metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C13⋊2D16, D8⋊1D13, C8.4D26, C26.8D8, C52.3D4, D104⋊3C2, C104.2C22, (C13×D8)⋊1C2, C13⋊2C16⋊1C2, C2.4(D4⋊D13), C4.1(C13⋊D4), SmallGroup(416,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13⋊D16
G = < a,b,c | a13=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C13⋊D16
►On 208 points
Generators in S208
(1 132 38 195 31 185 121 107 166 160 59 95 72)(2 73 96 60 145 167 108 122 186 32 196 39 133)(3 134 40 197 17 187 123 109 168 146 61 81 74)(4 75 82 62 147 169 110 124 188 18 198 41 135)(5 136 42 199 19 189 125 111 170 148 63 83 76)(6 77 84 64 149 171 112 126 190 20 200 43 137)(7 138 44 201 21 191 127 97 172 150 49 85 78)(8 79 86 50 151 173 98 128 192 22 202 45 139)(9 140 46 203 23 177 113 99 174 152 51 87 80)(10 65 88 52 153 175 100 114 178 24 204 47 141)(11 142 48 205 25 179 115 101 176 154 53 89 66)(12 67 90 54 155 161 102 116 180 26 206 33 143)(13 144 34 207 27 181 117 103 162 156 55 91 68)(14 69 92 56 157 163 104 118 182 28 208 35 129)(15 130 36 193 29 183 119 105 164 158 57 93 70)(16 71 94 58 159 165 106 120 184 30 194 37 131)
(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)(17 158)(18 157)(19 156)(20 155)(21 154)(22 153)(23 152)(24 151)(25 150)(26 149)(27 148)(28 147)(29 146)(30 145)(31 160)(32 159)(33 84)(34 83)(35 82)(36 81)(37 96)(38 95)(39 94)(40 93)(41 92)(42 91)(43 90)(44 89)(45 88)(46 87)(47 86)(48 85)(49 205)(50 204)(51 203)(52 202)(53 201)(54 200)(55 199)(56 198)(57 197)(58 196)(59 195)(60 194)(61 193)(62 208)(63 207)(64 206)(65 139)(66 138)(67 137)(68 136)(69 135)(70 134)(71 133)(72 132)(73 131)(74 130)(75 129)(76 144)(77 143)(78 142)(79 141)(80 140)(97 115)(98 114)(99 113)(100 128)(101 127)(102 126)(103 125)(104 124)(105 123)(106 122)(107 121)(108 120)(109 119)(110 118)(111 117)(112 116)(161 190)(162 189)(163 188)(164 187)(165 186)(166 185)(167 184)(168 183)(169 182)(170 181)(171 180)(172 179)(173 178)(174 177)(175 192)(176 191)
G:=sub<Sym(208)| (1,132,38,195,31,185,121,107,166,160,59,95,72)(2,73,96,60,145,167,108,122,186,32,196,39,133)(3,134,40,197,17,187,123,109,168,146,61,81,74)(4,75,82,62,147,169,110,124,188,18,198,41,135)(5,136,42,199,19,189,125,111,170,148,63,83,76)(6,77,84,64,149,171,112,126,190,20,200,43,137)(7,138,44,201,21,191,127,97,172,150,49,85,78)(8,79,86,50,151,173,98,128,192,22,202,45,139)(9,140,46,203,23,177,113,99,174,152,51,87,80)(10,65,88,52,153,175,100,114,178,24,204,47,141)(11,142,48,205,25,179,115,101,176,154,53,89,66)(12,67,90,54,155,161,102,116,180,26,206,33,143)(13,144,34,207,27,181,117,103,162,156,55,91,68)(14,69,92,56,157,163,104,118,182,28,208,35,129)(15,130,36,193,29,183,119,105,164,158,57,93,70)(16,71,94,58,159,165,106,120,184,30,194,37,131), (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)(17,158)(18,157)(19,156)(20,155)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,160)(32,159)(33,84)(34,83)(35,82)(36,81)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,205)(50,204)(51,203)(52,202)(53,201)(54,200)(55,199)(56,198)(57,197)(58,196)(59,195)(60,194)(61,193)(62,208)(63,207)(64,206)(65,139)(66,138)(67,137)(68,136)(69,135)(70,134)(71,133)(72,132)(73,131)(74,130)(75,129)(76,144)(77,143)(78,142)(79,141)(80,140)(97,115)(98,114)(99,113)(100,128)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)(107,121)(108,120)(109,119)(110,118)(111,117)(112,116)(161,190)(162,189)(163,188)(164,187)(165,186)(166,185)(167,184)(168,183)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,192)(176,191)>;
G:=Group( (1,132,38,195,31,185,121,107,166,160,59,95,72)(2,73,96,60,145,167,108,122,186,32,196,39,133)(3,134,40,197,17,187,123,109,168,146,61,81,74)(4,75,82,62,147,169,110,124,188,18,198,41,135)(5,136,42,199,19,189,125,111,170,148,63,83,76)(6,77,84,64,149,171,112,126,190,20,200,43,137)(7,138,44,201,21,191,127,97,172,150,49,85,78)(8,79,86,50,151,173,98,128,192,22,202,45,139)(9,140,46,203,23,177,113,99,174,152,51,87,80)(10,65,88,52,153,175,100,114,178,24,204,47,141)(11,142,48,205,25,179,115,101,176,154,53,89,66)(12,67,90,54,155,161,102,116,180,26,206,33,143)(13,144,34,207,27,181,117,103,162,156,55,91,68)(14,69,92,56,157,163,104,118,182,28,208,35,129)(15,130,36,193,29,183,119,105,164,158,57,93,70)(16,71,94,58,159,165,106,120,184,30,194,37,131), (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)(17,158)(18,157)(19,156)(20,155)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,160)(32,159)(33,84)(34,83)(35,82)(36,81)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,205)(50,204)(51,203)(52,202)(53,201)(54,200)(55,199)(56,198)(57,197)(58,196)(59,195)(60,194)(61,193)(62,208)(63,207)(64,206)(65,139)(66,138)(67,137)(68,136)(69,135)(70,134)(71,133)(72,132)(73,131)(74,130)(75,129)(76,144)(77,143)(78,142)(79,141)(80,140)(97,115)(98,114)(99,113)(100,128)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)(107,121)(108,120)(109,119)(110,118)(111,117)(112,116)(161,190)(162,189)(163,188)(164,187)(165,186)(166,185)(167,184)(168,183)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,192)(176,191) );
G=PermutationGroup([[(1,132,38,195,31,185,121,107,166,160,59,95,72),(2,73,96,60,145,167,108,122,186,32,196,39,133),(3,134,40,197,17,187,123,109,168,146,61,81,74),(4,75,82,62,147,169,110,124,188,18,198,41,135),(5,136,42,199,19,189,125,111,170,148,63,83,76),(6,77,84,64,149,171,112,126,190,20,200,43,137),(7,138,44,201,21,191,127,97,172,150,49,85,78),(8,79,86,50,151,173,98,128,192,22,202,45,139),(9,140,46,203,23,177,113,99,174,152,51,87,80),(10,65,88,52,153,175,100,114,178,24,204,47,141),(11,142,48,205,25,179,115,101,176,154,53,89,66),(12,67,90,54,155,161,102,116,180,26,206,33,143),(13,144,34,207,27,181,117,103,162,156,55,91,68),(14,69,92,56,157,163,104,118,182,28,208,35,129),(15,130,36,193,29,183,119,105,164,158,57,93,70),(16,71,94,58,159,165,106,120,184,30,194,37,131)], [(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),(17,158),(18,157),(19,156),(20,155),(21,154),(22,153),(23,152),(24,151),(25,150),(26,149),(27,148),(28,147),(29,146),(30,145),(31,160),(32,159),(33,84),(34,83),(35,82),(36,81),(37,96),(38,95),(39,94),(40,93),(41,92),(42,91),(43,90),(44,89),(45,88),(46,87),(47,86),(48,85),(49,205),(50,204),(51,203),(52,202),(53,201),(54,200),(55,199),(56,198),(57,197),(58,196),(59,195),(60,194),(61,193),(62,208),(63,207),(64,206),(65,139),(66,138),(67,137),(68,136),(69,135),(70,134),(71,133),(72,132),(73,131),(74,130),(75,129),(76,144),(77,143),(78,142),(79,141),(80,140),(97,115),(98,114),(99,113),(100,128),(101,127),(102,126),(103,125),(104,124),(105,123),(106,122),(107,121),(108,120),(109,119),(110,118),(111,117),(112,116),(161,190),(162,189),(163,188),(164,187),(165,186),(166,185),(167,184),(168,183),(169,182),(170,181),(171,180),(172,179),(173,178),(174,177),(175,192),(176,191)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 13A | ··· | 13F | 16A | 16B | 16C | 16D | 26A | ··· | 26F | 26G | ··· | 26R | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 13 | ··· | 13 | 16 | 16 | 16 | 16 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 8 | 104 | 2 | 2 | 2 | 2 | ··· | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D8 | D13 | D16 | D26 | C13⋊D4 | D4⋊D13 | C13⋊D16 |
| kernel | C13⋊D16 | C13⋊2C16 | D104 | C13×D8 | C52 | C26 | D8 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 4 | 6 | 12 | 6 | 12 |
Matrix representation of C13⋊D16 ►in GL4(𝔽1249) generated by
| 1064 | 1 | 0 | 0 |
| 895 | 461 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 419 | 241 | 0 | 0 |
| 23 | 830 | 0 | 0 |
| 0 | 0 | 536 | 571 |
| 0 | 0 | 339 | 1214 |
| 680 | 209 | 0 | 0 |
| 1176 | 569 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1248 | 1248 |
G:=sub<GL(4,GF(1249))| [1064,895,0,0,1,461,0,0,0,0,1,0,0,0,0,1],[419,23,0,0,241,830,0,0,0,0,536,339,0,0,571,1214],[680,1176,0,0,209,569,0,0,0,0,1,1248,0,0,0,1248] >;
C13⋊D16 in GAP, Magma, Sage, TeX
C_{13}\rtimes D_{16} % in TeX
G:=Group("C13:D16"); // GroupNames label
G:=SmallGroup(416,33);
// by ID
G=gap.SmallGroup(416,33);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,73,218,116,122,579,297,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^13=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export