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 |
| C1 — C4 |
| C1 — C4 |
Generators and relations for C4
G = < a | a4=1 >
Quotients: C1, C2, C4
Character table of C4
| class | 1 | 2 | 4A | 4B | |
| size | 1 | 1 | 1 | 1 | |
| ρ1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | -i | i | linear of order 4 faithful |
| ρ4 | 1 | -1 | i | -i | linear of order 4 faithful |
►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 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 4T1 | x4-5x2+5 | 24·53 |
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
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