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Properties

Label 16T994
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 $t$ :  $994$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $6$
Generators:  (1,16,8,10,5,12,4,13,2,15,7,9,6,11,3,14), (1,13,6,10,2,14,5,9)(3,12,7,16,4,11,8,15), (1,3)(2,4)(5,7)(6,8)(9,11,13,15,10,12,14,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 3
32:  $C_2^2 \wr C_2$
64:  $(((C_4 \times C_2): C_2):C_2):C_2$
128:  $C_2 \wr C_2\wr C_2$
256:  16T660

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

16T994 x 3, 32T10984 x 2, 32T10985 x 2, 32T10986 x 2, 32T10987 x 2, 32T10988 x 2, 32T10989 x 2, 32T10990 x 2, 32T21597, 32T21641, 32T21927 x 2, 32T21951

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $4$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 8, 8 $ $4$ $8$ $( 1, 8, 5, 4, 2, 7, 6, 3)( 9,11,14,16,10,12,13,15)$
$ 8, 8 $ $4$ $8$ $( 1, 7, 5, 3, 2, 8, 6, 4)( 9,12,14,15,10,11,13,16)$
$ 8, 8 $ $8$ $8$ $( 1, 8, 5, 4, 2, 7, 6, 3)( 9,12,14,15,10,11,13,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $4$ $( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14,10,13)(11,16,12,15)$
$ 16 $ $32$ $16$ $( 1,16, 8,10, 5,12, 4,13, 2,15, 7, 9, 6,11, 3,14)$
$ 16 $ $32$ $16$ $( 1,12, 7,14, 5,15, 3,10, 2,11, 8,13, 6,16, 4, 9)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $2$ $( 3, 7)( 4, 8)( 5, 6)(11,16)(12,15)(13,14)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,11,10,12)(13,16,14,15)$
$ 8, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $8$ $( 9,15,13,12,10,16,14,11)$
$ 8, 2, 2, 2, 2 $ $8$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16,13,11,10,15,14,12)$
$ 8, 4, 4 $ $8$ $8$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,11,14,16,10,12,13,15)$
$ 8, 4, 4 $ $8$ $8$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,12,14,15,10,11,13,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7, 9)( 8,10)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1,12, 2,11)( 3,10, 4, 9)( 5,15, 6,16)( 7,14, 8,13)$
$ 8, 8 $ $32$ $8$ $( 1,10, 5,13, 2, 9, 6,14)( 3,16, 8,12, 4,15, 7,11)$
$ 4, 4, 2, 2, 2, 1, 1 $ $32$ $4$ $( 3, 7)( 4, 8)( 5, 6)( 9,12,10,11)(13,15,14,16)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $16$ $2$ $( 3, 8)( 4, 7)( 5, 6)(11,12)(15,16)$
$ 4, 4, 2, 2, 2, 1, 1 $ $16$ $4$ $( 1, 5)( 2, 6)( 3, 4)( 9,14,10,13)(11,15,12,16)$
$ 8, 2, 2, 2, 2 $ $32$ $8$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,11,13,15,10,12,14,16)$
$ 8, 8 $ $64$ $8$ $( 1,16, 7,13, 2,15, 8,14)( 3,10, 5,11, 4, 9, 6,12)$
$ 8, 2, 2, 2, 1, 1 $ $32$ $8$ $( 3, 8)( 4, 7)( 5, 6)( 9,15,14,11,10,16,13,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $2$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)(11,12)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,14,10,13)(11,15,12,16)$
$ 4, 4, 4, 2, 2 $ $64$ $4$ $( 1,16, 2,15)( 3,10, 8,14)( 4, 9, 7,13)( 5,11)( 6,12)$

Group invariants

Order:  $512=2^{9}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [512, 60839]
Character table: Data not available.