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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

LabelCycle 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:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A 7B 7C 7D 7E 14A 14B
Size 1 7 7 49 49 49 49 49 49 49 49 49 6 6 12 12 12 42 42
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A1 3A-1 3A-1 3A1 7A 7B 7E 7C 7D 7A 7B
3 P 1A 2A 2B 2C 1A 1A 2B 2B 2C 2C 2A 2A 7A 7B 7E 7C 7D 14A 14B
7 P 1A 2A 2B 2C 3A1 3A-1 6B-1 6B1 6C1 6C-1 6A-1 6A1 1A 1A 1A 1A 1A 2B 2A
Type
588.37.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
588.37.1b R 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 −1 −1
588.37.1c R 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 −1
588.37.1d R 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 1
588.37.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1
588.37.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1
588.37.1f1 C 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 1 1 1 1 1 −1 −1
588.37.1f2 C 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 1 1 1 1 1 −1 −1
588.37.1g1 C 1 −1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1 1 1 −1
588.37.1g2 C 1 −1 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 1 −1
588.37.1h1 C 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 1 1 1 −1 1
588.37.1h2 C 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 −1 1
588.37.6a R 6 0 6 0 0 0 0 0 0 0 0 0 −1 6 −1 −1 −1 −1 0
588.37.6b R 6 6 0 0 0 0 0 0 0 0 0 0 6 −1 −1 −1 −1 0 −1
588.37.6c R 6 −6 0 0 0 0 0 0 0 0 0 0 6 −1 −1 −1 −1 0 1
588.37.6d R 6 0 −6 0 0 0 0 0 0 0 0 0 −1 6 −1 −1 −1 1 0
588.37.12a R 12 0 0 0 0 0 0 0 0 0 0 0 −2 −2 −2 −2 5 0 0
588.37.12b R 12 0 0 0 0 0 0 0 0 0 0 0 −2 −2 −2 5 −2 0 0
588.37.12c R 12 0 0 0 0 0 0 0 0 0 0 0 −2 −2 5 −2 −2 0 0

magma: CharacterTable(G);