Properties

Label 10T4
Degree $10$
Order $20$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_5$

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Show commands: Magma

magma: G := TransitiveGroup(10, 4);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_5$
CHM label:   $1/2[F(5)]2$
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:  (1,3,5,7,9)(2,4,6,8,10), (1,2,9,8)(3,6,7,4)(5,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $F_5$

Low degree siblings

5T3, 20T5

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
1A $1^{10}$ $1$ $1$ $()$
2A $2^{4},1^{2}$ $5$ $2$ $(1,9)(2,8)(3,7)(4,6)$
4A1 $4^{2},2$ $5$ $4$ $( 1, 2, 9, 8)( 3, 6, 7, 4)( 5,10)$
4A-1 $4^{2},2$ $5$ $4$ $( 1, 8, 9, 2)( 3, 4, 7, 6)( 5,10)$
5A $5^{2}$ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$

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:  20.3
magma: IdentifyGroup(G);
 
Character table:

1A 2A 4A1 4A-1 5A
Size 1 5 5 5 4
2 P 1A 1A 2A 2A 5A
5 P 1A 2A 4A1 4A-1 1A
Type
20.3.1a R 1 1 1 1 1
20.3.1b R 1 1 1 1 1
20.3.1c1 C 1 1 i i 1
20.3.1c2 C 1 1 i i 1
20.3.4a R 4 0 0 0 1

magma: CharacterTable(G);