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

Cycle 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);
 
Label:  60.5
magma: IdentifyGroup(G);
 
Character table:    
     2  2  .  2  .  .
     3  1  1  .  .  .
     5  1  .  .  1  1

       1a 3a 2a 5a 5b
    2P 1a 3a 1a 5b 5a
    3P 1a 1a 2a 5b 5a
    5P 1a 3a 2a 1a 1a

X.1     1  1  1  1  1
X.2     3  . -1  A *A
X.3     3  . -1 *A  A
X.4     4  1  . -1 -1
X.5     5 -1  1  .  .

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5

magma: CharacterTable(G);