direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C16×D13, C208⋊3C2, D26.4C8, C8.19D26, Dic13.4C8, C104.19C22, C13⋊3(C2×C16), C2.1(C8×D13), C13⋊2C16⋊6C2, C13⋊2C8.7C4, C26.10(C2×C8), C52.42(C2×C4), (C4×D13).9C4, C4.16(C4×D13), (C8×D13).11C2, SmallGroup(416,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C16×D13 |
Generators and relations for C16×D13
G = < a,b,c | a16=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C16×D13
►On 208 points
Generators in S208
(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)
(1 91 150 110 41 21 141 207 63 189 75 120 169)(2 92 151 111 42 22 142 208 64 190 76 121 170)(3 93 152 112 43 23 143 193 49 191 77 122 171)(4 94 153 97 44 24 144 194 50 192 78 123 172)(5 95 154 98 45 25 129 195 51 177 79 124 173)(6 96 155 99 46 26 130 196 52 178 80 125 174)(7 81 156 100 47 27 131 197 53 179 65 126 175)(8 82 157 101 48 28 132 198 54 180 66 127 176)(9 83 158 102 33 29 133 199 55 181 67 128 161)(10 84 159 103 34 30 134 200 56 182 68 113 162)(11 85 160 104 35 31 135 201 57 183 69 114 163)(12 86 145 105 36 32 136 202 58 184 70 115 164)(13 87 146 106 37 17 137 203 59 185 71 116 165)(14 88 147 107 38 18 138 204 60 186 72 117 166)(15 89 148 108 39 19 139 205 61 187 73 118 167)(16 90 149 109 40 20 140 206 62 188 74 119 168)
(1 161)(2 162)(3 163)(4 164)(5 165)(6 166)(7 167)(8 168)(9 169)(10 170)(11 171)(12 172)(13 173)(14 174)(15 175)(16 176)(17 195)(18 196)(19 197)(20 198)(21 199)(22 200)(23 201)(24 202)(25 203)(26 204)(27 205)(28 206)(29 207)(30 208)(31 193)(32 194)(33 63)(34 64)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(65 148)(66 149)(67 150)(68 151)(69 152)(70 153)(71 154)(72 155)(73 156)(74 157)(75 158)(76 159)(77 160)(78 145)(79 146)(80 147)(81 118)(82 119)(83 120)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 128)(92 113)(93 114)(94 115)(95 116)(96 117)(97 184)(98 185)(99 186)(100 187)(101 188)(102 189)(103 190)(104 191)(105 192)(106 177)(107 178)(108 179)(109 180)(110 181)(111 182)(112 183)(129 137)(130 138)(131 139)(132 140)(133 141)(134 142)(135 143)(136 144)
G:=sub<Sym(208)| (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), (1,91,150,110,41,21,141,207,63,189,75,120,169)(2,92,151,111,42,22,142,208,64,190,76,121,170)(3,93,152,112,43,23,143,193,49,191,77,122,171)(4,94,153,97,44,24,144,194,50,192,78,123,172)(5,95,154,98,45,25,129,195,51,177,79,124,173)(6,96,155,99,46,26,130,196,52,178,80,125,174)(7,81,156,100,47,27,131,197,53,179,65,126,175)(8,82,157,101,48,28,132,198,54,180,66,127,176)(9,83,158,102,33,29,133,199,55,181,67,128,161)(10,84,159,103,34,30,134,200,56,182,68,113,162)(11,85,160,104,35,31,135,201,57,183,69,114,163)(12,86,145,105,36,32,136,202,58,184,70,115,164)(13,87,146,106,37,17,137,203,59,185,71,116,165)(14,88,147,107,38,18,138,204,60,186,72,117,166)(15,89,148,108,39,19,139,205,61,187,73,118,167)(16,90,149,109,40,20,140,206,62,188,74,119,168), (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,169)(10,170)(11,171)(12,172)(13,173)(14,174)(15,175)(16,176)(17,195)(18,196)(19,197)(20,198)(21,199)(22,200)(23,201)(24,202)(25,203)(26,204)(27,205)(28,206)(29,207)(30,208)(31,193)(32,194)(33,63)(34,64)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,159)(77,160)(78,145)(79,146)(80,147)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,113)(93,114)(94,115)(95,116)(96,117)(97,184)(98,185)(99,186)(100,187)(101,188)(102,189)(103,190)(104,191)(105,192)(106,177)(107,178)(108,179)(109,180)(110,181)(111,182)(112,183)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144)>;
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), (1,91,150,110,41,21,141,207,63,189,75,120,169)(2,92,151,111,42,22,142,208,64,190,76,121,170)(3,93,152,112,43,23,143,193,49,191,77,122,171)(4,94,153,97,44,24,144,194,50,192,78,123,172)(5,95,154,98,45,25,129,195,51,177,79,124,173)(6,96,155,99,46,26,130,196,52,178,80,125,174)(7,81,156,100,47,27,131,197,53,179,65,126,175)(8,82,157,101,48,28,132,198,54,180,66,127,176)(9,83,158,102,33,29,133,199,55,181,67,128,161)(10,84,159,103,34,30,134,200,56,182,68,113,162)(11,85,160,104,35,31,135,201,57,183,69,114,163)(12,86,145,105,36,32,136,202,58,184,70,115,164)(13,87,146,106,37,17,137,203,59,185,71,116,165)(14,88,147,107,38,18,138,204,60,186,72,117,166)(15,89,148,108,39,19,139,205,61,187,73,118,167)(16,90,149,109,40,20,140,206,62,188,74,119,168), (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,169)(10,170)(11,171)(12,172)(13,173)(14,174)(15,175)(16,176)(17,195)(18,196)(19,197)(20,198)(21,199)(22,200)(23,201)(24,202)(25,203)(26,204)(27,205)(28,206)(29,207)(30,208)(31,193)(32,194)(33,63)(34,64)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,159)(77,160)(78,145)(79,146)(80,147)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,113)(93,114)(94,115)(95,116)(96,117)(97,184)(98,185)(99,186)(100,187)(101,188)(102,189)(103,190)(104,191)(105,192)(106,177)(107,178)(108,179)(109,180)(110,181)(111,182)(112,183)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144) );
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)], [(1,91,150,110,41,21,141,207,63,189,75,120,169),(2,92,151,111,42,22,142,208,64,190,76,121,170),(3,93,152,112,43,23,143,193,49,191,77,122,171),(4,94,153,97,44,24,144,194,50,192,78,123,172),(5,95,154,98,45,25,129,195,51,177,79,124,173),(6,96,155,99,46,26,130,196,52,178,80,125,174),(7,81,156,100,47,27,131,197,53,179,65,126,175),(8,82,157,101,48,28,132,198,54,180,66,127,176),(9,83,158,102,33,29,133,199,55,181,67,128,161),(10,84,159,103,34,30,134,200,56,182,68,113,162),(11,85,160,104,35,31,135,201,57,183,69,114,163),(12,86,145,105,36,32,136,202,58,184,70,115,164),(13,87,146,106,37,17,137,203,59,185,71,116,165),(14,88,147,107,38,18,138,204,60,186,72,117,166),(15,89,148,108,39,19,139,205,61,187,73,118,167),(16,90,149,109,40,20,140,206,62,188,74,119,168)], [(1,161),(2,162),(3,163),(4,164),(5,165),(6,166),(7,167),(8,168),(9,169),(10,170),(11,171),(12,172),(13,173),(14,174),(15,175),(16,176),(17,195),(18,196),(19,197),(20,198),(21,199),(22,200),(23,201),(24,202),(25,203),(26,204),(27,205),(28,206),(29,207),(30,208),(31,193),(32,194),(33,63),(34,64),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(65,148),(66,149),(67,150),(68,151),(69,152),(70,153),(71,154),(72,155),(73,156),(74,157),(75,158),(76,159),(77,160),(78,145),(79,146),(80,147),(81,118),(82,119),(83,120),(84,121),(85,122),(86,123),(87,124),(88,125),(89,126),(90,127),(91,128),(92,113),(93,114),(94,115),(95,116),(96,117),(97,184),(98,185),(99,186),(100,187),(101,188),(102,189),(103,190),(104,191),(105,192),(106,177),(107,178),(108,179),(109,180),(110,181),(111,182),(112,183),(129,137),(130,138),(131,139),(132,140),(133,141),(134,142),(135,143),(136,144)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 13A | ··· | 13F | 16A | ··· | 16H | 16I | ··· | 16P | 26A | ··· | 26F | 52A | ··· | 52L | 104A | ··· | 104X | 208A | ··· | 208AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 16 | ··· | 16 | 16 | ··· | 16 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 | 208 | ··· | 208 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 2 | ··· | 2 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D13 | D26 | C4×D13 | C8×D13 | C16×D13 |
| kernel | C16×D13 | C13⋊2C16 | C208 | C8×D13 | C13⋊2C8 | C4×D13 | Dic13 | D26 | D13 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 6 | 6 | 12 | 24 | 48 |
Matrix representation of C16×D13 ►in GL3(𝔽1249) generated by
| 98 | 0 | 0 |
| 0 | 585 | 0 |
| 0 | 0 | 585 |
| 1 | 0 | 0 |
| 0 | 1025 | 1 |
| 0 | 1008 | 804 |
| 1 | 0 | 0 |
| 0 | 807 | 580 |
| 0 | 152 | 442 |
G:=sub<GL(3,GF(1249))| [98,0,0,0,585,0,0,0,585],[1,0,0,0,1025,1008,0,1,804],[1,0,0,0,807,152,0,580,442] >;
C16×D13 in GAP, Magma, Sage, TeX
C_{16}\times D_{13} % in TeX
G:=Group("C16xD13"); // GroupNames label
G:=SmallGroup(416,4);
// by ID
G=gap.SmallGroup(416,4);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,31,50,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^16=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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