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G = C60order 60 = 22·3·5

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C60, also denoted Z60, SmallGroup(60,4)

Series: Derived Chief Lower central Upper central

C1 — C60
C1C2C10C30 — C60
C1 — C60
C1 — C60

Generators and relations for C60
 G = < a | a60=1 >


Smallest permutation representation of C60
Regular action on 60 points
Generators in S60
(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)

G:=sub<Sym(60)| (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)>;

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) );

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)])

60 conjugacy classes

class 1  2 3A3B4A4B5A5B5C5D6A6B10A10B10C10D12A12B12C12D15A···15H20A···20H30A···30H60A···60P
order123344555566101010101212121215···1520···2030···3060···60
size111111111111111111111···11···11···11···1

60 irreducible representations

dim111111111111
type++
imageC1C2C3C4C5C6C10C12C15C20C30C60
kernelC60C30C20C15C12C10C6C5C4C3C2C1
# reps1122424488816

Matrix representation of C60 in GL1(𝔽61) generated by

2
G:=sub<GL(1,GF(61))| [2] >;

C60 in GAP, Magma, Sage, TeX

C_{60}
% in TeX

G:=Group("C60");
// GroupNames label

G:=SmallGroup(60,4);
// by ID

G=gap.SmallGroup(60,4);
# by ID

G:=PCGroup([4,-2,-3,-5,-2,120]);
// Polycyclic

G:=Group<a|a^60=1>;
// generators/relations

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