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Properties

Label 16T817
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 $t$ :  $817$
Parity:  $1$
Primitive:  No
Nilpotency class:  $6$
Generators:  (1,10,4,11,5,14,8,16)(2,9,3,12,6,13,7,15), (1,10,4,12,5,13,7,15)(2,9,3,11,6,14,8,16)
$|\Aut(F/K)|$:  $2$

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

16T817 x 3, 16T840 x 4, 16T924 x 8, 32T9872 x 8, 32T9873 x 4, 32T9874 x 2, 32T9875 x 4, 32T9876 x 4, 32T9877 x 2, 32T9878 x 8, 32T9879 x 8, 32T9880 x 8, 32T9881 x 4, 32T9882 x 4, 32T9883 x 4, 32T9884 x 4, 32T10088 x 2, 32T10089 x 8, 32T10090 x 2, 32T10091 x 2, 32T10092 x 4, 32T10093 x 4, 32T10094 x 2, 32T10095 x 2, 32T10096 x 4, 32T10097 x 2, 32T10599 x 4, 32T10600 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 $ $4$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)$
$ 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, 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 $ $16$ $4$ $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,12,13,15)(10,11,14,16)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 8, 5, 4)( 2, 7, 6, 3)( 9,15,13,12)(10,16,14,11)$
$ 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, 2, 2 $ $8$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,13)(10,14)(11,15)(12,16)$
$ 8, 8 $ $32$ $8$ $( 1,10, 4,11, 5,14, 8,16)( 2, 9, 3,12, 6,13, 7,15)$
$ 8, 8 $ $32$ $8$ $( 1,14, 4,16, 5,10, 8,11)( 2,13, 3,15, 6, 9, 7,12)$
$ 8, 8 $ $32$ $8$ $( 1,11, 8,10, 5,16, 4,14)( 2,12, 7, 9, 6,15, 3,13)$
$ 8, 8 $ $32$ $8$ $( 1,16, 8,14, 5,11, 4,10)( 2,15, 7,13, 6,12, 3, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,13,10,14)(11,16,12,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 4, 5, 7)( 2, 3, 6, 8)( 9,12,13,16)(10,11,14,15)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 8, 6, 4)( 2, 7, 5, 3)( 9,15,14,12)(10,16,13,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $(13,14)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13,10,14)(11,16,12,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,12,13,16)(10,11,14,15)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 8, 5, 4)( 2, 7, 6, 3)( 9,15,14,12)(10,16,13,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)(13,14)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,13,10,14)(11,16,12,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 5, 6)(13,14)(15,16)$
$ 8, 8 $ $32$ $8$ $( 1,10, 4,11, 5,14, 7,15)( 2, 9, 3,12, 6,13, 8,16)$
$ 8, 8 $ $32$ $8$ $( 1,14, 3,15, 5,10, 8,11)( 2,13, 4,16, 6, 9, 7,12)$
$ 8, 8 $ $32$ $8$ $( 1,11, 8,10, 5,16, 3,13)( 2,12, 7, 9, 6,15, 4,14)$
$ 8, 8 $ $32$ $8$ $( 1,16, 7,13, 5,11, 4,10)( 2,15, 8,14, 6,12, 3, 9)$

Group invariants

Order:  $512=2^{9}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [512, 1710]
Character table: Data not available.