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Properties

Label 14T24
Degree $14$
Order $588$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_7:F_7$

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Show commands: Magma

magma: G := TransitiveGroup(14, 24);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $24$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_7:F_7$
CHM label:  $[7^{2}:6]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,4,6,8,10,12,14), (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $C_6\times C_2$
$42$:  $F_7$ x 2
$84$:  $F_7 \times C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T24 x 2, 28T77 x 3, 42T121 x 3

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 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,14,12,10, 8, 6, 4)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$
$ 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 14 $ $42$ $14$ $( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
$ 6, 6, 2 $ $49$ $6$ $( 1,14,13,10, 5, 8)( 2, 3, 4, 7,12, 9)( 6,11)$
$ 6, 6, 2 $ $49$ $6$ $( 1,12, 3, 6, 7, 8)( 2, 5,14,11,10, 9)( 4,13)$
$ 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 $ $49$ $6$ $( 3,11, 9,13, 5, 7)( 4,12,10,14, 6, 8)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3, 7, 5,13, 9,11)( 4, 8, 6,14,10,12)$
$ 14 $ $42$ $14$ $( 1, 8, 3, 6, 5, 4, 7, 2, 9,14,11,12,13,10)$
$ 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,14)(12,13)$
$ 6, 6, 2 $ $49$ $6$ $( 1,14, 7, 2, 3,10)( 4,13)( 5, 6, 9,12,11, 8)$
$ 6, 6, 2 $ $49$ $6$ $( 1,12, 7, 2, 5,10)( 3, 4,11,14,13, 6)( 8, 9)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $588=2^{2} \cdot 3 \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  588.37
magma: IdentifyGroup(G);
 
Character table:    
      2  2  .  1  .  .  1  2  2  2   1   2   2  2   2   2   1  2   2   2
      3  1  .  .  .  .  .  1  1  1   .   1   1  1   1   1   .  1   1   1
      7  2  2  2  2  2  2  .  .  1   1   .   .  .   .   .   1  1   .   .

        1a 7a 7b 7c 7d 7e 3a 3b 2a 14a  6a  6b 2b  6c  6d 14b 2c  6e  6f
     2P 1a 7a 7b 7c 7d 7e 3b 3a 1a  7b  3b  3a 1a  3b  3a  7e 1a  3b  3a
     3P 1a 7a 7b 7c 7d 7e 1a 1a 2a 14a  2a  2a 2b  2b  2b 14b 2c  2c  2c
     5P 1a 7a 7b 7c 7d 7e 3b 3a 2a 14a  6b  6a 2b  6d  6c 14b 2c  6f  6e
     7P 1a 1a 1a 1a 1a 1a 3a 3b 2a  2a  6a  6b 2b  6c  6d  2c 2c  6e  6f
    11P 1a 7a 7b 7c 7d 7e 3b 3a 2a 14a  6b  6a 2b  6d  6c 14b 2c  6f  6e
    13P 1a 7a 7b 7c 7d 7e 3a 3b 2a 14a  6a  6b 2b  6c  6d 14b 2c  6e  6f

X.1      1  1  1  1  1  1  1  1  1   1   1   1  1   1   1   1  1   1   1
X.2      1  1  1  1  1  1  1  1 -1  -1  -1  -1 -1  -1  -1   1  1   1   1
X.3      1  1  1  1  1  1  1  1 -1  -1  -1  -1  1   1   1  -1 -1  -1  -1
X.4      1  1  1  1  1  1  1  1  1   1   1   1 -1  -1  -1  -1 -1  -1  -1
X.5      1  1  1  1  1  1  A /A -1  -1  -A -/A -1  -A -/A   1  1   A  /A
X.6      1  1  1  1  1  1 /A  A -1  -1 -/A  -A -1 -/A  -A   1  1  /A   A
X.7      1  1  1  1  1  1  A /A -1  -1  -A -/A  1   A  /A  -1 -1  -A -/A
X.8      1  1  1  1  1  1 /A  A -1  -1 -/A  -A  1  /A   A  -1 -1 -/A  -A
X.9      1  1  1  1  1  1  A /A  1   1   A  /A -1  -A -/A  -1 -1  -A -/A
X.10     1  1  1  1  1  1 /A  A  1   1  /A   A -1 -/A  -A  -1 -1 -/A  -A
X.11     1  1  1  1  1  1  A /A  1   1   A  /A  1   A  /A   1  1   A  /A
X.12     1  1  1  1  1  1 /A  A  1   1  /A   A  1  /A   A   1  1  /A   A
X.13     6 -1  6 -1 -1 -1  .  .  .   .   .   .  .   .   .  -1  6   .   .
X.14     6 -1  6 -1 -1 -1  .  .  .   .   .   .  .   .   .   1 -6   .   .
X.15     6 -1 -1 -1 -1  6  .  . -6   1   .   .  .   .   .   .  .   .   .
X.16     6 -1 -1 -1 -1  6  .  .  6  -1   .   .  .   .   .   .  .   .   .
X.17    12  5 -2 -2 -2 -2  .  .  .   .   .   .  .   .   .   .  .   .   .
X.18    12 -2 -2 -2  5 -2  .  .  .   .   .   .  .   .   .   .  .   .   .
X.19    12 -2 -2  5 -2 -2  .  .  .   .   .   .  .   .   .   .  .   .   .

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

magma: CharacterTable(G);