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G = C54order 54 = 2·33

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C54, also denoted Z54, SmallGroup(54,2)

Series: Derived Chief Lower central Upper central

C1 — C54
C1C3C9C27 — C54
C1 — C54
C1 — C54

Generators and relations for C54
 G = < a | a54=1 >


Smallest permutation representation of C54
Regular action on 54 points
Generators in S54
(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)

G:=sub<Sym(54)| (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)>;

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) );

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)])

54 conjugacy classes

class 1  2 3A3B6A6B9A···9F18A···18F27A···27R54A···54R
order1233669···918···1827···2754···54
size1111111···11···11···11···1

54 irreducible representations

dim11111111
type++
imageC1C2C3C6C9C18C27C54
kernelC54C27C18C9C6C3C2C1
# reps1122661818

Matrix representation of C54 in GL1(𝔽109) generated by

36
G:=sub<GL(1,GF(109))| [36] >;

C54 in GAP, Magma, Sage, TeX

C_{54}
% in TeX

G:=Group("C54");
// GroupNames label

G:=SmallGroup(54,2);
// by ID

G=gap.SmallGroup(54,2);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,29,46]);
// Polycyclic

G:=Group<a|a^54=1>;
// generators/relations

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