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Properties

Label 16T47
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, 47);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $47$
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:  (1,13)(2,14)(3,12)(4,11)(5,9)(6,10)(7,15)(8,16), (1,8,6,4,2,7,5,3)(9,16,14,12,10,15,13,11), (1,10,2,9)(3,16,4,15)(5,13,6,14)(7,12,8,11)
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: $D_4$

Low degree siblings

16T44 x 2, 32T26

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

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:  32.42
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 4A1 4A-1 4B 4C 4D 8A1 8A3 8B1 8B-1
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 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−ζ83 ζ8+ζ83
32.42.2c2 C 2 −2 0 0 0 2ζ82 −2ζ82 0 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 ζ8+ζ83 −ζ8−ζ83
32.42.2c3 C 2 −2 0 0 0 −2ζ82 2ζ82 0 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8+ζ83 −ζ8−ζ83
32.42.2c4 C 2 −2 0 0 0 2ζ82 −2ζ82 0 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 −ζ8−ζ83 ζ8+ζ83

magma: CharacterTable(G);