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Properties

Label 10T7
Degree $10$
Order $60$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_{5}$

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Show commands: Magma

magma: G := TransitiveGroup(10, 7);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_{5}$
CHM label:   $A_{5}(10)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,9)(3,4)(5,10)(6,7), (1,3,5,7,9)(2,4,6,8,10)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: None

Low degree siblings

5T4, 6T12, 12T33, 15T5, 20T15, 30T9

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 $ $1$ $1$ $()$
$ 3, 3, 3, 1 $ $20$ $3$ $( 2, 4, 5)( 3, 6, 9)( 7, 8,10)$
$ 2, 2, 2, 2, 1, 1 $ $15$ $2$ $( 2, 7)( 4,10)( 5, 8)( 6, 9)$
$ 5, 5 $ $12$ $5$ $( 1, 2, 3, 6, 8)( 4, 9, 7, 5,10)$
$ 5, 5 $ $12$ $5$ $( 1, 2, 9, 6, 7)( 3, 8, 4,10, 5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  60.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 5A1 5A2
Size 1 15 20 12 12
2 P 1A 1A 3A 5A2 5A1
3 P 1A 2A 1A 5A2 5A1
5 P 1A 2A 3A 1A 1A
Type
60.5.1a R 1 1 1 1 1
60.5.3a1 R 3 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52
60.5.3a2 R 3 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5
60.5.4a R 4 0 1 −1 −1
60.5.5a R 5 1 −1 0 0

magma: CharacterTable(G);