metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C208⋊2C2, C16⋊2D13, C26.2D8, C4.2D52, C13⋊1SD32, C2.4D104, C52.25D4, C8.14D26, Dic52⋊1C2, D104.1C2, C104.15C22, SmallGroup(416,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16⋊D13
G = < a,b,c | a16=b13=c2=1, ab=ba, cac=a7, 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 83 32 140 34 180 166 114 111 62 147 195 75)(2 84 17 141 35 181 167 115 112 63 148 196 76)(3 85 18 142 36 182 168 116 97 64 149 197 77)(4 86 19 143 37 183 169 117 98 49 150 198 78)(5 87 20 144 38 184 170 118 99 50 151 199 79)(6 88 21 129 39 185 171 119 100 51 152 200 80)(7 89 22 130 40 186 172 120 101 52 153 201 65)(8 90 23 131 41 187 173 121 102 53 154 202 66)(9 91 24 132 42 188 174 122 103 54 155 203 67)(10 92 25 133 43 189 175 123 104 55 156 204 68)(11 93 26 134 44 190 176 124 105 56 157 205 69)(12 94 27 135 45 191 161 125 106 57 158 206 70)(13 95 28 136 46 192 162 126 107 58 159 207 71)(14 96 29 137 47 177 163 127 108 59 160 208 72)(15 81 30 138 48 178 164 128 109 60 145 193 73)(16 82 31 139 33 179 165 113 110 61 146 194 74)
(1 75)(2 66)(3 73)(4 80)(5 71)(6 78)(7 69)(8 76)(9 67)(10 74)(11 65)(12 72)(13 79)(14 70)(15 77)(16 68)(17 154)(18 145)(19 152)(20 159)(21 150)(22 157)(23 148)(24 155)(25 146)(26 153)(27 160)(28 151)(29 158)(30 149)(31 156)(32 147)(33 104)(34 111)(35 102)(36 109)(37 100)(38 107)(39 98)(40 105)(41 112)(42 103)(43 110)(44 101)(45 108)(46 99)(47 106)(48 97)(49 129)(50 136)(51 143)(52 134)(53 141)(54 132)(55 139)(56 130)(57 137)(58 144)(59 135)(60 142)(61 133)(62 140)(63 131)(64 138)(81 197)(82 204)(83 195)(84 202)(85 193)(86 200)(87 207)(88 198)(89 205)(90 196)(91 203)(92 194)(93 201)(94 208)(95 199)(96 206)(113 189)(114 180)(115 187)(116 178)(117 185)(118 192)(119 183)(120 190)(121 181)(122 188)(123 179)(124 186)(125 177)(126 184)(127 191)(128 182)(161 163)(162 170)(164 168)(165 175)(167 173)(169 171)(172 176)
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), (1,83,32,140,34,180,166,114,111,62,147,195,75)(2,84,17,141,35,181,167,115,112,63,148,196,76)(3,85,18,142,36,182,168,116,97,64,149,197,77)(4,86,19,143,37,183,169,117,98,49,150,198,78)(5,87,20,144,38,184,170,118,99,50,151,199,79)(6,88,21,129,39,185,171,119,100,51,152,200,80)(7,89,22,130,40,186,172,120,101,52,153,201,65)(8,90,23,131,41,187,173,121,102,53,154,202,66)(9,91,24,132,42,188,174,122,103,54,155,203,67)(10,92,25,133,43,189,175,123,104,55,156,204,68)(11,93,26,134,44,190,176,124,105,56,157,205,69)(12,94,27,135,45,191,161,125,106,57,158,206,70)(13,95,28,136,46,192,162,126,107,58,159,207,71)(14,96,29,137,47,177,163,127,108,59,160,208,72)(15,81,30,138,48,178,164,128,109,60,145,193,73)(16,82,31,139,33,179,165,113,110,61,146,194,74), (1,75)(2,66)(3,73)(4,80)(5,71)(6,78)(7,69)(8,76)(9,67)(10,74)(11,65)(12,72)(13,79)(14,70)(15,77)(16,68)(17,154)(18,145)(19,152)(20,159)(21,150)(22,157)(23,148)(24,155)(25,146)(26,153)(27,160)(28,151)(29,158)(30,149)(31,156)(32,147)(33,104)(34,111)(35,102)(36,109)(37,100)(38,107)(39,98)(40,105)(41,112)(42,103)(43,110)(44,101)(45,108)(46,99)(47,106)(48,97)(49,129)(50,136)(51,143)(52,134)(53,141)(54,132)(55,139)(56,130)(57,137)(58,144)(59,135)(60,142)(61,133)(62,140)(63,131)(64,138)(81,197)(82,204)(83,195)(84,202)(85,193)(86,200)(87,207)(88,198)(89,205)(90,196)(91,203)(92,194)(93,201)(94,208)(95,199)(96,206)(113,189)(114,180)(115,187)(116,178)(117,185)(118,192)(119,183)(120,190)(121,181)(122,188)(123,179)(124,186)(125,177)(126,184)(127,191)(128,182)(161,163)(162,170)(164,168)(165,175)(167,173)(169,171)(172,176)>;
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), (1,83,32,140,34,180,166,114,111,62,147,195,75)(2,84,17,141,35,181,167,115,112,63,148,196,76)(3,85,18,142,36,182,168,116,97,64,149,197,77)(4,86,19,143,37,183,169,117,98,49,150,198,78)(5,87,20,144,38,184,170,118,99,50,151,199,79)(6,88,21,129,39,185,171,119,100,51,152,200,80)(7,89,22,130,40,186,172,120,101,52,153,201,65)(8,90,23,131,41,187,173,121,102,53,154,202,66)(9,91,24,132,42,188,174,122,103,54,155,203,67)(10,92,25,133,43,189,175,123,104,55,156,204,68)(11,93,26,134,44,190,176,124,105,56,157,205,69)(12,94,27,135,45,191,161,125,106,57,158,206,70)(13,95,28,136,46,192,162,126,107,58,159,207,71)(14,96,29,137,47,177,163,127,108,59,160,208,72)(15,81,30,138,48,178,164,128,109,60,145,193,73)(16,82,31,139,33,179,165,113,110,61,146,194,74), (1,75)(2,66)(3,73)(4,80)(5,71)(6,78)(7,69)(8,76)(9,67)(10,74)(11,65)(12,72)(13,79)(14,70)(15,77)(16,68)(17,154)(18,145)(19,152)(20,159)(21,150)(22,157)(23,148)(24,155)(25,146)(26,153)(27,160)(28,151)(29,158)(30,149)(31,156)(32,147)(33,104)(34,111)(35,102)(36,109)(37,100)(38,107)(39,98)(40,105)(41,112)(42,103)(43,110)(44,101)(45,108)(46,99)(47,106)(48,97)(49,129)(50,136)(51,143)(52,134)(53,141)(54,132)(55,139)(56,130)(57,137)(58,144)(59,135)(60,142)(61,133)(62,140)(63,131)(64,138)(81,197)(82,204)(83,195)(84,202)(85,193)(86,200)(87,207)(88,198)(89,205)(90,196)(91,203)(92,194)(93,201)(94,208)(95,199)(96,206)(113,189)(114,180)(115,187)(116,178)(117,185)(118,192)(119,183)(120,190)(121,181)(122,188)(123,179)(124,186)(125,177)(126,184)(127,191)(128,182)(161,163)(162,170)(164,168)(165,175)(167,173)(169,171)(172,176) );
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)], [(1,83,32,140,34,180,166,114,111,62,147,195,75),(2,84,17,141,35,181,167,115,112,63,148,196,76),(3,85,18,142,36,182,168,116,97,64,149,197,77),(4,86,19,143,37,183,169,117,98,49,150,198,78),(5,87,20,144,38,184,170,118,99,50,151,199,79),(6,88,21,129,39,185,171,119,100,51,152,200,80),(7,89,22,130,40,186,172,120,101,52,153,201,65),(8,90,23,131,41,187,173,121,102,53,154,202,66),(9,91,24,132,42,188,174,122,103,54,155,203,67),(10,92,25,133,43,189,175,123,104,55,156,204,68),(11,93,26,134,44,190,176,124,105,56,157,205,69),(12,94,27,135,45,191,161,125,106,57,158,206,70),(13,95,28,136,46,192,162,126,107,58,159,207,71),(14,96,29,137,47,177,163,127,108,59,160,208,72),(15,81,30,138,48,178,164,128,109,60,145,193,73),(16,82,31,139,33,179,165,113,110,61,146,194,74)], [(1,75),(2,66),(3,73),(4,80),(5,71),(6,78),(7,69),(8,76),(9,67),(10,74),(11,65),(12,72),(13,79),(14,70),(15,77),(16,68),(17,154),(18,145),(19,152),(20,159),(21,150),(22,157),(23,148),(24,155),(25,146),(26,153),(27,160),(28,151),(29,158),(30,149),(31,156),(32,147),(33,104),(34,111),(35,102),(36,109),(37,100),(38,107),(39,98),(40,105),(41,112),(42,103),(43,110),(44,101),(45,108),(46,99),(47,106),(48,97),(49,129),(50,136),(51,143),(52,134),(53,141),(54,132),(55,139),(56,130),(57,137),(58,144),(59,135),(60,142),(61,133),(62,140),(63,131),(64,138),(81,197),(82,204),(83,195),(84,202),(85,193),(86,200),(87,207),(88,198),(89,205),(90,196),(91,203),(92,194),(93,201),(94,208),(95,199),(96,206),(113,189),(114,180),(115,187),(116,178),(117,185),(118,192),(119,183),(120,190),(121,181),(122,188),(123,179),(124,186),(125,177),(126,184),(127,191),(128,182),(161,163),(162,170),(164,168),(165,175),(167,173),(169,171),(172,176)]])
107 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 13A | ··· | 13F | 16A | 16B | 16C | 16D | 26A | ··· | 26F | 52A | ··· | 52L | 104A | ··· | 104X | 208A | ··· | 208AV |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 16 | 16 | 16 | 16 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 | 208 | ··· | 208 |
| size | 1 | 1 | 104 | 2 | 104 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
107 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D8 | D13 | SD32 | D26 | D52 | D104 | C16⋊D13 |
| kernel | C16⋊D13 | C208 | D104 | Dic52 | C52 | C26 | C16 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 4 | 6 | 12 | 24 | 48 |
Matrix representation of C16⋊D13 ►in GL2(𝔽1249) generated by
| 69 | 358 |
| 251 | 361 |
| 393 | 1 |
| 688 | 24 |
| 1235 | 972 |
| 898 | 14 |
G:=sub<GL(2,GF(1249))| [69,251,358,361],[393,688,1,24],[1235,898,972,14] >;
C16⋊D13 in GAP, Magma, Sage, TeX
C_{16}\rtimes D_{13} % in TeX
G:=Group("C16:D13"); // GroupNames label
G:=SmallGroup(416,7);
// by ID
G=gap.SmallGroup(416,7);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,73,79,506,50,579,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^16=b^13=c^2=1,a*b=b*a,c*a*c=a^7,c*b*c=b^-1>;
// generators/relations
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