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Properties

Label 10T3
Degree $10$
Order $20$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(10, 3);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$10$:  $D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $D_{5}$

Low degree siblings

10T3, 20T4

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $20=2^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  20.4
magma: IdentifyGroup(G);
 
Character table:    
     2  2  2  2   1  1   1  1  2
     5  1  .  .   1  1   1  1  1

       1a 2a 2b 10a 5a 10b 5b 2c
    2P 1a 1a 1a  5a 5b  5b 5a 1a
    3P 1a 2a 2b 10b 5b 10a 5a 2c
    5P 1a 2a 2b  2c 1a  2c 1a 2c
    7P 1a 2a 2b 10b 5b 10a 5a 2c

X.1     1  1  1   1  1   1  1  1
X.2     1 -1 -1   1  1   1  1  1
X.3     1 -1  1  -1  1  -1  1 -1
X.4     1  1 -1  -1  1  -1  1 -1
X.5     2  .  .   A *A  *A  A  2
X.6     2  .  .  *A  A   A *A  2
X.7     2  .  .  -A *A -*A  A -2
X.8     2  .  . -*A  A  -A *A -2

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5

magma: CharacterTable(G);