direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D130, C2×D65, C26⋊D5, C10⋊D13, C5⋊2D26, C13⋊2D10, C130⋊1C2, C65⋊2C22, sometimes denoted D260 or Dih130 or Dih260, SmallGroup(260,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C65 — D130 |
Generators and relations for D130
G = < a,b | a130=b2=1, bab=a-1 >
Smallest permutation representation of D130
►On 130 points
Generators in S130
(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)
(1 130)(2 129)(3 128)(4 127)(5 126)(6 125)(7 124)(8 123)(9 122)(10 121)(11 120)(12 119)(13 118)(14 117)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 110)(22 109)(23 108)(24 107)(25 106)(26 105)(27 104)(28 103)(29 102)(30 101)(31 100)(32 99)(33 98)(34 97)(35 96)(36 95)(37 94)(38 93)(39 92)(40 91)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(61 70)(62 69)(63 68)(64 67)(65 66)
G:=sub<Sym(130)| (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), (1,130)(2,129)(3,128)(4,127)(5,126)(6,125)(7,124)(8,123)(9,122)(10,121)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)>;
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), (1,130)(2,129)(3,128)(4,127)(5,126)(6,125)(7,124)(8,123)(9,122)(10,121)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,112)(20,111)(21,110)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66) );
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)], [(1,130),(2,129),(3,128),(4,127),(5,126),(6,125),(7,124),(8,123),(9,122),(10,121),(11,120),(12,119),(13,118),(14,117),(15,116),(16,115),(17,114),(18,113),(19,112),(20,111),(21,110),(22,109),(23,108),(24,107),(25,106),(26,105),(27,104),(28,103),(29,102),(30,101),(31,100),(32,99),(33,98),(34,97),(35,96),(36,95),(37,94),(38,93),(39,92),(40,91),(41,90),(42,89),(43,88),(44,87),(45,86),(46,85),(47,84),(48,83),(49,82),(50,81),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,72),(60,71),(61,70),(62,69),(63,68),(64,67),(65,66)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 10A | 10B | 13A | ··· | 13F | 26A | ··· | 26F | 65A | ··· | 65X | 130A | ··· | 130X |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 13 | ··· | 13 | 26 | ··· | 26 | 65 | ··· | 65 | 130 | ··· | 130 |
| size | 1 | 1 | 65 | 65 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D5 | D10 | D13 | D26 | D65 | D130 |
| kernel | D130 | D65 | C130 | C26 | C13 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 6 | 6 | 24 | 24 |
Matrix representation of D130 ►in GL3(𝔽131) generated by
| 130 | 0 | 0 |
| 0 | 83 | 89 |
| 0 | 93 | 95 |
| 1 | 0 | 0 |
| 0 | 56 | 52 |
| 0 | 43 | 75 |
G:=sub<GL(3,GF(131))| [130,0,0,0,83,93,0,89,95],[1,0,0,0,56,43,0,52,75] >;
D130 in GAP, Magma, Sage, TeX
D_{130} % in TeX
G:=Group("D130"); // GroupNames label
G:=SmallGroup(260,14);
// by ID
G=gap.SmallGroup(260,14);
# by ID
G:=PCGroup([4,-2,-2,-5,-13,194,3843]);
// Polycyclic
G:=Group<a,b|a^130=b^2=1,b*a*b=a^-1>;
// generators/relations
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