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Properties

Label 5T2
Order \(10\)
n \(5\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $D_{5}$

Related objects

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Group action invariants

Degree $n$ :  $5$
Transitive number $t$ :  $2$
Group :  $D_{5}$
CHM label :  $D(5) = 5:2$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3,4,5), (1,4)(2,3)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|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

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1 $ $5$ $2$ $(2,5)(3,4)$
$ 5 $ $2$ $5$ $(1,2,3,4,5)$
$ 5 $ $2$ $5$ $(1,3,5,2,4)$

Group invariants

Order:  $10=2 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [10, 1]
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

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