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G = C51order 51 = 3·17

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C51, also denoted Z51, SmallGroup(51,1)

Series: Derived Chief Lower central Upper central

C1 — C51
C1C17 — C51
C1 — C51
C1 — C51

Generators and relations for C51
 G = < a | a51=1 >


Smallest permutation representation of C51
Regular action on 51 points
Generators in S51
(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)

G:=sub<Sym(51)| (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)>;

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) );

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)])

51 conjugacy classes

class 1 3A3B17A···17P51A···51AF
order13317···1751···51
size1111···11···1

51 irreducible representations

dim1111
type+
imageC1C3C17C51
kernelC51C17C3C1
# reps121632

Matrix representation of C51 in GL1(𝔽103) generated by

38
G:=sub<GL(1,GF(103))| [38] >;

C51 in GAP, Magma, Sage, TeX

C_{51}
% in TeX

G:=Group("C51");
// GroupNames label

G:=SmallGroup(51,1);
// by ID

G=gap.SmallGroup(51,1);
# by ID

G:=PCGroup([2,-3,-17]);
// Polycyclic

G:=Group<a|a^51=1>;
// generators/relations

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