direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C13×D16, C208⋊5C2, C16⋊1C26, D8⋊1C26, C26.15D8, C52.36D4, C104.24C22, (C13×D8)⋊5C2, C8.2(C2×C26), C4.1(D4×C13), C2.3(C13×D8), SmallGroup(416,61)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×D16
G = < a,b,c | a13=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C13×D16
►On 208 points
Generators in S208
(1 171 87 135 201 112 125 40 67 185 62 151 22)(2 172 88 136 202 97 126 41 68 186 63 152 23)(3 173 89 137 203 98 127 42 69 187 64 153 24)(4 174 90 138 204 99 128 43 70 188 49 154 25)(5 175 91 139 205 100 113 44 71 189 50 155 26)(6 176 92 140 206 101 114 45 72 190 51 156 27)(7 161 93 141 207 102 115 46 73 191 52 157 28)(8 162 94 142 208 103 116 47 74 192 53 158 29)(9 163 95 143 193 104 117 48 75 177 54 159 30)(10 164 96 144 194 105 118 33 76 178 55 160 31)(11 165 81 129 195 106 119 34 77 179 56 145 32)(12 166 82 130 196 107 120 35 78 180 57 146 17)(13 167 83 131 197 108 121 36 79 181 58 147 18)(14 168 84 132 198 109 122 37 80 182 59 148 19)(15 169 85 133 199 110 123 38 65 183 60 149 20)(16 170 86 134 200 111 124 39 66 184 61 150 21)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 27)(18 26)(19 25)(20 24)(21 23)(28 32)(29 31)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(49 59)(50 58)(51 57)(52 56)(53 55)(60 64)(61 63)(65 69)(66 68)(70 80)(71 79)(72 78)(73 77)(74 76)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 96)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)(113 121)(114 120)(115 119)(116 118)(122 128)(123 127)(124 126)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(142 144)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)(158 160)(161 165)(162 164)(166 176)(167 175)(168 174)(169 173)(170 172)(178 192)(179 191)(180 190)(181 189)(182 188)(183 187)(184 186)(194 208)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)
G:=sub<Sym(208)| (1,171,87,135,201,112,125,40,67,185,62,151,22)(2,172,88,136,202,97,126,41,68,186,63,152,23)(3,173,89,137,203,98,127,42,69,187,64,153,24)(4,174,90,138,204,99,128,43,70,188,49,154,25)(5,175,91,139,205,100,113,44,71,189,50,155,26)(6,176,92,140,206,101,114,45,72,190,51,156,27)(7,161,93,141,207,102,115,46,73,191,52,157,28)(8,162,94,142,208,103,116,47,74,192,53,158,29)(9,163,95,143,193,104,117,48,75,177,54,159,30)(10,164,96,144,194,105,118,33,76,178,55,160,31)(11,165,81,129,195,106,119,34,77,179,56,145,32)(12,166,82,130,196,107,120,35,78,180,57,146,17)(13,167,83,131,197,108,121,36,79,181,58,147,18)(14,168,84,132,198,109,122,37,80,182,59,148,19)(15,169,85,133,199,110,123,38,65,183,60,149,20)(16,170,86,134,200,111,124,39,66,184,61,150,21), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,27)(18,26)(19,25)(20,24)(21,23)(28,32)(29,31)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(49,59)(50,58)(51,57)(52,56)(53,55)(60,64)(61,63)(65,69)(66,68)(70,80)(71,79)(72,78)(73,77)(74,76)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,96)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(113,121)(114,120)(115,119)(116,118)(122,128)(123,127)(124,126)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136)(142,144)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152)(158,160)(161,165)(162,164)(166,176)(167,175)(168,174)(169,173)(170,172)(178,192)(179,191)(180,190)(181,189)(182,188)(183,187)(184,186)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)>;
G:=Group( (1,171,87,135,201,112,125,40,67,185,62,151,22)(2,172,88,136,202,97,126,41,68,186,63,152,23)(3,173,89,137,203,98,127,42,69,187,64,153,24)(4,174,90,138,204,99,128,43,70,188,49,154,25)(5,175,91,139,205,100,113,44,71,189,50,155,26)(6,176,92,140,206,101,114,45,72,190,51,156,27)(7,161,93,141,207,102,115,46,73,191,52,157,28)(8,162,94,142,208,103,116,47,74,192,53,158,29)(9,163,95,143,193,104,117,48,75,177,54,159,30)(10,164,96,144,194,105,118,33,76,178,55,160,31)(11,165,81,129,195,106,119,34,77,179,56,145,32)(12,166,82,130,196,107,120,35,78,180,57,146,17)(13,167,83,131,197,108,121,36,79,181,58,147,18)(14,168,84,132,198,109,122,37,80,182,59,148,19)(15,169,85,133,199,110,123,38,65,183,60,149,20)(16,170,86,134,200,111,124,39,66,184,61,150,21), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,27)(18,26)(19,25)(20,24)(21,23)(28,32)(29,31)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(49,59)(50,58)(51,57)(52,56)(53,55)(60,64)(61,63)(65,69)(66,68)(70,80)(71,79)(72,78)(73,77)(74,76)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,96)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(113,121)(114,120)(115,119)(116,118)(122,128)(123,127)(124,126)(129,141)(130,140)(131,139)(132,138)(133,137)(134,136)(142,144)(145,157)(146,156)(147,155)(148,154)(149,153)(150,152)(158,160)(161,165)(162,164)(166,176)(167,175)(168,174)(169,173)(170,172)(178,192)(179,191)(180,190)(181,189)(182,188)(183,187)(184,186)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202) );
G=PermutationGroup([[(1,171,87,135,201,112,125,40,67,185,62,151,22),(2,172,88,136,202,97,126,41,68,186,63,152,23),(3,173,89,137,203,98,127,42,69,187,64,153,24),(4,174,90,138,204,99,128,43,70,188,49,154,25),(5,175,91,139,205,100,113,44,71,189,50,155,26),(6,176,92,140,206,101,114,45,72,190,51,156,27),(7,161,93,141,207,102,115,46,73,191,52,157,28),(8,162,94,142,208,103,116,47,74,192,53,158,29),(9,163,95,143,193,104,117,48,75,177,54,159,30),(10,164,96,144,194,105,118,33,76,178,55,160,31),(11,165,81,129,195,106,119,34,77,179,56,145,32),(12,166,82,130,196,107,120,35,78,180,57,146,17),(13,167,83,131,197,108,121,36,79,181,58,147,18),(14,168,84,132,198,109,122,37,80,182,59,148,19),(15,169,85,133,199,110,123,38,65,183,60,149,20),(16,170,86,134,200,111,124,39,66,184,61,150,21)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,27),(18,26),(19,25),(20,24),(21,23),(28,32),(29,31),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(49,59),(50,58),(51,57),(52,56),(53,55),(60,64),(61,63),(65,69),(66,68),(70,80),(71,79),(72,78),(73,77),(74,76),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(94,96),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,105),(113,121),(114,120),(115,119),(116,118),(122,128),(123,127),(124,126),(129,141),(130,140),(131,139),(132,138),(133,137),(134,136),(142,144),(145,157),(146,156),(147,155),(148,154),(149,153),(150,152),(158,160),(161,165),(162,164),(166,176),(167,175),(168,174),(169,173),(170,172),(178,192),(179,191),(180,190),(181,189),(182,188),(183,187),(184,186),(194,208),(195,207),(196,206),(197,205),(198,204),(199,203),(200,202)]])
143 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 13A | ··· | 13L | 16A | 16B | 16C | 16D | 26A | ··· | 26L | 26M | ··· | 26AJ | 52A | ··· | 52L | 104A | ··· | 104X | 208A | ··· | 208AV |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 13 | ··· | 13 | 16 | 16 | 16 | 16 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 | 208 | ··· | 208 |
| size | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | D4 | D8 | D16 | D4×C13 | C13×D8 | C13×D16 |
| kernel | C13×D16 | C208 | C13×D8 | D16 | C16 | D8 | C52 | C26 | C13 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 12 | 12 | 24 | 1 | 2 | 4 | 12 | 24 | 48 |
Matrix representation of C13×D16 ►in GL4(𝔽1249) generated by
| 933 | 0 | 0 | 0 |
| 0 | 933 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1248 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 35 | 678 |
| 0 | 0 | 910 | 713 |
| 1 | 0 | 0 | 0 |
| 0 | 1248 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1248 |
G:=sub<GL(4,GF(1249))| [933,0,0,0,0,933,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1248,0,0,0,0,0,35,910,0,0,678,713],[1,0,0,0,0,1248,0,0,0,0,1,1,0,0,0,1248] >;
C13×D16 in GAP, Magma, Sage, TeX
C_{13}\times D_{16} % in TeX
G:=Group("C13xD16"); // GroupNames label
G:=SmallGroup(416,61);
// by ID
G=gap.SmallGroup(416,61);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,649,3747,1881,165,9364,4690,88]);
// Polycyclic
G:=Group<a,b,c|a^13=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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