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G = D161order 322 = 2·7·23

Dihedral group

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

Aliases: D161, C23⋊D7, C7⋊D23, C1611C2, sometimes denoted D322 or Dih161 or Dih322, SmallGroup(322,3)

Series: Derived Chief Lower central Upper central

C1C161 — D161
C1C23C161 — D161
C161 — D161
C1

Generators and relations for D161
 G = < a,b | a161=b2=1, bab=a-1 >

161C2
23D7
7D23

Smallest permutation representation of D161
On 161 points
Generators in S161
(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)
(1 161)(2 160)(3 159)(4 158)(5 157)(6 156)(7 155)(8 154)(9 153)(10 152)(11 151)(12 150)(13 149)(14 148)(15 147)(16 146)(17 145)(18 144)(19 143)(20 142)(21 141)(22 140)(23 139)(24 138)(25 137)(26 136)(27 135)(28 134)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 125)(38 124)(39 123)(40 122)(41 121)(42 120)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(49 113)(50 112)(51 111)(52 110)(53 109)(54 108)(55 107)(56 106)(57 105)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)

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

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

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

82 conjugacy classes

class 1  2 7A7B7C23A···23K161A···161BN
order1277723···23161···161
size11612222···22···2

82 irreducible representations

dim11222
type+++++
imageC1C2D7D23D161
kernelD161C161C23C7C1
# reps1131166

Matrix representation of D161 in GL2(𝔽967) generated by

11905
268951
,
248707
575719
G:=sub<GL(2,GF(967))| [11,268,905,951],[248,575,707,719] >;

D161 in GAP, Magma, Sage, TeX

D_{161}
% in TeX

G:=Group("D161");
// GroupNames label

G:=SmallGroup(322,3);
// by ID

G=gap.SmallGroup(322,3);
# by ID

G:=PCGroup([3,-2,-7,-23,73,2774]);
// Polycyclic

G:=Group<a,b|a^161=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D161 in TeX

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