↑

Properties

Label 16T49
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_8.C_4$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(16, 49);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $49$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_8.C_4$
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:  (1,4,6,8,2,3,5,7)(9,12,14,16,10,11,13,15), (1,12,5,16,2,11,6,15)(3,10,8,13,4,9,7,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$, $C_4\times C_2$, $Q_8$
$16$:  $C_4:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8$

Low degree siblings

32T28

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

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:  32.15
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 4A1 4A-1 4B 8A1 8A3 8B1 8B-1 8C1 8C-1 8D1 8D-1
Size 1 1 2 1 1 2 2 2 2 2 4 4 4 4
2 P 1A 1A 1A 2A 2A 2A 4B 4B 4B 4B 4A1 4A-1 4A1 4A-1
Type
32.15.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.15.1b R 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1
32.15.1c R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1
32.15.1d R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
32.15.1e1 C 1 1 −1 −1 −1 1 −1 −1 1 1 −i i −i i
32.15.1e2 C 1 1 −1 −1 −1 1 −1 −1 1 1 i −i i −i
32.15.1f1 C 1 1 −1 −1 −1 1 1 1 −1 −1 −i i i −i
32.15.1f2 C 1 1 −1 −1 −1 1 1 1 −1 −1 i −i −i i
32.15.2a R 2 2 −2 2 2 −2 0 0 0 0 0 0 0 0
32.15.2b S 2 2 2 −2 −2 −2 0 0 0 0 0 0 0 0
32.15.2c1 C 2 −2 0 −2ζ82 2ζ82 0 −ζ8−1−ζ8 ζ8−1+ζ8 ζ8+ζ83 −ζ8−ζ83 0 0 0 0
32.15.2c2 C 2 −2 0 2ζ82 −2ζ82 0 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−ζ83 ζ8+ζ83 0 0 0 0
32.15.2c3 C 2 −2 0 −2ζ82 2ζ82 0 ζ8−1+ζ8 −ζ8−1−ζ8 −ζ8−ζ83 ζ8+ζ83 0 0 0 0
32.15.2c4 C 2 −2 0 2ζ82 −2ζ82 0 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8+ζ83 −ζ8−ζ83 0 0 0 0

magma: CharacterTable(G);