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G = C₃₂  order 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

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

(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  C₃²⋊₂C₃₂  C₃²⋊C₃₂
 C_2p.C₁₆: M₆(2)  C₃⋊C₃₂  C₅⋊₂C₃₂  C₅⋊C₃₂  C₇⋊C₃₂  C₁₁⋊C₃₂  C₁₃⋊₂C₃₂  C₁₃⋊C₃₂ ...
C₃₂ is a maximal quotient of
 C₆₄  C₃²⋊₂C₃₂  C₃²⋊C₃₂
 C_p⋊C₃₂: C₃⋊C₃₂  C₅⋊₂C₃₂  C₅⋊C₃₂  C₇⋊C₃₂  C₁₁⋊C₃₂  C₁₃⋊₂C₃₂  C₁₃⋊C₃₂ ...

32 conjugacy classes

32 irreducible representations

Matrix representation of C₃₂ ►in GL₁(𝔽₉₇) generated by

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

Export

Subgroup lattice of C₃₂ in TeX