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
| C67 — D134 |
Generators and relations for D134
G = < a,b | a134=b2=1, bab=a-1 >
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 | 2A | 2B | 2C | 67A | ··· | 67AG | 134A | ··· | 134AG |
| order | 1 | 2 | 2 | 2 | 67 | ··· | 67 | 134 | ··· | 134 |
| size | 1 | 1 | 67 | 67 | 2 | ··· | 2 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D67 | D134 |
| kernel | D134 | D67 | C134 | C2 | C1 |
| # reps | 1 | 2 | 1 | 33 | 33 |
Matrix representation of D134 ►in GL3(𝔽269) generated by
| 268 | 0 | 0 |
| 0 | 93 | 261 |
| 0 | 8 | 8 |
| 1 | 0 | 0 |
| 0 | 93 | 261 |
| 0 | 5 | 176 |
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
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