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Properties

Label 16T1049
Order \(768\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $1049$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,2,16)(3,9,4,10)(5,12,6,11)(7,14,8,13), (1,9,2,10)(3,12)(4,11)(5,13)(6,14)(7,16,8,15), (1,9,3,16,7,14)(2,10,4,15,8,13)(5,11)(6,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $C_2^3$
12:  $D_{6}$ x 3
24:  $S_4$, $S_3 \times C_2^2$
48:  $S_4\times C_2$ x 3
96:  12T48
192:  $V_4^2:(S_3\times C_2)$
384:  12T136

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $S_4$

Degree 8: $S_4\times C_2$

Low degree siblings

16T1049, 16T1051 x 2, 16T1058 x 4, 32T34676, 32T34677, 32T34678 x 2, 32T34681, 32T34682, 32T34700 x 2, 32T34701 x 2, 32T34702 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, 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 $ $12$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,15)(10,16)(11,14)(12,13)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $32$ $3$ $( 3, 6, 8)( 4, 5, 7)( 9,13,12)(10,14,11)$
$ 6, 6, 2, 2 $ $32$ $6$ $( 1, 2)( 3, 5, 8, 4, 6, 7)( 9,14,12,10,13,11)(15,16)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1,15, 2,16)( 3, 9, 4,10)( 5,12, 6,11)( 7,14, 8,13)$
$ 8, 8 $ $48$ $8$ $( 1,14, 6, 9, 2,13, 5,10)( 3,12, 7,16, 4,11, 8,15)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1,15, 2,16)( 3,10)( 4, 9)( 5,11, 6,12)( 7,14)( 8,13)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 3, 7)( 4, 8)( 9,14)(10,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,13)(10,14)(11,12)(15,16)$
$ 4, 4, 4, 4 $ $48$ $4$ $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,16,13,11)(10,15,14,12)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $24$ $4$ $( 3, 8, 4, 7)( 5, 6)( 9,14,10,13)(11,12)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,15, 2,16)( 3,14, 4,13)( 5,12, 6,11)( 7, 9, 8,10)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1,14, 2,13)( 3,16, 4,15)( 5,10)( 6, 9)( 7,12)( 8,11)$
$ 12, 4 $ $64$ $12$ $( 1,15, 2,16)( 3,11, 7,13, 6,10, 4,12, 8,14, 5, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,16)(10,15)(11,13)(12,14)$
$ 6, 3, 3, 2, 1, 1 $ $64$ $6$ $( 3, 6, 8)( 4, 5, 7)( 9,14,12,10,13,11)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1,15)( 2,16)( 3, 9)( 4,10)( 5,12)( 6,11)( 7,14)( 8,13)$
$ 8, 8 $ $48$ $8$ $( 1,14, 5,10, 2,13, 6, 9)( 3,12, 8,15, 4,11, 7,16)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1,15)( 2,16)( 3,10, 4, 9)( 5,11)( 6,12)( 7,14, 8,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $24$ $2$ $( 3, 7)( 4, 8)( 9,13)(10,14)(11,12)(15,16)$
$ 4, 4, 4, 4 $ $48$ $4$ $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,15,13,12)(10,16,14,11)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $24$ $4$ $( 3, 8, 4, 7)( 5, 6)( 9,13,10,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5,12)( 6,11)( 7, 9)( 8,10)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,10, 6, 9)( 7,12, 8,11)$
$ 6, 6, 2, 2 $ $64$ $6$ $( 1,15)( 2,16)( 3,11, 8,14, 6,10)( 4,12, 7,13, 5, 9)$

Group invariants

Order:  $768=2^{8} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [768, 1088551]
Character table: Data not available.