metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D131, C131⋊C2, sometimes denoted D262 or Dih131 or Dih262, SmallGroup(262,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C131 — D131 |
Generators and relations for D131
G = < a,b | a131=b2=1, bab=a-1 >
Smallest permutation representation of D131
►On 131 points: primitive
Generators in S131
(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)
(1 131)(2 130)(3 129)(4 128)(5 127)(6 126)(7 125)(8 124)(9 123)(10 122)(11 121)(12 120)(13 119)(14 118)(15 117)(16 116)(17 115)(18 114)(19 113)(20 112)(21 111)(22 110)(23 109)(24 108)(25 107)(26 106)(27 105)(28 104)(29 103)(30 102)(31 101)(32 100)(33 99)(34 98)(35 97)(36 96)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 84)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)
G:=sub<Sym(131)| (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), (1,131)(2,130)(3,129)(4,128)(5,127)(6,126)(7,125)(8,124)(9,123)(10,122)(11,121)(12,120)(13,119)(14,118)(15,117)(16,116)(17,115)(18,114)(19,113)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,103)(30,102)(31,101)(32,100)(33,99)(34,98)(35,97)(36,96)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)>;
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), (1,131)(2,130)(3,129)(4,128)(5,127)(6,126)(7,125)(8,124)(9,123)(10,122)(11,121)(12,120)(13,119)(14,118)(15,117)(16,116)(17,115)(18,114)(19,113)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,103)(30,102)(31,101)(32,100)(33,99)(34,98)(35,97)(36,96)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67) );
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)], [(1,131),(2,130),(3,129),(4,128),(5,127),(6,126),(7,125),(8,124),(9,123),(10,122),(11,121),(12,120),(13,119),(14,118),(15,117),(16,116),(17,115),(18,114),(19,113),(20,112),(21,111),(22,110),(23,109),(24,108),(25,107),(26,106),(27,105),(28,104),(29,103),(30,102),(31,101),(32,100),(33,99),(34,98),(35,97),(36,96),(37,95),(38,94),(39,93),(40,92),(41,91),(42,90),(43,89),(44,88),(45,87),(46,86),(47,85),(48,84),(49,83),(50,82),(51,81),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67)]])
67 conjugacy classes
| class | 1 | 2 | 131A | ··· | 131BM |
| order | 1 | 2 | 131 | ··· | 131 |
| size | 1 | 131 | 2 | ··· | 2 |
67 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D131 |
| kernel | D131 | C131 | C1 |
| # reps | 1 | 1 | 65 |
Matrix representation of D131 ►in GL2(𝔽263) generated by
| 238 | 262 |
| 8 | 116 |
| 153 | 136 |
| 29 | 110 |
G:=sub<GL(2,GF(263))| [238,8,262,116],[153,29,136,110] >;
D131 in GAP, Magma, Sage, TeX
D_{131} % in TeX
G:=Group("D131"); // GroupNames label
G:=SmallGroup(262,1);
// by ID
G=gap.SmallGroup(262,1);
# by ID
G:=PCGroup([2,-2,-131,1041]);
// Polycyclic
G:=Group<a,b|a^131=b^2=1,b*a*b=a^-1>;
// generators/relations
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