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Properties

Label 14T8
Degree $14$
Order $98$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7 \wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(14, 8);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7 \wr C_2$
CHM label:   $[7^{2}]2=7wr2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $7$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,4,6,8,10,12,14), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$7$:  $C_7$
$14$:  $D_{7}$, $C_{14}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T8 x 2

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $98=2 \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  98.3
magma: IdentifyGroup(G);
 
Character table:    35 x 35 character table

magma: CharacterTable(G);