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G = C57order 57 = 3·19

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C57, also denoted Z57, SmallGroup(57,2)

Series: Derived Chief Lower central Upper central

C1 — C57
C1C19 — C57
C1 — C57
C1 — C57

Generators and relations for C57
 G = < a | a57=1 >


Smallest permutation representation of C57
Regular action on 57 points
Generators in S57
(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)

G:=sub<Sym(57)| (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)>;

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) );

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)])

57 conjugacy classes

class 1 3A3B19A···19R57A···57AJ
order13319···1957···57
size1111···11···1

57 irreducible representations

dim1111
type+
imageC1C3C19C57
kernelC57C19C3C1
# reps121836

Matrix representation of C57 in GL1(𝔽229) generated by

193
G:=sub<GL(1,GF(229))| [193] >;

C57 in GAP, Magma, Sage, TeX

C_{57}
% in TeX

G:=Group("C57");
// GroupNames label

G:=SmallGroup(57,2);
// by ID

G=gap.SmallGroup(57,2);
# by ID

G:=PCGroup([2,-3,-19]);
// Polycyclic

G:=Group<a|a^57=1>;
// generators/relations

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