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Properties

Label 16T1776
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 $t$ :  $1776$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $8$
Generators:  (1,11)(2,12)(3,10,4,9)(5,15,14,7)(6,16,13,8), (1,3,14,8,10,12,6,15)(2,4,13,7,9,11,5,16), (1,9)(2,10)(3,11)(4,12)(5,6)(7,8)(13,14)
$|\Aut(F/K)|$:  $2$

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:  32T520843

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T1773 x 16, 16T1776 x 15, 32T723802 x 8, 32T723803 x 8, 32T723804 x 16, 32T723805 x 8, 32T723806 x 8, 32T723807 x 8, 32T723808 x 8, 32T723809 x 8, 32T723810 x 8, 32T723811 x 8, 32T723812 x 8, 32T723813 x 8, 32T723814 x 8, 32T723815 x 8, 32T723816 x 8, 32T723817 x 8, 32T723818 x 8, 32T723819 x 8, 32T723820 x 8, 32T723821 x 8, 32T723822 x 8, 32T723823 x 8, 32T723824 x 8, 32T723825 x 8, 32T723826 x 8, 32T723827 x 8, 32T723828 x 8, 32T723829 x 8, 32T723830 x 8, 32T723831 x 8, 32T723832 x 8, 32T723895 x 8, 32T723896 x 8, 32T723897 x 8, 32T723898 x 8, 32T723899 x 8, 32T723900 x 8, 32T723901 x 8, 32T723902 x 8, 32T723903 x 8, 32T723904 x 8, 32T723905 x 8, 32T723906 x 8, 32T723907 x 8, 32T723908 x 8, 32T723909 x 8, 32T723910 x 8, 32T723911 x 8, 32T723912 x 8, 32T723913 x 8, 32T723914 x 8, 32T723915 x 8, 32T723916 x 8, 32T723917 x 8, 32T723918 x 8, 32T723919 x 8, 32T723920 x 8, 32T723921 x 8, 32T723922 x 8, 32T723923 x 8, 32T723924 x 8, 32T726566 x 8, 32T726567 x 8, 32T728487 x 8, 32T744912 x 4, 32T744926 x 4, 32T744962 x 4, 32T745622 x 4, 32T745653 x 4, 32T745669 x 4, 32T842697 x 4, 32T842711 x 4, 32T874202 x 4, 32T874543 x 4, 32T968055 x 4, 32T968172 x 4, 32T1038020 x 4, 32T1038021 x 4, 32T1061691 x 4, 32T1061694 x 4, 32T1098797 x 4, 32T1098799 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 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.