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G = D199order 398 = 2·199

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D199, C199⋊C2, sometimes denoted D398 or Dih199 or Dih398, SmallGroup(398,1)

Series: Derived Chief Lower central Upper central

C1C199 — D199
C1C199 — D199
C199 — D199
C1

Generators and relations for D199
 G = < a,b | a199=b2=1, bab=a-1 >

199C2

Smallest permutation representation of D199
On 199 points: primitive
Generators in S199
(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)
(1 199)(2 198)(3 197)(4 196)(5 195)(6 194)(7 193)(8 192)(9 191)(10 190)(11 189)(12 188)(13 187)(14 186)(15 185)(16 184)(17 183)(18 182)(19 181)(20 180)(21 179)(22 178)(23 177)(24 176)(25 175)(26 174)(27 173)(28 172)(29 171)(30 170)(31 169)(32 168)(33 167)(34 166)(35 165)(36 164)(37 163)(38 162)(39 161)(40 160)(41 159)(42 158)(43 157)(44 156)(45 155)(46 154)(47 153)(48 152)(49 151)(50 150)(51 149)(52 148)(53 147)(54 146)(55 145)(56 144)(57 143)(58 142)(59 141)(60 140)(61 139)(62 138)(63 137)(64 136)(65 135)(66 134)(67 133)(68 132)(69 131)(70 130)(71 129)(72 128)(73 127)(74 126)(75 125)(76 124)(77 123)(78 122)(79 121)(80 120)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)

G:=sub<Sym(199)| (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), (1,199)(2,198)(3,197)(4,196)(5,195)(6,194)(7,193)(8,192)(9,191)(10,190)(11,189)(12,188)(13,187)(14,186)(15,185)(16,184)(17,183)(18,182)(19,181)(20,180)(21,179)(22,178)(23,177)(24,176)(25,175)(26,174)(27,173)(28,172)(29,171)(30,170)(31,169)(32,168)(33,167)(34,166)(35,165)(36,164)(37,163)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,148)(53,147)(54,146)(55,145)(56,144)(57,143)(58,142)(59,141)(60,140)(61,139)(62,138)(63,137)(64,136)(65,135)(66,134)(67,133)(68,132)(69,131)(70,130)(71,129)(72,128)(73,127)(74,126)(75,125)(76,124)(77,123)(78,122)(79,121)(80,120)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)>;

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), (1,199)(2,198)(3,197)(4,196)(5,195)(6,194)(7,193)(8,192)(9,191)(10,190)(11,189)(12,188)(13,187)(14,186)(15,185)(16,184)(17,183)(18,182)(19,181)(20,180)(21,179)(22,178)(23,177)(24,176)(25,175)(26,174)(27,173)(28,172)(29,171)(30,170)(31,169)(32,168)(33,167)(34,166)(35,165)(36,164)(37,163)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,156)(45,155)(46,154)(47,153)(48,152)(49,151)(50,150)(51,149)(52,148)(53,147)(54,146)(55,145)(56,144)(57,143)(58,142)(59,141)(60,140)(61,139)(62,138)(63,137)(64,136)(65,135)(66,134)(67,133)(68,132)(69,131)(70,130)(71,129)(72,128)(73,127)(74,126)(75,125)(76,124)(77,123)(78,122)(79,121)(80,120)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101) );

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)], [(1,199),(2,198),(3,197),(4,196),(5,195),(6,194),(7,193),(8,192),(9,191),(10,190),(11,189),(12,188),(13,187),(14,186),(15,185),(16,184),(17,183),(18,182),(19,181),(20,180),(21,179),(22,178),(23,177),(24,176),(25,175),(26,174),(27,173),(28,172),(29,171),(30,170),(31,169),(32,168),(33,167),(34,166),(35,165),(36,164),(37,163),(38,162),(39,161),(40,160),(41,159),(42,158),(43,157),(44,156),(45,155),(46,154),(47,153),(48,152),(49,151),(50,150),(51,149),(52,148),(53,147),(54,146),(55,145),(56,144),(57,143),(58,142),(59,141),(60,140),(61,139),(62,138),(63,137),(64,136),(65,135),(66,134),(67,133),(68,132),(69,131),(70,130),(71,129),(72,128),(73,127),(74,126),(75,125),(76,124),(77,123),(78,122),(79,121),(80,120),(81,119),(82,118),(83,117),(84,116),(85,115),(86,114),(87,113),(88,112),(89,111),(90,110),(91,109),(92,108),(93,107),(94,106),(95,105),(96,104),(97,103),(98,102),(99,101)])

101 conjugacy classes

class 1  2 199A···199CU
order12199···199
size11992···2

101 irreducible representations

dim112
type+++
imageC1C2D199
kernelD199C199C1
# reps1199

Matrix representation of D199 in GL2(𝔽797) generated by

200796
500663
,
37444
40760
G:=sub<GL(2,GF(797))| [200,500,796,663],[37,40,444,760] >;

D199 in GAP, Magma, Sage, TeX

D_{199}
% in TeX

G:=Group("D199");
// GroupNames label

G:=SmallGroup(398,1);
// by ID

G=gap.SmallGroup(398,1);
# by ID

G:=PCGroup([2,-2,-199,1585]);
// Polycyclic

G:=Group<a,b|a^199=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D199 in TeX

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