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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 invariants

Abstract group:  $Q_{16}:C_2$
magma: IdentifyGroup(G);
 
Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $3$
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $32$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(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

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{16}$ $1$ $1$ $0$ $()$
2A $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B $2^{8}$ $2$ $2$ $8$ $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,12)( 8,11)( 9,14)(10,13)$
2C $2^{6},1^{4}$ $4$ $2$ $6$ $( 1,16)( 2,15)( 3, 6)( 4, 5)( 9,10)(11,12)$
4A $4^{4}$ $2$ $4$ $12$ $( 1,16, 2,15)( 3, 6, 4, 5)( 7, 9, 8,10)(11,13,12,14)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 4, 2, 3)( 5,15, 6,16)( 7,14, 8,13)( 9,11,10,12)$
4C $4^{4}$ $4$ $4$ $12$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,13, 6,14)( 7,15, 8,16)$
4D $4^{4}$ $4$ $4$ $12$ $( 1,14, 2,13)( 3, 8, 4, 7)( 5,10, 6, 9)(11,16,12,15)$
4E $4^{4}$ $4$ $4$ $12$ $( 1, 3, 2, 4)( 5,15, 6,16)( 7,11, 8,12)( 9,14,10,13)$
8A $8^{2}$ $4$ $8$ $14$ $( 1, 9,16, 8, 2,10,15, 7)( 3,11, 6,13, 4,12, 5,14)$
8B $8^{2}$ $4$ $8$ $14$ $( 1,13,15,11, 2,14,16,12)( 3, 7, 5,10, 4, 8, 6, 9)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(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);
 

Regular extensions

Data not computed