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Properties

Label 5T2
Degree $5$
Order $10$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $D_{5}$

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Show commands: Magma

magma: G := TransitiveGroup(5, 2);
 

Group action invariants

Degree $n$:  $5$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{5}$
CHM label:   $D(5) = 5:2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5), (1,4)(2,3)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

10T2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{5}$ $1$ $1$ $0$ $()$
2A $2^{2},1$ $5$ $2$ $2$ $(2,5)(3,4)$
5A1 $5$ $2$ $5$ $4$ $(1,2,3,4,5)$
5A2 $5$ $2$ $5$ $4$ $(1,3,5,2,4)$

Malle's constant $a(G)$:     $1/2$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $10=2 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  10.1
magma: IdentifyGroup(G);
 
Character table:

1A 2A 5A1 5A2
Size 1 5 2 2
2 P 1A 1A 5A2 5A1
5 P 1A 2A 1A 1A
Type
10.1.1a R 1 1 1 1
10.1.1b R 1 −1 1 1
10.1.2a1 R 2 0 ζ5−2+ζ52 ζ5−1+ζ5
10.1.2a2 R 2 0 ζ5−1+ζ5 ζ5−2+ζ52

magma: CharacterTable(G);
 

Indecomposable integral representations

Complete list of indecomposable integral representations:

Name Dim $(1,2,3,4,5) \mapsto $ $(2,5)(3,4) \mapsto $
Triv $1$ $\left(\begin{array}{r}1\end{array}\right)$ $\left(\begin{array}{r}1\end{array}\right)$
Sign $1$ $\left(\begin{array}{r}1\end{array}\right)$ $\left(\begin{array}{r}-1\end{array}\right)$
$L$ $2$ $\left(\begin{array}{rr}1 & 0\\0 & 1\end{array}\right)$ $\left(\begin{array}{rr}0 & 1\\1 & 0\end{array}\right)$
$A$ $4$ $\left(\begin{array}{rrrr}0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\end{array}\right)$ $\left(\begin{array}{rrrr}1 & 0 & 0 & 0\\-1 & -1 & -1 & -1\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\end{array}\right)$
$A'$ $4$ $\left(\begin{array}{rrrr}0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\\-1 & -1 & -1 & -1\end{array}\right)$ $\left(\begin{array}{rrrr}-1 & 0 & 0 & 0\\1 & 1 & 1 & 1\\0 & 0 & 0 & -1\\0 & 0 & -1 & 0\end{array}\right)$
$(A,\textrm{Sign})$ $5$ $\left(\begin{array}{rrrrr}0 & 1 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 1 & 0\\-1 & -1 & -1 & -1 & 0\\1 & 0 & 0 & 0 & 1\end{array}\right)$ $\left(\begin{array}{rrrrr}1 & 0 & 0 & 0 & 0\\-1 & -1 & -1 & -1 & 0\\0 & 0 & 0 & 1 & 0\\0 & 0 & 1 & 0 & 0\\-1 & 0 & 0 & 0 & -1\end{array}\right)$
$(A',\textrm{Triv})$ $5$ $\left(\begin{array}{rrrrr}0 & 1 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 1 & 0\\-1 & -1 & -1 & -1 & 0\\1 & 0 & 0 & 0 & 1\end{array}\right)$ $\left(\begin{array}{rrrrr}-1 & 0 & 0 & 0 & 0\\1 & 1 & 1 & 1 & 0\\0 & 0 & 0 & -1 & 0\\0 & 0 & -1 & 0 & 0\\1 & 0 & 0 & 0 & 1\end{array}\right)$
$(A,L)$ $6$ $\left(\begin{array}{rrrrrr}0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 1 & 0 & 0\\-1 & -1 & -1 & -1 & 0 & 0\\-1 & 0 & 0 & 0 & 1 & 0\\1 & 0 & 0 & 0 & 0 & 1\end{array}\right)$ $\left(\begin{array}{rrrrrr}1 & 0 & 0 & 0 & 0 & 0\\-1 & -1 & -1 & -1 & 0 & 0\\0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0\\1 & 0 & 0 & 0 & 0 & 1\\-1 & 0 & 0 & 0 & 1 & 0\end{array}\right)$
$(A',L)$ $6$ $\left(\begin{array}{rrrrrr}0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 1 & 0 & 0\\-1 & -1 & -1 & -1 & 0 & 0\\1 & 0 & 0 & 0 & 1 & 0\\1 & 0 & 0 & 0 & 0 & 1\end{array}\right)$ $\left(\begin{array}{rrrrrr}-1 & 0 & 0 & 0 & 0 & 0\\1 & 1 & 1 & 1 & 0 & 0\\0 & 0 & 0 & -1 & 0 & 0\\0 & 0 & -1 & 0 & 0 & 0\\1 & 0 & 0 & 0 & 0 & 1\\1 & 0 & 0 & 0 & 1 & 0\end{array}\right)$
$(A+A',L)$ $10$ $\left(\begin{array}{rrrrrrrrrr}0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{array}\right)$ $\left(\begin{array}{rrrrrrrrrr}0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right)$
The decomposition of an arbitrary integral representation as a direct sum of indecomposables is not unique, in general. It is unique up to the following isomorphisms:
Triv $\oplus$ $(A',L)$ $\cong$ $L$ $\oplus$ $(A',\textrm{Triv})$
Sign $\oplus$ $(A,L)$ $\cong$ $L$ $\oplus$ $(A,\textrm{Sign})$
Triv $\oplus$ $(A+A',L)$ $\cong$ $(A,L)$ $\oplus$ $(A',L)$