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G = D180  order 360 = 23·32·5

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D180, C4⋊D45, C5⋊1D36, C9⋊1D20, C45⋊4D4, C3.D60, C36⋊1D5, C20⋊1D9, C180⋊1C2, D90⋊1C2, C60.2S3, C2.4D90, C6.10D30, C30.42D6, C15.2D12, C12.2D15, C10.10D18, C18.10D10, C90.10C22, sometimes denoted D360 or Dih180 or Dih360, SmallGroup(360,27)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C90 — D180
Chief series
C1 — C3 — C15 — C45 — C90 — D90 — D180
Lower central
C45 — C90 — D180
Upper central
C1 — C2 — C4

Generators and relations for D180
 G = < a,b | a180=b2=1, bab=a-1 >

C1
C2
90C2
90C2
C3
C5
C4
45C22
45C22
C6
30S3
30S3
C9
C10
18D5
18D5
C15
45D4
C12
15D6
15D6
C18
10D9
10D9
C20
9D10
9D10
C30
6D15
6D15
C45
15D12
C36
5D18
5D18
9D20
C60
3D30
3D30
C90
2D45
2D45
5D36
3D60
D90
D90
C180
D180

Smallest permutation representation of D180
►On 180 points
Generators in S180
(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)

(1 180)(2 179)(3 178)(4 177)(5 176)(6 175)(7 174)(8 173)(9 172)(10 171)(11 170)(12 169)(13 168)(14 167)(15 166)(16 165)(17 164)(18 163)(19 162)(20 161)(21 160)(22 159)(23 158)(24 157)(25 156)(26 155)(27 154)(28 153)(29 152)(30 151)(31 150)(32 149)(33 148)(34 147)(35 146)(36 145)(37 144)(38 143)(39 142)(40 141)(41 140)(42 139)(43 138)(44 137)(45 136)(46 135)(47 134)(48 133)(49 132)(50 131)(51 130)(52 129)(53 128)(54 127)(55 126)(56 125)(57 124)(58 123)(59 122)(60 121)(61 120)(62 119)(63 118)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)
 
G:=sub<Sym(180)| (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), (1,180)(2,179)(3,178)(4,177)(5,176)(6,175)(7,174)(8,173)(9,172)(10,171)(11,170)(12,169)(13,168)(14,167)(15,166)(16,165)(17,164)(18,163)(19,162)(20,161)(21,160)(22,159)(23,158)(24,157)(25,156)(26,155)(27,154)(28,153)(29,152)(30,151)(31,150)(32,149)(33,148)(34,147)(35,146)(36,145)(37,144)(38,143)(39,142)(40,141)(41,140)(42,139)(43,138)(44,137)(45,136)(46,135)(47,134)(48,133)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,126)(56,125)(57,124)(58,123)(59,122)(60,121)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)>;
 
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), (1,180)(2,179)(3,178)(4,177)(5,176)(6,175)(7,174)(8,173)(9,172)(10,171)(11,170)(12,169)(13,168)(14,167)(15,166)(16,165)(17,164)(18,163)(19,162)(20,161)(21,160)(22,159)(23,158)(24,157)(25,156)(26,155)(27,154)(28,153)(29,152)(30,151)(31,150)(32,149)(33,148)(34,147)(35,146)(36,145)(37,144)(38,143)(39,142)(40,141)(41,140)(42,139)(43,138)(44,137)(45,136)(46,135)(47,134)(48,133)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,126)(56,125)(57,124)(58,123)(59,122)(60,121)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91) );
 
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)], [(1,180),(2,179),(3,178),(4,177),(5,176),(6,175),(7,174),(8,173),(9,172),(10,171),(11,170),(12,169),(13,168),(14,167),(15,166),(16,165),(17,164),(18,163),(19,162),(20,161),(21,160),(22,159),(23,158),(24,157),(25,156),(26,155),(27,154),(28,153),(29,152),(30,151),(31,150),(32,149),(33,148),(34,147),(35,146),(36,145),(37,144),(38,143),(39,142),(40,141),(41,140),(42,139),(43,138),(44,137),(45,136),(46,135),(47,134),(48,133),(49,132),(50,131),(51,130),(52,129),(53,128),(54,127),(55,126),(56,125),(57,124),(58,123),(59,122),(60,121),(61,120),(62,119),(63,118),(64,117),(65,116),(66,115),(67,114),(68,113),(69,112),(70,111),(71,110),(72,109),(73,108),(74,107),(75,106),(76,105),(77,104),(78,103),(79,102),(80,101),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91)]])
 

93 conjugacy classes

class 1 2A2B2C 3  4 5A5B 6 9A9B9C10A10B12A12B15A15B15C15D18A18B18C20A20B20C20D30A30B30C30D36A···36F45A···45L60A···60H90A···90L180A···180X
order1222345569991010121215151515181818202020203030303036···3645···4560···6090···90180···180
size1190902222222222222222222222222222···22···22···22···22···2

93 irreducible representations

dim1112222222222222222
type+++++++++++++++++++
imageC1C2C2S3D4D5D6D9D10D12D15D18D20D30D36D45D60D90D180
kernelD180C180D90C60C45C36C30C20C18C15C12C10C9C6C5C4C3C2C1
# reps1121121322434461281224

Matrix representation of D180 ►in GL2(𝔽181) generated by

6034
14726
,
12337
9558
G:=sub<GL(2,GF(181))| [60,147,34,26],[123,95,37,58] >;
 

D180 in GAP, Magma, Sage, TeX 

D_{180}
 
% in TeX
 
G:=Group("D180");
 
// GroupNames label
 
G:=SmallGroup(360,27);
 
// by ID
 
G=gap.SmallGroup(360,27);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,73,31,3267,741,2884,8645]);
 
// Polycyclic
 
G:=Group<a,b|a^180=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D180 in TeX

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