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Properties

Label 16T659
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2.D_4^2.C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $659$
Group :  $C_2.D_4^2.C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $5$
Generators:  (1,10,2,9)(3,11,4,12)(5,14,6,13)(7,15,8,16), (3,7)(4,8)(5,6)(11,12)(15,16), (1,15,2,16)(3,14,4,13)(5,11,6,12)(7,10,8,9)
$|\Aut(F/K)|$:  $2$

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 6, $C_2^3$
16:  $D_4\times C_2$ x 3
32:  $C_2^2 \wr C_2$
64:  $(((C_4 \times C_2): C_2):C_2):C_2$
128:  $C_2 \wr C_2\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T659, 16T665 x 2, 16T696 x 2, 32T3217, 32T3218 x 4, 32T3219, 32T3220 x 4, 32T3221, 32T3241 x 2, 32T3242, 32T3243, 32T3344, 32T3345, 32T6289 x 4, 32T7363

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

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 6661]
Character table: Data not available.