# Properties

 Label 3T1 Degree $3$ Order $3$ Cyclic yes Abelian yes Solvable yes Primitive yes $p$-group yes Group: $C_3$

# Related objects

## Group action invariants

 Degree $n$: $3$ Transitive number $t$: $1$ Group: $C_3$ CHM label: $A3$ Parity: $1$ Primitive: yes Nilpotency class: $1$ $|\Aut(F/K)|$: $3$ Generators: (1,2,3)

## 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 $1, 1, 1$ $1$ $1$ $()$ $3$ $1$ $3$ $(1,2,3)$ $3$ $1$ $3$ $(1,3,2)$

## Group invariants

 Order: $3$ (is prime) Cyclic: yes Abelian: yes Solvable: yes GAP id: [3, 1]
 Character table:  3 1 1 1 1a 3a 3b X.1 1 1 1 X.2 1 A /A X.3 1 /A A A = E(3) = (-1+Sqrt(-3))/2 = b3 

## Indecomposable integral representations

Complete list of indecomposable integral representations:

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