Properties

Label 34T45
Degree $34$
Order $17408$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no

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Group action invariants

Degree $n$:  $34$
Transitive number $t$:  $45$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $2$
Generators:  (1,21,8,28,13,34,19,5,25,12,32,18,3,24,10,29,16,2,22,7,27,14,33,20,6,26,11,31,17,4,23,9,30,15), (1,31)(2,32)(3,29)(4,30)(5,27,6,28)(7,25)(8,26)(9,23)(10,24)(11,22)(12,21)(13,20,14,19)(15,17,16,18)(33,34)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$34$:  $D_{17}$
$68$:  $D_{34}$
$8704$:  34T30

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 17: $D_{17}$

Low degree siblings

34T45 x 29

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

There are 80 conjugacy classes of elements. Data not shown.

Group invariants

Order:  $17408=2^{10} \cdot 17$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  not available
Character table: not available.