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Properties

Label 16T209
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4.C_2^2:D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $209$
Group :  $C_4.C_2^2:D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,7,15,12,2,8,16,11)(3,5,13,10,4,6,14,9), (1,15)(2,16)(3,13)(4,14)(5,8,6,7)(9,11,10,12), (1,13,2,14)(3,16,4,15)(5,12,6,11)(7,9,8,10)
$|\Aut(F/K)|$:  $4$

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 8, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$
32:  $C_2^2 \wr C_2$, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
64:  32T239

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_2^2:C_4$

Low degree siblings

16T209 x 3, 16T279 x 2, 32T456 x 2, 32T457 x 2, 32T458 x 2, 32T647 x 2, 32T648 x 4, 32T649, 32T650

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

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 623]
Character table: Data not available.