↑

Properties

Label 16T48
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2\times \SD_{16}$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(16, 48);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $48$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times \SD_{16}$
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:  (1,10)(2,9)(3,15)(4,16)(5,6)(7,11)(8,12)(13,14), (1,16,14,12,9,7,6,3)(2,15,13,11,10,8,5,4), (1,3,9,12)(2,4,10,11)(5,15,13,8)(6,16,14,7)
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, $C_2^3$
$16$:  $QD_{16}$ x 2, $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $QD_{16}$ x 2, $D_4\times C_2$

Low degree siblings

16T48, 32T27

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  32.40
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A 4B 4C 4D 8A1 8A-1 8B1 8B-1
Size 1 1 1 1 4 4 2 2 4 4 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 4A 4A 4A 4A
Type
32.40.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.40.1b R 1 1 −1 −1 −1 1 1 −1 −1 1 −1 −1 1 1
32.40.1c R 1 1 −1 −1 −1 1 1 −1 1 −1 1 1 −1 −1
32.40.1d R 1 1 −1 −1 1 −1 1 −1 −1 1 1 1 −1 −1
32.40.1e R 1 1 −1 −1 1 −1 1 −1 1 −1 −1 −1 1 1
32.40.1f R 1 1 1 1 −1 −1 1 1 −1 −1 1 1 1 1
32.40.1g R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
32.40.1h R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
32.40.2a R 2 2 −2 −2 0 0 −2 2 0 0 0 0 0 0
32.40.2b R 2 2 2 2 0 0 −2 −2 0 0 0 0 0 0
32.40.2c1 C 2 −2 −2 2 0 0 0 0 0 0 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83
32.40.2c2 C 2 −2 −2 2 0 0 0 0 0 0 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83
32.40.2d1 C 2 −2 2 −2 0 0 0 0 0 0 −ζ8−ζ83 ζ8+ζ83 ζ8+ζ83 −ζ8−ζ83
32.40.2d2 C 2 −2 2 −2 0 0 0 0 0 0 ζ8+ζ83 −ζ8−ζ83 −ζ8−ζ83 ζ8+ζ83

magma: CharacterTable(G);