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Properties

Label 16T1
Degree $16$
Order $16$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_{16}$

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Show commands: Magma

magma: G := TransitiveGroup(16, 1);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8$

Low degree siblings

There are no siblings 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, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 16 $ $1$ $16$ $( 1, 3, 5, 7,10,11,14,16, 2, 4, 6, 8, 9,12,13,15)$
$ 16 $ $1$ $16$ $( 1, 4, 5, 8,10,12,14,15, 2, 3, 6, 7, 9,11,13,16)$
$ 8, 8 $ $1$ $8$ $( 1, 5,10,14, 2, 6, 9,13)( 3, 7,11,16, 4, 8,12,15)$
$ 8, 8 $ $1$ $8$ $( 1, 6,10,13, 2, 5, 9,14)( 3, 8,11,15, 4, 7,12,16)$
$ 16 $ $1$ $16$ $( 1, 7,14, 4, 9,15, 5,11, 2, 8,13, 3,10,16, 6,12)$
$ 16 $ $1$ $16$ $( 1, 8,14, 3, 9,16, 5,12, 2, 7,13, 4,10,15, 6,11)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 6,14)( 7,15, 8,16)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,14, 6,13)( 7,16, 8,15)$
$ 16 $ $1$ $16$ $( 1,11, 6,15,10, 4,13, 7, 2,12, 5,16, 9, 3,14, 8)$
$ 16 $ $1$ $16$ $( 1,12, 6,16,10, 3,13, 8, 2,11, 5,15, 9, 4,14, 7)$
$ 8, 8 $ $1$ $8$ $( 1,13, 9, 6, 2,14,10, 5)( 3,15,12, 8, 4,16,11, 7)$
$ 8, 8 $ $1$ $8$ $( 1,14, 9, 5, 2,13,10, 6)( 3,16,12, 7, 4,15,11, 8)$
$ 16 $ $1$ $16$ $( 1,15,13,12, 9, 8, 6, 4, 2,16,14,11,10, 7, 5, 3)$
$ 16 $ $1$ $16$ $( 1,16,13,11, 9, 7, 6, 3, 2,15,14,12,10, 8, 5, 4)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $16=2^{4}$
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  16.1
magma: IdentifyGroup(G);
 
Character table:

1A 2A 4A1 4A-1 8A1 8A-1 8A3 8A-3 16A1 16A-1 16A3 16A-3 16A5 16A-5 16A7 16A-7
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 2A 2A 4A-1 4A1 4A1 4A-1 8A1 8A-1 8A-3 8A3 8A-3 8A3 8A1 8A-1
Type
16.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16.1.1b R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
16.1.1c1 C 1 1 1 1 −1 −1 −1 −1 −i i i −i −i i i −i
16.1.1c2 C 1 1 1 1 −1 −1 −1 −1 i −i −i i i −i −i i
16.1.1d1 C 1 1 −1 −1 −ζ82 ζ82 ζ82 −ζ82 ζ83 −ζ8 ζ8 −ζ83 −ζ83 ζ8 −ζ8 ζ83
16.1.1d2 C 1 1 −1 −1 ζ82 −ζ82 −ζ82 ζ82 −ζ8 ζ83 −ζ83 ζ8 ζ8 −ζ83 ζ83 −ζ8
16.1.1d3 C 1 1 −1 −1 −ζ82 ζ82 ζ82 −ζ82 −ζ83 ζ8 −ζ8 ζ83 ζ83 −ζ8 ζ8 −ζ83
16.1.1d4 C 1 1 −1 −1 ζ82 −ζ82 −ζ82 ζ82 ζ8 −ζ83 ζ83 −ζ8 −ζ8 ζ83 −ζ83 ζ8
16.1.1e1 C 1 −1 −ζ164 ζ164 ζ166 −ζ162 ζ162 −ζ166 −ζ163 ζ165 ζ16 −ζ167 ζ167 −ζ16 −ζ165 ζ163
16.1.1e2 C 1 −1 ζ164 −ζ164 −ζ162 ζ166 −ζ166 ζ162 ζ165 −ζ163 −ζ167 ζ16 −ζ16 ζ167 ζ163 −ζ165
16.1.1e3 C 1 −1 −ζ164 ζ164 ζ166 −ζ162 ζ162 −ζ166 ζ163 −ζ165 −ζ16 ζ167 −ζ167 ζ16 ζ165 −ζ163
16.1.1e4 C 1 −1 ζ164 −ζ164 −ζ162 ζ166 −ζ166 ζ162 −ζ165 ζ163 ζ167 −ζ16 ζ16 −ζ167 −ζ163 ζ165
16.1.1e5 C 1 −1 −ζ164 ζ164 −ζ166 ζ162 −ζ162 ζ166 ζ167 −ζ16 ζ165 −ζ163 ζ163 −ζ165 ζ16 −ζ167
16.1.1e6 C 1 −1 ζ164 −ζ164 ζ162 −ζ166 ζ166 −ζ162 −ζ16 ζ167 −ζ163 ζ165 −ζ165 ζ163 −ζ167 ζ16
16.1.1e7 C 1 −1 −ζ164 ζ164 −ζ166 ζ162 −ζ162 ζ166 −ζ167 ζ16 −ζ165 ζ163 −ζ163 ζ165 −ζ16 ζ167
16.1.1e8 C 1 −1 ζ164 −ζ164 ζ162 −ζ166 ζ166 −ζ162 ζ16 −ζ167 ζ163 −ζ165 ζ165 −ζ163 ζ167 −ζ16

magma: CharacterTable(G);