↑

Properties

Label 14T4
Order \(42\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_7$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $4$
Group :  $F_7$
CHM label :  $2[1/2]F_{42}(7)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,6)(2,5)(3,4)(7,14)(8,13)(9,12)(10,11), (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:  $C_2$
3:  $C_3$
6:  $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $F_7$

Low degree siblings

7T4, 21T4, 42T4

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 1, 1 $ $7$ $3$ $( 2,10,12)( 3, 5, 9)( 4,14, 6)( 7,13,11)$
$ 3, 3, 3, 3, 1, 1 $ $7$ $3$ $( 2,12,10)( 3, 9, 5)( 4, 6,14)( 7,11,13)$
$ 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1, 2)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$
$ 6, 6, 2 $ $7$ $6$ $( 1, 2, 5,14,13,10)( 3, 8, 9,12, 7, 6)( 4,11)$
$ 6, 6, 2 $ $7$ $6$ $( 1, 2, 7, 4, 3,12)( 5, 8, 9,14,11,10)( 6,13)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$

Group invariants

Order:  $42=2 \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [42, 1]
Character table:    
     2  1  1  1  1   1   1  .
     3  1  1  1  1   1   1  .
     7  1  .  .  .   .   .  1

       1a 3a 3b 2a  6a  6b 7a
    2P 1a 3b 3a 1a  3a  3b 7a
    3P 1a 1a 1a 2a  2a  2a 7a
    5P 1a 3b 3a 2a  6b  6a 7a
    7P 1a 3a 3b 2a  6a  6b 1a

X.1     1  1  1  1   1   1  1
X.2     1  1  1 -1  -1  -1  1
X.3     1  A /A -1 -/A  -A  1
X.4     1 /A  A -1  -A -/A  1
X.5     1  A /A  1  /A   A  1
X.6     1 /A  A  1   A  /A  1
X.7     6  .  .  .   .   . -1

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3