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Properties

Label 16T32
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $Q_{16}:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 32);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $32$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $Q_{16}:C_2$
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:  (1,15)(2,16)(3,5)(4,6)(7,8)(13,14), (1,14,15,12,2,13,16,11)(3,8,5,9,4,7,6,10), (1,6)(2,5)(3,15)(4,16)(7,12)(8,11)(9,14)(10,13)
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: $D_4\times C_2$

Low degree siblings

16T50, 32T18

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  32.44
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B
Size 1 1 2 4 2 2 4 4 4 4 4
2 P 1A 1A 1A 1A 2A 2A 2A 2A 2A 4A 4A
Type
32.44.1a R 1 1 1 1 1 1 1 1 1 1 1
32.44.1b R 1 1 −1 −1 1 −1 1 −1 1 1 −1
32.44.1c R 1 1 −1 −1 1 −1 1 1 −1 −1 1
32.44.1d R 1 1 −1 1 1 −1 −1 −1 1 −1 1
32.44.1e R 1 1 −1 1 1 −1 −1 1 −1 1 −1
32.44.1f R 1 1 1 −1 1 1 −1 −1 −1 1 1
32.44.1g R 1 1 1 −1 1 1 −1 1 1 −1 −1
32.44.1h R 1 1 1 1 1 1 1 −1 −1 −1 −1
32.44.2a R 2 2 −2 0 −2 2 0 0 0 0 0
32.44.2b R 2 2 2 0 −2 −2 0 0 0 0 0
32.44.4a S 4 −4 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);