# Properties

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

# Related objects

## 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 Type Size Order Representative $1, 1, 1, 1, 1$ $1$ $1$ $()$ $5$ $1$ $5$ $(1,2,3,4,5)$ $5$ $1$ $5$ $(1,3,5,2,4)$ $5$ $1$ $5$ $(1,4,2,5,3)$ $5$ $1$ $5$ $(1,5,4,3,2)$

## 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.