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Properties

Label 14T49
Degree $14$
Order $10080$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_7\times C_2$

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Show commands: Magma

magma: G := TransitiveGroup(14, 49);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $49$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_7\times C_2$
CHM label:   $2[x]S(7)$
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:  (3,5)(10,12), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (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)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5040$:  $S_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $S_7$

Low degree siblings

14T49, 28T363, 42T549 x 2, 42T550 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, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 5, 7)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,14)( 6,13)( 7,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $70$ $3$ $( 1, 9, 3)( 2,10, 8)$
$ 6, 2, 2, 2, 2 $ $70$ $6$ $( 1, 2, 3, 8, 9,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 3, 3, 2, 2, 1, 1, 1, 1 $ $420$ $6$ $( 1, 3, 9)( 2, 8,10)( 5, 7)(12,14)$
$ 6, 2, 2, 2, 2 $ $420$ $6$ $( 1,10, 9, 8, 3, 2)( 4,11)( 5,14)( 6,13)( 7,12)$
$ 7, 7 $ $720$ $7$ $( 1, 7,13, 9,11, 3, 5)( 2, 4,10,12, 8,14, 6)$
$ 14 $ $720$ $14$ $( 1,14,13, 2,11,10, 5, 8, 7, 6, 9, 4, 3,12)$
$ 5, 5, 1, 1, 1, 1 $ $504$ $5$ $( 1, 3, 9,13, 7)( 2, 6,14, 8,10)$
$ 10, 2, 2 $ $504$ $10$ $( 1,10, 9, 6, 7, 8, 3, 2,13,14)( 4,11)( 5,12)$
$ 5, 5, 2, 2 $ $504$ $10$ $( 1,13, 3, 7, 9)( 2, 8, 6,10,14)( 4,12)( 5,11)$
$ 10, 2, 2 $ $504$ $10$ $( 1, 6, 3,14, 9, 8,13,10, 7, 2)( 4, 5)(11,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $105$ $2$ $( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2 $ $105$ $2$ $( 1, 4)( 2, 3)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $105$ $2$ $( 2,10)( 3, 9)( 5, 7)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2 $ $105$ $2$ $( 1, 8)( 2, 3)( 4,11)( 5,14)( 6,13)( 7,12)( 9,10)$
$ 4, 4, 1, 1, 1, 1, 1, 1 $ $210$ $4$ $( 2,12,10,14)( 3, 7, 9, 5)$
$ 4, 4, 2, 2, 2 $ $210$ $4$ $( 1, 8)( 2, 5,10, 7)( 3,14, 9,12)( 4,11)( 6,13)$
$ 3, 3, 2, 2, 2, 2 $ $210$ $6$ $( 1,13,11)( 2,10)( 3, 9)( 4, 8, 6)( 5, 7)(12,14)$
$ 6, 2, 2, 2, 2 $ $210$ $6$ $( 1, 6,11, 8,13, 4)( 2, 3)( 5,14)( 7,12)( 9,10)$
$ 4, 4, 3, 3 $ $420$ $12$ $( 1,11,13)( 2,14,10,12)( 3, 5, 9, 7)( 4, 6, 8)$
$ 6, 4, 4 $ $420$ $12$ $( 1, 4,13, 8,11, 6)( 2, 7,10, 5)( 3,12, 9,14)$
$ 4, 4, 2, 2, 1, 1 $ $630$ $4$ $( 1,13)( 2,12,10,14)( 3, 7, 9, 5)( 6, 8)$
$ 4, 4, 2, 2, 2 $ $630$ $4$ $( 1, 6)( 2, 5,10, 7)( 3,14, 9,12)( 4,11)( 8,13)$
$ 3, 3, 3, 3, 1, 1 $ $280$ $3$ $( 1, 7, 3)( 2, 4,12)( 5, 9,11)( 8,14,10)$
$ 6, 6, 2 $ $280$ $6$ $( 1,14, 3, 8, 7,10)( 2,11,12, 9, 4, 5)( 6,13)$
$ 6, 6, 1, 1 $ $840$ $6$ $( 1, 9, 7,11, 3, 5)( 2,14, 4,10,12, 8)$
$ 6, 6, 2 $ $840$ $6$ $( 1, 2, 7, 4, 3,12)( 5, 8, 9,14,11,10)( 6,13)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $10080=2^{5} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  10080.l
magma: IdentifyGroup(G);
 
Character table:

Size
2 P
3 P
5 P
7 P
Type

magma: CharacterTable(G);