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G = C55order 55 = 5·11

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C55, also denoted Z55, SmallGroup(55,2)

Series: Derived Chief Lower central Upper central

C1 — C55
C1C11 — C55
C1 — C55
C1 — C55

Generators and relations for C55
 G = < a | a55=1 >


Smallest permutation representation of C55
Regular action on 55 points
Generators in S55
(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)

G:=sub<Sym(55)| (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)>;

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) );

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)])

55 conjugacy classes

class 1 5A5B5C5D11A···11J55A···55AN
order1555511···1155···55
size111111···11···1

55 irreducible representations

dim1111
type+
imageC1C5C11C55
kernelC55C11C5C1
# reps141040

Matrix representation of C55 in GL1(𝔽331) generated by

258
G:=sub<GL(1,GF(331))| [258] >;

C55 in GAP, Magma, Sage, TeX

C_{55}
% in TeX

G:=Group("C55");
// GroupNames label

G:=SmallGroup(55,2);
// by ID

G=gap.SmallGroup(55,2);
# by ID

G:=PCGroup([2,-5,-11]);
// Polycyclic

G:=Group<a|a^55=1>;
// generators/relations

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