↑

Properties

Label 16T859
Order \(512\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes

Related objects

Learn more about

Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $859$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $6$
Generators:  (1,3,2,4)(5,7)(6,8)(9,12,10,11)(15,16), (1,7,12,14,2,8,11,13)(3,5,10,16,4,6,9,15), (1,5,11,15,2,6,12,16)(3,8,9,14,4,7,10,13)
$|\Aut(F/K)|$:  $2$

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 4, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
32:  $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$
64:  $((C_8 : C_2):C_2):C_2$ x 2, 16T76
128:  16T227
256:  16T567

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $((C_8 : C_2):C_2):C_2$

Low degree siblings

16T859 x 7, 32T10225 x 4, 32T10226 x 4, 32T10227 x 8, 32T10228 x 16, 32T10229 x 4, 32T10230 x 4, 32T10231 x 8, 32T10232 x 4, 32T10233 x 4, 32T10234 x 4, 32T20547 x 4

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

Group invariants

Order:  $512=2^{9}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [512, 60663]
Character table: Data not available.