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Properties

Label 6T15
Degree $6$
Order $360$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_6$

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Show commands: Magma

magma: G := TransitiveGroup(6, 15);
 

Group action invariants

Degree $n$:  $6$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $15$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_6$
CHM label:   $A6$
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:  (1,2,3), (1,2)(3,4,5,6)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Low degree siblings

6T15, 10T26, 15T20 x 2, 20T89, 30T88 x 2, 36T555, 40T304, 45T49

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 $ $1$ $1$ $()$
$ 3, 1, 1, 1 $ $40$ $3$ $(4,5,6)$
$ 2, 2, 1, 1 $ $45$ $2$ $(3,4)(5,6)$
$ 5, 1 $ $72$ $5$ $(2,3,4,5,6)$
$ 5, 1 $ $72$ $5$ $(2,3,4,6,5)$
$ 4, 2 $ $90$ $4$ $(1,2)(3,4,5,6)$
$ 3, 3 $ $40$ $3$ $(1,2,3)(4,5,6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  360.118
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B 4A 5A1 5A2
Size 1 45 40 40 90 72 72
2 P 1A 1A 3A 3B 2A 5A2 5A1
3 P 1A 2A 1A 1A 4A 5A2 5A1
5 P 1A 2A 3A 3B 4A 1A 1A
Type
360.118.1a R 1 1 1 1 1 1 1
360.118.5a R 5 1 −1 2 −1 0 0
360.118.5b R 5 1 2 −1 −1 0 0
360.118.8a1 R 8 0 −1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52
360.118.8a2 R 8 0 −1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5
360.118.9a R 9 1 0 0 1 −1 −1
360.118.10a R 10 −2 1 1 0 0 0

magma: CharacterTable(G);