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Properties

Label 12T33
Degree $12$
Order $60$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $A_5$

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Show commands: Magma

magma: G := TransitiveGroup(12, 33);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_5$
CHM label:   $A_{5}(12)$
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)(2,4,6,8,12), (1,11,5)(2,7,9)(3,6,8)(4,12,10)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: $\PSL(2,5)$

Low degree siblings

5T4, 6T12, 10T7, 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 TypeSizeOrderIndexRepresentative
1A $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $15$ $2$ $6$ $( 1, 6)( 2, 5)( 3, 4)( 7,12)( 8, 9)(10,11)$
3A $3^{4}$ $20$ $3$ $8$ $( 1,11, 5)( 2, 7, 9)( 3, 6, 8)( 4,12,10)$
5A1 $5^{2},1^{2}$ $12$ $5$ $8$ $( 2, 7, 5, 8,11)( 3, 6, 4, 9,10)$
5A2 $5^{2},1^{2}$ $12$ $5$ $8$ $( 1, 5, 6,10, 8)( 4, 7,11, 9,12)$

Malle's constant $a(G)$:     $1/6$

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);