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G = D200order 400 = 24·52

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D200, C251D8, C5.D40, C81D25, C2001C2, C50.3D4, C40.2D5, D1001C2, C4.10D50, C10.3D20, C2.5D100, C20.42D10, C100.10C22, sometimes denoted D400 or Dih200 or Dih400, SmallGroup(400,8)

Series: Derived Chief Lower central Upper central

C1C100 — D200
C1C5C25C50C100D100 — D200
C25C50C100 — D200
C1C2C4C8

Generators and relations for D200
 G = < a,b | a200=b2=1, bab=a-1 >

100C2
100C2
50C22
50C22
20D5
20D5
25D4
25D4
10D10
10D10
4D25
4D25
25D8
5D20
5D20
2D50
2D50
5D40

Smallest permutation representation of D200
On 200 points
Generators in S200
(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)
(1 175)(2 174)(3 173)(4 172)(5 171)(6 170)(7 169)(8 168)(9 167)(10 166)(11 165)(12 164)(13 163)(14 162)(15 161)(16 160)(17 159)(18 158)(19 157)(20 156)(21 155)(22 154)(23 153)(24 152)(25 151)(26 150)(27 149)(28 148)(29 147)(30 146)(31 145)(32 144)(33 143)(34 142)(35 141)(36 140)(37 139)(38 138)(39 137)(40 136)(41 135)(42 134)(43 133)(44 132)(45 131)(46 130)(47 129)(48 128)(49 127)(50 126)(51 125)(52 124)(53 123)(54 122)(55 121)(56 120)(57 119)(58 118)(59 117)(60 116)(61 115)(62 114)(63 113)(64 112)(65 111)(66 110)(67 109)(68 108)(69 107)(70 106)(71 105)(72 104)(73 103)(74 102)(75 101)(76 100)(77 99)(78 98)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)(85 91)(86 90)(87 89)(176 200)(177 199)(178 198)(179 197)(180 196)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)

G:=sub<Sym(200)| (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), (1,175)(2,174)(3,173)(4,172)(5,171)(6,170)(7,169)(8,168)(9,167)(10,166)(11,165)(12,164)(13,163)(14,162)(15,161)(16,160)(17,159)(18,158)(19,157)(20,156)(21,155)(22,154)(23,153)(24,152)(25,151)(26,150)(27,149)(28,148)(29,147)(30,146)(31,145)(32,144)(33,143)(34,142)(35,141)(36,140)(37,139)(38,138)(39,137)(40,136)(41,135)(42,134)(43,133)(44,132)(45,131)(46,130)(47,129)(48,128)(49,127)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,120)(57,119)(58,118)(59,117)(60,116)(61,115)(62,114)(63,113)(64,112)(65,111)(66,110)(67,109)(68,108)(69,107)(70,106)(71,105)(72,104)(73,103)(74,102)(75,101)(76,100)(77,99)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89)(176,200)(177,199)(178,198)(179,197)(180,196)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189)>;

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), (1,175)(2,174)(3,173)(4,172)(5,171)(6,170)(7,169)(8,168)(9,167)(10,166)(11,165)(12,164)(13,163)(14,162)(15,161)(16,160)(17,159)(18,158)(19,157)(20,156)(21,155)(22,154)(23,153)(24,152)(25,151)(26,150)(27,149)(28,148)(29,147)(30,146)(31,145)(32,144)(33,143)(34,142)(35,141)(36,140)(37,139)(38,138)(39,137)(40,136)(41,135)(42,134)(43,133)(44,132)(45,131)(46,130)(47,129)(48,128)(49,127)(50,126)(51,125)(52,124)(53,123)(54,122)(55,121)(56,120)(57,119)(58,118)(59,117)(60,116)(61,115)(62,114)(63,113)(64,112)(65,111)(66,110)(67,109)(68,108)(69,107)(70,106)(71,105)(72,104)(73,103)(74,102)(75,101)(76,100)(77,99)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89)(176,200)(177,199)(178,198)(179,197)(180,196)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189) );

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)], [(1,175),(2,174),(3,173),(4,172),(5,171),(6,170),(7,169),(8,168),(9,167),(10,166),(11,165),(12,164),(13,163),(14,162),(15,161),(16,160),(17,159),(18,158),(19,157),(20,156),(21,155),(22,154),(23,153),(24,152),(25,151),(26,150),(27,149),(28,148),(29,147),(30,146),(31,145),(32,144),(33,143),(34,142),(35,141),(36,140),(37,139),(38,138),(39,137),(40,136),(41,135),(42,134),(43,133),(44,132),(45,131),(46,130),(47,129),(48,128),(49,127),(50,126),(51,125),(52,124),(53,123),(54,122),(55,121),(56,120),(57,119),(58,118),(59,117),(60,116),(61,115),(62,114),(63,113),(64,112),(65,111),(66,110),(67,109),(68,108),(69,107),(70,106),(71,105),(72,104),(73,103),(74,102),(75,101),(76,100),(77,99),(78,98),(79,97),(80,96),(81,95),(82,94),(83,93),(84,92),(85,91),(86,90),(87,89),(176,200),(177,199),(178,198),(179,197),(180,196),(181,195),(182,194),(183,193),(184,192),(185,191),(186,190),(187,189)])

103 conjugacy classes

class 1 2A2B2C 4 5A5B8A8B10A10B20A20B20C20D25A···25J40A···40H50A···50J100A···100T200A···200AN
order12224558810102020202025···2540···4050···50100···100200···200
size11100100222222222222···22···22···22···22···2

103 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2D4D5D8D10D20D25D40D50D100D200
kernelD200C200D100C50C40C25C20C10C8C5C4C2C1
# reps11212224108102040

Matrix representation of D200 in GL2(𝔽401) generated by

22265
13616
,
19075
48211
G:=sub<GL(2,GF(401))| [22,136,265,16],[190,48,75,211] >;

D200 in GAP, Magma, Sage, TeX

D_{200}
% in TeX

G:=Group("D200");
// GroupNames label

G:=SmallGroup(400,8);
// by ID

G=gap.SmallGroup(400,8);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,79,218,50,4324,628,11525]);
// Polycyclic

G:=Group<a,b|a^200=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D200 in TeX

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