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Properties

Label 16T289
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4.D_4:C_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $289$
Group :  $C_4.D_4:C_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,2)(3,4)(5,6)(7,8)(9,15,13,12,10,16,14,11), (1,15,5,11,2,16,6,12)(3,9,7,14,4,10,8,13)
$|\Aut(F/K)|$:  $8$

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:  $D_{8}$, $C_8:C_2$, $QD_{16}$, $C_2^2:C_4$, $C_8\times C_2$
32:  $C_4\wr C_2$, $C_2^2 : C_8$, 16T26
64:  32T272

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_4\wr C_2$

Low degree siblings

16T289 x 3, 32T667 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, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 8, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $8$ $( 9,11,14,16,10,12,13,15)$
$ 8, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $8$ $( 9,12,14,15,10,11,13,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $4$ $( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $4$ $( 9,14,10,13)(11,16,12,15)$
$ 8, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $8$ $( 9,15,13,12,10,16,14,11)$
$ 8, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $8$ $( 9,16,13,11,10,15,14,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 8, 2, 2, 2, 2 $ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11,14,16,10,12,13,15)$
$ 8, 2, 2, 2, 2 $ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,14,15,10,11,13,16)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14,10,13)(11,16,12,15)$
$ 8, 2, 2, 2, 2 $ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,15,13,12,10,16,14,11)$
$ 8, 2, 2, 2, 2 $ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16,13,11,10,15,14,12)$
$ 8, 8 $ $1$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,11,14,16,10,12,13,15)$
$ 8, 8 $ $2$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,12,14,15,10,11,13,16)$
$ 8, 4, 4 $ $2$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,13,10,14)(11,15,12,16)$
$ 8, 4, 4 $ $2$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,14,10,13)(11,16,12,15)$
$ 8, 8 $ $2$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,15,13,12,10,16,14,11)$
$ 8, 8 $ $2$ $8$ $( 1, 3, 5, 7, 2, 4, 6, 8)( 9,16,13,11,10,15,14,12)$
$ 8, 8 $ $1$ $8$ $( 1, 4, 5, 8, 2, 3, 6, 7)( 9,12,14,15,10,11,13,16)$
$ 8, 4, 4 $ $2$ $8$ $( 1, 4, 5, 8, 2, 3, 6, 7)( 9,13,10,14)(11,15,12,16)$
$ 8, 4, 4 $ $2$ $8$ $( 1, 4, 5, 8, 2, 3, 6, 7)( 9,14,10,13)(11,16,12,15)$
$ 8, 8 $ $2$ $8$ $( 1, 4, 5, 8, 2, 3, 6, 7)( 9,15,13,12,10,16,14,11)$
$ 8, 8 $ $2$ $8$ $( 1, 4, 5, 8, 2, 3, 6, 7)( 9,16,13,11,10,15,14,12)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,14,10,13)(11,16,12,15)$
$ 8, 4, 4 $ $2$ $8$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,15,13,12,10,16,14,11)$
$ 8, 4, 4 $ $2$ $8$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,16,13,11,10,15,14,12)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$
$ 8, 4, 4 $ $2$ $8$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,15,13,12,10,16,14,11)$
$ 8, 4, 4 $ $2$ $8$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,16,13,11,10,15,14,12)$
$ 8, 8 $ $2$ $8$ $( 1, 7, 6, 3, 2, 8, 5, 4)( 9,15,13,12,10,16,14,11)$
$ 8, 8 $ $1$ $8$ $( 1, 7, 6, 3, 2, 8, 5, 4)( 9,16,13,11,10,15,14,12)$
$ 8, 8 $ $1$ $8$ $( 1, 8, 6, 4, 2, 7, 5, 3)( 9,15,13,12,10,16,14,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,14, 6,13)( 7,16, 8,15)$
$ 16 $ $8$ $16$ $( 1, 9, 3,11, 5,14, 7,16, 2,10, 4,12, 6,13, 8,15)$
$ 16 $ $8$ $16$ $( 1, 9, 4,12, 5,14, 8,15, 2,10, 3,11, 6,13, 7,16)$
$ 8, 8 $ $8$ $8$ $( 1, 9, 5,14, 2,10, 6,13)( 3,11, 7,16, 4,12, 8,15)$
$ 8, 8 $ $8$ $8$ $( 1, 9, 6,13, 2,10, 5,14)( 3,11, 8,15, 4,12, 7,16)$
$ 16 $ $8$ $16$ $( 1, 9, 7,16, 6,13, 3,11, 2,10, 8,15, 5,14, 4,12)$
$ 16 $ $8$ $16$ $( 1, 9, 8,15, 6,13, 4,12, 2,10, 7,16, 5,14, 3,11)$

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 67]
Character table: Data not available.