↑

Properties

Label 16T104
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^3.C_8$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $104$
Group :  $C_2^3.C_8$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,16,9,7,4,14,12,5,2,15,10,8,3,13,11,6), (1,14,10,5,4,15,11,8,2,13,9,6,3,16,12,7)
$|\Aut(F/K)|$:  $4$

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:  $C_2^2 : C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8$

Low degree siblings

32T107, 32T108

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 30]
Character table: Data not available.