direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D126, C2×D63, C14⋊D9, C18⋊D7, C7⋊2D18, C9⋊2D14, C3.D42, C126⋊1C2, C42.2S3, C63⋊2C22, C6.2D21, C21.2D6, sometimes denoted D252 or Dih126 or Dih252, SmallGroup(252,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C63 — D126 |
Generators and relations for D126
G = < a,b | a126=b2=1, bab=a-1 >
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 | 2A | 2B | 2C | 3 | 6 | 7A | 7B | 7C | 9A | 9B | 9C | 14A | 14B | 14C | 18A | 18B | 18C | 21A | ··· | 21F | 42A | ··· | 42F | 63A | ··· | 63R | 126A | ··· | 126R |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 7 | 7 | 7 | 9 | 9 | 9 | 14 | 14 | 14 | 18 | 18 | 18 | 21 | ··· | 21 | 42 | ··· | 42 | 63 | ··· | 63 | 126 | ··· | 126 |
| size | 1 | 1 | 63 | 63 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D6 | D7 | D9 | D14 | D18 | D21 | D42 | D63 | D126 |
| kernel | D126 | D63 | C126 | C42 | C21 | C18 | C14 | C9 | C7 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 18 | 18 |
Matrix representation of D126 ►in GL3(𝔽127) generated by
| 126 | 0 | 0 |
| 0 | 50 | 108 |
| 0 | 19 | 69 |
| 1 | 0 | 0 |
| 0 | 50 | 108 |
| 0 | 58 | 77 |
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
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