Properties

Label 18T776
Degree $18$
Order $82944$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4^3.(C_2\times S_4)$

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Show commands: Magma

magma: G := TransitiveGroup(18, 776);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $776$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4^3.(C_2\times S_4)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7,13,6,12,17,2,8,14,5,11,18)(3,10,15,4,9,16), (1,5)(2,6)(7,16,8,15)(9,14,10,13)(11,17)(12,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$
$48$:  $S_4\times C_2$
$1296$:  $S_3\wr S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: None

Degree 9: $S_3\wr S_3$

Low degree siblings

12T294, 18T770, 18T773, 18T777, 18T779, 18T780, 18T783, 18T785, 24T16593, 24T16594, 24T16595, 24T16596, 24T16597, 24T16598, 24T16599, 36T19182, 36T19183, 36T19184, 36T19185, 36T19192, 36T19193, 36T19194, 36T19195, 36T19211, 36T19212, 36T19213, 36T19214, 36T19215, 36T19216, 36T19219, 36T19220, 36T19221, 36T19222, 36T19229, 36T19230, 36T19231, 36T19232, 36T19247, 36T19248, 36T19249, 36T19250, 36T19251, 36T19252, 36T19254, 36T19255, 36T19256, 36T19257, 36T19420, 36T19421, 36T19422, 36T19423

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

There are 65 conjugacy classes of elements. Data not shown.

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $82944=2^{10} \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  82944.d
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);