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Properties

Label 14T16
Degree $14$
Order $336$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\PGL(2,7)$

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Show commands: Magma

magma: G := TransitiveGroup(14, 16);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $16$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PGL(2,7)$
CHM label:  $L_{7}:2(14)=[L(7)_%]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,13,11,9,7,5,3)(2,4,6,8,10,12,14), (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (2,4)(5,13)(6,12)(9,11)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

8T43, 16T713, 21T20, 24T707, 28T42, 28T46, 42T81, 42T82, 42T83

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 5,11)( 6,12)( 9,13)(10,14)$
$ 3, 3, 3, 3, 1, 1 $ $56$ $3$ $( 3, 5,11)( 4,12, 6)( 7,13, 9)( 8,10,14)$
$ 4, 4, 2, 2, 1, 1 $ $42$ $4$ $( 2, 4,12, 6)( 3, 9, 7,11)( 5,13)( 8,14)$
$ 2, 2, 2, 2, 2, 2, 2 $ $28$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$
$ 6, 6, 2 $ $56$ $6$ $( 1, 2)( 3, 6, 5, 4,11,12)( 7,14,13, 8, 9,10)$
$ 8, 4, 2 $ $42$ $8$ $( 1, 2, 3, 4)( 5,10,13,12,11,14, 9, 6)( 7, 8)$
$ 8, 4, 2 $ $42$ $8$ $( 1, 2, 3, 4)( 5,14,13, 6,11,10, 9,12)( 7, 8)$
$ 7, 7 $ $48$ $7$ $( 1, 3, 9,11,13, 7, 5)( 2, 4,10, 8, 6,14,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $336=2^{4} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  336.208
magma: IdentifyGroup(G);
 
Character table:    
     2  4  4  1  3  2  1  3  3  .
     3  1  .  1  .  1  1  .  .  .
     7  1  .  .  .  .  .  .  .  1

       1a 2a 3a 4a 2b 6a 8a 8b 7a
    2P 1a 1a 3a 2a 1a 3a 4a 4a 7a
    3P 1a 2a 1a 4a 2b 2b 8b 8a 7a
    5P 1a 2a 3a 4a 2b 6a 8b 8a 7a
    7P 1a 2a 3a 4a 2b 6a 8a 8b 1a

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

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2

magma: CharacterTable(G);