Properties

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

Related objects

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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 TypeSizeOrderRepresentative
$ 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.