Properties

Label 14T10
Order \(168\)
n \(14\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $\PSL(2,7)$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $10$
Group :  $\PSL(2,7)$
CHM label :  $L_{7}(14)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(2,13)(3,10)(4,12)(5,11)(6,9), (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $\GL(3,2)$

Low degree siblings

7T5 x 2, 8T37, 14T10, 21T14, 24T284, 28T32, 42T37, 42T38 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $56$ $3$ $( 2, 3, 6)( 4, 7,12)( 5,11,14)( 9,10,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $21$ $2$ $( 2, 4)( 3,10)( 5, 6)( 7,14)( 9,11)(12,13)$
$ 7, 7 $ $24$ $7$ $( 1, 2, 3, 7, 6, 4,12)( 5, 8, 9,10,14,13,11)$
$ 4, 4, 4, 2 $ $42$ $4$ $( 1, 2, 8, 9)( 3,12,13, 7)( 4,11)( 5, 6,14,10)$
$ 7, 7 $ $24$ $7$ $( 1, 2,12,11,10, 7,13)( 3,14, 6, 8, 9, 5, 4)$

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [168, 42]
Character table:   
     2  3  .  3  .  2  .
     3  1  1  .  .  .  .
     7  1  .  .  1  .  1

       1a 3a 2a 7a 4a 7b
    2P 1a 3a 1a 7a 2a 7b
    3P 1a 1a 2a 7b 4a 7a
    5P 1a 3a 2a 7b 4a 7a
    7P 1a 3a 2a 1a 4a 1a

X.1     1  1  1  1  1  1
X.2     3  . -1  A  1 /A
X.3     3  . -1 /A  1  A
X.4     6  .  2 -1  . -1
X.5     7  1 -1  . -1  .
X.6     8 -1  .  1  .  1

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7