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Properties

Label 14T6
Order \(56\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_8$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $6$
Group :  $F_8$
CHM label :  $[2^{3}]7$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,10)(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)
7:  $C_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7$

Low degree siblings

8T25, 28T11

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

Group invariants

Order:  $56=2^{3} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [56, 11]
Character table:    
     2  3  3  .  .  .  .  .  .
     7  1  .  1  1  1  1  1  1

       1a 2a 7a 7b 7c 7d 7e 7f
    2P 1a 1a 7b 7d 7f 7a 7c 7e
    3P 1a 2a 7c 7f 7b 7e 7a 7d
    5P 1a 2a 7e 7c 7a 7f 7d 7b
    7P 1a 2a 1a 1a 1a 1a 1a 1a

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

A = E(7)^6
B = E(7)^5
C = E(7)^4