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Properties

Label 14T50
Degree $14$
Order $10752$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2^6:\GL(3,2)$

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Show commands: Magma

magma: G := TransitiveGroup(14, 50);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $50$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^6:\GL(3,2)$
CHM label:   $[2^{6}]L(7)$
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:  (1,9,11)(2,4,8)(3,13,5)(6,12,10), (2,9)(7,14), (2,4)(5,13)(6,12)(9,11), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$168$:  $\GL(3,2)$
$1344$:  $C_2^3:\GL(3,2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $\GL(3,2)$

Low degree siblings

14T50, 28T388, 28T390 x 2, 42T613 x 2, 42T614 x 2, 42T615 x 2, 42T616 x 2

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, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 4,11)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $28$ $2$ $( 2, 9)( 4,11)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 2, 9)( 4,11)( 5,12)( 6,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $7$ $2$ $( 1, 8)( 2, 9)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 4, 2, 2, 2, 1, 1, 1, 1 $ $336$ $4$ $( 3, 5)( 4,11)( 6, 7,13,14)(10,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $84$ $2$ $( 3, 5)( 6,14)( 7,13)(10,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $168$ $2$ $( 2, 9)( 3, 5)( 4,11)( 6,14)( 7,13)(10,12)$
$ 4, 2, 2, 2, 1, 1, 1, 1 $ $168$ $4$ $( 2, 9)( 3, 5)( 6, 7,13,14)(10,12)$
$ 4, 4, 1, 1, 1, 1, 1, 1 $ $84$ $4$ $( 3,12,10, 5)( 6, 7,13,14)$
$ 4, 4, 2, 2, 1, 1 $ $168$ $4$ $( 2, 9)( 3,12,10, 5)( 4,11)( 6, 7,13,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $84$ $2$ $( 1, 8)( 3, 5)( 4,11)( 6,14)( 7,13)(10,12)$
$ 4, 2, 2, 2, 2, 2 $ $168$ $4$ $( 1, 8)( 2, 9)( 3, 5)( 4,11)( 6, 7,13,14)(10,12)$
$ 4, 4, 2, 2, 1, 1 $ $84$ $4$ $( 1, 8)( 3,12,10, 5)( 4,11)( 6, 7,13,14)$
$ 8, 4, 1, 1 $ $672$ $8$ $( 2, 3, 4, 7, 9,10,11,14)( 5,13,12, 6)$
$ 4, 4, 2, 2, 1, 1 $ $672$ $4$ $( 2, 3, 4,14)( 5, 6)( 7, 9,10,11)(12,13)$
$ 4, 4, 4, 2 $ $672$ $4$ $( 1, 8)( 2, 3, 4,14)( 5,13,12, 6)( 7, 9,10,11)$
$ 8, 2, 2, 2 $ $672$ $8$ $( 1, 8)( 2, 3, 4, 7, 9,10,11,14)( 5, 6)(12,13)$
$ 3, 3, 3, 3, 1, 1 $ $896$ $3$ $( 2, 3, 5)( 4, 7,13)( 6,11,14)( 9,10,12)$
$ 6, 6, 1, 1 $ $896$ $6$ $( 2, 3, 5, 9,10,12)( 4,14, 6,11, 7,13)$
$ 6, 3, 3, 2 $ $896$ $6$ $( 1, 8)( 2, 3, 5)( 4,14, 6,11, 7,13)( 9,10,12)$
$ 6, 3, 3, 2 $ $896$ $6$ $( 1, 8)( 2, 3, 5, 9,10,12)( 4, 7,13)( 6,11,14)$
$ 7, 7 $ $1536$ $7$ $( 1, 2, 3, 4,12,13, 7)( 5, 6,14, 8, 9,10,11)$
$ 7, 7 $ $1536$ $7$ $( 1, 2, 3, 7,13,11, 5)( 4,12, 8, 9,10,14, 6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $10752=2^{9} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  10752.k
magma: IdentifyGroup(G);
 
Character table:

Size
2 P
3 P
7 P
Type

magma: CharacterTable(G);