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G = D138order 276 = 22·3·23

Dihedral group

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

Aliases: D138, C2×D69, C46⋊S3, C6⋊D23, C32D46, C232D6, C1381C2, C692C22, sometimes denoted D276 or Dih138 or Dih276, SmallGroup(276,9)

Series: Derived Chief Lower central Upper central

C1C69 — D138
C1C23C69D69 — D138
C69 — D138
C1C2

Generators and relations for D138
 G = < a,b | a138=b2=1, bab=a-1 >

69C2
69C2
69C22
23S3
23S3
3D23
3D23
23D6
3D46

Smallest permutation representation of D138
On 138 points
Generators in S138
(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)
(1 138)(2 137)(3 136)(4 135)(5 134)(6 133)(7 132)(8 131)(9 130)(10 129)(11 128)(12 127)(13 126)(14 125)(15 124)(16 123)(17 122)(18 121)(19 120)(20 119)(21 118)(22 117)(23 116)(24 115)(25 114)(26 113)(27 112)(28 111)(29 110)(30 109)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(41 98)(42 97)(43 96)(44 95)(45 94)(46 93)(47 92)(48 91)(49 90)(50 89)(51 88)(52 87)(53 86)(54 85)(55 84)(56 83)(57 82)(58 81)(59 80)(60 79)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)

G:=sub<Sym(138)| (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), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,132)(8,131)(9,130)(10,129)(11,128)(12,127)(13,126)(14,125)(15,124)(16,123)(17,122)(18,121)(19,120)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)>;

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), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,132)(8,131)(9,130)(10,129)(11,128)(12,127)(13,126)(14,125)(15,124)(16,123)(17,122)(18,121)(19,120)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70) );

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)], [(1,138),(2,137),(3,136),(4,135),(5,134),(6,133),(7,132),(8,131),(9,130),(10,129),(11,128),(12,127),(13,126),(14,125),(15,124),(16,123),(17,122),(18,121),(19,120),(20,119),(21,118),(22,117),(23,116),(24,115),(25,114),(26,113),(27,112),(28,111),(29,110),(30,109),(31,108),(32,107),(33,106),(34,105),(35,104),(36,103),(37,102),(38,101),(39,100),(40,99),(41,98),(42,97),(43,96),(44,95),(45,94),(46,93),(47,92),(48,91),(49,90),(50,89),(51,88),(52,87),(53,86),(54,85),(55,84),(56,83),(57,82),(58,81),(59,80),(60,79),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70)])

72 conjugacy classes

class 1 2A2B2C 3  6 23A···23K46A···46K69A···69V138A···138V
order12223623···2346···4669···69138···138
size116969222···22···22···22···2

72 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D6D23D46D69D138
kernelD138D69C138C46C23C6C3C2C1
# reps1211111112222

Matrix representation of D138 in GL3(𝔽139) generated by

13800
061107
01014
,
100
0064
0630
G:=sub<GL(3,GF(139))| [138,0,0,0,61,101,0,107,4],[1,0,0,0,0,63,0,64,0] >;

D138 in GAP, Magma, Sage, TeX

D_{138}
% in TeX

G:=Group("D138");
// GroupNames label

G:=SmallGroup(276,9);
// by ID

G=gap.SmallGroup(276,9);
# by ID

G:=PCGroup([4,-2,-2,-3,-23,98,4227]);
// Polycyclic

G:=Group<a,b|a^138=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D138 in TeX

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