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Properties

Label 16T591
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2\times OD_{16}).D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $591$
Group :  $(C_2\times OD_{16}).D_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $5$
Generators:  (1,8,12,16,4,6,10,13,2,7,11,15,3,5,9,14), (1,11,4,10,2,12,3,9)(5,13,7,15,6,14,8,16)
$|\Aut(F/K)|$:  $2$

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8$

Low degree siblings

16T591 x 3, 32T2898 x 8, 32T2899 x 8, 32T2900 x 4, 32T2901 x 4, 32T2902 x 4, 32T2903 x 2

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

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 328]
Character table: Data not available.