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Properties

Label 16T1722
Order \(8192\)
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 $t$ :  $1722$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $8$
Generators:  (1,9,16,8,5,14,4,12)(2,10,15,7,6,13,3,11), (1,11,2,12)(3,13,4,14)(5,8)(6,7)(9,15)(10,16), (1,5,2,6)(3,16,4,15)(7,14,12,10,8,13,11,9)
$|\Aut(F/K)|$:  $2$

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 8, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$
32:  $C_4\wr C_2$ x 4, $C_2^2 \wr C_2$, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
64:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T111 x 2, 32T239
128:  $C_2 \wr C_2\wr C_2$ x 2, 16T208, 16T211, 16T222, 16T345 x 2
256:  32T3766, 32T4357 x 2
512:  16T876
1024:  32T58388
2048:  16T1361
4096:  32T316392

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_4\wr C_2$

Low degree siblings

16T1702 x 8, 16T1722 x 7, 32T399628 x 4, 32T399629 x 4, 32T399630 x 4, 32T399631 x 4, 32T399632 x 4, 32T399633 x 4, 32T399634 x 4, 32T399635 x 8, 32T399636 x 4, 32T399637 x 4, 32T399638 x 4, 32T399639 x 4, 32T399640 x 4, 32T399641 x 4, 32T399642 x 4, 32T400072 x 8, 32T400073 x 8, 32T400074 x 4, 32T400075 x 4, 32T400076 x 4, 32T400077 x 4, 32T400078 x 8, 32T400079 x 8, 32T400080 x 4, 32T400081 x 4, 32T400082 x 4, 32T400083 x 4, 32T400084 x 4, 32T400085 x 4, 32T400086 x 8, 32T400087 x 8, 32T400088 x 8, 32T400089 x 4, 32T400090 x 4, 32T400091 x 4, 32T400092 x 8, 32T400093 x 4, 32T408495 x 4, 32T433239 x 2, 32T433314 x 2, 32T433317 x 4, 32T433318 x 2, 32T433323 x 2, 32T540768 x 2, 32T540791 x 2, 32T547845 x 2, 32T549466 x 2, 32T549589 x 2, 32T549592 x 2, 32T549593 x 4, 32T702758 x 4, 32T715948 x 2, 32T715949 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy Classes

There are 104 conjugacy classes of elements. Data not shown.

Group invariants

Order:  $8192=2^{13}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.