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G = D181order 362 = 2·181

Dihedral group

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

Aliases: D181, C181⋊C2, sometimes denoted D362 or Dih181 or Dih362, SmallGroup(362,1)

Series: Derived Chief Lower central Upper central

C1C181 — D181
C1C181 — D181
C181 — D181
C1

Generators and relations for D181
 G = < a,b | a181=b2=1, bab=a-1 >

181C2

Smallest permutation representation of D181
On 181 points: primitive
Generators in S181
(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)
(1 181)(2 180)(3 179)(4 178)(5 177)(6 176)(7 175)(8 174)(9 173)(10 172)(11 171)(12 170)(13 169)(14 168)(15 167)(16 166)(17 165)(18 164)(19 163)(20 162)(21 161)(22 160)(23 159)(24 158)(25 157)(26 156)(27 155)(28 154)(29 153)(30 152)(31 151)(32 150)(33 149)(34 148)(35 147)(36 146)(37 145)(38 144)(39 143)(40 142)(41 141)(42 140)(43 139)(44 138)(45 137)(46 136)(47 135)(48 134)(49 133)(50 132)(51 131)(52 130)(53 129)(54 128)(55 127)(56 126)(57 125)(58 124)(59 123)(60 122)(61 121)(62 120)(63 119)(64 118)(65 117)(66 116)(67 115)(68 114)(69 113)(70 112)(71 111)(72 110)(73 109)(74 108)(75 107)(76 106)(77 105)(78 104)(79 103)(80 102)(81 101)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)

G:=sub<Sym(181)| (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), (1,181)(2,180)(3,179)(4,178)(5,177)(6,176)(7,175)(8,174)(9,173)(10,172)(11,171)(12,170)(13,169)(14,168)(15,167)(16,166)(17,165)(18,164)(19,163)(20,162)(21,161)(22,160)(23,159)(24,158)(25,157)(26,156)(27,155)(28,154)(29,153)(30,152)(31,151)(32,150)(33,149)(34,148)(35,147)(36,146)(37,145)(38,144)(39,143)(40,142)(41,141)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(61,121)(62,120)(63,119)(64,118)(65,117)(66,116)(67,115)(68,114)(69,113)(70,112)(71,111)(72,110)(73,109)(74,108)(75,107)(76,106)(77,105)(78,104)(79,103)(80,102)(81,101)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)>;

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), (1,181)(2,180)(3,179)(4,178)(5,177)(6,176)(7,175)(8,174)(9,173)(10,172)(11,171)(12,170)(13,169)(14,168)(15,167)(16,166)(17,165)(18,164)(19,163)(20,162)(21,161)(22,160)(23,159)(24,158)(25,157)(26,156)(27,155)(28,154)(29,153)(30,152)(31,151)(32,150)(33,149)(34,148)(35,147)(36,146)(37,145)(38,144)(39,143)(40,142)(41,141)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(61,121)(62,120)(63,119)(64,118)(65,117)(66,116)(67,115)(68,114)(69,113)(70,112)(71,111)(72,110)(73,109)(74,108)(75,107)(76,106)(77,105)(78,104)(79,103)(80,102)(81,101)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92) );

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)], [(1,181),(2,180),(3,179),(4,178),(5,177),(6,176),(7,175),(8,174),(9,173),(10,172),(11,171),(12,170),(13,169),(14,168),(15,167),(16,166),(17,165),(18,164),(19,163),(20,162),(21,161),(22,160),(23,159),(24,158),(25,157),(26,156),(27,155),(28,154),(29,153),(30,152),(31,151),(32,150),(33,149),(34,148),(35,147),(36,146),(37,145),(38,144),(39,143),(40,142),(41,141),(42,140),(43,139),(44,138),(45,137),(46,136),(47,135),(48,134),(49,133),(50,132),(51,131),(52,130),(53,129),(54,128),(55,127),(56,126),(57,125),(58,124),(59,123),(60,122),(61,121),(62,120),(63,119),(64,118),(65,117),(66,116),(67,115),(68,114),(69,113),(70,112),(71,111),(72,110),(73,109),(74,108),(75,107),(76,106),(77,105),(78,104),(79,103),(80,102),(81,101),(82,100),(83,99),(84,98),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92)])

92 conjugacy classes

class 1  2 181A···181CL
order12181···181
size11812···2

92 irreducible representations

dim112
type+++
imageC1C2D181
kernelD181C181C1
# reps1190

Matrix representation of D181 in GL2(𝔽1087) generated by

5631086
546418
,
158722
801929
G:=sub<GL(2,GF(1087))| [563,546,1086,418],[158,801,722,929] >;

D181 in GAP, Magma, Sage, TeX

D_{181}
% in TeX

G:=Group("D181");
// GroupNames label

G:=SmallGroup(362,1);
// by ID

G=gap.SmallGroup(362,1);
# by ID

G:=PCGroup([2,-2,-181,1441]);
// Polycyclic

G:=Group<a,b|a^181=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D181 in TeX

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