metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D135, C27⋊D5, C5⋊D27, C3.D45, C9.D15, C135⋊1C2, C45.1S3, C15.1D9, sometimes denoted D270 or Dih135 or Dih270, SmallGroup(270,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C135 — D135 |
Generators and relations for D135
G = < a,b | a135=b2=1, bab=a-1 >
Smallest permutation representation of D135
►On 135 points
Generators in S135
(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)
(2 135)(3 134)(4 133)(5 132)(6 131)(7 130)(8 129)(9 128)(10 127)(11 126)(12 125)(13 124)(14 123)(15 122)(16 121)(17 120)(18 119)(19 118)(20 117)(21 116)(22 115)(23 114)(24 113)(25 112)(26 111)(27 110)(28 109)(29 108)(30 107)(31 106)(32 105)(33 104)(34 103)(35 102)(36 101)(37 100)(38 99)(39 98)(40 97)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)
G:=sub<Sym(135)| (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), (2,135)(3,134)(4,133)(5,132)(6,131)(7,130)(8,129)(9,128)(10,127)(11,126)(12,125)(13,124)(14,123)(15,122)(16,121)(17,120)(18,119)(19,118)(20,117)(21,116)(22,115)(23,114)(24,113)(25,112)(26,111)(27,110)(28,109)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)>;
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), (2,135)(3,134)(4,133)(5,132)(6,131)(7,130)(8,129)(9,128)(10,127)(11,126)(12,125)(13,124)(14,123)(15,122)(16,121)(17,120)(18,119)(19,118)(20,117)(21,116)(22,115)(23,114)(24,113)(25,112)(26,111)(27,110)(28,109)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69) );
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)], [(2,135),(3,134),(4,133),(5,132),(6,131),(7,130),(8,129),(9,128),(10,127),(11,126),(12,125),(13,124),(14,123),(15,122),(16,121),(17,120),(18,119),(19,118),(20,117),(21,116),(22,115),(23,114),(24,113),(25,112),(26,111),(27,110),(28,109),(29,108),(30,107),(31,106),(32,105),(33,104),(34,103),(35,102),(36,101),(37,100),(38,99),(39,98),(40,97),(41,96),(42,95),(43,94),(44,93),(45,92),(46,91),(47,90),(48,89),(49,88),(50,87),(51,86),(52,85),(53,84),(54,83),(55,82),(56,81),(57,80),(58,79),(59,78),(60,77),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69)]])
69 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 9A | 9B | 9C | 15A | 15B | 15C | 15D | 27A | ··· | 27I | 45A | ··· | 45L | 135A | ··· | 135AJ |
| order | 1 | 2 | 3 | 5 | 5 | 9 | 9 | 9 | 15 | 15 | 15 | 15 | 27 | ··· | 27 | 45 | ··· | 45 | 135 | ··· | 135 |
| size | 1 | 135 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
69 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | D5 | D9 | D15 | D27 | D45 | D135 |
| kernel | D135 | C135 | C45 | C27 | C15 | C9 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 3 | 4 | 9 | 12 | 36 |
Matrix representation of D135 ►in GL2(𝔽271) generated by
| 264 | 245 |
| 26 | 19 |
| 1 | 0 |
| 270 | 270 |
G:=sub<GL(2,GF(271))| [264,26,245,19],[1,270,0,270] >;
D135 in GAP, Magma, Sage, TeX
D_{135} % in TeX
G:=Group("D135"); // GroupNames label
G:=SmallGroup(270,3);
// by ID
G=gap.SmallGroup(270,3);
# by ID
G:=PCGroup([5,-2,-3,-5,-3,-3,341,756,362,3003,138,4504]);
// Polycyclic
G:=Group<a,b|a^135=b^2=1,b*a*b=a^-1>;
// generators/relations
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