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Properties

Label 16T1048
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 $t$ :  $1048$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,16,3,7,14)(2,5,15,4,8,13)(9,12)(10,11), (1,8,15,2,7,16)(3,6,13)(4,5,14)(9,10), (1,16,8,12)(2,15,7,11)(3,13,5,10)(4,14,6,9)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
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

16T1048, 16T1054 x 2, 32T34673 x 2, 32T34674, 32T34675, 32T34689, 32T34690, 32T34787, 32T35040

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 $ $12$ $4$ $( 1,12, 2,11)( 3, 9, 4,10)( 5,14, 6,13)( 7,15, 8,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,14)(10,13)(11,15)(12,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $32$ $3$ $( 5,14, 9)( 6,13,10)( 7,15,11)( 8,16,12)$
$ 6, 6, 2, 2 $ $32$ $6$ $( 1, 2)( 3, 4)( 5,13, 9, 6,14,10)( 7,16,11, 8,15,12)$
$ 6, 6, 2, 2 $ $64$ $6$ $( 1, 6,16, 3, 7,14)( 2, 5,15, 4, 8,13)( 9,12)(10,11)$
$ 4, 4, 4, 4 $ $12$ $4$ $( 1,13, 2,14)( 3,15, 4,16)( 5,11, 6,12)( 7, 9, 8,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,16)(10,15)(11,13)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,14, 6,13)( 7,16, 8,15)$
$ 6, 3, 3, 2, 1, 1 $ $64$ $6$ $( 3, 4)( 5,14, 9)( 6,13,10)( 7,16,11, 8,15,12)$
$ 12, 4 $ $64$ $12$ $( 1, 6,16, 4, 7,13, 2, 5,15, 3, 8,14)( 9,12,10,11)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1,13, 2,14)( 3,16, 4,15)( 5,11)( 6,12)( 7,10)( 8, 9)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $24$ $4$ $( 3, 4)( 7, 8)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 4, 4 $ $48$ $4$ $( 1,12, 7,16)( 2,11, 8,15)( 3,10, 6,13)( 4, 9, 5,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $24$ $2$ $( 3, 4)( 5, 6)( 9,14)(10,13)(11,16)(12,15)$
$ 4, 4, 4, 4 $ $48$ $4$ $( 1, 6,16,10)( 2, 5,15, 9)( 3, 8,14,12)( 4, 7,13,11)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5, 8, 6, 7)( 9,11,10,12)$
$ 4, 4, 4, 4 $ $24$ $4$ $( 1, 4, 2, 3)( 5,11, 6,12)( 7,10, 8, 9)(13,15,14,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $4$ $( 9,13,10,14)(11,16,12,15)$
$ 4, 4, 2, 2, 2, 2 $ $12$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14,10,13)(11,15,12,16)$
$ 8, 8 $ $48$ $8$ $( 1,12, 8,16, 2,11, 7,15)( 3, 9, 5,14, 4,10, 6,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $24$ $2$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,10)(11,12)$
$ 8, 8 $ $48$ $8$ $( 1, 6,16, 9, 2, 5,15,10)( 3, 7,14,12, 4, 8,13,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5, 8)( 6, 7)( 9,11)(10,12)$
$ 4, 4, 2, 2, 2, 2 $ $24$ $4$ $( 1,14, 2,13)( 3,16, 4,15)( 5, 7)( 6, 8)( 9,11)(10,12)$

Group invariants

Order:  $768=2^{8} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [768, 1088563]
Character table: Data not available.