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G = D208order 416 = 25·13

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D208, C131D16, C2081C2, C161D13, C26.1D8, C4.1D52, D1041C2, C52.24D4, C2.3D104, C8.13D26, C104.14C22, sometimes denoted D416 or Dih208 or Dih416, SmallGroup(416,6)

Series: Derived Chief Lower central Upper central

C1C104 — D208
C1C13C26C52C104D104 — D208
C13C26C52C104 — D208
C1C2C4C8C16

Generators and relations for D208
 G = < a,b | a208=b2=1, bab=a-1 >

104C2
104C2
52C22
52C22
8D13
8D13
26D4
26D4
4D26
4D26
13D8
13D8
2D52
2D52
13D16

Smallest permutation representation of D208
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)
(1 143)(2 142)(3 141)(4 140)(5 139)(6 138)(7 137)(8 136)(9 135)(10 134)(11 133)(12 132)(13 131)(14 130)(15 129)(16 128)(17 127)(18 126)(19 125)(20 124)(21 123)(22 122)(23 121)(24 120)(25 119)(26 118)(27 117)(28 116)(29 115)(30 114)(31 113)(32 112)(33 111)(34 110)(35 109)(36 108)(37 107)(38 106)(39 105)(40 104)(41 103)(42 102)(43 101)(44 100)(45 99)(46 98)(47 97)(48 96)(49 95)(50 94)(51 93)(52 92)(53 91)(54 90)(55 89)(56 88)(57 87)(58 86)(59 85)(60 84)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(144 208)(145 207)(146 206)(147 205)(148 204)(149 203)(150 202)(151 201)(152 200)(153 199)(154 198)(155 197)(156 196)(157 195)(158 194)(159 193)(160 192)(161 191)(162 190)(163 189)(164 188)(165 187)(166 186)(167 185)(168 184)(169 183)(170 182)(171 181)(172 180)(173 179)(174 178)(175 177)

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), (1,143)(2,142)(3,141)(4,140)(5,139)(6,138)(7,137)(8,136)(9,135)(10,134)(11,133)(12,132)(13,131)(14,130)(15,129)(16,128)(17,127)(18,126)(19,125)(20,124)(21,123)(22,122)(23,121)(24,120)(25,119)(26,118)(27,117)(28,116)(29,115)(30,114)(31,113)(32,112)(33,111)(34,110)(35,109)(36,108)(37,107)(38,106)(39,105)(40,104)(41,103)(42,102)(43,101)(44,100)(45,99)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,88)(57,87)(58,86)(59,85)(60,84)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(144,208)(145,207)(146,206)(147,205)(148,204)(149,203)(150,202)(151,201)(152,200)(153,199)(154,198)(155,197)(156,196)(157,195)(158,194)(159,193)(160,192)(161,191)(162,190)(163,189)(164,188)(165,187)(166,186)(167,185)(168,184)(169,183)(170,182)(171,181)(172,180)(173,179)(174,178)(175,177)>;

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), (1,143)(2,142)(3,141)(4,140)(5,139)(6,138)(7,137)(8,136)(9,135)(10,134)(11,133)(12,132)(13,131)(14,130)(15,129)(16,128)(17,127)(18,126)(19,125)(20,124)(21,123)(22,122)(23,121)(24,120)(25,119)(26,118)(27,117)(28,116)(29,115)(30,114)(31,113)(32,112)(33,111)(34,110)(35,109)(36,108)(37,107)(38,106)(39,105)(40,104)(41,103)(42,102)(43,101)(44,100)(45,99)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,88)(57,87)(58,86)(59,85)(60,84)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(144,208)(145,207)(146,206)(147,205)(148,204)(149,203)(150,202)(151,201)(152,200)(153,199)(154,198)(155,197)(156,196)(157,195)(158,194)(159,193)(160,192)(161,191)(162,190)(163,189)(164,188)(165,187)(166,186)(167,185)(168,184)(169,183)(170,182)(171,181)(172,180)(173,179)(174,178)(175,177) );

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)], [(1,143),(2,142),(3,141),(4,140),(5,139),(6,138),(7,137),(8,136),(9,135),(10,134),(11,133),(12,132),(13,131),(14,130),(15,129),(16,128),(17,127),(18,126),(19,125),(20,124),(21,123),(22,122),(23,121),(24,120),(25,119),(26,118),(27,117),(28,116),(29,115),(30,114),(31,113),(32,112),(33,111),(34,110),(35,109),(36,108),(37,107),(38,106),(39,105),(40,104),(41,103),(42,102),(43,101),(44,100),(45,99),(46,98),(47,97),(48,96),(49,95),(50,94),(51,93),(52,92),(53,91),(54,90),(55,89),(56,88),(57,87),(58,86),(59,85),(60,84),(61,83),(62,82),(63,81),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(144,208),(145,207),(146,206),(147,205),(148,204),(149,203),(150,202),(151,201),(152,200),(153,199),(154,198),(155,197),(156,196),(157,195),(158,194),(159,193),(160,192),(161,191),(162,190),(163,189),(164,188),(165,187),(166,186),(167,185),(168,184),(169,183),(170,182),(171,181),(172,180),(173,179),(174,178),(175,177)])

107 conjugacy classes

class 1 2A2B2C 4 8A8B13A···13F16A16B16C16D26A···26F52A···52L104A···104X208A···208AV
order122248813···131616161626···2652···52104···104208···208
size111041042222···222222···22···22···22···2

107 irreducible representations

dim11122222222
type+++++++++++
imageC1C2C2D4D8D13D16D26D52D104D208
kernelD208C208D104C52C26C16C13C8C4C2C1
# reps11212646122448

Matrix representation of D208 in GL2(𝔽1249) generated by

362525
36518
,
733915
412516
G:=sub<GL(2,GF(1249))| [362,36,525,518],[733,412,915,516] >;

D208 in GAP, Magma, Sage, TeX

D_{208}
% in TeX

G:=Group("D208");
// GroupNames label

G:=SmallGroup(416,6);
// by ID

G=gap.SmallGroup(416,6);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,73,79,218,122,579,69,13829]);
// Polycyclic

G:=Group<a,b|a^208=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D208 in TeX

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