Galois Group: 3T1

• Label: 3T1
• Order: \(3\)
• n: \(3\)
• Cyclic: Yes
• Abelian: Yes
• Solvable: Yes
• Primitive: Yes
• $p$-group: Yes
• Group: $C_3$

Group action invariants

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

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.