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Properties

Label 15T20
Degree $15$
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(15, 20);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $20$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_6$
CHM label:  $A_{6}(15)$
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:  (1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,5)(2,7)(3,6)(4,15)(8,9)(12,13)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: None

Low degree siblings

6T15 x 2, 10T26, 15T20, 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

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 1, 1, 1 $ $40$ $3$ $( 3, 8,10)( 4,13, 6)( 5, 7,14)( 9,15,11)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $45$ $2$ $( 2, 5)( 3,13)( 4,10)( 7,14)( 9,15)(11,12)$
$ 4, 4, 4, 2, 1 $ $90$ $4$ $( 2, 9, 5,15)( 3,10,13, 4)( 6, 8)( 7,11,14,12)$
$ 5, 5, 5 $ $72$ $5$ $( 1, 2, 3, 4,11)( 5,12,14, 8,15)( 6,13, 9, 7,10)$
$ 5, 5, 5 $ $72$ $5$ $( 1, 2, 3, 6, 9)( 4,13,11, 5, 8)( 7,12,14,10,15)$
$ 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 2,12)( 3, 9, 5)( 4, 6,13)( 7, 8,15)(10,11,14)$

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:    
     2  3  .  3  2  .  .  .
     3  2  2  .  .  .  .  2
     5  1  .  .  .  1  1  .

       1a 3a 2a 4a 5a 5b 3b
    2P 1a 3a 1a 2a 5b 5a 3b
    3P 1a 1a 2a 4a 5b 5a 1a
    5P 1a 3a 2a 4a 1a 1a 3b

X.1     1  1  1  1  1  1  1
X.2     5  2  1 -1  .  . -1
X.3     5 -1  1 -1  .  .  2
X.4     8 -1  .  .  A *A -1
X.5     8 -1  .  . *A  A -1
X.6     9  .  1  1 -1 -1  .
X.7    10  1 -2  .  .  .  1

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

magma: CharacterTable(G);