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Properties

Label 16T841
Order \(512\)
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 $t$ :  $841$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $6$
Generators:  (1,14,4,15,6,10,7,12,2,13,3,16,5,9,8,11), (1,12,8,10,6,15,3,13,2,11,7,9,5,16,4,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
16:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$
32:  $(C_8:C_2):C_2$, $C_2^3 : C_4 $, $C_2^2 : C_8$
64:  $((C_8 : C_2):C_2):C_2$ x 2, 16T84
128:  16T228
256:  16T565

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8$

Low degree siblings

16T841 x 3, 16T917 x 4, 32T10098 x 8, 32T10099 x 2, 32T10100 x 4, 32T10101 x 8, 32T10102 x 8, 32T10103 x 2, 32T10104 x 8, 32T10105 x 4, 32T10106 x 4, 32T10107 x 4, 32T10108 x 4, 32T10109 x 4, 32T10110 x 4, 32T10559 x 4, 32T10560 x 4, 32T10561 x 2, 32T10562 x 4, 32T10563 x 4, 32T10564 x 4, 32T10565 x 2, 32T10566 x 4

Siblings are shown with degree $\leq 47$

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

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,14,10,13)(11,15,12,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 8, 8 $ $16$ $8$ $( 1, 4, 6, 7, 2, 3, 5, 8)( 9,11,14,15,10,12,13,16)$
$ 8, 8 $ $16$ $8$ $( 1, 7, 5, 4, 2, 8, 6, 3)( 9,15,13,11,10,16,14,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14,10,13)(11,16,12,15)$
$ 16 $ $32$ $16$ $( 1,14, 4,15, 6,10, 7,12, 2,13, 3,16, 5, 9, 8,11)$
$ 16 $ $32$ $16$ $( 1,10, 3,11, 6,13, 8,15, 2, 9, 4,12, 5,14, 7,16)$
$ 16 $ $32$ $16$ $( 1,15, 7,13, 5,11, 4,10, 2,16, 8,14, 6,12, 3, 9)$
$ 16 $ $32$ $16$ $( 1,12, 8,10, 5,15, 3,14, 2,11, 7, 9, 6,16, 4,13)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14)(10,13)(11,15)(12,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 8, 8 $ $16$ $8$ $( 1, 4, 6, 8, 2, 3, 5, 7)( 9,11,14,16,10,12,13,15)$
$ 8, 8 $ $16$ $8$ $( 1, 7, 6, 4, 2, 8, 5, 3)( 9,15,14,11,10,16,13,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,14)(10,13)(11,15)(12,16)$
$ 8, 8 $ $32$ $8$ $( 1, 4, 6, 7, 2, 3, 5, 8)( 9,11,14,16,10,12,13,15)$
$ 8, 8 $ $32$ $8$ $( 1, 7, 5, 4, 2, 8, 6, 3)( 9,15,14,11,10,16,13,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 5, 6)( 9,10)(11,12)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14)(10,13)(11,15)(12,16)$
$ 16 $ $32$ $16$ $( 1,14, 3,16, 6,10, 7,12, 2,13, 4,15, 5, 9, 8,11)$
$ 16 $ $32$ $16$ $( 1,10, 3,11, 6,13, 7,16, 2, 9, 4,12, 5,14, 8,15)$
$ 16 $ $32$ $16$ $( 1,15, 8,14, 5,11, 4,10, 2,16, 7,13, 6,12, 3, 9)$
$ 16 $ $32$ $16$ $( 1,12, 8,10, 5,15, 4,13, 2,11, 7, 9, 6,16, 3,14)$

Group invariants

Order:  $512=2^{9}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [512, 1714]
Character table: Data not available.