69,35),(28,70,36),(29,71,37),(30,72,38),(31,73,39),(32,74,40),(33,75,41),(34,76,42),(103,188,154),(104,189,155),(105,190,156),(106,191,157),(107,192,158),(108,193,159),(109,194,160),(110,195,161),(111,196,162),(112,197,163),(113,198,164),(114,199,165),(115,200,166),(116,201,167),(117,202,168),(118,203,169),(119,204,170),(120,171,137),(121,172,138),(122,173,139),(123,174,140),(124,175,141),(125,176,142),(126,177,143),(127,178,144),(128,179,145),(129,180,146),(130,181,147),(131,182,148),(132,183,149),(133,184,150),(134,185,151),(135,186,152),(136,187,153)], [(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)], [(1,120,18,103),(2,119,19,136),(3,118,20,135),(4,117,21,134),(5,116,22,133),(6,115,23,132),(7,114,24,131),(8,113,25,130),(9,112,26,129),(10,111,27,128),(11,110,28,127),(12,109,29,126),(13,108,30,125),(14,107,31,124),(15,106,32,123),(16,105,33,122),(17,104,34,121),(35,145,52,162),(36,144,53,161),(37,143,54,160),(38,142,55,159),(39,141,56,158),(40,140,57,157),(41,139,58,156),(42,138,59,155),(43,137,60,154),(44,170,61,153),(45,169,62,152),(46,168,63,151),(47,167,64,150),(48,166,65,149),(49,165,66,148),(50,164,67,147),(51,163,68,146),(69,179,86,196),(70,178,87,195),(71,177,88,194),(72,176,89,193),(73,175,90,192),(74,174,91,191),(75,173,92,190),(76,172,93,189),(77,171,94,188),(78,204,95,187),(79,203,96,186),(80,202,97,185),(81,201,98,184),(82,200,99,183),(83,199,100,182),(84,198,101,181),(85,197,102,180)]])
C3×Dic17 is a maximal subgroup of
C51⋊3C8 D51⋊2C4 C17⋊D12 C51⋊Q8 C12×D17
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51P | 102A | ··· | 102P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 1 | 1 | 17 | 17 | 1 | 1 | 17 | 17 | 17 | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D17 | Dic17 | C3×D17 | C3×Dic17 |
| kernel | C3×Dic17 | C102 | Dic17 | C51 | C34 | C17 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 16 | 16 |
Matrix representation of C3×Dic17 ►in GL3(𝔽409) generated by
| 53 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 404 | 0 |
| 0 | 0 | 327 |
| 408 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 408 | 0 |
G:=sub<GL(3,GF(409))| [53,0,0,0,1,0,0,0,1],[1,0,0,0,404,0,0,0,327],[408,0,0,0,0,408,0,1,0] >;
C3×Dic17 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{17} % in TeX
G:=Group("C3xDic17"); // GroupNames label
G:=SmallGroup(204,2);
// by ID
G=gap.SmallGroup(204,2);
# by ID
G:=PCGroup([4,-2,-3,-2,-17,24,3075]);
// Polycyclic
G:=Group<a,b,c|a^3=b^34=1,c^2=b^17,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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