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

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.