metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D133, C19⋊D7, C7⋊D19, C133⋊1C2, sometimes denoted D266 or Dih133 or Dih266, SmallGroup(266,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C133 — D133 |
Generators and relations for D133
G = < a,b | a133=b2=1, bab=a-1 >
Smallest permutation representation of D133
►On 133 points
Generators in S133
(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)
(1 133)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 126)(9 125)(10 124)(11 123)(12 122)(13 121)(14 120)(15 119)(16 118)(17 117)(18 116)(19 115)(20 114)(21 113)(22 112)(23 111)(24 110)(25 109)(26 108)(27 107)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)(49 85)(50 84)(51 83)(52 82)(53 81)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)
G:=sub<Sym(133)| (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), (1,133)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,120)(15,119)(16,118)(17,117)(18,116)(19,115)(20,114)(21,113)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)>;
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), (1,133)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,120)(15,119)(16,118)(17,117)(18,116)(19,115)(20,114)(21,113)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68) );
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)], [(1,133),(2,132),(3,131),(4,130),(5,129),(6,128),(7,127),(8,126),(9,125),(10,124),(11,123),(12,122),(13,121),(14,120),(15,119),(16,118),(17,117),(18,116),(19,115),(20,114),(21,113),(22,112),(23,111),(24,110),(25,109),(26,108),(27,107),(28,106),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,99),(36,98),(37,97),(38,96),(39,95),(40,94),(41,93),(42,92),(43,91),(44,90),(45,89),(46,88),(47,87),(48,86),(49,85),(50,84),(51,83),(52,82),(53,81),(54,80),(55,79),(56,78),(57,77),(58,76),(59,75),(60,74),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68)]])
68 conjugacy classes
| class | 1 | 2 | 7A | 7B | 7C | 19A | ··· | 19I | 133A | ··· | 133BB |
| order | 1 | 2 | 7 | 7 | 7 | 19 | ··· | 19 | 133 | ··· | 133 |
| size | 1 | 133 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | D7 | D19 | D133 |
| kernel | D133 | C133 | C19 | C7 | C1 |
| # reps | 1 | 1 | 3 | 9 | 54 |
Matrix representation of D133 ►in GL2(𝔽1597) generated by
| 1303 | 1004 |
| 593 | 1522 |
| 1303 | 1004 |
| 714 | 294 |
G:=sub<GL(2,GF(1597))| [1303,593,1004,1522],[1303,714,1004,294] >;
D133 in GAP, Magma, Sage, TeX
D_{133} % in TeX
G:=Group("D133"); // GroupNames label
G:=SmallGroup(266,3);
// by ID
G=gap.SmallGroup(266,3);
# by ID
G:=PCGroup([3,-2,-7,-19,73,2270]);
// Polycyclic
G:=Group<a,b|a^133=b^2=1,b*a*b=a^-1>;
// generators/relations
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