Properties

Label 14T37
Order \(1764\)
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 $t$ :  $37$
CHM label :  $[1/2.F_{42}(7)^{2}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (2,4,8)(6,12,10), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)
$|\Aut(F/K)|$:  $1$

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
12:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
36:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

21T29 x 2, 28T170, 42T223 x 2, 42T224 x 2, 42T225 x 2, 42T252, 42T253, 42T254, 42T255

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 $ $18$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $18$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 3, 3, 3, 3, 1, 1 $ $98$ $3$ $( 3, 9, 5)( 4, 6,10)( 7,11,13)( 8,14,12)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 14 $ $126$ $14$ $( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $3$ $( 4, 6,10)( 8,14,12)$
$ 7, 3, 3, 1 $ $84$ $21$ $( 1, 3, 5, 7, 9,11,13)( 4, 6,10)( 8,14,12)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$
$ 6, 6, 2 $ $147$ $6$ $( 1, 8, 7,14, 5,12)( 2, 9)( 3,10,11, 4,13, 6)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $3$ $( 4,10, 6)( 8,12,14)$
$ 7, 3, 3, 1 $ $84$ $21$ $( 1, 3, 5, 7, 9,11,13)( 4,10, 6)( 8,12,14)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$
$ 6, 6, 2 $ $147$ $6$ $( 1, 8, 5,12, 7,14)( 2, 9)( 3,10,13, 6,11, 4)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $49$ $2$ $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 6, 6, 1, 1 $ $98$ $6$ $( 3, 7, 5,13, 9,11)( 4,12,10,14, 6, 8)$
$ 14 $ $126$ $14$ $( 1, 8, 3, 6, 5, 4, 7, 2, 9,14,11,12,13,10)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,14)(12,13)$
$ 6, 2, 2, 2, 1, 1 $ $98$ $6$ $( 3,13)( 4,12,10,14, 6, 8)( 5,11)( 7, 9)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3, 7, 5,13, 9,11)( 4, 8, 6,14,10,12)$
$ 6, 6, 2 $ $147$ $6$ $( 1, 8,11,12, 3, 6)( 2, 9,14,13,10, 7)( 4, 5)$
$ 6, 2, 2, 2, 1, 1 $ $98$ $6$ $( 3,13)( 4, 8, 6,14,10,12)( 5,11)( 7, 9)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3,11, 9,13, 5, 7)( 4,12,10,14, 6, 8)$
$ 6, 6, 2 $ $147$ $6$ $( 1, 8,13,10, 5, 4)( 2, 9,14, 3, 6, 7)(11,12)$

Group invariants

Order:  $1764=2^{2} \cdot 3^{2} \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1764, 134]
Character table: Data not available.