Properties

Label 5T1
Degree $5$
Order $5$
Cyclic yes
Abelian yes
Solvable yes
Primitive yes
$p$-group yes
Group: $C_5$

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Show commands: Magma

magma: G := TransitiveGroup(5, 1);
 

Group action invariants

Degree $n$:  $5$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5$
CHM label:  $C(5) = 5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 5 $ $1$ $5$ $(1,2,3,4,5)$
$ 5 $ $1$ $5$ $(1,3,5,2,4)$
$ 5 $ $1$ $5$ $(1,4,2,5,3)$
$ 5 $ $1$ $5$ $(1,5,4,3,2)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $5$ (is prime)
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  5.1
magma: IdentifyGroup(G);
 
Character table:   
     5  1  1  1  1  1

       1a 5a 5b 5c 5d

X.1     1  1  1  1  1
X.2     1  A  B /B /A
X.3     1  B /A  A /B
X.4     1 /B  A /A  B
X.5     1 /A /B  B  A

A = E(5)
B = E(5)^2

magma: CharacterTable(G);
 

Indecomposable integral representations

Complete list of indecomposable integral representations:

Name Dim $(1,2,3,4,5) \mapsto $
Triv $1$ $\left(\begin{array}{r}1\end{array}\right)$
$J$ $4$ $\left(\begin{array}{rrrr}0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\end{array}\right)$
$R$ $5$ $\left(\begin{array}{rrrrr}0 & 1 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 1\\1 & 0 & 0 & 0 & 0\end{array}\right)$
The decomposition of an arbitrary integral representation as a direct sum of indecomposables is unique.