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Properties

Label 16T1584
Order \(4096\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $1584$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $8$
Generators:  (1,13,3,12)(2,14,4,11)(5,10,7,15,6,9,8,16), (1,9,4,11,6,14,7,16,2,10,3,12,5,13,8,15), (1,11,7,14,2,12,8,13)(3,10,5,16)(4,9,6,15)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
8:  $D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
16:  $C_8:C_2$ x 4, $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
32:  $(C_8:C_2):C_2$ x 2, $C_2^3 : C_4 $ x 2, $C_2 \times (C_8:C_2)$ x 2, $C_2 \times (C_2^2:C_4)$
64:  $((C_8 : C_2):C_2):C_2$ x 4, 16T72, 16T76, 16T95
128:  16T227 x 2, 16T252
256:  16T485
512:  32T22852
1024:  16T1134
2048:  32T128397

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

16T1584 x 15, 16T1604 x 16, 32T207414 x 8, 32T207415 x 16, 32T207416 x 16, 32T207417 x 16, 32T207418 x 8, 32T207419 x 8, 32T207420 x 16, 32T207421 x 8, 32T207422 x 16, 32T207423 x 16, 32T207424 x 8, 32T207425 x 16, 32T207426 x 8, 32T207427 x 8, 32T207428 x 16, 32T207429 x 16, 32T207430 x 8, 32T207431 x 8, 32T207432 x 8, 32T207433 x 8, 32T207434 x 8, 32T207435 x 8, 32T207436 x 8, 32T207706 x 8, 32T207707 x 32, 32T207708 x 16, 32T207709 x 16, 32T207710 x 16, 32T207711 x 8, 32T207712 x 32, 32T207713 x 16, 32T207714 x 8, 32T207715 x 16, 32T207716 x 8, 32T207717 x 16, 32T207718 x 16, 32T207719 x 32, 32T207720 x 16, 32T207721 x 32, 32T207722 x 8, 32T207723 x 16, 32T207724 x 8, 32T207725 x 16, 32T207726 x 8, 32T207727 x 16, 32T207728 x 16, 32T207729 x 16, 32T207730 x 16, 32T207731 x 8, 32T207732 x 8, 32T207733 x 8, 32T207734 x 8, 32T207735 x 16, 32T207736 x 16, 32T207737 x 8, 32T207738 x 8, 32T207739 x 8, 32T221217 x 8, 32T242706 x 4, 32T242822 x 4, 32T309802 x 4, 32T323743 x 4, 32T365379 x 4, 32T396479 x 4

Siblings are shown with degree $\leq 47$

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

Conjugacy Classes

There are 73 conjugacy classes of elements. Data not shown.

Group invariants

Order:  $4096=2^{12}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.