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G = D131order 262 = 2·131

Dihedral group

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

Aliases: D131, C131⋊C2, sometimes denoted D262 or Dih131 or Dih262, SmallGroup(262,1)

Series: Derived Chief Lower central Upper central

C1C131 — D131
C1C131 — D131
C131 — D131
C1

Generators and relations for D131
 G = < a,b | a131=b2=1, bab=a-1 >

131C2

Smallest permutation representation of D131
On 131 points: primitive
Generators in S131
(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)
(1 131)(2 130)(3 129)(4 128)(5 127)(6 126)(7 125)(8 124)(9 123)(10 122)(11 121)(12 120)(13 119)(14 118)(15 117)(16 116)(17 115)(18 114)(19 113)(20 112)(21 111)(22 110)(23 109)(24 108)(25 107)(26 106)(27 105)(28 104)(29 103)(30 102)(31 101)(32 100)(33 99)(34 98)(35 97)(36 96)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 84)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)

G:=sub<Sym(131)| (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), (1,131)(2,130)(3,129)(4,128)(5,127)(6,126)(7,125)(8,124)(9,123)(10,122)(11,121)(12,120)(13,119)(14,118)(15,117)(16,116)(17,115)(18,114)(19,113)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,103)(30,102)(31,101)(32,100)(33,99)(34,98)(35,97)(36,96)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)>;

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), (1,131)(2,130)(3,129)(4,128)(5,127)(6,126)(7,125)(8,124)(9,123)(10,122)(11,121)(12,120)(13,119)(14,118)(15,117)(16,116)(17,115)(18,114)(19,113)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,103)(30,102)(31,101)(32,100)(33,99)(34,98)(35,97)(36,96)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67) );

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)], [(1,131),(2,130),(3,129),(4,128),(5,127),(6,126),(7,125),(8,124),(9,123),(10,122),(11,121),(12,120),(13,119),(14,118),(15,117),(16,116),(17,115),(18,114),(19,113),(20,112),(21,111),(22,110),(23,109),(24,108),(25,107),(26,106),(27,105),(28,104),(29,103),(30,102),(31,101),(32,100),(33,99),(34,98),(35,97),(36,96),(37,95),(38,94),(39,93),(40,92),(41,91),(42,90),(43,89),(44,88),(45,87),(46,86),(47,85),(48,84),(49,83),(50,82),(51,81),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67)])

67 conjugacy classes

class 1  2 131A···131BM
order12131···131
size11312···2

67 irreducible representations

dim112
type+++
imageC1C2D131
kernelD131C131C1
# reps1165

Matrix representation of D131 in GL2(𝔽263) generated by

238262
8116
,
153136
29110
G:=sub<GL(2,GF(263))| [238,8,262,116],[153,29,136,110] >;

D131 in GAP, Magma, Sage, TeX

D_{131}
% in TeX

G:=Group("D131");
// GroupNames label

G:=SmallGroup(262,1);
// by ID

G=gap.SmallGroup(262,1);
# by ID

G:=PCGroup([2,-2,-131,1041]);
// Polycyclic

G:=Group<a,b|a^131=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D131 in TeX

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