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Properties

Label 16T1782
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 $t$ :  $1782$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $8$
Generators:  (1,15,13,4,2,16,14,3)(5,11,10,7,6,12,9,8), (1,12,2,11)(3,9)(4,10)(5,8,6,7)(13,16,14,15), (1,3,2,4)(5,16,13,7,6,15,14,8)(9,11,10,12)
$|\Aut(F/K)|$:  $2$

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:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, $(((C_4 \times C_2): C_2):C_2):C_2$ x 4, 16T76 x 2, 16T79, 16T146 x 2
128:  $C_2 \wr C_2\wr C_2$ x 4, 16T235 x 2, 16T240, 32T1151 x 2
256:  16T478, 16T482 x 2, 16T532, 16T537, 16T542 x 2
512:  32T12279, 32T12349 x 2
1024:  16T1178
2048:  32T159727
4096:  16T1589
8192:  32T519870

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

16T1774 x 16, 16T1782 x 15, 32T723833 x 8, 32T723834 x 8, 32T723835 x 8, 32T723836 x 8, 32T723837 x 8, 32T723838 x 8, 32T723839 x 8, 32T723840 x 8, 32T723841 x 8, 32T723842 x 8, 32T723843 x 8, 32T723844 x 8, 32T723845 x 8, 32T723846 x 8, 32T723847 x 8, 32T723848 x 8, 32T723849 x 8, 32T723850 x 8, 32T723851 x 8, 32T723852 x 8, 32T723853 x 8, 32T723854 x 8, 32T723855 x 8, 32T723856 x 16, 32T723857 x 8, 32T723858 x 8, 32T723859 x 8, 32T723860 x 8, 32T723861 x 8, 32T723862 x 8, 32T723863 x 8, 32T724078 x 8, 32T724079 x 8, 32T724080 x 8, 32T724081 x 8, 32T724082 x 8, 32T724083 x 8, 32T724084 x 8, 32T724085 x 8, 32T724086 x 8, 32T724087 x 8, 32T724088 x 8, 32T724089 x 8, 32T724090 x 8, 32T724091 x 8, 32T724092 x 8, 32T724093 x 8, 32T724094 x 8, 32T724095 x 8, 32T724096 x 8, 32T724097 x 8, 32T724098 x 8, 32T724099 x 8, 32T724100 x 8, 32T724101 x 8, 32T724102 x 8, 32T724103 x 8, 32T724104 x 8, 32T724105 x 8, 32T724106 x 8, 32T724107 x 8, 32T727430 x 8, 32T728359 x 8, 32T743938 x 4, 32T744295 x 4, 32T744317 x 4, 32T744471 x 4, 32T744761 x 4, 32T744794 x 4, 32T864191 x 4, 32T864193 x 4, 32T873932 x 4, 32T874540 x 4, 32T968054 x 4, 32T968176 x 4, 32T1038014 x 4, 32T1038027 x 4, 32T1061680 x 4, 32T1061683 x 4, 32T1099811 x 4, 32T1099814 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 130 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.