Properties

Label 14T49
Order \(10080\)
n \(14\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_7\times C_2$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $49$
Group :  $S_7\times C_2$
CHM label :  $2[x]S(7)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $2$

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

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 7, 7 $ $720$ $7$ $( 1,11, 9, 3,13, 5, 7)( 2,10, 6,12,14, 8, 4)$
$ 14 $ $720$ $14$ $( 1, 4, 9,10,13,12, 7, 8,11, 2, 3, 6, 5,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $105$ $2$ $( 1, 9)( 2, 8)( 5, 7)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2 $ $105$ $2$ $( 1, 2)( 3,10)( 4,11)( 5,14)( 6,13)( 7,12)( 8, 9)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $70$ $3$ $( 3,11,13)( 4, 6,10)$
$ 6, 2, 2, 2, 2 $ $70$ $6$ $( 1, 8)( 2, 9)( 3, 4,13,10,11, 6)( 5,12)( 7,14)$
$ 3, 3, 2, 2, 2, 2 $ $210$ $6$ $( 1, 9)( 2, 8)( 3,13,11)( 4,10, 6)( 5, 7)(12,14)$
$ 6, 2, 2, 2, 2 $ $210$ $6$ $( 1, 2)( 3, 6,11,10,13, 4)( 5,14)( 7,12)( 8, 9)$
$ 4, 4, 1, 1, 1, 1, 1, 1 $ $210$ $4$ $( 1, 7, 9, 5)( 2,12, 8,14)$
$ 4, 4, 2, 2, 2 $ $210$ $4$ $( 1,14, 9,12)( 2, 5, 8, 7)( 3,10)( 4,11)( 6,13)$
$ 4, 4, 3, 3 $ $420$ $12$ $( 1, 5, 9, 7)( 2,14, 8,12)( 3,13,11)( 4,10, 6)$
$ 6, 4, 4 $ $420$ $12$ $( 1,12, 9,14)( 2, 7, 8, 5)( 3, 6,11,10,13, 4)$
$ 5, 5, 1, 1, 1, 1 $ $504$ $5$ $( 2, 6, 4,14,10)( 3, 9,13,11, 7)$
$ 10, 2, 2 $ $504$ $10$ $( 1, 8)( 2,13, 4, 7,10, 9, 6,11,14, 3)( 5,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 1, 5)( 8,12)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1,12)( 2, 9)( 3,10)( 4,11)( 5, 8)( 6,13)( 7,14)$
$ 3, 3, 2, 2, 1, 1, 1, 1 $ $420$ $6$ $( 2,12)( 3,11,13)( 4, 6,10)( 5, 9)$
$ 6, 2, 2, 2, 2 $ $420$ $6$ $( 1, 8)( 2, 5)( 3, 4,13,10,11, 6)( 7,14)( 9,12)$
$ 5, 5, 2, 2 $ $504$ $10$ $( 1, 3, 9,11, 7)( 2, 4,14, 8,10)( 5,13)( 6,12)$
$ 10, 2, 2 $ $504$ $10$ $( 1,10, 9, 4, 7, 8, 3, 2,11,14)( 5, 6)(12,13)$
$ 4, 4, 2, 2, 1, 1 $ $630$ $4$ $( 1, 9, 3,11)( 2,10, 4, 8)( 5, 7)(12,14)$
$ 4, 4, 2, 2, 2 $ $630$ $4$ $( 1, 2, 3, 4)( 5,14)( 6,13)( 7,12)( 8, 9,10,11)$
$ 3, 3, 3, 3, 1, 1 $ $280$ $3$ $( 1, 7, 9)( 2, 8,14)( 3,13,11)( 4,10, 6)$
$ 6, 6, 2 $ $280$ $6$ $( 1,14, 9, 8, 7, 2)( 3, 6,11,10,13, 4)( 5,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $105$ $2$ $( 1,13)( 2,10)( 3, 9)( 4,14)( 6, 8)( 7,11)$
$ 2, 2, 2, 2, 2, 2, 2 $ $105$ $2$ $( 1, 6)( 2, 3)( 4, 7)( 5,12)( 8,13)( 9,10)(11,14)$
$ 6, 6, 1, 1 $ $840$ $6$ $( 1,11, 9,13, 7, 3)( 2, 6,14,10, 8, 4)$
$ 6, 6, 2 $ $840$ $6$ $( 1, 4, 9, 6, 7,10)( 2,13,14, 3, 8,11)( 5,12)$

Group invariants

Order:  $10080=2^{5} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.