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G = D129order 258 = 2·3·43

Dihedral group

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

Aliases: D129, C43⋊S3, C3⋊D43, C1291C2, sometimes denoted D258 or Dih129 or Dih258, SmallGroup(258,5)

Series: Derived Chief Lower central Upper central

C1C129 — D129
C1C43C129 — D129
C129 — D129
C1

Generators and relations for D129
 G = < a,b | a129=b2=1, bab=a-1 >

129C2
43S3
3D43

Smallest permutation representation of D129
On 129 points
Generators in S129
(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)
(1 129)(2 128)(3 127)(4 126)(5 125)(6 124)(7 123)(8 122)(9 121)(10 120)(11 119)(12 118)(13 117)(14 116)(15 115)(16 114)(17 113)(18 112)(19 111)(20 110)(21 109)(22 108)(23 107)(24 106)(25 105)(26 104)(27 103)(28 102)(29 101)(30 100)(31 99)(32 98)(33 97)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)

G:=sub<Sym(129)| (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), (1,129)(2,128)(3,127)(4,126)(5,125)(6,124)(7,123)(8,122)(9,121)(10,120)(11,119)(12,118)(13,117)(14,116)(15,115)(16,114)(17,113)(18,112)(19,111)(20,110)(21,109)(22,108)(23,107)(24,106)(25,105)(26,104)(27,103)(28,102)(29,101)(30,100)(31,99)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)>;

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), (1,129)(2,128)(3,127)(4,126)(5,125)(6,124)(7,123)(8,122)(9,121)(10,120)(11,119)(12,118)(13,117)(14,116)(15,115)(16,114)(17,113)(18,112)(19,111)(20,110)(21,109)(22,108)(23,107)(24,106)(25,105)(26,104)(27,103)(28,102)(29,101)(30,100)(31,99)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66) );

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)], [(1,129),(2,128),(3,127),(4,126),(5,125),(6,124),(7,123),(8,122),(9,121),(10,120),(11,119),(12,118),(13,117),(14,116),(15,115),(16,114),(17,113),(18,112),(19,111),(20,110),(21,109),(22,108),(23,107),(24,106),(25,105),(26,104),(27,103),(28,102),(29,101),(30,100),(31,99),(32,98),(33,97),(34,96),(35,95),(36,94),(37,93),(38,92),(39,91),(40,90),(41,89),(42,88),(43,87),(44,86),(45,85),(46,84),(47,83),(48,82),(49,81),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,72),(59,71),(60,70),(61,69),(62,68),(63,67),(64,66)])

66 conjugacy classes

class 1  2  3 43A···43U129A···129AP
order12343···43129···129
size112922···22···2

66 irreducible representations

dim11222
type+++++
imageC1C2S3D43D129
kernelD129C129C43C3C1
# reps1112142

Matrix representation of D129 in GL2(𝔽1033) generated by

500117
91647
,
765719
1015268
G:=sub<GL(2,GF(1033))| [500,916,117,47],[765,1015,719,268] >;

D129 in GAP, Magma, Sage, TeX

D_{129}
% in TeX

G:=Group("D129");
// GroupNames label

G:=SmallGroup(258,5);
// by ID

G=gap.SmallGroup(258,5);
# by ID

G:=PCGroup([3,-2,-3,-43,25,2270]);
// Polycyclic

G:=Group<a,b|a^129=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D129 in TeX

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