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G = D135order 270 = 2·33·5

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

C1C135 — D135
C1C3C9C45C135 — D135
C135 — D135
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 5A5B9A9B9C15A15B15C15D27A···27I45A···45L135A···135AJ
order123559991515151527···2745···45135···135
size113522222222222···22···22···2

69 irreducible representations

dim112222222
type+++++++++
imageC1C2S3D5D9D15D27D45D135
kernelD135C135C45C27C15C9C5C3C1
# reps11123491236

Matrix representation of D135 in GL2(𝔽271) generated by

264245
2619
,
10
270270
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

Export

Subgroup lattice of D135 in TeX

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