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G = D159order 318 = 2·3·53

Dihedral group

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

Aliases: D159, C53⋊S3, C3⋊D53, C1591C2, sometimes denoted D318 or Dih159 or Dih318, SmallGroup(318,3)

Series: Derived Chief Lower central Upper central

C1C159 — D159
C1C53C159 — D159
C159 — D159
C1

Generators and relations for D159
 G = < a,b | a159=b2=1, bab=a-1 >

159C2
53S3
3D53

Smallest permutation representation of D159
On 159 points
Generators in S159
(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)
(1 159)(2 158)(3 157)(4 156)(5 155)(6 154)(7 153)(8 152)(9 151)(10 150)(11 149)(12 148)(13 147)(14 146)(15 145)(16 144)(17 143)(18 142)(19 141)(20 140)(21 139)(22 138)(23 137)(24 136)(25 135)(26 134)(27 133)(28 132)(29 131)(30 130)(31 129)(32 128)(33 127)(34 126)(35 125)(36 124)(37 123)(38 122)(39 121)(40 120)(41 119)(42 118)(43 117)(44 116)(45 115)(46 114)(47 113)(48 112)(49 111)(50 110)(51 109)(52 108)(53 107)(54 106)(55 105)(56 104)(57 103)(58 102)(59 101)(60 100)(61 99)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)(79 81)

G:=sub<Sym(159)| (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), (1,159)(2,158)(3,157)(4,156)(5,155)(6,154)(7,153)(8,152)(9,151)(10,150)(11,149)(12,148)(13,147)(14,146)(15,145)(16,144)(17,143)(18,142)(19,141)(20,140)(21,139)(22,138)(23,137)(24,136)(25,135)(26,134)(27,133)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,120)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,110)(51,109)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,100)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)>;

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), (1,159)(2,158)(3,157)(4,156)(5,155)(6,154)(7,153)(8,152)(9,151)(10,150)(11,149)(12,148)(13,147)(14,146)(15,145)(16,144)(17,143)(18,142)(19,141)(20,140)(21,139)(22,138)(23,137)(24,136)(25,135)(26,134)(27,133)(28,132)(29,131)(30,130)(31,129)(32,128)(33,127)(34,126)(35,125)(36,124)(37,123)(38,122)(39,121)(40,120)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,110)(51,109)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,100)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81) );

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)], [(1,159),(2,158),(3,157),(4,156),(5,155),(6,154),(7,153),(8,152),(9,151),(10,150),(11,149),(12,148),(13,147),(14,146),(15,145),(16,144),(17,143),(18,142),(19,141),(20,140),(21,139),(22,138),(23,137),(24,136),(25,135),(26,134),(27,133),(28,132),(29,131),(30,130),(31,129),(32,128),(33,127),(34,126),(35,125),(36,124),(37,123),(38,122),(39,121),(40,120),(41,119),(42,118),(43,117),(44,116),(45,115),(46,114),(47,113),(48,112),(49,111),(50,110),(51,109),(52,108),(53,107),(54,106),(55,105),(56,104),(57,103),(58,102),(59,101),(60,100),(61,99),(62,98),(63,97),(64,96),(65,95),(66,94),(67,93),(68,92),(69,91),(70,90),(71,89),(72,88),(73,87),(74,86),(75,85),(76,84),(77,83),(78,82),(79,81)])

81 conjugacy classes

class 1  2  3 53A···53Z159A···159AZ
order12353···53159···159
size115922···22···2

81 irreducible representations

dim11222
type+++++
imageC1C2S3D53D159
kernelD159C159C53C3C1
# reps1112652

Matrix representation of D159 in GL2(𝔽3181) generated by

12122505
6761741
,
12122505
31001969
G:=sub<GL(2,GF(3181))| [1212,676,2505,1741],[1212,3100,2505,1969] >;

D159 in GAP, Magma, Sage, TeX

D_{159}
% in TeX

G:=Group("D159");
// GroupNames label

G:=SmallGroup(318,3);
// by ID

G=gap.SmallGroup(318,3);
# by ID

G:=PCGroup([3,-2,-3,-53,25,2810]);
// Polycyclic

G:=Group<a,b|a^159=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D159 in TeX

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