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Properties

Label 10T2
Degree $10$
Order $10$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5$

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Show commands: Magma

magma: G := TransitiveGroup(10, 2);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5$
CHM label:   $D(10)=5:2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,4)(2,3)(5,10)(6,9)(7,8), (1,3,5,7,9)(2,4,6,8,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $D_{5}$

Low degree siblings

5T2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7)$
$ 5, 5 $ $2$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 5, 5 $ $2$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$

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, 3, 5, 7, 9)( 2, 4, 6, 8,10) \mapsto $ $(1,4)(2,3)(5,10)(6,9)(7,8) \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 $\oplus$ $(A+A',L)$ $\cong$ $(A,L)$ $\oplus$ $(A',L)$