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G = C32order 32 = 25

Cyclic group

p-group, cyclic, abelian, monomial

Aliases: C32, also denoted Z32, SmallGroup(32,1)

Series: Derived Chief Lower central Upper central Jennings

C1 — C32
C1C2C4C8C16 — C32
C1 — C32
C1 — C32
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — C32

Generators and relations for C32
 G = < a | a32=1 >


Smallest permutation representation of C32
Regular action on 32 points
Generators in S32
(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)

G:=sub<Sym(32)| (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)>;

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) );

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)])

32 conjugacy classes

class 1  2 4A4B8A8B8C8D16A···16H32A···32P
order1244888816···1632···32
size111111111···11···1

32 irreducible representations

dim111111
type++
imageC1C2C4C8C16C32
kernelC32C16C8C4C2C1
# reps1124816

Matrix representation of C32 in GL1(𝔽97) generated by

20
G:=sub<GL(1,GF(97))| [20] >;

C32 in GAP, Magma, Sage, TeX

C_{32}
% in TeX

G:=Group("C32");
// GroupNames label

G:=SmallGroup(32,1);
// by ID

G=gap.SmallGroup(32,1);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-2,10,26,42,58]);
// Polycyclic

G:=Group<a|a^32=1>;
// generators/relations

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