↑

Properties

Label 16T35
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_8:C_2$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(16, 35);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $35$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_8:C_2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16), (1,4,6,7,9,12,14,15)(2,3,5,8,10,11,13,16), (1,9)(2,10)(5,13)(6,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$16$:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$, $Z_8 : Z_8^\times$ x 2

Low degree siblings

8T15 x 2, 16T38 x 2, 16T45, 32T21

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $3$
Label:  32.43
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B
Size 1 1 2 4 4 4 2 2 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2A 2A 2A 4A 4A
Type
32.43.1a R 1 1 1 1 1 1 1 1 1 1 1
32.43.1b R 1 1 −1 −1 1 −1 1 −1 1 1 −1
32.43.1c R 1 1 −1 −1 1 1 1 −1 −1 −1 1
32.43.1d R 1 1 −1 1 −1 −1 1 −1 1 −1 1
32.43.1e R 1 1 −1 1 −1 1 1 −1 −1 1 −1
32.43.1f R 1 1 1 −1 −1 −1 1 1 −1 1 1
32.43.1g R 1 1 1 −1 −1 1 1 1 1 −1 −1
32.43.1h R 1 1 1 1 1 −1 1 1 −1 −1 −1
32.43.2a R 2 2 −2 0 0 0 −2 2 0 0 0
32.43.2b R 2 2 2 0 0 0 −2 −2 0 0 0
32.43.4a R 4 −4 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);