direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D138, C2×D69, C46⋊S3, C6⋊D23, C3⋊2D46, C23⋊2D6, C138⋊1C2, C69⋊2C22, sometimes denoted D276 or Dih138 or Dih276, SmallGroup(276,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C69 — D138 |
Generators and relations for D138
G = < a,b | a138=b2=1, bab=a-1 >
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 | 2A | 2B | 2C | 3 | 6 | 23A | ··· | 23K | 46A | ··· | 46K | 69A | ··· | 69V | 138A | ··· | 138V |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 | 138 | ··· | 138 |
| size | 1 | 1 | 69 | 69 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D6 | D23 | D46 | D69 | D138 |
| kernel | D138 | D69 | C138 | C46 | C23 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 11 | 11 | 22 | 22 |
Matrix representation of D138 ►in GL3(𝔽139) generated by
| 138 | 0 | 0 |
| 0 | 61 | 107 |
| 0 | 101 | 4 |
| 1 | 0 | 0 |
| 0 | 0 | 64 |
| 0 | 63 | 0 |
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
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