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Properties

Label 16T813
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 $t$ :  $813$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,10)(2,9)(3,11,4,12)(5,15)(6,16)(7,14,8,13), (1,4)(2,3)(5,6)(7,8)(11,12)(13,15,14,16), (1,5)(2,6)(3,7,4,8)(9,15,10,16)(11,13)(12,14)
$|\Aut(F/K)|$:  $2$

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, $Q_8:C_2$ x 4, $C_4\times C_2^2$
32:  $C_2^3 : C_4 $ x 4, $C_4^2:C_2$ x 2, $C_2 \times (C_2^2:C_4)$, 16T37 x 4
64:  16T76 x 2, 32T197
128:  16T212 x 2, 16T274
256:  32T3799

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$

Low degree siblings

16T813, 16T833 x 2, 16T874 x 4, 32T9839 x 2, 32T9840 x 2, 32T9841, 32T9842 x 4, 32T9843, 32T9998 x 2, 32T9999, 32T10000, 32T10001 x 2, 32T10316 x 2, 32T10317 x 2, 32T19556, 32T19557, 32T19558, 32T19559, 32T19829, 32T19830, 32T20150, 32T20151, 32T26475 x 2, 32T28630, 32T33256 x 2, 32T33746, 32T33771

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

Group invariants

Order:  $512=2^{9}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [512, 42712]
Character table: Data not available.