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## G = C32order 32 = 25

### Cyclic group

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32
 Chief series C1 — C2 — C4 — C8 — C16 — C32
 Lower central C1 — C32
 Upper central C1 — C32
 Jennings C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C16 — 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)])

C32 is a maximal subgroup of
C64  D32  SD64  Q64  C322C32  C32⋊C32
C2p.C16: M6(2)  C3⋊C32  C52C32  C5⋊C32  C7⋊C32  C11⋊C32  C132C32  C13⋊C32 ...
C32 is a maximal quotient of
C64  C322C32  C32⋊C32
Cp⋊C32: C3⋊C32  C52C32  C5⋊C32  C7⋊C32  C11⋊C32  C132C32  C13⋊C32 ...

32 conjugacy classes

 class 1 2 4A 4B 8A 8B 8C 8D 16A ··· 16H 32A ··· 32P order 1 2 4 4 8 8 8 8 16 ··· 16 32 ··· 32 size 1 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1

32 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C4 C8 C16 C32 kernel C32 C16 C8 C4 C2 C1 # reps 1 1 2 4 8 16

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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