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Properties

Label 16T260
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_8:C_8$

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Show commands: Magma

magma: G := TransitiveGroup(16, 260);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $260$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_8:C_8$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,16)(2,15)(3,9)(4,10)(5,12)(6,11)(7,14)(8,13), (1,10,7,16,6,13,4,11,2,9,8,15,5,14,3,12)
magma: Generators(G);
 

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

16T260, 32T600 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $128=2^{7}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $4$
Label:  128.68
magma: IdentifyGroup(G);
 
Character table:    32 x 32 character table

magma: CharacterTable(G);