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Properties

Label 16T50
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, 50);
 

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 $t$:  $50$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,8,2,7)(3,6,4,5)(9,12,10,11)(13,15,14,16)$, $(9,10)(11,12)(13,14)(15,16)$, $(1,13)(2,14)(3,11)(4,12)(5,15)(6,16)(7,10)(8,9)$
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: $D_4$

Low degree siblings

16T32, 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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2C $2^{8}$ $4$ $2$ $8$ $( 1,13)( 2,14)( 3,11)( 4,12)( 5,15)( 6,16)( 7,10)( 8, 9)$
4A $4^{4}$ $2$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,16,10,15)(11,13,12,14)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,15,10,16)(11,14,12,13)$
4C $4^{4}$ $4$ $4$ $12$ $( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,11,10,12)(13,16,14,15)$
4D $4^{4}$ $4$ $4$ $12$ $( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,12,10,11)(13,15,14,16)$
4E $4^{4}$ $4$ $4$ $12$ $( 1,13, 2,14)( 3,11, 4,12)( 5,15, 6,16)( 7,10, 8, 9)$
8A $8^{2}$ $4$ $8$ $14$ $( 1,10, 3,15, 2, 9, 4,16)( 5,12, 7,14, 6,11, 8,13)$
8B $8^{2}$ $4$ $8$ $14$ $( 1,10, 4,16, 2, 9, 3,15)( 5,12, 8,13, 6,11, 7,14)$

Malle's constant $a(G)$:     $1/4$

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);