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G = C4order 4 = 22

Cyclic group

p-group, cyclic, abelian, monomial

Aliases: C4, also denoted Z4, rotations of a square, SmallGroup(4,1)

Series: Derived Chief Lower central Upper central Jennings

C1 — C4
C1C2 — C4
C1 — C4
C1 — C4
C1C2 — C4

Generators and relations for C4
 G = < a | a4=1 >


Character table of C4

 class 124A4B
 size 1111
ρ11111    trivial
ρ211-1-1    linear of order 2
ρ31-1-ii    linear of order 4 faithful
ρ41-1i-i    linear of order 4 faithful

Permutation representations of C4
Regular action on 4 points - transitive group 4T1
Generators in S4
(1 2 3 4)

G:=sub<Sym(4)| (1,2,3,4)>;

G:=Group( (1,2,3,4) );

G=PermutationGroup([(1,2,3,4)])

G:=TransitiveGroup(4,1);

C4 is a maximal subgroup of
C8  D4  C32⋊C4  C72⋊C4  C112⋊C4
 Dicp: Q8  Dic3  Dic5  Dic7  Dic11  Dic13  Dic17  Dic19 ...
 Cp⋊C4, p=1 mod 4: F5  C13⋊C4  C17⋊C4  C29⋊C4  C37⋊C4  C41⋊C4  C53⋊C4  C61⋊C4 ...
C4 is a maximal quotient of
C8  C32⋊C4  C72⋊C4  A5⋊C4  C112⋊C4
 Dicp: Dic3  Dic5  Dic7  Dic11  Dic13  Dic17  Dic19  Dic23 ...
 Cp⋊C4, p=1 mod 4: F5  C13⋊C4  C17⋊C4  C29⋊C4  C37⋊C4  C41⋊C4  C53⋊C4  C61⋊C4 ...

Polynomial with Galois group C4 over ℚ
actionf(x)Disc(f)
4T1x4-5x2+524·53

Matrix representation of C4 in GL1(𝔽5) generated by

2
G:=sub<GL(1,GF(5))| [2] >;

C4 in GAP, Magma, Sage, TeX

C_4
% in TeX

G:=Group("C4");
// GroupNames label

G:=SmallGroup(4,1);
// by ID

G=gap.SmallGroup(4,1);
# by ID

G:=PCGroup([2,-2,-2,4]:ExponentLimit:=1);
// Polycyclic

G:=Group<a|a^4=1>;
// generators/relations

Export

Subgroup lattice of C4 in TeX
Character table of C4 in TeX

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