Properties

Label 5T1
Order \(5\)
n \(5\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive Yes
$p$-group Yes
Group: $C_5$

Related objects

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Group action invariants

Degree $n$ :  $5$
Transitive number $t$ :  $1$
Group :  $C_5$
CHM label :  $C(5) = 5$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $1$
Generators:  (1,2,3,4,5)
$|\Aut(F/K)|$:  $5$

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

Group invariants

Order:  $5$ (is prime)
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [5, 1]
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

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.