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Properties

Label 14T45
Order \(3528\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $45$
CHM label :  $[F_{42}(7)^{2}]2=F_{42}(7)wr2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14), (2,12)(4,10)(6,8), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (2,4,8)(6,12,10)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
8:  $D_{4}$
12:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
24:  $(C_6\times C_2):C_2$, $D_4 \times C_3$
36:  $C_6\times S_3$
72:  12T42

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

28T251, 28T252, 28T253, 42T368, 42T369, 42T370, 42T371, 42T372

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 $ $1$ $1$ $()$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $12$ $7$ $( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $36$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $2$ $( 4,14)( 6,12)( 8,10)$
$ 7, 2, 2, 2, 1 $ $84$ $14$ $( 1, 3, 5, 7, 9,11,13)( 4,14)( 6,12)( 8,10)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $49$ $2$ $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 3, 3, 3, 3, 1, 1 $ $98$ $3$ $( 3, 9, 5)( 4, 6,10)( 7,11,13)( 8,14,12)$
$ 6, 3, 3, 1, 1 $ $98$ $6$ $( 3, 9, 5)( 4,12,10,14, 6, 8)( 7,11,13)$
$ 6, 3, 3, 1, 1 $ $98$ $6$ $( 3, 7, 5,13, 9,11)( 4, 6,10)( 8,14,12)$
$ 6, 6, 1, 1 $ $98$ $6$ $( 3, 7, 5,13, 9,11)( 4,12,10,14, 6, 8)$
$ 2, 2, 2, 2, 2, 2, 2 $ $42$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 14 $ $252$ $14$ $( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
$ 4, 4, 4, 2 $ $294$ $4$ $( 1,10, 3, 8)( 2, 9)( 4,11,14, 7)( 5, 6,13,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $3$ $( 3, 5, 9)( 7,13,11)$
$ 7, 3, 3, 1 $ $84$ $21$ $( 2, 4, 6, 8,10,12,14)( 3, 5, 9)( 7,13,11)$
$ 3, 3, 2, 2, 2, 1, 1 $ $98$ $6$ $( 3, 5, 9)( 4,14)( 6,12)( 7,13,11)( 8,10)$
$ 6, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $6$ $( 3,11, 9,13, 5, 7)$
$ 7, 6, 1 $ $84$ $42$ $( 2, 4, 6, 8,10,12,14)( 3,11, 9,13, 5, 7)$
$ 6, 2, 2, 2, 1, 1 $ $98$ $6$ $( 3,11, 9,13, 5, 7)( 4,14)( 6,12)( 8,10)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$
$ 6, 3, 3, 1, 1 $ $98$ $6$ $( 3, 9, 5)( 4, 8, 6,14,10,12)( 7,11,13)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3, 7, 5,13, 9,11)( 4, 8, 6,14,10,12)$
$ 6, 6, 2 $ $294$ $6$ $( 1, 8)( 2, 9,10, 3,12, 5)( 4,11,14, 7, 6,13)$
$ 12, 2 $ $294$ $12$ $( 1,10, 3, 6,13,14, 7,12, 5, 2, 9, 8)( 4,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $3$ $( 3, 9, 5)( 7,11,13)$
$ 7, 3, 3, 1 $ $84$ $21$ $( 2, 4, 6, 8,10,12,14)( 3, 9, 5)( 7,11,13)$
$ 3, 3, 2, 2, 2, 1, 1 $ $98$ $6$ $( 3, 9, 5)( 4,14)( 6,12)( 7,11,13)( 8,10)$
$ 6, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $6$ $( 3, 7, 5,13, 9,11)$
$ 7, 6, 1 $ $84$ $42$ $( 2, 4, 6, 8,10,12,14)( 3, 7, 5,13, 9,11)$
$ 6, 2, 2, 2, 1, 1 $ $98$ $6$ $( 3, 7, 5,13, 9,11)( 4,14)( 6,12)( 8,10)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$
$ 6, 3, 3, 1, 1 $ $98$ $6$ $( 3, 5, 9)( 4,12,10,14, 6, 8)( 7,13,11)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3,11, 9,13, 5, 7)( 4,12,10,14, 6, 8)$
$ 6, 6, 2 $ $294$ $6$ $( 1, 8)( 2, 9,12, 5,10, 3)( 4,11, 6,13,14, 7)$
$ 12, 2 $ $294$ $12$ $( 1,10, 3, 2, 9, 6,13, 4,11,12, 5, 8)( 7,14)$

Group invariants

Order:  $3528=2^{3} \cdot 3^{2} \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.