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Properties

Label 16T44
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_8:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 44);
 

Group invariants

Abstract group:  $D_8:C_2$
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 $t$:  $44$
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:  $(1,7,16,9,2,8,15,10)(3,13,6,12,4,14,5,11)$, $(1,8)(2,7)(3,14)(4,13)(5,11)(6,12)(9,16)(10,15)$, $(1,14,2,13)(3,7,4,8)(5,9,6,10)(11,16,12,15)$
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

16T44, 16T47, 32T26

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 $2^{8}$ $2$ $2$ $8$ $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,11)( 8,12)( 9,13)(10,14)$
2C $2^{6},1^{4}$ $4$ $2$ $6$ $( 5, 6)( 7,10)( 8, 9)(11,13)(12,14)(15,16)$
2D $2^{8}$ $4$ $2$ $8$ $( 1, 7)( 2, 8)( 3,13)( 4,14)( 5,12)( 6,11)( 9,15)(10,16)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 4, 2, 3)( 5,15, 6,16)( 7,14, 8,13)( 9,11,10,12)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1, 3, 2, 4)( 5,16, 6,15)( 7,13, 8,14)( 9,12,10,11)$
4B $4^{4}$ $2$ $4$ $12$ $( 1,15, 2,16)( 3, 5, 4, 6)( 7,10, 8, 9)(11,14,12,13)$
4C $4^{4}$ $4$ $4$ $12$ $( 1, 6, 2, 5)( 3,15, 4,16)( 7,13, 8,14)( 9,11,10,12)$
4D $4^{4}$ $4$ $4$ $12$ $( 1,12, 2,11)( 3,10, 4, 9)( 5, 7, 6, 8)(13,15,14,16)$
8A1 $8^{2}$ $2$ $8$ $14$ $( 1,14,15,12, 2,13,16,11)( 3, 7, 5,10, 4, 8, 6, 9)$
8A-1 $8^{2}$ $2$ $8$ $14$ $( 1,11,16,13, 2,12,15,14)( 3, 9, 6, 8, 4,10, 5, 7)$
8B1 $8^{2}$ $2$ $8$ $14$ $( 1,10,15, 8, 2, 9,16, 7)( 3,11, 5,14, 4,12, 6,13)$
8B3 $8^{2}$ $2$ $8$ $14$ $( 1, 8,16,10, 2, 7,15, 9)( 3,14, 6,11, 4,13, 5,12)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 4A1 4A-1 4B 4C 4D 8A1 8A-1 8B1 8B3
Size 1 1 2 4 4 1 1 2 4 4 2 2 2 2
2 P 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 4B 4B 4B 4B
Type
32.42.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.42.1b R 1 1 −1 −1 −1 −1 −1 1 1 1 −1 1 −1 1
32.42.1c R 1 1 −1 −1 1 −1 −1 1 1 −1 1 −1 1 −1
32.42.1d R 1 1 −1 1 −1 −1 −1 1 −1 1 1 −1 1 −1
32.42.1e R 1 1 −1 1 1 −1 −1 1 −1 −1 −1 1 −1 1
32.42.1f R 1 1 1 −1 −1 1 1 1 −1 −1 1 1 1 1
32.42.1g R 1 1 1 −1 1 1 1 1 −1 1 −1 −1 −1 −1
32.42.1h R 1 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 −1
32.42.2a R 2 2 −2 0 0 2 2 −2 0 0 0 0 0 0
32.42.2b R 2 2 2 0 0 −2 −2 −2 0 0 0 0 0 0
32.42.2c1 C 2 −2 0 0 0 −2ζ82 2ζ82 0 0 0 −ζ8−ζ83 −ζ8−1−ζ8 ζ8+ζ83 ζ8−1+ζ8
32.42.2c2 C 2 −2 0 0 0 2ζ82 −2ζ82 0 0 0 ζ8+ζ83 −ζ8−1−ζ8 −ζ8−ζ83 ζ8−1+ζ8
32.42.2c3 C 2 −2 0 0 0 −2ζ82 2ζ82 0 0 0 ζ8+ζ83 ζ8−1+ζ8 −ζ8−ζ83 −ζ8−1−ζ8
32.42.2c4 C 2 −2 0 0 0 2ζ82 −2ζ82 0 0 0 −ζ8−ζ83 ζ8−1+ζ8 ζ8+ζ83 −ζ8−1−ζ8

magma: CharacterTable(G);
 

Regular extensions

Data not computed