↑

Properties

Label 16T1851
Degree $16$
Order $49152$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^7.C_2\wr S_4$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(16, 1851);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1851$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^7.C_2\wr S_4$
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:  (1,11,14)(2,12,13)(3,5,9)(4,6,10), (1,14,8,10,5,16)(2,13,7,9,6,15)(11,12), (3,6,4,5)(9,10)(11,13,12,14)(15,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$ x 3
$48$:  $S_4\times C_2$ x 3
$96$:  $V_4^2:S_3$
$192$:  $C_2^3:S_4$ x 2, $V_4^2:(S_3\times C_2)$ x 2, 12T100
$384$:  $C_2 \wr S_4$ x 2, 16T747
$768$:  16T1068
$1536$:  24T3293, 24T3382 x 2
$3072$:  16T1521 x 2, 16T1538
$6144$:  32T397650
$24576$:  48T?

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Degree 8: $C_2 \wr S_4$

Low degree siblings

16T1851 x 7, 32T1515657 x 4, 32T1515658 x 4, 32T1515659 x 4, 32T1515660 x 4, 32T1515661 x 4, 32T1515662 x 4, 32T1515663 x 4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

The 116 conjugacy class representatives for $C_2^7.C_2\wr S_4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $49152=2^{14} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  49152.z
magma: IdentifyGroup(G);
 
Character table:    116 x 116 character table

magma: CharacterTable(G);