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## G = D135order 270 = 2·33·5

### Dihedral group

Aliases: D135, C27⋊D5, C5⋊D27, C3.D45, C9.D15, C1351C2, C45.1S3, C15.1D9, sometimes denoted D270 or Dih135 or Dih270, SmallGroup(270,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C135 — D135
 Chief series C1 — C3 — C9 — C45 — C135 — D135
 Lower central C135 — D135
 Upper central C1

Generators and relations for D135
G = < a,b | a135=b2=1, bab=a-1 >

135C2
45S3
27D5
15D9
9D15
5D27
3D45

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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