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Properties

Label 16T984
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 $t$ :  $984$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,14)(2,13)(3,11,4,12)(5,10)(6,9)(7,15,8,16), (1,16)(2,15)(3,14)(4,13)(5,12,6,11)(7,9,8,10), (1,11,2,12)(3,9)(4,10)(5,15,6,16)(7,14)(8,13)
$|\Aut(F/K)|$:  $2$

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 12, $C_2^3$
16:  $D_4\times C_2$ x 6, $Q_8:C_2$
32:  $C_2^2 \wr C_2$ x 3, 16T34 x 3, $C_4^2:C_2$
64:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 6, 32T320
128:  16T350 x 3
256:  32T6030

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$

Low degree siblings

16T965 x 16, 16T966 x 4, 16T984 x 3, 32T10815 x 16, 32T10816 x 16, 32T10817 x 8, 32T10818 x 8, 32T10819 x 16, 32T10820 x 4, 32T10821 x 4, 32T10822 x 8, 32T10823 x 2, 32T10824 x 4, 32T10825 x 8, 32T10826 x 2, 32T10922 x 8, 32T10923 x 8, 32T10924 x 16, 32T10925 x 2, 32T10926 x 4, 32T10927 x 8, 32T10928 x 2, 32T10929 x 4, 32T20084 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 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, 2, 2, 2, 2 $ $16$ $4$ $( 1,14)( 2,13)( 3,11, 4,12)( 5,10)( 6, 9)( 7,15, 8,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 5, 6)( 7, 8)(11,12)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 5, 6)(13,14)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1,14, 2,13)( 3,11)( 4,12)( 5, 9, 6,10)( 7,16)( 8,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,16,12,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3, 4)( 7, 8)( 9,10)(13,14)$
$ 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 $ $4$ $2$ $( 1, 2)( 5, 6)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1,14)( 2,13)( 3,11)( 4,12)( 5,10)( 6, 9)( 7,15)( 8,16)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 9, 2,10)( 3,16, 4,15)( 5,13, 6,14)( 7,12, 8,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,13,10,14)(11,16,12,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12, 6,11)( 7, 9, 8,10)$
$ 8, 8 $ $32$ $8$ $( 1, 7, 6, 4, 2, 8, 5, 3)( 9,16,14,11,10,15,13,12)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1,16)( 2,15)( 3,13, 4,14)( 5,11, 6,12)( 7, 9)( 8,10)$
$ 8, 8 $ $32$ $8$ $( 1, 7, 6, 3, 2, 8, 5, 4)( 9,15,13,12,10,16,14,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)(11,12)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 5, 6)( 9,10)(15,16)$
$ 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 $ $32$ $4$ $( 1,14, 2,13)( 3,11, 4,12)( 5,10)( 6, 9)( 7,15)( 8,16)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1, 9, 2,10)( 3,16, 4,15)( 5,13)( 6,14)( 7,12)( 8,11)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,13,10,14)(11,16,12,15)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1,16, 2,15)( 3,14, 4,13)( 5,12, 6,11)( 7, 9, 8,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 8, 8 $ $32$ $8$ $( 1, 7, 6, 4, 2, 8, 5, 3)( 9,16,13,11,10,15,14,12)$
$ 8, 8 $ $32$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,11,13,16,10,12,14,15)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1,11, 2,12)( 3,10)( 4, 9)( 5,15, 6,16)( 7,13)( 8,14)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1,12)( 2,11)( 3,10, 4, 9)( 5,16)( 6,15)( 7,13, 8,14)$

Group invariants

Order:  $512=2^{9}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [512, 51578]
Character table: Data not available.