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G = C45order 45 = 32·5

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C45, also denoted Z45, SmallGroup(45,1)

Series: Derived Chief Lower central Upper central

C1 — C45
C1C3C15 — C45
C1 — C45
C1 — C45

Generators and relations for C45
 G = < a | a45=1 >


Smallest permutation representation of C45
Regular action on 45 points
Generators in S45
(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)

G:=sub<Sym(45)| (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)>;

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) );

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)])

45 conjugacy classes

class 1 3A3B5A5B5C5D9A···9F15A···15H45A···45X
order13355559···915···1545···45
size11111111···11···11···1

45 irreducible representations

dim111111
type+
imageC1C3C5C9C15C45
kernelC45C15C9C5C3C1
# reps1246824

Matrix representation of C45 in GL1(𝔽181) generated by

87
G:=sub<GL(1,GF(181))| [87] >;

C45 in GAP, Magma, Sage, TeX

C_{45}
% in TeX

G:=Group("C45");
// GroupNames label

G:=SmallGroup(45,1);
// by ID

G=gap.SmallGroup(45,1);
# by ID

G:=PCGroup([3,-3,-5,-3,45]);
// Polycyclic

G:=Group<a|a^45=1>;
// generators/relations

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