Properties

Label 16T1192
Degree $16$
Order $1024$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4\wr C_4$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(16, 1192);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1192$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4\wr C_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,7,10,2,13,8,9)(3,15,6,12,4,16,5,11), (1,5,2,6)(3,7,4,8)(9,15,12,14,10,16,11,13)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 6, $C_2^2$
$8$:  $D_{4}$ x 3, $C_4\times C_2$ x 3, $Q_8$
$16$:  $C_2^2:C_4$ x 3, $C_4^2$, $C_4:C_4$ x 3
$32$:  $C_4\wr C_2$ x 2, $C_2^3 : C_4 $ x 2, 32T41
$64$:  $((C_8 : C_2):C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T74, 16T77, 16T121, 16T140, 16T154
$128$:  32T1095, 32T1311, 32T1313
$256$:  16T521, 16T563, 16T590
$512$:  32T13179

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $((C_8 : C_2):C_2):C_2$

Low degree siblings

16T1192 x 15, 32T36166 x 8, 32T36167 x 8, 32T36168 x 16, 32T36169 x 16, 32T36170 x 8, 32T56480 x 8

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

There are 88 conjugacy classes of elements. Data not shown.

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1024=2^{10}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $6$
Label:  1024.dgg
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);