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G = D134order 268 = 22·67

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D134, C2×D67, C134⋊C2, C67⋊C22, sometimes denoted D268 or Dih134 or Dih268, SmallGroup(268,3)

Series: Derived Chief Lower central Upper central

C1C67 — D134
C1C67D67 — D134
C67 — D134
C1C2

Generators and relations for D134
 G = < a,b | a134=b2=1, bab=a-1 >

67C2
67C2
67C22

Smallest permutation representation of D134
On 134 points
Generators in S134
(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)
(1 134)(2 133)(3 132)(4 131)(5 130)(6 129)(7 128)(8 127)(9 126)(10 125)(11 124)(12 123)(13 122)(14 121)(15 120)(16 119)(17 118)(18 117)(19 116)(20 115)(21 114)(22 113)(23 112)(24 111)(25 110)(26 109)(27 108)(28 107)(29 106)(30 105)(31 104)(32 103)(33 102)(34 101)(35 100)(36 99)(37 98)(38 97)(39 96)(40 95)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)

G:=sub<Sym(134)| (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), (1,134)(2,133)(3,132)(4,131)(5,130)(6,129)(7,128)(8,127)(9,126)(10,125)(11,124)(12,123)(13,122)(14,121)(15,120)(16,119)(17,118)(18,117)(19,116)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)>;

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), (1,134)(2,133)(3,132)(4,131)(5,130)(6,129)(7,128)(8,127)(9,126)(10,125)(11,124)(12,123)(13,122)(14,121)(15,120)(16,119)(17,118)(18,117)(19,116)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68) );

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)], [(1,134),(2,133),(3,132),(4,131),(5,130),(6,129),(7,128),(8,127),(9,126),(10,125),(11,124),(12,123),(13,122),(14,121),(15,120),(16,119),(17,118),(18,117),(19,116),(20,115),(21,114),(22,113),(23,112),(24,111),(25,110),(26,109),(27,108),(28,107),(29,106),(30,105),(31,104),(32,103),(33,102),(34,101),(35,100),(36,99),(37,98),(38,97),(39,96),(40,95),(41,94),(42,93),(43,92),(44,91),(45,90),(46,89),(47,88),(48,87),(49,86),(50,85),(51,84),(52,83),(53,82),(54,81),(55,80),(56,79),(57,78),(58,77),(59,76),(60,75),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68)])

70 conjugacy classes

class 1 2A2B2C67A···67AG134A···134AG
order122267···67134···134
size1167672···22···2

70 irreducible representations

dim11122
type+++++
imageC1C2C2D67D134
kernelD134D67C134C2C1
# reps1213333

Matrix representation of D134 in GL3(𝔽269) generated by

26800
093261
088
,
100
093261
05176
G:=sub<GL(3,GF(269))| [268,0,0,0,93,8,0,261,8],[1,0,0,0,93,5,0,261,176] >;

D134 in GAP, Magma, Sage, TeX

D_{134}
% in TeX

G:=Group("D134");
// GroupNames label

G:=SmallGroup(268,3);
// by ID

G=gap.SmallGroup(268,3);
# by ID

G:=PCGroup([3,-2,-2,-67,2378]);
// Polycyclic

G:=Group<a,b|a^134=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D134 in TeX

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