Copied to
clipboard

G = D165order 330 = 2·3·5·11

Dihedral group

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

Aliases: D165, C5⋊D33, C3⋊D55, C11⋊D15, C551S3, C331D5, C1651C2, C151D11, sometimes denoted D330 or Dih165 or Dih330, SmallGroup(330,11)

Series: Derived Chief Lower central Upper central

C1C165 — D165
C1C11C55C165 — D165
C165 — D165
C1

Generators and relations for D165
 G = < a,b | a165=b2=1, bab=a-1 >

165C2
55S3
33D5
15D11
11D15
5D33
3D55

Smallest permutation representation of D165
On 165 points
Generators in S165
(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)
(2 165)(3 164)(4 163)(5 162)(6 161)(7 160)(8 159)(9 158)(10 157)(11 156)(12 155)(13 154)(14 153)(15 152)(16 151)(17 150)(18 149)(19 148)(20 147)(21 146)(22 145)(23 144)(24 143)(25 142)(26 141)(27 140)(28 139)(29 138)(30 137)(31 136)(32 135)(33 134)(34 133)(35 132)(36 131)(37 130)(38 129)(39 128)(40 127)(41 126)(42 125)(43 124)(44 123)(45 122)(46 121)(47 120)(48 119)(49 118)(50 117)(51 116)(52 115)(53 114)(54 113)(55 112)(56 111)(57 110)(58 109)(59 108)(60 107)(61 106)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)

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

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

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

84 conjugacy classes

class 1  2  3 5A5B11A···11E15A15B15C15D33A···33J55A···55T165A···165AN
order1235511···111515151533···3355···55165···165
size11652222···222222···22···22···2

84 irreducible representations

dim112222222
type+++++++++
imageC1C2S3D5D11D15D33D55D165
kernelD165C165C55C33C15C11C5C3C1
# reps111254102040

Matrix representation of D165 in GL2(𝔽331) generated by

70156
2941
,
2071
182124
G:=sub<GL(2,GF(331))| [70,29,156,41],[207,182,1,124] >;

D165 in GAP, Magma, Sage, TeX

D_{165}
% in TeX

G:=Group("D165");
// GroupNames label

G:=SmallGroup(330,11);
// by ID

G=gap.SmallGroup(330,11);
# by ID

G:=PCGroup([4,-2,-3,-5,-11,33,290,4803]);
// Polycyclic

G:=Group<a,b|a^165=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D165 in TeX

׿
×
𝔽