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Properties

Label 16T5
Degree $16$
Order $16$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_8\times C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 5);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_8\times C_2$
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:  (1,4,5,8,9,11,13,16)(2,3,6,7,10,12,14,15), (1,15,13,12,9,7,5,3)(2,16,14,11,10,8,6,4)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_8$ x 2, $C_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_8$ x 2, $C_4\times C_2$

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 TypeSizeOrder IndexRepresentative
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 $2^{8}$ $1$ $2$ $8$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,15)( 8,16)$
2C $2^{8}$ $1$ $2$ $8$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,12,15)( 4, 8,11,16)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1,13, 9, 5)( 2,14,10, 6)( 3,15,12, 7)( 4,16,11, 8)$
4B1 $4^{4}$ $1$ $4$ $12$ $( 1,14, 9, 6)( 2,13,10, 5)( 3,16,12, 8)( 4,15,11, 7)$
4B-1 $4^{4}$ $1$ $4$ $12$ $( 1, 6, 9,14)( 2, 5,10,13)( 3, 8,12,16)( 4, 7,11,15)$
8A1 $8^{2}$ $1$ $8$ $14$ $( 1,11, 5,16, 9, 4,13, 8)( 2,12, 6,15,10, 3,14, 7)$
8A-1 $8^{2}$ $1$ $8$ $14$ $( 1, 4, 5, 8, 9,11,13,16)( 2, 3, 6, 7,10,12,14,15)$
8A3 $8^{2}$ $1$ $8$ $14$ $( 1, 7,13, 3, 9,15, 5,12)( 2, 8,14, 4,10,16, 6,11)$
8A-3 $8^{2}$ $1$ $8$ $14$ $( 1, 3, 5, 7, 9,12,13,15)( 2, 4, 6, 8,10,11,14,16)$
8B1 $8^{2}$ $1$ $8$ $14$ $( 1,15,13,12, 9, 7, 5, 3)( 2,16,14,11,10, 8, 6, 4)$
8B-1 $8^{2}$ $1$ $8$ $14$ $( 1,16,13,11, 9, 8, 5, 4)( 2,15,14,12,10, 7, 6, 3)$
8B3 $8^{2}$ $1$ $8$ $14$ $( 1,12, 5,15, 9, 3,13, 7)( 2,11, 6,16,10, 4,14, 8)$
8B-3 $8^{2}$ $1$ $8$ $14$ $( 1, 8,13, 4, 9,16, 5,11)( 2, 7,14, 3,10,15, 6,12)$

magma: ConjugacyClasses(G);
 

Malle's constant $a(G)$:     $1/8$

Group invariants

Order:  $16=2^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  16.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 8A1 8A-1 8A3 8A-3 8B1 8B-1 8B3 8B-3
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 1A 1A 2C 2C 2C 2C 4A1 4A1 4A-1 4A1 4A-1 4A-1 4A1 4A-1
Type
16.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16.5.1b R 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1
16.5.1c R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
16.5.1d R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
16.5.1e1 C 1 1 −1 −1 −1 −1 1 1 −i i i −i −i i i −i
16.5.1e2 C 1 1 −1 −1 −1 −1 1 1 i −i −i i i −i −i i
16.5.1f1 C 1 1 1 1 −1 −1 −1 −1 −i i i −i i −i −i i
16.5.1f2 C 1 1 1 1 −1 −1 −1 −1 i −i −i i −i i i −i
16.5.1g1 C 1 −1 −1 1 −ζ82 ζ82 −ζ82 ζ82 ζ83 −ζ8 ζ8 −ζ83 ζ8 −ζ83 ζ83 −ζ8
16.5.1g2 C 1 −1 −1 1 ζ82 −ζ82 ζ82 −ζ82 −ζ8 ζ83 −ζ83 ζ8 −ζ83 ζ8 −ζ8 ζ83
16.5.1g3 C 1 −1 −1 1 −ζ82 ζ82 −ζ82 ζ82 −ζ83 ζ8 −ζ8 ζ83 −ζ8 ζ83 −ζ83 ζ8
16.5.1g4 C 1 −1 −1 1 ζ82 −ζ82 ζ82 −ζ82 ζ8 −ζ83 ζ83 −ζ8 ζ83 −ζ8 ζ8 −ζ83
16.5.1h1 C 1 −1 1 −1 −ζ82 ζ82 ζ82 −ζ82 ζ83 −ζ8 ζ8 −ζ83 −ζ8 ζ83 −ζ83 ζ8
16.5.1h2 C 1 −1 1 −1 ζ82 −ζ82 −ζ82 ζ82 −ζ8 ζ83 −ζ83 ζ8 ζ83 −ζ8 ζ8 −ζ83
16.5.1h3 C 1 −1 1 −1 −ζ82 ζ82 ζ82 −ζ82 −ζ83 ζ8 −ζ8 ζ83 ζ8 −ζ83 ζ83 −ζ8
16.5.1h4 C 1 −1 1 −1 ζ82 −ζ82 −ζ82 ζ82 ζ8 −ζ83 ζ83 −ζ8 −ζ83 ζ8 −ζ8 ζ83

magma: CharacterTable(G);