Properties

Label 16T1664
Degree $16$
Order $6144$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no

Related objects

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Group action invariants

Degree $n$:  $16$
Transitive number $t$:  $1664$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $4$
Generators:  (1,9,3,12)(2,10,4,11)(13,16)(14,15), (1,15,8,10,3,13,6,11)(2,16,7,9,4,14,5,12), (1,15,8,12)(2,16,7,11)(3,13,6,9)(4,14,5,10)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$24$:  $S_4$, $S_3 \times C_2^2$
$48$:  $S_4\times C_2$ x 3
$96$:  12T48
$192$:  $V_4^2:(S_3\times C_2)$ x 3
$384$:  $C_2 \wr S_4$ x 6, 12T136 x 3
$768$:  16T1045 x 3
$1536$:  24T4591
$3072$:  24T5576 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Degree 8: $C_2 \wr S_4$ x 3

Low degree siblings

16T1664 x 63, 32T397419 x 16, 32T397420 x 48, 32T397421 x 48

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

There are 105 conjugacy classes of elements. Data not shown.

Group invariants

Order:  $6144=2^{11} \cdot 3$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  not available
Character table: not available.