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Properties

Label 15T47
Degree $15$
Order $2520$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_7$

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Show commands: Magma

magma: G := TransitiveGroup(15, 47);
 

Group action invariants

Degree $n$:  $15$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $47$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_7$
CHM label:  $A_{7}(15)$
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), (1,2,3)(5,6,7)(8,10,9)(12,14,13)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: None

Low degree siblings

7T6, 15T47, 21T33, 35T28, 42T294, 42T299

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 5, 5, 5 $ $504$ $5$ $( 1,14,11, 6,13)( 2, 4, 3,15,10)( 5, 7, 9, 8,12)$
$ 7, 7, 1 $ $360$ $7$ $( 1,15, 9, 2, 8,11, 3)( 4, 5,12, 7,13,14, 6)$
$ 7, 7, 1 $ $360$ $7$ $( 1, 3,11, 8, 2, 9,15)( 4, 6,14,13, 7,12, 5)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $105$ $2$ $( 1, 4)( 2, 8)( 3,12)( 6, 9)( 7,13)(11,14)$
$ 3, 3, 3, 3, 3 $ $70$ $3$ $( 1, 9, 7)( 2,14, 3)( 4, 6,13)( 5,10,15)( 8,11,12)$
$ 6, 6, 3 $ $210$ $6$ $( 1,13, 9, 4, 7, 6)( 2,12,14, 8, 3,11)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $280$ $3$ $( 1, 7,15)( 2, 3, 5)( 8,14, 9)(10,12,11)$
$ 4, 4, 4, 2, 1 $ $630$ $4$ $( 1,14, 4,11)( 2, 6, 8, 9)( 3,13,12, 7)( 5,15)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  2520.a
magma: IdentifyGroup(G);
 
Character table:    
     2  3  3  2  2  2  .  .  .  .
     3  2  1  .  2  1  .  .  2  .
     5  1  .  .  .  .  .  .  .  1
     7  1  .  .  .  .  1  1  .  .

       1a 2a 4a 3a 6a 7a 7b 3b 5a
    2P 1a 1a 2a 3a 3a 7a 7b 3b 5a
    3P 1a 2a 4a 1a 2a 7b 7a 1a 5a
    5P 1a 2a 4a 3a 6a 7b 7a 3b 1a
    7P 1a 2a 4a 3a 6a 1a 1a 3b 5a

X.1     1  1  1  1  1  1  1  1  1
X.2     6  2  .  3 -1 -1 -1  .  1
X.3    10 -2  .  1  1  A /A  1  .
X.4    10 -2  .  1  1 /A  A  1  .
X.5    14  2  .  2  2  .  . -1 -1
X.6    14  2  . -1 -1  .  .  2 -1
X.7    15 -1 -1  3 -1  1  1  .  .
X.8    21  1 -1 -3  1  .  .  .  1
X.9    35 -1  1 -1 -1  .  . -1  .

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7

magma: CharacterTable(G);