Properties

Label 16T1905
Degree $16$
Order $294912$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^6.S_4^2:D_4$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(16, 1905);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1905$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^6.S_4^2:D_4$
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:  (1,11,3,13,16,9,2,12,4,14,15,10)(5,8)(6,7), (1,16,3)(2,15,4)(7,12,13,9,8,11,14,10), (1,3)(2,4)(5,16,6,15)(7,12,13,9,8,11,14,10)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$
$72$:  $C_3^2:D_4$
$144$:  12T77
$288$:  12T125
$1152$:  $S_4\wr C_2$
$2304$:  12T235
$4608$:  12T260
$73728$:  16T1864
$147456$:  32T2077237

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 8: $S_4\wr C_2$

Low degree siblings

16T1905 x 15, 32T2265666 x 8, 32T2265667 x 8, 32T2265668 x 8, 32T2265669 x 8, 32T2265670 x 8, 32T2265671 x 8, 32T2265672 x 8, 32T2265673 x 8, 32T2265674 x 8, 32T2265675 x 8, 32T2265676 x 8, 32T2265677 x 8, 32T2265678 x 8, 32T2265679 x 8, 32T2265680 x 8, 32T2265730 x 8

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Conjugacy classes not computed

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $294912=2^{15} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  294912.a
magma: IdentifyGroup(G);
 
Character table:    not computed

magma: CharacterTable(G);