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Group action invariants

Degree $n$:		$4$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$2$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_2^2$
CHM label:		$E(4) = 2[x]2$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$4$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,2)(3,4), (1,4)(2,3)	magma: Generators(G);

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{4}$	$1$	$1$	$0$	$()$
2A	$2^{2}$	$1$	$2$	$2$	$(1,2)(3,4)$
2B	$2^{2}$	$1$	$2$	$2$	$(1,4)(2,3)$
2C	$2^{2}$	$1$	$2$	$2$	$(1,3)(2,4)$

Malle's constant $a(G)$:     $1/2$

Group invariants

		1A	2A	2B	2C
	Size	1	1	1	1
	2 P	1A	1A	1A	1A
	Type
4.2.1a	R	$1$	$1$	$1$	$1$
4.2.1b	R	$1$	$-1$	$-1$	$1$
4.2.1c	R	$1$	$-1$	$1$	$-1$
4.2.1d	R	$1$	$1$	$-1$	$-1$

Indecomposable integral representations

Partial list of indecomposable integral representations: Name Dim $(1,2)(3,4) \mapsto $ $(1,4)(2,3) \mapsto $ Triv $1$ $\left(\begin{array}{r}1\end{array}\right)$ $\left(\begin{array}{r}1\end{array}\right)$ $A_1$ $1$ $\left(\begin{array}{r}1\end{array}\right)$ $\left(\begin{array}{r}-1\end{array}\right)$ $A_2$ $1$ $\left(\begin{array}{r}-1\end{array}\right)$ $\left(\begin{array}{r}1\end{array}\right)$ $A_3$ $1$ $\left(\begin{array}{r}-1\end{array}\right)$ $\left(\begin{array}{r}-1\end{array}\right)$ $B_1$ $2$ $\left(\begin{array}{rr}-1 & 0\\0 & -1\end{array}\right)$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ $B_2$ $2$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ $\left(\begin{array}{rr}-1 & 0\\0 & -1\end{array}\right)$ $B_3$ $2$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ $\left(\begin{array}{rr}0 & -1\\-1 & 0\end{array}\right)$ $A_2B_1$ $2$ $\left(\begin{array}{rr}1 & 0\\0 & 1\end{array}\right)$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ $A_3B_1$ $2$ $\left(\begin{array}{rr}1 & 0\\0 & 1\end{array}\right)$ $\left(\begin{array}{rr}0 & -1\\-1 & 0\end{array}\right)$ $A_1B_2$ $2$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ $\left(\begin{array}{rr}1 & 0\\0 & 1\end{array}\right)$ $A_3B_2$ $2$ $\left(\begin{array}{rr}0 & -1\\-1 & 0\end{array}\right)$ $\left(\begin{array}{rr}1 & 0\\0 & 1\end{array}\right)$ $A_1B_3$ $2$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$ $A_2B_3$ $2$ $\left(\begin{array}{rr}0 & -1\\-1 & 0\end{array}\right)$ $\left(\begin{array}{rr}0 & -1\\-1 & 0\end{array}\right)$ $J$ $3$ $\left(\begin{array}{rrr}0 & -1 & 1\\0 & -1 & 0\\1 & -1 & 0\end{array}\right)$ $\left(\begin{array}{rrr}0 & 1 & -1\\1 & 0 & -1\\0 & 0 & -1\end{array}\right)$ $J'$ $3$ $\left(\begin{array}{rrr}0 & 0 & 1\\-1 & -1 & -1\\1 & 0 & 0\end{array}\right)$ $\left(\begin{array}{rrr}0 & 1 & 0\\1 & 0 & 0\\-1 & -1 & -1\end{array}\right)$

The decomposition of an arbitrary integral representation as a direct sum of indecomposables is unique.

Additional information

This is the smallest transitive permutation group on $n$ elements which does not contain an $n$-cycle.

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