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Properties

Label 16T1770
Order \(16384\)
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 $t$ :  $1770$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $8$
Generators:  (1,8)(2,7)(3,5,11,13)(4,6,12,14)(9,16)(10,15), (1,12,6,16,10,3,14,7)(2,11,5,15,9,4,13,8), (1,13,10,5)(2,14,9,6)(3,8)(4,7)(11,15,12,16)
$|\Aut(F/K)|$:  $2$

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 12, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 6, $C_2^2:C_4$ x 12, $C_4\times C_2^2$
32:  $C_2^2 \wr C_2$ x 4, $C_2^3 : C_4 $ x 4, $C_2 \times (C_2^2:C_4)$ x 3
64:  $((C_8 : C_2):C_2):C_2$ x 4, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T76 x 2, 16T79, 16T146 x 2
128:  $C_2 \wr C_2\wr C_2$ x 4, 16T227 x 2, 16T240, 32T1151 x 2
256:  16T482 x 2, 16T502, 16T532, 16T542 x 2, 16T543
512:  32T12349 x 2, 32T13346
1024:  16T1174
2048:  32T126555
4096:  16T1559
8192:  32T520838

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_2 \wr C_2\wr C_2$

Low degree siblings

16T1770 x 31, 32T723661 x 16, 32T723662 x 16, 32T723663 x 16, 32T723664 x 16, 32T723665 x 16, 32T723666 x 16, 32T723667 x 16, 32T723668 x 16, 32T723669 x 16, 32T723670 x 16, 32T723671 x 16, 32T723672 x 16, 32T723673 x 16, 32T723674 x 16, 32T723675 x 16, 32T723676 x 16, 32T723677 x 16, 32T723678 x 16, 32T723679 x 16, 32T723680 x 16, 32T723681 x 16, 32T723682 x 16, 32T723683 x 16, 32T723684 x 16, 32T723685 x 16, 32T723686 x 16, 32T723687 x 16, 32T723688 x 16, 32T723689 x 16, 32T723690 x 16, 32T723691 x 16, 32T728468 x 16, 32T745010 x 8, 32T745105 x 8, 32T745182 x 8, 32T874081 x 8, 32T874400 x 8, 32T968025 x 8, 32T968169 x 8, 32T1038013 x 8, 32T1038028 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 136 conjugacy classes of elements. Data not shown.

Group invariants

Order:  $16384=2^{14}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.