Properties

Label 22T57
Degree $22$
Order $3.187\times 10^{15}$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_{11}\wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(22, 57);
 

Group action invariants

Degree $n$:  $22$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $57$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_{11}\wr C_2$
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:  (1,12,10,15,4,16,3,22,9,13,8,21)(2,19,6,14,7,18)(5,17,11,20), (1,19,11,22,9,18)(2,15,4,21,3,16,8,13,10,20,7,17,5,14,6,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: None

Low degree siblings

44T1966, 44T1967, 44T1968

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

There are 1,652 conjugacy classes of elements. Data not shown.

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $3186701844480000=2^{17} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  3186701844480000.a
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);