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G = C135order 135 = 33·5

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C135, also denoted Z135, SmallGroup(135,1)

Series: Derived Chief Lower central Upper central

C1 — C135
C1C3C9C45 — C135
C1 — C135
C1 — C135

Generators and relations for C135
 G = < a | a135=1 >


Smallest permutation representation of C135
Regular action 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)

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)>;

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) );

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)])

C135 is a maximal subgroup of   D135

135 conjugacy classes

class 1 3A3B5A5B5C5D9A···9F15A···15H27A···27R45A···45X135A···135BT
order13355559···915···1527···2745···45135···135
size11111111···11···11···11···11···1

135 irreducible representations

dim11111111
type+
imageC1C3C5C9C15C27C45C135
kernelC135C45C27C15C9C5C3C1
# reps12468182472

Matrix representation of C135 in GL1(𝔽271) generated by

163
G:=sub<GL(1,GF(271))| [163] >;

C135 in GAP, Magma, Sage, TeX

C_{135}
% in TeX

G:=Group("C135");
// GroupNames label

G:=SmallGroup(135,1);
// by ID

G=gap.SmallGroup(135,1);
# by ID

G:=PCGroup([4,-3,-5,-3,-3,60,46]);
// Polycyclic

G:=Group<a|a^135=1>;
// generators/relations

Export

Subgroup lattice of C135 in TeX

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