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G = D126order 252 = 22·32·7

Dihedral group

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

Aliases: D126, C2×D63, C14⋊D9, C18⋊D7, C72D18, C92D14, C3.D42, C1261C2, C42.2S3, C632C22, C6.2D21, C21.2D6, sometimes denoted D252 or Dih126 or Dih252, SmallGroup(252,14)

Series: Derived Chief Lower central Upper central

C1C63 — D126
C1C3C21C63D63 — D126
C63 — D126
C1C2

Generators and relations for D126
 G = < a,b | a126=b2=1, bab=a-1 >

63C2
63C2
63C22
21S3
21S3
9D7
9D7
21D6
7D9
7D9
9D14
3D21
3D21
7D18
3D42

Smallest permutation representation of D126
On 126 points
Generators in S126
(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)
(1 126)(2 125)(3 124)(4 123)(5 122)(6 121)(7 120)(8 119)(9 118)(10 117)(11 116)(12 115)(13 114)(14 113)(15 112)(16 111)(17 110)(18 109)(19 108)(20 107)(21 106)(22 105)(23 104)(24 103)(25 102)(26 101)(27 100)(28 99)(29 98)(30 97)(31 96)(32 95)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)

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

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

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

66 conjugacy classes

class 1 2A2B2C 3  6 7A7B7C9A9B9C14A14B14C18A18B18C21A···21F42A···42F63A···63R126A···126R
order12223677799914141418181821···2142···4263···63126···126
size116363222222222222222···22···22···22···2

66 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3D6D7D9D14D18D21D42D63D126
kernelD126D63C126C42C21C18C14C9C7C6C3C2C1
# reps121113333661818

Matrix representation of D126 in GL3(𝔽127) generated by

12600
050108
01969
,
100
050108
05877
G:=sub<GL(3,GF(127))| [126,0,0,0,50,19,0,108,69],[1,0,0,0,50,58,0,108,77] >;

D126 in GAP, Magma, Sage, TeX

D_{126}
% in TeX

G:=Group("D126");
// GroupNames label

G:=SmallGroup(252,14);
// by ID

G=gap.SmallGroup(252,14);
# by ID

G:=PCGroup([5,-2,-2,-3,-7,-3,1382,642,1443,4204]);
// Polycyclic

G:=Group<a,b|a^126=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D126 in TeX

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