metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D13⋊C16, C104.4C4, D26.2C8, Dic13.2C8, C13⋊C16⋊3C2, C13⋊1(C2×C16), C8.5(C13⋊C4), C26.1(C2×C8), C52.13(C2×C4), (C8×D13).7C2, (C4×D13).5C4, C2.1(D13⋊C8), C13⋊2C8.14C22, C4.14(C2×C13⋊C4), SmallGroup(416,64)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — D13⋊C16 |
Generators and relations for D13⋊C16
G = < a,b,c | a13=b2=c16=1, bab=a-1, cac-1=a5, cbc-1=a4b >
Smallest permutation representation of D13⋊C16
►On 208 points
Generators in S208
(1 18 104 117 69 200 91 182 144 61 164 39 147)(2 201 165 105 183 148 70 62 19 92 40 118 129)(3 149 41 166 63 130 184 93 202 71 119 106 20)(4 131 120 42 94 21 64 72 150 185 107 167 203)(5 22 108 121 73 204 95 186 132 49 168 43 151)(6 205 169 109 187 152 74 50 23 96 44 122 133)(7 153 45 170 51 134 188 81 206 75 123 110 24)(8 135 124 46 82 25 52 76 154 189 111 171 207)(9 26 112 125 77 208 83 190 136 53 172 47 155)(10 193 173 97 191 156 78 54 27 84 48 126 137)(11 157 33 174 55 138 192 85 194 79 127 98 28)(12 139 128 34 86 29 56 80 158 177 99 175 195)(13 30 100 113 65 196 87 178 140 57 176 35 159)(14 197 161 101 179 160 66 58 31 88 36 114 141)(15 145 37 162 59 142 180 89 198 67 115 102 32)(16 143 116 38 90 17 60 68 146 181 103 163 199)
(1 147)(2 129)(3 20)(4 203)(5 151)(6 133)(7 24)(8 207)(9 155)(10 137)(11 28)(12 195)(13 159)(14 141)(15 32)(16 199)(17 68)(18 39)(19 183)(21 72)(22 43)(23 187)(25 76)(26 47)(27 191)(29 80)(30 35)(31 179)(33 127)(34 177)(36 161)(37 115)(38 181)(40 165)(41 119)(42 185)(44 169)(45 123)(46 189)(48 173)(49 121)(50 152)(51 206)(53 125)(54 156)(55 194)(57 113)(58 160)(59 198)(61 117)(62 148)(63 202)(65 140)(67 162)(69 144)(71 166)(73 132)(75 170)(77 136)(79 174)(81 134)(82 154)(84 97)(85 138)(86 158)(88 101)(89 142)(90 146)(92 105)(93 130)(94 150)(96 109)(98 157)(99 128)(100 176)(102 145)(103 116)(104 164)(106 149)(107 120)(108 168)(110 153)(111 124)(112 172)(114 197)(118 201)(122 205)(126 193)(131 167)(135 171)(139 175)(143 163)(178 196)(182 200)(186 204)(190 208)
(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)
G:=sub<Sym(208)| (1,18,104,117,69,200,91,182,144,61,164,39,147)(2,201,165,105,183,148,70,62,19,92,40,118,129)(3,149,41,166,63,130,184,93,202,71,119,106,20)(4,131,120,42,94,21,64,72,150,185,107,167,203)(5,22,108,121,73,204,95,186,132,49,168,43,151)(6,205,169,109,187,152,74,50,23,96,44,122,133)(7,153,45,170,51,134,188,81,206,75,123,110,24)(8,135,124,46,82,25,52,76,154,189,111,171,207)(9,26,112,125,77,208,83,190,136,53,172,47,155)(10,193,173,97,191,156,78,54,27,84,48,126,137)(11,157,33,174,55,138,192,85,194,79,127,98,28)(12,139,128,34,86,29,56,80,158,177,99,175,195)(13,30,100,113,65,196,87,178,140,57,176,35,159)(14,197,161,101,179,160,66,58,31,88,36,114,141)(15,145,37,162,59,142,180,89,198,67,115,102,32)(16,143,116,38,90,17,60,68,146,181,103,163,199), (1,147)(2,129)(3,20)(4,203)(5,151)(6,133)(7,24)(8,207)(9,155)(10,137)(11,28)(12,195)(13,159)(14,141)(15,32)(16,199)(17,68)(18,39)(19,183)(21,72)(22,43)(23,187)(25,76)(26,47)(27,191)(29,80)(30,35)(31,179)(33,127)(34,177)(36,161)(37,115)(38,181)(40,165)(41,119)(42,185)(44,169)(45,123)(46,189)(48,173)(49,121)(50,152)(51,206)(53,125)(54,156)(55,194)(57,113)(58,160)(59,198)(61,117)(62,148)(63,202)(65,140)(67,162)(69,144)(71,166)(73,132)(75,170)(77,136)(79,174)(81,134)(82,154)(84,97)(85,138)(86,158)(88,101)(89,142)(90,146)(92,105)(93,130)(94,150)(96,109)(98,157)(99,128)(100,176)(102,145)(103,116)(104,164)(106,149)(107,120)(108,168)(110,153)(111,124)(112,172)(114,197)(118,201)(122,205)(126,193)(131,167)(135,171)(139,175)(143,163)(178,196)(182,200)(186,204)(190,208), (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)>;
G:=Group( (1,18,104,117,69,200,91,182,144,61,164,39,147)(2,201,165,105,183,148,70,62,19,92,40,118,129)(3,149,41,166,63,130,184,93,202,71,119,106,20)(4,131,120,42,94,21,64,72,150,185,107,167,203)(5,22,108,121,73,204,95,186,132,49,168,43,151)(6,205,169,109,187,152,74,50,23,96,44,122,133)(7,153,45,170,51,134,188,81,206,75,123,110,24)(8,135,124,46,82,25,52,76,154,189,111,171,207)(9,26,112,125,77,208,83,190,136,53,172,47,155)(10,193,173,97,191,156,78,54,27,84,48,126,137)(11,157,33,174,55,138,192,85,194,79,127,98,28)(12,139,128,34,86,29,56,80,158,177,99,175,195)(13,30,100,113,65,196,87,178,140,57,176,35,159)(14,197,161,101,179,160,66,58,31,88,36,114,141)(15,145,37,162,59,142,180,89,198,67,115,102,32)(16,143,116,38,90,17,60,68,146,181,103,163,199), (1,147)(2,129)(3,20)(4,203)(5,151)(6,133)(7,24)(8,207)(9,155)(10,137)(11,28)(12,195)(13,159)(14,141)(15,32)(16,199)(17,68)(18,39)(19,183)(21,72)(22,43)(23,187)(25,76)(26,47)(27,191)(29,80)(30,35)(31,179)(33,127)(34,177)(36,161)(37,115)(38,181)(40,165)(41,119)(42,185)(44,169)(45,123)(46,189)(48,173)(49,121)(50,152)(51,206)(53,125)(54,156)(55,194)(57,113)(58,160)(59,198)(61,117)(62,148)(63,202)(65,140)(67,162)(69,144)(71,166)(73,132)(75,170)(77,136)(79,174)(81,134)(82,154)(84,97)(85,138)(86,158)(88,101)(89,142)(90,146)(92,105)(93,130)(94,150)(96,109)(98,157)(99,128)(100,176)(102,145)(103,116)(104,164)(106,149)(107,120)(108,168)(110,153)(111,124)(112,172)(114,197)(118,201)(122,205)(126,193)(131,167)(135,171)(139,175)(143,163)(178,196)(182,200)(186,204)(190,208), (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) );
G=PermutationGroup([[(1,18,104,117,69,200,91,182,144,61,164,39,147),(2,201,165,105,183,148,70,62,19,92,40,118,129),(3,149,41,166,63,130,184,93,202,71,119,106,20),(4,131,120,42,94,21,64,72,150,185,107,167,203),(5,22,108,121,73,204,95,186,132,49,168,43,151),(6,205,169,109,187,152,74,50,23,96,44,122,133),(7,153,45,170,51,134,188,81,206,75,123,110,24),(8,135,124,46,82,25,52,76,154,189,111,171,207),(9,26,112,125,77,208,83,190,136,53,172,47,155),(10,193,173,97,191,156,78,54,27,84,48,126,137),(11,157,33,174,55,138,192,85,194,79,127,98,28),(12,139,128,34,86,29,56,80,158,177,99,175,195),(13,30,100,113,65,196,87,178,140,57,176,35,159),(14,197,161,101,179,160,66,58,31,88,36,114,141),(15,145,37,162,59,142,180,89,198,67,115,102,32),(16,143,116,38,90,17,60,68,146,181,103,163,199)], [(1,147),(2,129),(3,20),(4,203),(5,151),(6,133),(7,24),(8,207),(9,155),(10,137),(11,28),(12,195),(13,159),(14,141),(15,32),(16,199),(17,68),(18,39),(19,183),(21,72),(22,43),(23,187),(25,76),(26,47),(27,191),(29,80),(30,35),(31,179),(33,127),(34,177),(36,161),(37,115),(38,181),(40,165),(41,119),(42,185),(44,169),(45,123),(46,189),(48,173),(49,121),(50,152),(51,206),(53,125),(54,156),(55,194),(57,113),(58,160),(59,198),(61,117),(62,148),(63,202),(65,140),(67,162),(69,144),(71,166),(73,132),(75,170),(77,136),(79,174),(81,134),(82,154),(84,97),(85,138),(86,158),(88,101),(89,142),(90,146),(92,105),(93,130),(94,150),(96,109),(98,157),(99,128),(100,176),(102,145),(103,116),(104,164),(106,149),(107,120),(108,168),(110,153),(111,124),(112,172),(114,197),(118,201),(122,205),(126,193),(131,167),(135,171),(139,175),(143,163),(178,196),(182,200),(186,204),(190,208)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 13A | 13B | 13C | 16A | ··· | 16P | 26A | 26B | 26C | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | 13 | 13 | 16 | ··· | 16 | 26 | 26 | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 4 | 4 | 4 | 13 | ··· | 13 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C13⋊C4 | C2×C13⋊C4 | D13⋊C8 | D13⋊C16 |
| kernel | D13⋊C16 | C13⋊C16 | C8×D13 | C104 | C4×D13 | Dic13 | D26 | D13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 3 | 3 | 6 | 12 |
Matrix representation of D13⋊C16 ►in GL4(𝔽1249) generated by
| 1207 | 1208 | 1207 | 1207 |
| 42 | 42 | 43 | 42 |
| 641 | 641 | 641 | 642 |
| 1248 | 1248 | 1248 | 1248 |
| 1248 | 41 | 650 | 1247 |
| 0 | 1207 | 1164 | 1206 |
| 0 | 608 | 41 | 608 |
| 0 | 1 | 641 | 2 |
| 978 | 387 | 915 | 1247 |
| 505 | 862 | 1139 | 1139 |
| 935 | 1099 | 87 | 1004 |
| 121 | 0 | 894 | 571 |
G:=sub<GL(4,GF(1249))| [1207,42,641,1248,1208,42,641,1248,1207,43,641,1248,1207,42,642,1248],[1248,0,0,0,41,1207,608,1,650,1164,41,641,1247,1206,608,2],[978,505,935,121,387,862,1099,0,915,1139,87,894,1247,1139,1004,571] >;
D13⋊C16 in GAP, Magma, Sage, TeX
D_{13}\rtimes C_{16} % in TeX
G:=Group("D13:C16"); // GroupNames label
G:=SmallGroup(416,64);
// by ID
G=gap.SmallGroup(416,64);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,55,50,69,9221,3473]);
// Polycyclic
G:=Group<a,b,c|a^13=b^2=c^16=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^4*b>;
// generators/relations
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