# Properties

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

# Related objects

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); Nilpotency class: $-1$ (not nilpotent) 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

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2$ $5$ $2$ $( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7)$ $5, 5$ $2$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$ $5, 5$ $2$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$

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); Label: 10.1 magma: IdentifyGroup(G);
 Character table:  2 1 1 . . 5 1 . 1 1 1a 2a 5a 5b 2P 1a 1a 5b 5a 3P 1a 2a 5b 5a 5P 1a 2a 1a 1a X.1 1 1 1 1 X.2 1 -1 1 1 X.3 2 . A *A X.4 2 . *A A A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 

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 $\oplus$ $(A+A',L)$ $\cong$ $(A,L)$ $\oplus$ $(A',L)$