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G = C50order 50 = 2·52

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C50, also denoted Z50, SmallGroup(50,2)

Series: Derived Chief Lower central Upper central

C1 — C50
C1C5C25 — C50
C1 — C50
C1 — C50

Generators and relations for C50
 G = < a | a50=1 >


Smallest permutation representation of C50
Regular action on 50 points
Generators in S50
(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)

G:=sub<Sym(50)| (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)>;

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) );

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)])

50 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D25A···25T50A···50T
order1255551010101025···2550···50
size11111111111···11···1

50 irreducible representations

dim111111
type++
imageC1C2C5C10C25C50
kernelC50C25C10C5C2C1
# reps11442020

Matrix representation of C50 in GL2(𝔽101) generated by

140
088
G:=sub<GL(2,GF(101))| [14,0,0,88] >;

C50 in GAP, Magma, Sage, TeX

C_{50}
% in TeX

G:=Group("C50");
// GroupNames label

G:=SmallGroup(50,2);
// by ID

G=gap.SmallGroup(50,2);
# by ID

G:=PCGroup([3,-2,-5,-5,34]);
// Polycyclic

G:=Group<a|a^50=1>;
// generators/relations

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