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G = C52order 52 = 22·13

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C52, also denoted Z52, SmallGroup(52,2)

Series: Derived Chief Lower central Upper central

C1 — C52
C1C2C26 — C52
C1 — C52
C1 — C52

Generators and relations for C52
 G = < a | a52=1 >


Smallest permutation representation of C52
Regular action on 52 points
Generators in S52
(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)

G:=sub<Sym(52)| (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)>;

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) );

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)])

52 conjugacy classes

class 1  2 4A4B13A···13L26A···26L52A···52X
order124413···1326···2652···52
size11111···11···11···1

52 irreducible representations

dim111111
type++
imageC1C2C4C13C26C52
kernelC52C26C13C4C2C1
# reps112121224

Matrix representation of C52 in GL1(𝔽53) generated by

14
G:=sub<GL(1,GF(53))| [14] >;

C52 in GAP, Magma, Sage, TeX

C_{52}
% in TeX

G:=Group("C52");
// GroupNames label

G:=SmallGroup(52,2);
// by ID

G=gap.SmallGroup(52,2);
# by ID

G:=PCGroup([3,-2,-13,-2,78]);
// Polycyclic

G:=Group<a|a^52=1>;
// generators/relations

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